Interactions of Flexible

Macromolecules with Surfaces and

Their Role in Viral Assembly

Thesis Submitted for the Degree

Doctor of Philosophy

by

Shelly Tzlil

Submitted to the Hebrew University Senate

December 2006

This work was carried out under the supervision of

Professor Avinoam Ben-Shaul

Acknowledgments

I would like to thank Professor Avinoam Ben-Shaul for his endless support

and guidance, for giving me the passion for science and for educating me to

think.

I thank Professor William Gelbart for our collaboration, for his advices and for

giving LA a friendly face.

Special thanks go to Dr. Daniel Harries for endless discussions and for a very

meaningful friendship.

I thank Dr. James Kindt and Dr. Markus Deserno for fruitful collaborations.

I thank also Professor David Andelman and Professor Diana Murray for

challenging discussions.

i

Abstract The main motivation for this work is to study the physical principles underlying viral

assembly and budding. Since viral assembly essentially involves interactions between

flexible polymers (e.g., nucleic acids) and surfaces (e.g., protein shells and lipid membranes),

this work is concerned with describing these interactions and their role in viral assembly.

In the Introduction, we discuss the physical characteristics of biological membranes

and polyelectrolytes (such as viral ssRNA and dsDNA) and their role in viral assembly. The

flexibility of biopolymers enables them to change their configuration as they adsorb onto a

surface, thereby optimizing the interaction. Since a biological membrane is a two-

dimensional fluid mixture of various lipid species and proteins, it too can respond

dynamically to macromolecule adsorption by bringing either lipids or proteins with high

polymer affinity into the interaction zone. Lipid diffusion toward the interaction zone results

in the formation of a domain whose lipid/protein composition is different from the membrane

average. Furthermore, a lipid membrane is sufficiently elastic to go through both curvature

and stretching deformations without losing its integrity. This capability is crucial to processes

such as budding, where the membrane becomes wrapped around viral nucleocapsids.

In Chapter 2, we consider the infection mechanism of bacterial viruses

(bacteriophages). Upon binding to a bacterial cell, the genome of these viruses (usually

dsDNA) is ejected into the bacterial cell, while the capsid is left outside. We provide an

analytical theory, featuring the energetic and structural aspects of DNA ejection from a

bacteriophage capsid. Viewed in the reverse direction, we model DNA loading into the

capsid. The free energy of the DNA chain can be separated into contributions from its

encapsidated and released portions. Each is expressed as a sum of the bending and

interhelical interaction energies, but the two chain portions are subject to different boundary

conditions. For each ejected length, the equilibrium structure and energy are determined

simultaneously for both chain portions by a variational approach, minimizing the free energy

with respect to the shape profile and interstrand spacing. Numerical calculations are

performed using the genome length and capsid dimensions of a λ phage. We find that the

fully encapsidated genome is highly compressed and strongly bent into a spool-like

condensate, storing an enormous amount of elastic energy. The elastic stress is rapidly

released during the first stage of DNA injection which, therefore, occurs spontaneously. The

second injection stage sets in after ~1/3 of the genome has been released, and the interhelical

ii

distance has nearly reached its equilibrium value (corresponding to that of a relaxed torus in

solution). During this stage the encapsidated chain begins a gradual morphological

transformation from a spool to a torus. When the capsid walls no longer constrain the DNA

condensate, it forms a torus identical to that which would have formed in solution. Since

most of the pressure has been released by the end of the first stage of ejection, a delicate

balance between surface and bending energies motivates the second stage of release. Thus,

small changes in either the experimental or the model system (e.g., the existence of osmotic

pressure in the bacterial cell due to the presence of PEG) might arrest the ejection process. In

addition to the structure and free energy of the condensate, we also calculate the loading

force, the average pressure on the capsid walls, and the anisotropic pressure profile within the

capsid. The results are shown to be in good agreement with available experimental data.

Next we turn our discussion to animal viruses. There are two possible pathways for

the assembly and budding of animal viruses. In the first pathway, characterizing Alpha

viruses, the viral proteins and genome (usually ssRNA) are assembled to form a

nucleocapsid. Then, the nucleocapsids move to the plasma membrane. Viral glycoproteins

(spikes) embedded at the membrane are served to anchor and wrap it around the viral capsid.

In Chapter 3, we present a statistical-thermodynamic model for the budding viral

nucleocapsids at the cell membrane. The free energy of a single bud is expressed as a sum of

the bending energy of its membrane coating, the spike-mediated capsid-membrane adhesion

energy, and the line energy associated with the bud’s rim. All three terms depend on the

extent of the wrapping (i.e., the bud size) and the density of spikes in the curved membrane.

This self-energy is incorporated into a simple free energy functional for the many-bud

system, allowing for different spike densities in the curved (budding) and planar membrane

regions, as well as for a distribution of bud sizes. Thus, this free energy functional includes

the spikes entropy as well as the configurational entropy of the polydisperse bud population.

The equilibrium spike densities in the coexisting, curved, and planar membrane regions and

the bud size distribution are calculated by free energy minimization. We show that complete

budding (full wrapping of nucleocapsids) can only take place if the adhesion energy exceeds

a certain, threshold, bending energy (needed to bend a spike adhesion site in the membrane

around the nucleocapsid). Wherever budding takes place, every virion contains an identical

number of spikes, which corresponds to the occupation of all spike adhesion sites. The rim

destabilizes partially wrapped buds and thus promotes the production of mature virions. The

fraction of fully wrapped buds increases as this energy increases, eventually resulting in an

iii

“all-or-none” mechanism where nucleocapsids at the plasma membrane are either fully

enveloped or completely naked (just touching the membrane). We also find that at low

concentrations all capsids arriving at the membrane become tightly and fully enveloped.

Beyond a certain concentration corresponding approximately to a stoichiometric spike/capsid

ratio, newly arriving capsids cannot be fully wrapped and the budding yield decreases.

The second pathway occurs in more complex animal viruses such as retroviruses (e.g.,

HIV-1). In this pathway the processes of assembly and budding occur simultaneously at the

plasma membrane. The complexity of this process is reflected in its many degrees of freedom

and the couplings between them. The relevant degrees of freedom include viral genome and

membrane flexibility, lipid mobility, and the mobility of structural viral proteins adsorbed

onto the membrane. As a first step toward modeling this system, Chapter 4 deals with the

interactions between polyelectrolytes and charged fluid membranes. We develop an extended

version of the Rosenbluth simulation method, enabling the simultaneous generation of

polymer and membrane configurations. One of our main findings is that lipid mobility makes

an important contribution to the adsorption energy as well as to the structural characteristics

of both polymer and membrane. Adsorption onto a fluid membrane is much stronger than

adsorption onto an equally charged quenched or uniform membrane, due to the diffusion of

lipids with preferable polymer interactions towards the interaction zone. Although the

properties of a fluid membrane and a quenched membrane are significantly different, we

show that the thermodynamics of adsorption onto a fluid membrane can be derived from a

weighted average of an ensemble of quenched membranes. In this average the weight of each

quenched membrane configuration is given by the product of the fraction of such membranes

and the statistical weight of all polymer conformations on such membranes. Using a simple

cell model, we are able to account for the dependence of adsorption properties on polymer

concentration in solution and show that the average probabilities of adsorption onto annealed

and quenched membranes coincide at vanishing surface concentrations. The simulation in

Chapter 4 is performed for a flexible cationic polyelectrolyte interacting with membranes

containing neutral lipids (e.g., PC), 1% tetravalent anionic lipids (e.g., PIP2), and either 1% or

10% monovalent anionic lipids (e.g., PS). The polymer and lipid membrane interact via a

Debye-Huckel potential and a short-range repulsion. The tetravalent lipids are found to

concentrate in the polymer region, while the monovalent lipids do not. Tetravalent

segregation is preferable in terms of entropy, since the electrostatic gain from localizing a

single tetravalent lipid is equal to the electrostatic gain from localizing four monovalent

iv

lipids. The presence of tetravalent lipids appears to be crucial to the adsorption of flexible

polymers, which lose much of their conformational entropy upon adsorption. The attraction

between the polymer and membrane is electrostatic; however, the simulation’s main results

can be extended to any kind of interaction.

In Appendix A, we use the simulation to study the MARCKS protein adsorption-

desorption mechanism in biological membranes (often referred to as the “electrostatic

switch” mechanism). The biological function of the MARCKS protein is attributed to its

ability to bind PIP2 lipids, thus regulating the number of free PIP2s in the membrane plane.

These lipids are known to play an important role in the regulation of calcium channels

opening and membrane-cytoskeleton attachment.

Chapter 5 Concludes with a short summary.

Two major themes run throughout this work. The first concerns with the statistical mechanics

of flexible polymers, fluid membranes, and their mutual interaction. The second is the

relationship between the biological mechanism by which a virus infects cells and the physical

properties of the biological macromolecules involved. The introduction (Chapter 1) focuses

on the first theme, while the summary (Chapter 5) focuses on the second.

The Statistical Mechanics of Flexible Polymers and Fluid Membranes The first chapter

considers the statistical mechanics of confining a semi-flexible polymer inside a protein shell

whose dimensions are comparable to the polymer’s persistence length yet hundreds of times

smaller than its contour length. The genomes of animal viruses, as introduced in Chapter 3,

are single-stranded rather than double-stranded and thus much more flexible. As a result the

conformational entropy involved in their encapsidation becomes significant. The fact that the

genome is a single stranded RNA introduces an additional degree of freedom: the ability to

form secondary structures by base pairing. This makes the statistical properties of the

polymer much more complicated. The simulation described in Chapter 4 is capable of

studying this kind of polymer. While lipid mobility plays an important role as illustrate in

Chapter 4 and Appendix A, it is taken into account only implicitly in Chapter 3 by

introducing a free energy term associated with the interface between the bud and the planar

membrane (the line energy). This line energy term arises from several reasons, one of which

is the differences in the lipid-protein composition across the boundary separating the bud and

planar membrane regions. The mobility of spikes and the membrane elasticity are both

v

explicitly taken into account by this model. Chapter 4 deals entirely with statistical-

thermodynamic aspects of the adsorption process; the mobility of lipids as well as the

flexibility of the polymer are taken into account explicitly. As has already been mentioned,

the conclusions of Chapter 4 can be extended to any kind of interaction, not necessarily an

electrostatic one.

The Relationship Between the Biological Mechanism used by a Virus to Infect Cells and the

Physical Properties of the Macromolecules Involved – Since infection occurs spontaneously

after a bacteriophage binds to a receptor on the bacterial cell, the genome inside the capsid

must be under stress. That is, the genome must be compressed into the protein shell in the

process of packaging, by the end of which the stored pressure is sufficient to drive DNA

ejection when the capsid is eventually opened. This implies that loading must be an active

process. Indeed, a motor protein is required for pushing the phage genome into the capsid.

Animal viruses, on the other hand, infect cells first by binding to receptors on the surface and

then by fusion of their membrane with the cell membrane. In this process, a high pressure in

the viral capsid is unnecessary and animal virus assembly is entirely spontaneous. Viral

spikes are essential for the mature virion, since their extracellular domain binds to host cell

receptors upon infection. Therefore, to ensure their presence in the mature virions, the spikes

create the driving force for budding by providing the capsid-membrane adhesion energy. The

localization of proteins and lipids with “favorable” interactions in the budding region results

in a free energy penalty associated with the interface between the viral and cell membranes

(the line energy). This line energy term actually leads to a more efficient viral release,

however, since partial wrapping of viral capsids becomes energetically unfavorable. The

assembly and budding of retroviruses such as HIV are beyond the scope of this thesis.

Nevertheless, the trademark physical properties of the macromolecules involved in these

processes will probably also show up in the particular mechanism chosen by the HIV virus.

vi

Contents 1 Introduction……………………………………………………………………... 1

1.1 Preface……………………………………………………………………… 1

1.2 Biopolymers………………………………………………………………... 2

1.2.1 Radius of Gyration…………………………………………………. 4

1.2.2 Persistence Length…………………………………………………. 5

1.2.3 Polymer Elasticity…………………………………………………. 7

1.2.4 Polyelectrolyte condensation………………………………………. 11

1.3 Lipid Membrane……………………………………………………………. 11

1.4 Biopolymer-Surface Interactions…………………………………………... 14

1.5 Viral Systems………………………………………………………………. 15

1.6 Overview…………………………………………………………………… 17

2 Packaging and Ejection of DNA from Bacteriophages………………………… 18

2.1 Introduction………………………………………………………………… 18

2.2 Theory……………………………………………………………………… 22

2.2.1 Model and Free Energy…………………………………………….. 22

2.2.2 DNA-DNA interaction Potential…………………………………… 25

2.2.3 Analytical Model for a Toroidal DNA in Solution………………… 26

2.2.4 Method of Solution………………………………………………… 27

2.3 Results and Discussion…………………………………………………….. 28

2.3.1 DNA Condensate Structure………………………………………… 28

2.3.2 Ejection Mechanism – Forces and Energetics……………………... 32

2.3.3 An Approximate "Two State Model"……………………………… 34

2.3.4 Pressure…………………………………………………………….. 36

2.3.5 Incomplete Ejection………………………………………………... 40

2.4 Concluding Remarks……………………………………………………….. 42

3 Viral Budding……………………………………………………………………... 44

3.1 Introduction………………………………………………………………… 44

3.2 Model………………………………………………………………………. 46

vii

3.2.1 Macroscopic Phase Approximation………………………………... 48

3.2.2 Bud Size Distribution………………………………………………. 50

3.3 Results……………………………………………………………………… 54

3.3.1 Choice of Parameters………………………………………………. 54

3.3.2 Spike Partitioning………………………………………………….. 56

3.3.3 Bud Size Distribution………………………………………………. 59

3.3.4 Mature Buds………………………………………………………... 61

3.4 Concluding Remarks……………………………………………………….. 63

4 Adsorption of Flexible Macromolecules on Fluid Membranes………………… 66

4.1 Introduction………………………………………………………………… 66

4.2 Adsorption Thermodynamics……………………………………………….70

4.2.1 Single Polymer Adsorption………………………………………… 70

4.2.2 Finite Polymer Concentration……………………………………… 73

4.2.3 Adsorbed State Definition………………………………………….. 76

4.3 The Model System…………………………………………………………. 80

4.4 Simulation Method………………………………………………………… 81

4.4.1 Quenched Membrane………………………………………………. 82

4.4.2 Fluid Membrane……………………………………………………. 84

4.4.3 Free Energies of Adsorption……………………………………….. 86

4.5 Results……………………………………………………………………… 87

4.5.1 Structural Properties………………………………………………... 88

4.5.2 Adsorption Thermodynamics……………………………………….95

4.6 Concluding Remarks……………………………………………………….. 102

5 Summary…………………………………………………………………………... 104

Appendix A - MARCKS Protein - The “Electrostatic-Switch Mechanism”………….. 107

A.1 Introduction………………………………………………………………… 107

A.2 Model………………………………………………………………………. 108

A.2.1 Excluded Volume Interactions……………………………………... 111

A.2.2 Electrostatic Interactions…………………………………………… 111

A.2.3 Hydrophobic Interactions…………………………………………...112

viii

A.3 Results and Discussion…………………………………………………….. 112

A.3.1 Lipid Distribution………………………………………………….. 114

A.3.2 Adsorption Free Energies…………………………………………..116

A.3.3 Adsorption Isotherm- "Electrostatic-Switch Mechanism"…………. 120

A.3.4 Concluding Remarks……………………………………………….. 121

References ………………………………………………………………………………… 123

1

Chapter 1 Introduction

1.1 Preface This work is concerned with the interactions between flexible macromolecules and surfaces

and their role in a variety of biological process, focusing mainly on viral assembly.

Viral systems are of enormous importance for several reasons. First, the

understanding of viral infection mechanisms will enable the development of strategies for

infection inhibition. Second, viral infection could be mimicked for gene therapy and drug

delivery purposes. Gene therapy is a strategy for treating disease caused by missing or

corrupted genes by addressing the source of the problem at the genome level. This can be

done directly by utilizing the virus itself and its ability to integrate its genome into the host

cell genome (e.g., engineering a retrovirus whose genome has been replaced by the desired

gene). Alternatively, one can mimic certain aspects of viral infection (e.g., directing an empty

liposome loaded with the desired gene into the target cell). To do so, one needs either to

control the virus in order to utilize it for his own needs or to design a synthetic system

capable of mimicking its behavior. Both require a profound understanding of the physical

mechanisms that govern viral infection. A third and no less significant reason for our interest

in viral systems, is the physical problems that arise out of it. The assembly and budding of

viruses is a complex self-assembly process which involves an interplay between various

physico-chemical forces and deals with fundamental issues in soft condensed matter.

Viral systems, like most biological systems, are complicated and have many degrees

of freedom. When beginning the theoretical study of such a system, one should ask himself

which principal physical forces govern its behavior. In other words, what are the real

essential features that one should incorporate in its model in order to be able to elucidate the

physical mechanism that underlies the system’s behavior. A too detailed model will be

difficult to analyze and understand almost as much as the experimental system itself. A too

simplified model might miss the real physics involved and thus be incapable of giving

accurate predictions. An understanding of the physical characteristics of the macromolecules

which participate in the process, their intrinsic degrees of freedom and the coupling between

2

them upon interaction, can give us clues of the relative importance of these degrees of

freedom. Such an understanding can give us an insight into the relationship between physical

properties and biological function and equip us with the necessary tools for modeling and

designing complex systems.

We will begin the introduction by describing the physical characteristics of the

biopolymers studied in this work (DNA, RNA and proteins) and of the surfaces they interact

with, mostly mixed fluid membranes. Then, we will discuss the ability of both biopolymers

and surfaces containing mobile charges to change their properties upon interaction with one

another. We will emphasize how the various physical principles described are expressed in

viral systems. We will finish the introduction with an overview of the work.

1.2 Biopolymers A polymer is a macromolecule composed of a large number of simpler units (monomers)

covalently bonded together. A single monomer can be as small as a few atoms or much

larger, like an amino acid or a nucleic acid. An important class of polymers of relevance here

is biopolymers (e.g., DNA, RNA and proteins). Their structure is usually discussed on three

distinct levels. The sequence of the polymer’s building blocks is called the primary structure.

Short-range order, which usually arises because of interactions between nearby segments, is

referred to as the secondary structure and the whole spatial arrangement is called tertiary

structure.

DNA (deoxyribonucleic acid) and RNA (ribonucleic acid) carry the genetic

information of the cell. They are chemically very similar. The primary structure of both these

biopolymers is a linear polymer composed of monomers called nucleotides. All of the

nucleotides share a similar structure: a phosphate group linked by a phosphoester bond to a

pentose (a five-carbon sugar molecule) that is linked in turn to a nitrogen- and carbon-

containing ring structure commonly referred to as a base [1]. In RNA, the pentose is ribose

while in DNA it is deoxyribose. The bases involved in forming DNA are: adenine (A) and

guanine (G), which are derivatives of purine, and thymine (T) and cytosine (C) , which are

derivatives of pyrimidine. For RNA, thymine is replaced by uracil (U) (Figure 1.1).

Proteins are the most abundant biological macromolecules and their functions are

diverse. There are structural proteins which provide structural rigidity to the cell, transport

proteins which control the flow of materials across cellular membranes, signaling proteins

which are responsible for regulating protein function and transmit external signals into the

3

cell and more [1,2]. Proteins are linear polymers; their monomeric building blocks are twenty

amino acids joined by peptide bonds, see Figure 1.2. All of the amino acids have a

characteristic structure which consists of a central α-carbon atom bonded to four different

chemical groups: an amino (NH2) group, a carboxyl (COOH) group, a hydrogen (H) atom

and one variable group, R, called “the side chain”. Amino acids are classified as polar,

charged or non-polar according to their R group properties. (Their charging state is

determined at pH=7), see Figure 1.3.

Figure 1.1 (a): Ribose and deoxyribose, the pentoses in RNA and DNA, respectively. (b): One strand of DNA

is composed of a phosphate sugar backbone which can carry four types of bases, adenine (A), thymine (T),

guanine (G) and cytosine (C). (c): Four RNA nucleotides. The bases are identical to the DNA bases, except for

uracil (U) which takes the place of thymine (T).

Figure 1.2 A protein backbone composed of amino acids with different side chains. The individual amino

acids are connected to each other by peptide bonds.

( )a( )b

©1999 Addison Wesley Longman, Inc.

( )c

Peptide bond

4

Figure 1.3 The 20 common amino acids.

The structural and physical properties of biopolymers have a direct influence on their

biological functions. Two important physical properties of polymers are their effective size

and flexibility.

1.2.1 Radius of Gyration The polymer is neither completely straight nor fully collapsed because of entropic

considerations and hardcore interactions. It has a characteristic size that is usually defined in

one of two ways.

Alanine

(Ala,A)

Glycine

(Gly,G)

Valine

(Val,V)

Leucine

(Leu,L)

Isoleucine

(Ile,I)

Methionine

(Met,M)

Proline

(Pro,P)

Phenylalanine

(Phe,F)

Tryptophan

(Trp,W)

Serine

(Ser,S)

Threonine

(Thr,T)

Glutamine

(Gln,Q)

Asparagine

(Asn,N)

Thyrosine

(Tyr,Y)

Cysteine

(Cys,C)

Lysine

(Lys,K)

Arginine

(Arg,R)

Histidine

(His,H)

Aspartate

(Asp,D)

Glutamate

(Glu,E)

5

The Root Mean Square End to End Distance

The root mean square end-to-end distance 0R roughly defines the coil diameter. It is defined

as the average size of the end-to-end vector R over all polymer configurations.

1/ 22

1/ 220

1

N

ii

R R b=

⎛ ⎞= = ⎜ ⎟⎝ ⎠∑ (1.1)

where 1i i ib r r+≡ − is the bond vector, defined as the vector connecting the i monomer with the

i+1 monomer; ir and 1ir + denoting the positions of the i and i+1 monomers respectively.

Figure 1.4 shows these variables for a given polymer configuration.

Radius of Gyration

The Radius of Gyration is defined as

( )22

1 1

N N

G i i CM ii i

R m r r m= =

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑ (1.2)

When all the polymer segments have the same mass, we have

( )22

1

1 N

G i CMi

R r rN =

= −∑ (1.3)

For an ideal chain, 0 6GR R = , [6]. Since a polymer coil is not spherical, the

definitions of its size and diameter are not very precise and the two definitions are acceptable.

The scaling behavior with the polymer parameters (such as its length) is similar for both of

them.

1.2.2 Persistence Length A measure of the flexibility (or rigidity) of a polymer chain is its persistence length. It can be

thought of as the distance over which the polymer is kept straight. Viewed differently, it is

the distance beyond which thermal fluctuations erase orientational correlations.

More specifically, consider a polymer of length L which consists of N segments each with a

length l, i.e., L Nl= . Let us denote by ijθ , the angle between two vectors, ib and jb , tangent

to the molecule axis (“bond vectors”) and separated by a distance s along the molecule

contour (see Figure 1.4). The angular correlation function between polymer segments can be

written as

2 cos ( )i jb b l sθ⋅ = (1.4)

where the averaging is over all pairs ( ,i j ) with /j i s l− = .

6

Figure 1.4 Illustration of a polymer chain. On left: Illustration of a freely jointed chain model, with two bond

vectors, ib and jb that create an angle ijθ between them. Also marked is the end-to-end distance, R . On right:

Schematic drawing of a continuous model for a polymer chain, with two tangent vectors at two points separated

by a distance s along the chain contour, forming an angle θ between them.

Similarly, if the polymer is modeled as a continuous curve of length L parameterized by a

variable s that follows the contour from 0 to L, we can take any two points that are a contour

length s apart and compute the dot product of their tangents, averaging over all pairs of points

of the same distance (see Figure 1.4). Both correlation functions decay exponentially [3]

/cos ( ) ps ls eθ −∼ (1.5)

The decay length, pl , is known as the persistence length.

Biopolymers are semi-flexible polymers. That is, their flexibility comes into play only

over length scales that are much larger than several monomers. There are two limits to the

semi-flexible polymer’s behavior. In the limit pL l<< , the polymer behaves as a rigid body,

whereas, in the limit pL l>> the polymer is essentially a flexible chain.

For a freely jointed chain, there are no correlations between polymer segments ( 0i jb b = for

i j≠ ) and using Eq. (1.1), the average end-to-end distance obtained in this case is

2

2 2 20

1 1

N N

i i i ji i i j

R b b b b Nl= = ≠

⎛ ⎞= = + =⎜ ⎟⎝ ⎠∑ ∑ ∑ (1.6)

Thus, if we redefine a segment as a group of monomers whose collective length is 20Kl R L=

(“an effective monomer”), we get a chain composed of / KN L l′ = “effective monomers”

which behaves as a freely jointed chain. That is, each “effective monomer” can move freely

with respect to its neighbors and the average end-to-end distance behaves according to Eq.

7

(1.6). Kl is called the Kuhn length and the “effective monomers” are called Kuhn segments.

The Kuhn length is comparable to the persistence length, K pl l∼ . Thus, in the limit pL l>> ,

the polymer can be treated as a freely jointed chain of / KN L l′ = Khun segments.

For a simple synthetic polymer, such as polystyrene, 1.0 1.4pl ≈ − Å corresponds

approximately to the length of four to five chain bonds. In contrast, the persistence length of

biopolymers is of the order of nanometers [3]. RNA is usually found in the cell in its single-

stranded (ssRNA) form; its persistent length is 5pl nm≈ . Cellular DNA, however, is not

found as a single strand, but consists of two associated strands that wind together to form a

double helix. This double helix is generally referred to as double-stranded DNA (dsDNA).

The two sugar phosphate backbones are on the outside of the double helix and the bases face

the interior, see Figure 1.5. The strands are held together by hydrogen bonds between bases

on opposed strands (called base pairs, bp) and hydrophobic stacking interactions between

adjacent bases on the same strand. In natural DNA, hydrogen bonds are always formed

between A and T (two hydrogen bonds) and between G and C (three hydrogen bonds). In

this sense, the two strands are complementary [1]. However, in synthetic DNA, other base

pairs can be formed. For dsDNA, 50pl nm≈ (approximately 150 bps) [3]. There are two

main reasons for the large persistence length of DNA. First, the double helical structure,

particularly the stacking interactions between adjacent bases, makes the DNA hard to bend

along its axis. Second, the repulsion between negative charges along the chain increases the

rigidity of the molecule. The distance between bases on a chain is 3.4∼ Å. Since there are

two phosphate groups associated with each base pair, the average distance between two

charges along the DNA axis is 1.7∼ Å. This makes DNA a highly charged polymer. The

contribution of electrostatic repulsion to the persistence length depends on the salt

concentration in solution (which screen the interactions between charges) and can be

estimated theoretically [4,5].

1.2.3 Polymer Elasticity A simple expression for the elasticity of a biopolymer can be obtained in the two limits

introduced previously. We will give a qualitative description followed by a more quantitative

one.

Qualitatively, in the first limit pL l>> the polymer behaves as a flexible chain. Thus,

stretching the polymer results in an entropic penalty. The entropy loss upon stretching is a

8

result of the reduction in the number of conformations. In the second limit pL l<< the

polymer behaves as a rigid body and changing its conformation by bending it causes an

energetic penalty. The energy loss upon bending arises from the interactions between

polymer segments. These interactions cause preferable distances between the segments and

thus are responsible for the polymer tendency to be straight in lengths comparable to (or

much smaller than) the persistence length.

Let us look at these limits more closely. To demonstrate the physical origin of

polymer elasticity in the limit pL l>> , let us look on an ideal chain. Ideal chains have no

interactions between monomers that are far apart along the chain, even if they approach each

other in space. The obtained distribution function of end-to-end distances in this case is

Gaussian (for 1N >> ),

( ) ( )3/ 22 2 2( ) 2 3 exp 3 2P R R R Rπ− ⎡ ⎤= −⎣ ⎦ (1.7)

Remember that ( ) ( ) ( )R

P R R R= Ω Ω∑ where ( )RΩ is the number of chain conformations of a

polymer whose end-to-end distance vector is R . Hence, the entropy of a polymer with end-to-

end distance R can be written as

2

0

3( ) ln ( ) (0)2B

RS R k R SR

⎛ ⎞= Ω = − ⎜ ⎟

⎝ ⎠ (1.8)

This result can be interpreted as if the polymer behaves as an entropic spring. Its elasticity is

a direct result of the reduction in the number of conformations upon stretching the polymer

from its optimal size. Following Eq. (1.8), the free energy increase upon stretching an ideal

polymer chain can be expressed as

2

0

3( )2 B

RF R k TR

⎛ ⎞Δ = ⎜ ⎟

⎝ ⎠ (1.9)

Note that excluded volume interactions will change the asymptotic polymer behavior and

result in a different functional dependence on R .

Now consider the second limit pL l<< . Following the discussion of Grosberg and

Khokhlov [3], we consider a short polymer section s such that ps l<< for which, following

Eq. (1.5), we get

cos ( ) 1 / ps s lθ ≅ − (1.10)

9

Since the section length is smaller than the persistence length, we can make the additional

assumption that ( )sθ is small and thus,

2cos ( ) 1 ( ) 2s sθ θ≅ − (1.11)

Using Eqs. (1.10) and (1.11), we get

2 ( ) 2 ps s lθ ≅ (1.12)

Since ( )sθ is the local curvature, the elastic free energy upon bending can be written as

2

21 22elasticF s s

sθκ κθ⎛ ⎞= =⎜ ⎟⎝ ⎠

(1.13)

Where κ is the 1D bending rigidity.

Using Eq. (1.13), we can write

( )2 2( ) 2 exp 2elastic B Bs F k T d sk Tθ θ θ κ= − =∫ (1.14)

Comparing Eqs. (1.12) and (1.14), we get

pB

lk Tκ

= (1.15)

Thus, the energy per unit length necessary to bend a polymer to a radius R (where R is on the

order of its persistence length) can be written as ( ) ( )21/ 2 1elasticF Rκ= . Obviously, if R is

much larger than the polymer’s persistence length, this term goes to zero and there is no

energy penalty for bending.

Before continuing, let us make two remarks about ssRNA and proteins.

Single-Stranded RNA - Unlike DNA, which exists primarily as long double helices, most

cellular RNA is single- stranded. However, intra-strand base pairing produces a complex

secondary structure. RNA secondary structure is generally divided into helices (contiguous

base pairs) and various kinds of loops (unpaired nucleotides surrounded by helices), see

Figure 1.6. In addition to the purine-pyrimidine base pairing (i.e., Watson-Crick base pairs,

A=U, G C≡ ) which are the standard DNA base pairs, G U= is also quite common in RNA.

Figure 1.6c shows a model for the structure of ribosomal RNA. As shown by the figure, the

structure is not at all linear and so it is expected to behave more like a branched polymer with

dynamic branches (the base pair interaction is on the order of Bk T ). Therefore, the statistics

of ssRNA (e.g., its radius of gyration) are different than those of simple, linear polymers.

10

Hydrogen Bonds

Stacking

Interactions

Figure 1.5

Figure 1.6 (a) A schematic drawing of RNA secondary structure. The paired regions generally create a right-

handed helix, as shown for a hairpin loop in (b). ( (a) and (b) are taken from [1]. (c) The secondary structure of

E. coli ribosomal RNA (from http://rna.ucsc.edu/rnacenter/ribosome_images.html).

( )a

( )b

( )c

A schematic representation of a double

helical DNA structure. Base pairing is

created between the bases

A=T and G C≡ through two and three

hydrogen bonds, respectively.

11

Proteins - The absolute majority of proteins spontaneously fold into characteristic compact

shapes which determine their biological functions and depend in a complicated way, on their

primary structures. Thus, usually proteins are rigid and do not behave according to polymer

statistics. There are, however, a small number of proteins which are known to be naturally

unfolded. This very unusual property of proteins is assumed to be directly connected to their

biological functions. Most of these molecules are known to be signaling proteins. In this

work, we study one such an example (see Appendix A – the MARCKS protein).

1.2.4 Polyelectrolyte condensation Macromolecules containing charged segments are called polyelectrolytes. In solution, they

dissociate to form charged segments and low molecular weight counterions. The number of

counterions equals the number of charged segments, so that the whole polymer solution is

electrically neutral. The counterions are attracted to the charged polymers via long-range

Coulomb interactions, which typically lead to rather loosely bound counterion clouds around

the polyelectrolyte chains [7]. In a mean field treatment (i.e., Poisson-Boltzmann equation in

electrostatics), negatively charged macromolecules will always repel each other [8].

However, experimentally, the presence of polyvalent ions makes highly charged

polyelectrolytes, such as dsDNA, self- attract. This attractive interaction is responsible for the

condensation of dsDNA into compact, typically toroidal aggregates [9-11], see Figure 2.3

and compare to Figure 2.2 where no polyvalent ions are present. DNA condensation can be

explained by correlated fluctuations in the counterion density around the DNA [12]. An

alternative mechanism was suggested in which ions firmly bound to one DNA molecule may

attract a similar, correlated array on another DNA molecule (Wigner crystal-type attractions)

[13,14]. An important difference between the two mechanisms is that the former is based on

thermal fluctuations and, therefore, the resulting attraction becomes stronger at higher

temperatures. In contrast, the second mechanism arises from correlations between bound

counterions which are weaker bound at higher temperatures. Therefore, these mechanisms

represent different regimes of polyelectrolyte behavior.

1.3 Lipid Membrane In this work, we study biopolymers interacting either with viral protein shells or with lipid

membranes. Since the protein shell is rigid, it does not have additional degrees of freedom

and we consider it as a confinement to the polymer with or without fixed charges on its

12

surface. Lipid membranes, however, are much more complex. Biological membranes form

closed structures to separate the cell from its environment. These structures are mainly made

up of phospholipids. Phospholipids are amphiphilic molecules. That is, each phospholipid

consists of a hydrophilic, polar “head” attached to a hydrophobic “tail” consisting of fatty

acid residues. Structures of three such phospholipids are shown in Figure 1.7. All of them

share the same “tail” unit, but have very different headgroups that can be either negatively

charged (e.g., phophsatidylserine, PS) or zwitterionic (e.g., phosphatidylcholine, PC and

phosphatidylethanolamine, PE). Most biological membranes are negatively charged with

around 10% of their lipids carrying charges.

When mixed with water, lipid molecules tend to spontaneously form lipid aggregates

where the hydrophobic chains face each other to create an oily bulk, while the hydrophilic

headgroups reside at the interface separating the chains from the surrounding aqueous

solution, see Figure 1.8. The stability of these aggregates is due to the effective attractive

forces between hydrocarbon tails (hydrophobic interaction) resulting from a tendency to

minimize the hydrocarbon-water contact area. The planar bilayer is just one of several

possible aggregates geometries that satisfy the hydrophobic effect [15,16].

Two main characteristics of lipid membranes are their two-dimensional fluidity and curvature

elasticity. In biological membranes, the lipid bilayer serves as a solvent within which integral

proteins are embedded. Both proteins and lipids are free to move laterally within the

membrane plane (see Figure 1.9). The importance of membrane fluidity is demonstrated in

Chapter 4.

One can describe the free energy of a lipid layer in terms of its elastic properties. In

general, three elastic terms arise for a lipid layer, lateral stretching, bending elastic energy

and Gaussian curvature. The energy associated with stretching is very large and the Gaussian

curvature is just a constant1. Thus, the only elastic term we consider is the curvature elastic

energy. The curvature elastic energy for small bending deformations of a lipid monolayer can

be expanded up to quadratic terms in the curvature

21 2 0

1 ( )2elastic cF k c c c= + − (1.16)

� 1 Due to the Gauss-Bonnet theorem, which states that the integral of Gaussian curvature is a topological

invariant, and since all states considered in this work have the same number of membrane pieces and the same

number of handles, we shall not consider this term further.

13

where 0c is the spontaneous curvature, a local property that depends on membrane

composition of the lipids forming it, and ck is the elastic bending modulus [17]. A connection

between the molecular picture of amphiphile molecules and the elastic properties of a lipid

membrane can be formulated [15].

Figure 1.7 The common phospholipids, phosphatidylcholine (PC) and phosphatidylethanolamine (PE), are

neutral (zwitterionic) lipids. Phosphatidylserine (PS) is a negatively charged phospholipid.

Figure 1.8 Illustrations of a lipid molecule consisting of a polar head and a hydrophobic tail and two of the self

assembled structures it might form.

Figure 1.9 A schematic drawing of a biological membrane.

14

1.4 Biopolymer-Surface Interactions A system which involves all the degrees of freedom introduced so far is a polyelectrolyte

(charged polymer) interacting with a fluid membrane. Such a system involves polymer

flexibility and lipid mobility as well as nonspecific interactions (long-range electrostatics and

excluded volume). The interplay between entropy loss and energy gain can be studied

thoroughly in this system, so as the coupling between the different degrees of freedom.

As we investigate specific systems such as the MARCKS flexible protein or viral

RNA, we must notice that the biological function (biochemistry) of these macromolecules is

often governed by short range – specific interactions that should be taken into account. To

avoid atomic details, we mainly use experimental measurements to extract the intermolecular

potentials.

Whenever a polymer adsorbs on a surface, it looses entropy as a consequence of the

reduction in its configurational space. Interaction energy with the surface can overcome this

entropic barrier and make the adsorption favorable. Not only the presence of a surface, but

also the distribution of ligands on its face and the distribution of interaction points (e.g.,

charges) on the polymer, can give rise to differences in polymer configuration. Notice that the

ligands might be either lipids or proteins which are embedded at the membrane. The

interaction of the ligands with the polymer can be electrostatic (e.g., lipids whose charge is

opposite to the polymer segment charge) or of another nature (e.g., hydrophobic). The

adsorption process might also involve a change in the lipid distribution profile from a

uniform one to one where “favorable” lipids are segregated into the interaction zone. Thus,

the binding of a flexible polymer to a fluid lipid membrane is partially opposed by two kinds

of entropy losses. The first associated with the localization of “favorable” lipids into the

interaction zone and the second associated with the lower conformational freedom of the

polymer. Clearly, these entropy losses will occur only if the system can compensate for them

through gains in binding energy. This energetic-entropic balance is in the basis of these

adsorption processes.

Lipid Rafts - Lately, the term “lipid rafts” has been used for microdomains in the

membrane whose lipid composition differs from the average composition (mostly rich in

sphingolipids, highly charged lipids and cholesterol). These areas are usually “liquid-

ordered” domains and are assumed to serve as sites for signal transduction events [18-20].

There are a number of proteins which are known to be associated with membrane rafts and it

15

is not farfetched to assume that the segregation of lipids to create these domains is highly

encouraged by the presence of “raft” proteins which are more likely to interact with specific

lipids.

1.5 Viral Systems Let us now return to our discussion of viral systems and demonstrate how the various

properties of biopolymers and membrane surface outlined so far come into play.

Virtually all viruses, whether they infect bacteria, plants or animals, have a common

fundamental structure that involves the viral genome (RNA or DNA) being encapsidated by a

rigid protein shell (capsid). Since viruses cannot actively reproduce themselves, they utilize

the biochemical machinery of infected host cells to replicate their genomes and synthesize

their proteins, thereby acquiring the necessary ingredients for propagation (assembly and

leaving the host cell). In almost all cases of plant and animal viral infections, the entire virus

particle, capsid and all, enters the cell cytoplasm. The genome ends up being de-encapsidated

and thereby made available for integration into the host cell machinery, through a variety of

scenarios [1,21]. The infection mechanism of bacterial viruses (bacteriophages), on the other

hand, is unique in that, with few exceptions, only the genome enters the host cell, while the

capsid remains outside. This suggests that there are different physical mechanisms for these

infection processes.

Bacteriophages - The fact that the bacterial virus leaves its protein capsid outside the

cell it infects implies that its genome (usually dsDNA) is sufficiently pressurized in the

capsid to initiate the ejection process. A major theoretical challenge is to account for how the

genome, a semi-flexible, highly charged chain, can be confined in dimensions comparable to

its persistence length and yet hundreds of times smaller than its contour length. Obviously,

the loading of DNA into the viral capsid, must be an active process since it involves both

large elastic (bending) energy and repulsion due to the squeezing of the genome into a capsid

whose size is small compared to the genome length. Under biological conditions, there are

usually polyvalent counterions (e.g., spermidine, spermine) present in the bacteria cell [22]

whose presence, as explained before, makes the DNA self-attracting. Still, over short

distances like those inside the viral capsid, dsDNA repels itself strongly even in the presence

of polyvalent ions and thus is strongly pressurized inside the viral protein shell.

16

Figure 1.10 A schematic drawing of a CAN-like viral assembly (left) and a MAN-like viral assembly (right).

Enveloped Animal Viruses – Most animal viruses acquire lipid-protein membrane envelopes

upon their exocytosis through the plasma membrane of the infected cell. Viral exocytosis, or

budding, begins with a local bending of the lipid membrane around the nucleocapsid,

followed by complete wrapping of the capsid and ending when the enveloped virion pinches

off from the plasma membrane into the intracellular space. There are two possible pathways

for the combined processes of virus assembly and budding. The first pathway, capsid

assembled nucleocapsids (CAN, e.g., Alpha viruses) consists of two stages. First, the viral

genome and capsid proteins coassemble into well defined nucleocapsids within the

cytoplasm. The pre-assembled nucleocapsids then migrate towards the lipid-protein

membrane where pre-adsorbed transmembrane “spike” proteins await their arrival. In the

second mechanism of viral exocytosis, membrane assembled nucleocapsids (MAN, e.g.,

HIV-1), the genomic material coassembles with the viral proteins (Gag polyproteins for

retroviruses) on the membrane surface, concomitantly with membrane bending and budding,

see Figure 1.10.

We start by developing a phenomenological model for budding of viral nucleocapsids

at the cell membrane (CAN pathway). Here, the pre-assembled nucleocapsids are rigid bodies

interacting with a fluid, flexible membrane. The binding energy between the capsid and the

spike glycoproteins, which are embedded in the membrane, is opposed by the elastic energy

associated with the bending of the membrane around the nucleocapsid and by the loss of

spikes entropy when sequestered into the bud region. The fluid nature of the membrane is

also implied by the fact that the composition of lipids in the bud region, which later becomes

the viral lipid envelope, is different than the average composition of the membrane. Some

studies suggest that the chemical compositions of various viral membranes resemble the

composition of lipid rafts [23,24]. Therefore, the creation of a boundary between the bud and

the planar membrane is associated with a line energy.

17

The simulation method described in Chapter 4 enables us to study more complex

systems, such as the arrangement of ssRNA inside a rigid protein shell (nucleocapsid

assembly) and the assembly and budding of retroviruses (e.g., HIV-1). The HIV-1 assembly

process involves all the degrees of freedom introduced so far, the mobility of the lipids and

Gag proteins, as well as the conformational freedom of the ssRNA and of the Gag

polyprotein (which consists of three folded domains connected by flexible chains). These

studies are the next logical step which is, however, beyond the scope of this work.

1.6 Overview The research described below is organized as follows. In Chapter 2, we study DNA

packaging and ejection from a bacteriophage capsid. We study the structural evolution of the

dsDNA inside the viral capsid as well as the energies, forces and pressures associated with

the packaging and ejection processes. In Chapter 3, we will study the budding of

nucleocapsids from the cell membrane. To enable the study of more complex viral assembly

processes, we develop a simulation method presented in Chapter 4. Using the simulation, we

will study a general system that involves all the degrees of freedom of interest to us, a

polyelectrolyte interacting with a mixed fluid membrane. To demonstrate the important rule

of lipid mobility, our results for the fluid membrane are compared to those obtained for

quenched and uniform membranes with the same average lipid composition. To demonstrate

the capability of the simulation to study complex systems, we will consider in Appendix A,

the myristoylated alanine-rich C kinase substrate (MARCKS) protein. The “electrostatic-

switch” mechanism, underlying the operation of this protein, is governed by a delicate

balance between the energetic and entropic contributions to the adsorption free energy. A

short summary will follow in Chapter 5.

18

Chapter 2

Packaging and Ejection of DNA from

Bacteriophages

2.1 Introduction

In the present work, we formulate a model for the processes of DNA packaging and ejection

from a bacteriophage capsid1. Bacteriophages are viruses that infect bacteria. They consist of

a protein shell where their genome is encapsulated, and a long protein tail (Figure 2.1). Upon

binding to its receptor protein in the outer membrane of the bacterial cell (LamB in the case

of a λ phage), the viral capsid opens and the viral genome (usually dsDNA) is ejected into the

cytoplasm. The protein shell is left outside the cell. This mechanism suggests that the genome

must be strongly stressed inside the capsid, with an associated pressure sufficient to inject the

genome into the host cell.

Figure 2.2 illustrates the huge amount of dsDNA found inside a viral capsid, in

comparison with the capsid dimensions. The charge associated with the high density of

phosphate groups makes these DNA chains strongly self-repelling. Indeed, the presence of

multivalent ions (present in bacterial viruses) induces an effective attraction between DNA

double-strands, as explained in Sec. 1.2.4. However, these attractive forces result in an

equilibrium inter-strand distance, d0=28Å [9,10,27], much larger than the interhelical

separation inside viral capsids

In bacteriophage T7, for example, it has been explicitly demonstrated that the dsDNA chain

is organized in a spool-like configuration (see Figure 2.6) with an average interhelical

separation as small as 25Å [28]. If we compare this number to the hard core double-helix

diameter of 20Å and the interhelical distance of ~28Å in a relaxed toroidal DNA condensate,

it becomes clear that there must be a strong repulsive force between neighboring chain

segments throughout the capsid. A short calculation can illustrate the dramatic crowding of

DNA inside the capsid. Consider for instance the λ phage. The total length of its genome is

� 1 The results presented in this chapter were previously reported in [25] and [26].

19

L = 330ξ (ξ = 500Å is the dsDNA persistence length under physiological conditions). The

radius of its capsid (which is approximately spherical) is RC = 275Å = 0.55ξ, implying a

capsid volume of 3 34 / 3 0.679C CV Rπ ξ= ≈ . If the entire λ-DNA were packed uniformly and

hexagonally within the capsid interior, then the interhelical distance d would be given by

( )( )22 CL d Vπ γ = , where 0.91γ = is the maximal packing fraction (i.e., the volume

fraction of hexagonally close-packed cylinders). This organization thus implies

0.0495 24.75d ξ= = Å, which is much smaller than d0 = 28Å and well within the repulsive

range of the interhelical potential (see Figure 2.5).

Figure 2.1 Bacteriophage lambda. Electron micrograph image from Electron Micrograph Library - Virus &

Bacteriophage, Institute of Molecular Virology, University of Wisconsin-Madison.

http://www.biochem.wisc.edu/inman/empics/virus.htm.

Figure 2.2 One-step release of bacteriophage T4 DNA by osmotic pressure. Electron micrograph image from:

Light and Electron Microscopy Atlas, http://pages.unibas.ch/zmb/ATLAS/htm/t54.htm

Micrograph courtesy of Dr. Jurg Meyer (1991), Institute of Dentistry, University of Basel.

20

In addition to its high density inside the viral capsid, the DNA is also strongly bent therein.

The capsid size (hundreds of Angstroms) is comparable to the persistence length (ξ = 500Å)

of the genome, so large elastic deformation energies are necessarily involved (see Sec. 1.2.3).

In the case of T7, the average radius of curvature of the circumferentially-wound chain is as

small as 100Å near the hollow core of the packaged genome. The stress associated with this

strong curvature drives the chain outward, resulting in further crowding and an even smaller

interhelical spacing [29]. The balance between the bending and interhelical repulsion forces

dictates the structural characteristics of the encapsidated chain, and the pressure exerted by

this chain on the capsid wall. This balance of forces will be discussed theoretically and

demonstrated numerically in the following sections.

DNA packaging is an active process; a motor protein [1,30] is responsible for pushing

the phage genome into the capsid. This motor protein appears in the infected bacterial cell as

one of the viral gene products. The ejection process, on the other hand, is to a large extent

spontaneous. Ejection is essentially driven by the work of packaging, which is stored as

elastic energy in the genome itself.

Theoretical models for bacteriophage DNA packaging and ejection have previously

been proposed [31-34]. Reimer and Bloomfield provided the first systematic estimates of

several free energy components in DNA packaging. Gabashvili et al. discussed the interaction

between the packaged chain and the inner wall of the capsid. In particular, they argued that

the ejection of DNA should in general be incomplete rather than all-or-none, due not only to

the possibility of chain-wall attractions but also to changes in capsid size and/or the poor

quality of solvent. Subsequent work by this group [34] studied the effects of friction on the

rate of ejection. Specifically, various kinetic scenarios were examined as possible sources of

friction: reptation of the chain along its length within the spool, rotation of the spool with

respect to the inner capsid walls, the translational motion of the dsDNA through the phage’s

hollow cylindrical tail, etc.

In the present work, we provide a model which address the processes of DNA

packaging and ejection from a bacteriophage capsid. The free energy of the DNA chain is

divided into contributions from its encapsidated and released portions, each of which is

expressed as a sum of bending and interaction energies. The free energy of the whole system

is described in Section 2.2. The interactions between neighboring chains, take into account

the fact that dsDNA repels itself strongly at small interhelical spacings, even in the presence

of polyvalent ions. These short range repulsions are modeled using a simple functional form

21

derived from the osmotic stress measurements of Rau and Parsegian [27], see Section 2.2.2.

In a previous theoretical study, Odijk [29] modeled the encapsidated chain of bacteriophage

T7 as a perfect spool, with the hexagonally packed DNA circumferentially wound around the

main (tail) axis. He derived the interhelical distance by balancing the bending and interstrand

interaction components of the packing energy, finding good agreement with the experimental

results of Cerritelli [28]. Our model is similar to Odijk’s in that bending and interhelical

repulsion are treated as the most important components of the DNA packing energy.

However, our analysis focuses on structural and energetic changes during the injection (or

loading) process, as well as the forces and pressures involved. Our DNA-DNA interaction

potential is also quite different from Odijk’s, involving an attractive minimum and an

exponential repulsion. Furthermore, unlike Odijk we do not assume that the encapsidated

aggregate is a perfect spool (which is, however, an excellent approximation for highly loaded

nucleocapsids or a purely repulsive interhelical potential). Rather, we assume an arbitrary

uniaxial profile and allow it to evolve continuously. The structure of the encapsidated chain is

determined by functional minimization of the free energy with respect to the profile and

interhelical distance, subject to boundary conditions imposed by the presence of an

impenetrable capsid wall (Sec. 2.2.4).

In Section 2.3, we present the main results of our theory. Section 2.3.1 deals with the

structural evolution of the condensate structure, Section 2.3.2 describes the contributions of

various forces to the ejection, Section 2.3.3 provides an approximate model, and 2.3.4

proposes that the ejection process can be controlled by osmotic pressure. Our central finding

is that the average pressure of the DNA is a strongly increasing function of its encapsidated

length. More explicitly, for a typical phage (such as λ) whose capsid size is comparable to the

genome’s (dsDNA) persistence length (ξ≈500Å) yet hundreds of times smaller than the

DNA contour length (~15μm), we find that the pressure increases sharply from just a few

atmospheres when half the genome is packaged to ~50 atm when the capsid is fully loaded.

The largest rate of increase occurs at a loading fraction of ~3/4, at which point the interhelical

spacing begins to drop sharply. We show in particular that this is the point where the

packaged DNA is no longer able to fill the capsid by winding into loops of smaller radii of

curvature due to the prohibitive elastic energy cost that would have to be paid by bending on

these small length scales. Beyond this point, additional DNA must be accommodated by

crowding the chain (i.e., by decreasing the separation between neighboring chain segments)

and thereby experiencing the energy increases associated with short-range repulsions.

22

Looking in the opposite direction, we find that the ejection process consists of two stages.

The first stage is associated with a rapid release of DNA compression. The second stage set

in after about third of the genome has been released and the interhelical distance has nearly

reached its optimal value. Thus, the second stage progresses more slowly, and is associated

with the aggregation of both condensates (one outside in solution, and one within the capsid)

into one aggregate via an “Ostwald-like” mechanism. The second stage of ejection is mild

enough to be arrested by influencing the details of the system (such as the osmotic pressure in

solution). This prediction was recently confirmed by experiments. If the loading and ejection

of DNA into phages (or alternatively liposomes) can be controlled, it may enable us to

harness the process for gene/drug delivery purposes.

Figure 2.3 The DNA condensate in solution. (a) AFM image taken from Golan, Pietrasanta, Hsieh and

Hansma, Biochemistry 38:14069, 1999 [11] (b) Electron microscopy image taken from Hud and Downing,

PNAS 98:14925, 2001 [10]. The interstrand distance is 0 28d = Å in this image. (c) A schematic drawing of a

DNA torus. R is the average radius of the torus, r is the radius of the torus cross-section, and 0d is the

interstrand distance.

2.2 Theory

2.2.1 Model and Free Energy

As already mentioned in the Introduction, dsDNA in a solution containing multivalent ions,

such as polyamines (e.g., spermine and spermidine), is known to condense into toroids with

an optimal interhelical distance of d0=28Å. Inside the capsid, however, the DNA can not be

packed in its optimal structure and density. Therefore, the viral genome fills the capsid with

much larger density than in solution and creates a spool, (see Figure 2.6). As the ejection

proceeds, part of the DNA is ejected into solution and creates a torus outside, while the DNA

structure inside the capsid starts to relax. The dsDNA is ejected through a long protein tail of

a cb

23

~150 nm in length and with a diameter which is only slightly larger than the dsDNA

diameter. As a result, it is reasonable to assume that the DNA translocation through the tail is

slow enough and thus both the internal and external chains have sufficient time to relax

during every stage of the ejection process.

Therefore, the full free energy of the system can be written as a sum of the free energies of

the DNA within the capsid and that in solution.

( ) ( ) ( )total out capsid out solution outF L F L L F L= − + (2.1)

We use in outL L L= + to denote the total length of the dsDNA chain, with outL denoting the

length of DNA found in solution (ejected) and inL denoting the length of DNA remaining in

the capsid (or loaded).

The free energy of a DNA condensate, either within the capsid or in solution, can be written

as a sum of three terms

cohesion surface elasticF F F F= + + (2.2)

The first term is the cohesion energy, ( )cohesionF d Lε= − where ( )dε is the interaction energy

per unit length of DNA in the bulk of the hexagonally packed aggregate. It accounts for the

effective attraction between DNA helices. The next two terms are relevant for a finite size

aggregate, such as the toroidal or spool-like condensates of interest here. The second term,

represents the surface correction to the free energy, ( )surface surfaceF L dα ε= where surfaceL is the

total DNA length at the condensate’s surface, and α is a geometrical factor expressing the

fraction of DNA-DNA contacts lost upon creating a surface. In all calculations, we use

1/ 2α = (choosing a somewhat smallerα , e.g., 1/ 3α = , does not affect our results). The last

term, elasticF is the elastic bending energy associated with the semi-flexible DNA chain. It is

an integral of local contributions, ( )elastic elasticF f s ds= ∫ , with 2

1( )2 ( )elasticf s

R sκ

= the 1D

bending energy per unit length at point s along the DNA contour, and ( )R s the local radius

of curvature at s . Bk Tκ ξ= is the 1D bending rigidity, with 500ξ = Å the persistence length

of dsDNA (see Sec. 1.2.3). Since the DNA chain is condensed inside the capsid as well as

outside - in solution, the configurational entropy of the chains is not taken into account.

Structural measurements on various viruses indicate that the symmetry of the DNA

condensate inside the phage capsid is uniaxial rather than spherical [28,35], presumably

because the stiff DNA chain is packaged through a unique entry hole (the portal). Thus, we

24

assume that the DNA condensate in solution as well as in the capsid posses cylindrical

symmetry, as shown schematically in Figure 2.4. Following Ubbink and Odijk, [29,36] we

describe the shape of the condensate in terms of the condensate cross-section, the profile

function, ( )h r , shown in Figure 2.4.

Using the profile function, the free energy terms can be written as

( ) ( )1/ 2 1/ 22 2( ) ( ) ( ) 4 ( )2 2 1 1

2 2 2 2

out out

in in

R Rsurface

surfaceR R

d L d Area d dF r h dr r h drd d d

ε ε ε πεπ ⎡ ⎤ ⎡ ⎤′ ′= = = + = +⎣ ⎦ ⎣ ⎦∫ ∫ (2.3)

2 2 2

( ) 2 2 ( ) ( )42 2 3

out out out

in in in

R R R

elasticR R R

L r r h r h rF dr dr drr r S rd

κ κ π κπ⋅= = =

⋅∫ ∫ ∫ (2.4)

Where 23 / 2S d= is the area of a unit cell in a hexagonal lattice and the integrals in

equations (2.3) and (2.4) extend from the inner to the outer radius of the condensate.

The full free energy functional of a DNA condensate is therefore

( )2

2

( ) ( )( ( ), ( ), ; ) ( ) 4 1 ( )23

out

in

R

DNAR

h r dF h r h r r d d L r h r drdrd

κ εε π ⎛ ⎞′ ′= − + + +⎜ ⎟⎝ ⎠∫ (2.5)

The first term is the bulk cohesive energy of the condensate, as if it were a portion of an

infinite hexagonal array of DNA rods with an inter-strand distance d . The second term

accounts for the finite size of the curved condensate, (i.e., elastic and surface free energies).

For any given d , the profile function must satisfy the volume conservation constraint,

2

4 ( ) ( )2

out

in

R

DNAR

dh r rdr L V dπγ π ⎛ ⎞= =⎜ ⎟⎝ ⎠∫ (2.6)

Where ( )DNAV d denoting the volume of the condensate and 0.91γ = is the maximal packing

fraction introduced in Section 2.1.

Throughout this chapter, energies are measured in units of Bk T and length in units of ξ .

Figure 2.4 A schematic drawing of the DNA condensate. d is the inter-strand distance, r is the distance from

the center of the condensate and h(r) is the profile function.

25

2.2.2 DNA-DNA Interaction Potential We still miss the dependence of the interaction energy on the inter-helical distance, i.e., the

function ( )dε . The minimal hard core distance between neighboring dsDNA molecules is

min 20d = Å. However, strong electrostatic repulsion and hydration forces induce an inter-

strand repulsion at larger distances ( 25 30≈ − Å). The van der Waals attraction is generally

weak compared to these forces. Yet, polyvalent cations can act as condensing agents and

induce an attractive minimum in ( )dε .

In order to account for the correct interaction potential, we used experimental results

reported by Rau and Parsegian [27] where the interhelical distance between hexagonally

packed DNAs was measured as a function of the osmotic pressure operated on them, ( )dΠ .

The equilibrium interhelical distance was found by extrapolating the curve to zero osmotic

pressure ( 0( ) 0dΠ = ) to get 0 28d = Å. (Recently, in a cryelectron microscopy study, Hud and

Dowing [10], found that, in solution containing polyvalent cations, the dsDNA of the λ-phage

condeses into a well defined torus with an interhelical spacing of 0 28d = Å).

Figure 2.5 The cohesive energy per unit length of DNA packed in a hexagonal array, as a function of the

interstrand distance. The inset illustrates hexagonal packing of dsDNA rods.

We fitted the experimental data to the form: [ ]{ }0 0( ) exp ( ) / 1d d d cΠ = Π − − − with

0 28d = Å, 40 1.2 10 /Bk T−Π = × Å3 and 1.4c = Å.

Integration of the full pressure curve gives the repulsive part of the potential. That is, 0

0( ) ( ) 2 ( )dd

d

d d d d dε ε π ′ ′ ′− = − Π∫ . The obtained potential is shown in Figure 2.5.

26

The attractive part of the potential (i.e., ( )dε for 0d d> ) turns out to play no role in our

analysis or calculations. This is not a surprise remembering that the DNA is highly

compressed inside the capsid while optimally packed in solution and so the regime of interest

is 0d d≤ . The attractive part of the potential is shown (dashed curve) in Figure 2.5 only for

visual purposes.

2.2.3 Analytical Model for a Toroidal DNA in Solution

In order to estimate the minimum of the cohesion potential, we derived a simple approximate

analytical model for a DNA torus in solution. Using the structural values of the condensed

torus known from experiments, we can estimate the potential minima.

As shown schematically in Figure 2.3c, we denote the major radius of the torus is by R and

the radius of the torus cross-section by r . Thus, 2/ 2 2 / 4 /surfaceL Area d r R d rR dπ π π= = ⋅ = .

Using the volume constraint, 2 2( / 2) 2V L d R rπ γ π π= = ⋅ ⋅ , we get ( )1/ 2/ / 8r d L Rπγ= and

thus, ( )1/ 224 / 8surfaceL RLπ πγ= . Therefore, the full free energy can be written as

22 2

1 1 1( , , ) ( ) ( ) ( ) 2 ( )2 2 8 2torus surface

L RL LF L R d d L d L d L dR R

ε ε κ ε π ε κπγ

= − + + = − + + (2.7)

The first two terms here, represent the bulk and surface contributions to the interaction free

energy and the third term is the average bending energy of a torus.

R and d are independent variables and hence the free energy has to be minimized with

respect to both of them. Minimizing the free energy with respect to d , i.e., 0torusF d∂ ∂ = , we

find 0d d= . Therefore, we can replace ( )dε by 0 0( )dε ε≡ . Minimizing the free energy with

respect to the average radius of the torus, i.e., 0torusF R∂ ∂ = , we get

2/5

1/5

0eqR c Lκ

ε⎛ ⎞

= ⎜ ⎟⎝ ⎠

(2.8)

with 0.75c = a numerical constant.

Using Bk Tκ ξ= and the experimentally known dimensions of toroidal condensates of DNA

in solution, [37,38], (for instance, for λ-DNA, 330L ξ= , 287 0.575eqR ξ≅ Α = ), we obtain

0 35.3 /Bk Tε ξ= .

Substituting eqR from Eq. (2.8) into Eq. (2.7), we obtain an approximate expression for the

torus free energy

27

1/5 4 /5 3/50 0( )torusF L L c Lε κ ε= − + (2.9)

Here, the first term is the cohesive energy which scales with L. The second term is the sum of

the surface and elastic free energies, both scaling with 3 5L . This is a first clue for the joint

role played by the surface and elastic free energies during the packaging/ejection processes.

We will elaborate on the relative contribution of the various free energy terms in Section

2.3.2.

2.2.4 Method of Solution

The equilibrium profile, corresponding to a given d is obtained by minimizing F subject to

Eq. (2.6), or equivalently, by minimizing DNAF F Vλ= + with λ denoting the Lagrange

multiplier conjugate to the volume conservation condition, Eq. (2.6).

For the condensate in solution, minimizing of F results in Euler-Lagrange equations [36],

i.e., d 0dh r h

∂ ∂⎛ ⎞− =⎜ ⎟′∂ ∂⎝ ⎠L L where ( )2

2

( ) ( ) 1 ( ) ( )23

h r d r h r h r rdrd

κ ε λ′≡ + + +L .

The solution of the Euler-Lagrange equations is:

( )1/ 22

2 ln( ) where ( )3 ( )1in

r

R

y r rd Ch r dr D yr d rd dy

λεε

= + = + +−

∫ (2.10)

The constants ,C D and the Lagrange multiplier λ, can be obtained from the boundary

conditions, ( ) ( ) 0in outh R h R= = ; ( )inh R′ = ∞ ; ( )outh R′ = −∞ and from the volume constraint,

Eq. (2.6).

As we shall see in the next section, the solution for Eq. (2.10) subject to the boundary

conditions indicates an essentially perfect toroidal shape for the outside condensate.

Furthermore, the condensate in solution is relaxed, i.e., 0d d= .

Minimizing F for the DNA condensate within the capsid is more complicated owing to the

additional boundary conditions imposed by the presence of the rigid, impenetrable, capsid

wall. This restriction (implying ( )1/ 22 2( ) Ch r R r≤ − ) prohibits analytical evaluation of ( )h r .

Consequently, F was minimized numerically, with { }( )( ) ,d h r r substituted from the

packing constraint, Eq. (2.6), subject to the condition that ( )h r cannot exceed the capsid

limits.

28

2.3 Results and Discussion

The numerical results presented and analyzed in this section are concerned with the structure,

energy, force and pressure characteristics of DNA packaging in viral capsids and its ejection

into solution. Most of the calculations were carried out for a model system, resembling the λ-

phage. Thus, the protein capsid is modeled as a spherical shell of radius 0.55 275CR ξ= = Å.

The total length of the viral genome is 330 16.5 mL ξ μ= = . For comparison, the structural

properties of the encapsidated T7 dsDNA, were analyzed for a fully loaded phage.

The free energy and structural properties of the partially loaded capsid, has been determined

by minimizing both contribution to totalF , i.e., ( )capsid in outF L L L= − and ( )solution outF L (see,

Eq. (2.1)), with respect to their DNA packing profiles, ( )h r .

Notice that since we assume a reversible process, we allow ourselves to refer to the

process either as an ejection process (where the reaction coordinate is outL ) or as a loading

process (where the reaction coordinate is in outL L L= − ) and we interchange between these

two viewpoints.

2.3.1 DNA Condensate Structure

The minimization of ( )solution outF L reveals that the DNA chain in solution organizes into a

relaxed 0( 28d d= = Å), perfectly shaped torus, for all values of outL . Different outL values

results in different torus sizes. This result is consistent with the toroidal structures found

experimentally in solution [9-11]. Inside the capsid, the condensate structure changes during

the ejection (loading) process. For small values of inL , we find that the encapsidated chain is

condensed into a perfect and relaxed 0( )d d= torus. This behavior prevails as long as the

loaded length of DNA, inL , is small enough to ensure that the relaxed torus can be

accommodated within the capsid. Namely, as long as ( )( ) 2eq in C out inR L R R R< − − . Figure

2.7 which shows DNA packing profiles obtained via full minimization of ( )capsid inF L

confirms this behavior for small loading fractions; 1 4inL L ≤ or so, in good agreement with

the value predicted by Eq. (2.8). As soon as the outer radius of the relaxed torus exceeds CR ,

the shape of the encapsidated condensate must deviate from that of a perfect torus. In our

treatment of the λ-phage this happens when 4inL L≈ . Then, a continuous transformation

begins going from a torus to a spool-like structure (see Figure 2.6 and Figure 2.7). In the

29

course of this continuous, torus-to-spool transition, the inner face of the condensate becomes

increasingly flatter whereas its outer face adopts the shape of the capsid’s wall. This

transformation evolves gradually with inL and is driven by the tendency of the DNA bundle to

minimize the bending energy penalty associated with lowering its inner radius, inR , in the

course of DNA loading.

Figure 2.6 A schematic drawing of a DNA spool

that is condensed inside the viral capsid. Rin is the

inner radius of the spool (where DNA is absent)

and Rc is the radius of the viral capsid.

Figure 2.7 DNA packing profiles within the viral

capsid. The figure shows the contour lines

corresponding to the top-right quarter of the

condensate’s cross section, for different values of

the loading fraction, /inL L , as labeled.

When Most of the DNA is found inside the capsid, i.e., for large values of inL , we find that

the DNA chain is condensed into a perfect spool, with the hexagonally packed DNA

circumferentially wound around the main (tail) axis. This result is consistent with the work

done by the group of Cerritelli, [28].

In addition to the change in condensate’s shape, there is another degree of freedom for

accommodating the increasing amount of DNA loaded into the capsid, namely, compressing

the bundle to higher densities than those corresponding to the optimal interhelical spacing,

0d d= . As discussed in Section 2.1, simple geometric considerations reveal that the maximal

length of hexagonally packed DNA which can be loaded into the λ capsid with using an

optimal interhelical spacing 0d d= , is 0.8inL L≈ which is significantly less than the total

genome length. Therefore, to complete the loading process, the DNA helices within the

condensate are compressed to a larger density than the optimal, i.e., 0d d< see Figure 2.8.

However, this compression involves a strong exponential increase in interhelical repulsion

30

fast(d<d0)

slow(d d0)≅

d

fast(d<d0)

slow(d d0)≅slow

(d d0)≅

d

and thus it becomes operative only when the bending energy penalty exceeds the increase in

interhelical compression energy. Before this crossover point, DNA packing into the growing

spool enables an increase in the condensate’s volume at a small change in inR and thus

tolerable bending energy cost. In Figure 2.7 we see that the torus-to-spool transition begins at

0.3inL L ≈ and is essentially completed when 0.7inL L ≈ , with 0d d= throughout this

range; see Figure 2.8. Thus, in this range the increase in bending energy is, indeed, still small

compared to that of compressing the bundle into the repulsive interaction regime ( 0d d< ).

DNA compression begins immediately afterwards, that is the interhelical distance begins

falling below 0d approximately at / 0.7inL L ≈ . Clearly then, above this loading fraction, the

bending energy penalty increases so steeply with the decreasing (already small) values of inR

that the DNA is compressed to lower values of d despite the substantial energetic cost. The

competition between the bending and repulsion forces is further discussed in Section 2.3.2.

Figure 2.8 The black curve describes

the total free energy of the DNA chain,

tot capsid solutionF F F= + , as a function of

the ejected length, outL ( 0totF ≡ at

0outL = ). The purple curve shows the

corresponding variation in the inter-

strand spacing d.

We compared our results for the interhelical distance with available experimental

data. For the λ-phage, Earnshaw and Harrison derived the interhelical spacing from

diffraction measurements. Their results are given in Table 1, including one value for an

overloaded capsid, (the experimental error is on the order of 2± Å). The results predicted by

our theory, corresponding to the capsid radius ( 0.55 275CR ξ= = Å) used in all our

calculations in this work, are given in the second column. The agreement between theory and

experiment is quite reasonable. Better agreement can be obtained by using a slightly smaller

capsid volume, 0.54 270CR ξ= = Å, as shown in the third column. It should be stressed,

however, that our main goal in showing this alternative calculation is to demonstrate how

sensitive the interhelical distance (and hence the compression energy) is to very small

variations in capsid volume, or more precisely to the ratio between capsid and DNA volumes.

31

More recently, Cerritelli et.al., determined the interhelical spacing for a T7 capsid for several

loading fractions, based on electron microscopy measurements, [28]. The results are shown in

Table 2. The T7 capsid, similar to that of λ, is nearly spherical, with a radius

0.55 275CR ζ= = Å, but it contains a protein connector of height CR and radius 105b = Å,

restricting the inner spool radius within one hemisphere of the capsid to inR b≥ . Its full

genome length is 13.6 m=272L μ ξ= . Taking these special characteristics into account, we

applied our free-energy minimization procedure to calculate the interhelical spacing d in T7

for the three loading fractions reported experimentally. The results, revealing very good

agreement with experiment, are given in Table 2. Similarly good agreement between

experiment and theory had previously been obtained by Odijk [29]. Our larger values for d

indicate the steeper short range repulsion of our interhelical interaction potential.

Interhelical distance [Å]

Theory

/inL L

Experiment

275CR = Å 270CR = Å

105% 23.2 24.0 23.4

100% 23.6 24.6 24.0

89% 24.6 25.9 25.3

88% 24.7 26.0 25.5

78% 25.8 27.1 26.7 Table 1 Interhelical spacing in the λ-phage for five (high) values of the loading fraction, Lin/L, as measured by

diffraction measurements [35], and calculated theoretically for two different capsid radii.

Interhelical distance [Å]

/inL L Experiment Odijk This work

100% 25.4 24.9 25.6

92.1% 26.4 25.9 26.5

84.4% 27.5 27.0 27.4

Table 2 Interhelical spacing in the T7-Phage for three (high) values of the loading fraction, Lin/L , as measured

by Cerritelli et.al. [28] and calculated theoretically by Odijk [29] and by the present work.

32

2.3.2 Ejection Mechanism – Forces and Energetics

Figure 2.8 displays the total free energy of the DNA chain, totalF , as a function of the ejected

length out inL L L= − . Remember that viewed in the opposite direction, i.e., as a function of

inL , the figure describes the loading free energy. As demonstrated in the figure, the ejection

process takes place in two stages. The first stage is very rapid and associated with a fast

release of the DNA compression. In the course of this stage, the DNA chain density is higher

than its optimal density, i.e., 0d d< and towards the end of it, the density reaches its optimal

value. During the second stage of the ejection process, the interhelical distance is already

optimal 0d d= . The second stage is slower and it is driven by interplay between bending and

surface free energies. The beginning of this stage involves an additional release of stress

stored in the DNA condensate inside the capsid. This stress, which is much smaller than the

stress associated with DNA compression, is released while the DNA condensate shape slowly

transforms from a spool (at optimal inter-helical distance) to a torus. When both the

condensate inside the capsid and outside in solution are optimized toroids, they aggregate to

create a single condensate via an “Ostwald-ripening-like” mechanism. We refer to the two

stages as “fast” and “slow” since using linear force approach, the velocity is proportional to

the free energy slope (that is, v F∝∇ where v is the velocity). Viewed in the reverse

direction, only a small force needed for DNA loading as long as 0d d= , i.e., as long as the

loading fraction is less than / 0.7inL L ≈ . At somewhat higher loading fractions, the force

needed to load the DNA increases dramatically. Quantitatively, the loading force is defined

by total inf F L= ∂ ∂ . The individual surface, bending and interstrand repulsion contributions to

the loading force are shown in Figure 2.9. Note that the bending and surface contributions are

relatively small throughout the loading process. Thus, the total force curve essentially

overlaps the DNA-DNA interaction contributions. This fact is somewhat misleading. It

suggests that the role played by the bending energy is small. As a matter of fact, the opposite

is true, the bending energy is so large that it is much more favorable to compress the DNA

and pay in repulsion energy than to keep the optimal density constant and decrease the inner

radius (thereby increasing the curvature and thus the elastic energy). To illustrate this

interpretation, we show in Figure 2.9 the results of a calculation corresponding to a loading

process where d is not allowed to decrease below a certain interhelical distance, say d=27Å

(That is an infinite repulsive wall has been superimposed on our ( )dε at d=27Å). For this

33

system, once the interhelical spacing reaches this limiting value, additional DNA length can

be loaded only by decreasing the spool inner radius. In that stage, the repulsive force

becomes constant, whereas the bending force increases extremely rapidly. More significantly,

the increase in the bending force is steeper than the increase in the repulsive force in the

unconstraint system.

Using optical tweezers to pull on the dsDNA genome of the bacteriophage φ 29,

Smith et al. [39] have recently measured the force necessary to stop (stall) the loading of

DNA by the portal motor protein of this phage. The capsid of this virus is, roughly, a 420 Å ×

540 Å prolate ellipsoid and its available volume is 1/ 2∼ the volume of the λ-capsid.

Correspondingly, its genome length (19.3 kilobases 65ξ≈ ) is less than half that of λ.

Notwithstanding these differences, the loading mechanism in φ 29 appears to involve two

regimes: a fast stage (i.e., small stalling force) followed by a slow loading stage, indicating

that the action of the motor protein is progressively resisted by an opposite force exerted by

the packaged genome portion.

Figure 2.9 The surface, bending and DNA-DNA repulsion components of the loading force, as a function of the

loaded genome length. The total force curve overlaps the repulsive component. The dashed curves describe the

repulsion and bending forces corresponding to a model calculation in which d is not allowed to fall below 27Å:

the effect of this constraint on the surface term is negligible and therefore not shown.

From their stalling force measurements, Smith et.al [39] have concluded that the internal,

opposing force starts increasing when 1/ 2∼ of the genome is packed, reaching 50 pN∼

toward the end of the loading process. These values are of the same order of magnitude as

those derived from our model, see Figure 2.9.

A quantitative comparison between theory and experiment is not warranted here

because the structural characteristics of the φ 29 phage are quite different from those of our

present model. Also, the force measurements were carried out in solutions containing

34

counterions (sodium and magnesium) which do not mediate DNA attraction, suggesting the

need for a larger loading force. It may also be noted that the loading process may involve

dissipative losses associated with the dynamical character of the experiment, suggesting the

measured force is an upper bound to the calculated, statistical-thermodynamical force.

2.3.3 An Approximate “Two State Model”

The behavior described in Sections 2.3.1 and 2.3.2, can be modeled to a good approximation as

a “two state model”. Assuming that the DNA can be either a torus or a spool, we can write an

expression for the free energy of these two structures.

The free energy of a DNA torus

An analytical expression for a DNA torus was given in Section 2.2.3 (Eq. (2.7)). The

expression given is correct as long as the DNA is not constrained by the capsid walls. As soon

as the DNA condensate becomes a “squeezed torus”, it is useful to convert to inR and outR

instead of r and R . The relations between these variables are: ( ) / 2in outR R R= + and

( ) / 2in C inr R R R R= − = − . For DNA lengths such that the capsid walls act as a constraint,

out CR R= and there is only one degree of freedom. Therefore, the interhelical distance d (or

alternatively the inner radius inR ) is allowed to adjust so as to minimize the free energy of the

DNA condensate. Using Eq. (2.7), and changing variables, we get that the free energy of the

torus is:

22

( ) 2( , ) ( ) 2 ( )16 ( )

C intorus in

C in

L R R LF L R d L dR R

κε π επγ+

= − + ++

(2.11)

where the interstrand distance d is dictated by the volume constraint,

2 2( / 2) 2V L d R rπ γ π π= = ⋅ ⋅ to give ( )( ) C inC in

R Rd R RL

πγ+= − .

The free energy of a DNA spool

A schematic drawing of a spool is shown in Figure 2.6. Following Section 2.2, the free

energy of a spool-like condensate can be written as

( , ) ( ) surface elasticspool in spool spoolF L R d L F Fε= − + + (2.12)

where

35

0 100 200 300 L

out [ξ]

-11000

-10000

-9000

-8000

-7000

Ftot

[kT] torus spool

( ){ }

( )

2 2 1/ 2 2

1/ 22 2

( ) 4 ( ) 4 sin arccos /

2

4 ( )

2

in C in C in Csurfacespool

C in in C

d R R R R R RF

d

d R R R Rd

ε π π

πε

− + ⎡ ⎤⎣ ⎦=

− +⎛ ⎞= ⎜ ⎟⎝ ⎠

(2.13)

2 2 2 2 2 22 2

2 2 2

2 2 4 ln2 3 3

C C

in in

R RC C C C inelastic

spool C C ininR R

r R r R r R R RF dr dr R R R

Sr r Rd dπκ πκ πκ ⎡ ⎤⎛ ⎞− − + −

⎢ ⎥⎜ ⎟= = = − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∫ ∫ (2.14)

As before, the volume constraint, Eq. (2.6) dictates a relation between d and inR . In this

case, the volume constraint can be expressed as: ( ) ( )3/ 22 2 22 4 / 3C inV L d R Rπ πγ= = − , and

thus, ( )3/ 22 216 3C ind R R Lγ= − . Using the two free energies (Eqs. (2.11), (2.12), (2.13) and

(2.14)), we obtain two curves for the system free energy by modeling the encapsidated chain

(i.e., capsidF contribution to totalF ) as being either a perfect torus or a perfect spool for all

0 inL L≤ ≤ . As explained, the interhelical distance is allowed to adjust so as to minimize

capsidF for any given inL . The DNA chain in solution is always assumed to condense into a

torus.

Figure 2.10 The total free energy of the DNA chain, tot capsid solutionF F F= + , as a function of the ejected length,

outL . The red and blue curves describe totF for a DNA chain whose encapsidated part, capsidF , is treated as a

perfect spool or a perfect torus (but adjustable d), respectively. The black dashed curve shows totF for the

entire range of possible outL values.

36

The results of this calculation are shown in Figure 2.10. As expected, the torus curve provides

a good model for the early stages of loading (large out inL L L= − ) but fails badly as soon as its

outer radius exceeds the capsid radius. The spool curve provides an excellent approximation

to the late stages of loading (early stages of ejection). The fact that the free energy curve for

the full minimization coincides with the “perfect torus” free energy curve for small outL

values and with the “perfect spool” free energy curve for large outL values is an additional

(indirect) proof of the torus-to-spool transition described in Section 2.3.1.

2.3.4 Pressure

The average pressure exerted on the capsid walls, for a given length of loaded DNA, inL ,

may be defined in analogy to the thermodynamic pressure:

capsid

C

FP

V∂

= −∂

(2.15)

With CV denoting the capsid’s volume. Notice, however, that the DNA condensate inside the

capsid is not isotropically packed and thus, in principle, the local pressure on the capsid wall

may vary both in magnitude and in direction. In other words, the derivative in Eq. (2.15) may

depend on how the volume is changed. In the following, we shall use P, as define in Eq.

(2.15), for the free energy change corresponding to a change in CV keeping the spherical

shape of the capsid, i.e., 24C C CdV R dRπ= ( CR denoting the radius of the spherical capsid

shell). We shall refer to P as the average isotropic pressure.

An alternative procedure for calculating the pressure inside the capsid, as well as its

anisotropic distribution, is to replace the rigid walls by a repulsive potential resisting the

expansion of the (highly compressed and strongly bent) DNA condensate. We do it by

representing the viral capsid by an elastic spherical shell of radius CR , consisting of a

continuous distribution of radial springs. More explicitly, let , ,R θ φ denote a system of polar

coordinates whose origin coincides with the center of the empty capsid and whose z-axis

coincides with the axis of the condensate. Consider now a small area element, 2 2 sinC CdA R d R d dθ θ φ= Ω = , on the capsid’s envelope at ( , )θ φ . If the direction

( , )θ φΩ = corresponds to a point where the outside of the packaged chain profile ( )h r lies at

a distance RΩ from the center such that CR RΩ > , then we associate a local harmonic restoring

37

force ( )Ck R R dAΩ− − with this area element. Since the DNA condensate is uniaxial, we are

only concerned with changes in capsid shape which are independent of the azimuthal angle

φ , i.e., C CR R R R R Rθ θδ δΩ Ω≡ − = − ≡ . The total elastic energy penalty associated with a

small ( C CR R Rθ − << for all θ ) but arbitrary (uniaxial) deformation of the capsid wall is

( )2 21 2 sin2wall C CU k R R R dθ π θ θ= −∫ (2.16)

The total free energy of the loaded virus, is now given by

capsid DNA wallF F U= + (2.17)

Where DNAF denotes the packing energy of the DNA condensate as given by Eq. (2.5).

We can express wallU as a functional of the profile function, ( )h r , by using the relation

2 2( )R h r rθ = + ( ( ) tanh r r θ= ). The equilibrium profile, is dictated by the minimum of

capsid DNA wallF F U= + with respect to { }( )h r , or equivalently { }Rθ , that is

0capsid DNA wallF F Uδ δ δ= + = for all possible variations in capsid and condensate shapes. The

equilibrium condition reads, ( ) ( )2 22 sin 0capsid DNA C CF d F R k R R R Rθ θ θδ π θ θ δ⎡ ⎤= ∂ ∂ + − =⎣ ⎦∫

for all { }Rθδ around the equilibrium configuration. Using these definitions, the local

pressure, ( ) ( )p p θΩ = , exerted by the condensate on the capsid’s wall along the direction

( , )θ φΩ = is given by

( )2

1( ) DNAC

C

Fp k R RR R θ

θ

θ ∂= − = −

∂ (2.18)

The equilibrium shape of the nucleocapsid is given by { }Rθ , which slightly deviates from a

sphere of radius CR , characterizing the empty capsid. Clearly, as k increases, the deviation

from the empty capsid dimensions, CR Rθ − decreases for all θ . However, the product

( ) ( )Ck R R pθ θ− = approaches a constant value, corresponding to ( )( )21 C DNAR F Rθ− ∂ ∂ with

the derivative evaluated at CR Rθ → . In this hard-wall capsid limit, we have 0wallU → , so

that all the energy of the loaded phage is stored within the DNA condensate, capsid DNAF F= . It

may be noted, however, that the procedure just outlined for evaluating Rθ and ( )p θ is also

applicable to DNA (or other) condensates trapped within compartments bounded by softer

38

walls such as lipid vesicles [22,40] or viral pro-capsids. It can also be applied to capsids

characterized by non-uniform k’s or non-spherical equilibrium shapes.

In order to connect between the pressures calculated in the two different approaches (Eq.

(2.15) and Eq. (2.18)), one should notice that the change in free energy of the condensate,

upon arbitrary deviation of the capsid’s shape is given by 22 sin ( )DNA CF d R p Rθδ θ π θ θ δ= ∫ ,

the corresponding change in volume is 22 sinC CV d R Rθδ θ π θ δ= ∫ . The average pressure on the

capsid walls can be defined by DNA CP F Vδ δ= − and will depend on the exact shape of the

deformation. The average isotropic pressure, defined in Eq. (2.15), was defined as the change

of capsidF upon a uniform spherical expansion of the hard capsid walls. For this case,

CR Rθδ δ= for all θ (A uniform spherical expansion changes from a spherical shell of radius

CR to a spherical shell of radius C CR Rδ+ ), and thus we find,

1 ( )sin2

P P p dθ θ θ= = ∫ (2.19)

with ( )( ) Cp k R Rθθ = − evaluated in the large k (hard-wall) limit.

The average pressure exerted by the DNA on the capsid wall as a function of the

loaded genome length is shown in Figure 2.11. The increase in the average pressure, P,

correlates with the decrease in the interhelical distance, d to values lower than the optimal

spacing, d0. This happens at 3 / 4inL L ≈ (see Figure 2.8), where the encapsidated chain

already forms a pool condensate, Figure 2.7. Qualitatively, the rapid increase of P with inL

results from the decrease in the interhelical distance (d) that implies a stronger DNA-DNA

repulsion and hence a larger pressure on the capsid walls. This can be further reinforced by

the following. At the late stages of loading, the capsid’s volume CV is mostly occupied by the

DNA spool, whose volume is 2DNA inV L d∼ . To a good approximation, a uniform increase in

CV implies a corresponding change in DNAV (with the DNA condensate staying as a spool

with the same cylindrical core dimension inR ). This approximation is reasonable since the

shape transformation into a spool has evolved in order to allow packing of a larger amount of

genome without a change in d, and for the late stages of loading, a small expansion of the

capsid will not change d back to the optimal spacing. Thus, for the late stages of loading, we

can write approximately dC DNA indV dV L d d≅ ∼ and ( ) ( )cap in inF L L dε−∼ .

39

Figure 2.11 Solid line: the average (thermodynamic)

pressure on the capsid wall , CcapsidP F V=−∂ ∂ as a

function of the length of DNA loaded into the capsid.

Dashed line: the average pressure calculated for a

capsid wall represented by a harmonic restoring

force with 7 410 Bk k T ξ= . For 9 410 Bk k T ξ≥ ,

the calculated pressure is indistinguishable from the

thermodynamic pressure. Dotted line: the osmotic

pressure in a macroscopic phase of hexagonally

packed DNA (taken from Parsegian et.al.).

Figure 2.12 The pressure profile along one

hemisphere of the viral capsid, for 290inL ξ= .

Therefore ( )( )1/capsid C capsid DNAP F V F V d dε= −∂ ∂ ≅ −∂ ∂ ∂ ∂∼ . In other words, the average

pressure inside the (nearly fully loaded) capsid is, to a good approximation, given by the

pressure ( )dΠ determined by osmotic stress measurements, [27]. Indeed, Figure 2.11

confirms that ( ) ( )d P dΠ ≈ . Also shown in Figure 2.11 is a pressure curve calculated for a

DNA condensate in a capsid bounded by elastic walls, represented by an harmonic restoring

potential with a force constant of either 710 Bk k T ξ= or 910 Bk k T ξ= . As demonstrated

in the figure, smaller values of k result in a lower P curves. This is because a softer wall

(lower k) allows the capsid to expand more easily beyond the equilibrium position of the

empty capsid. This expansion implies a lower volume fraction of the encapsidated chain and

hence a smaller pressure on the capsid wall. For a wall potential such that 910 Bk k T ξ≥ , the

pressure calculated for the elastic capsid converges to the isotropically averaged pressure

determined from Eq. (2.15) The average isotropic pressure corresponds to the hard-wall

(infinite k) limit, as discussed previously.

P

θ

40

Finally, in Figure 2.12 we show the pressure profile ( )p θ , in the northern hemisphere

of the capsid, as calculated from Eq. (2.18). Recall that ( )p θ is the local radial force per unit

area acting at the point ( ), ,CR θ φ on the wall of the capsid. Its angular average (see Eq.

(2.19)) is the average isotropic pressure. Our calculation reveals that the force is nearly

constant for a wide range of polar angles, falling sharply to zero at some finite angle which

corresponds to the cylindrical hole in the center of the spool. For 290inL ξ= this happens at

* 0.05θ ≈ as shown in Figure 2.12. Not surprisingly, larger inL ’s correspond to smaller *θ

values and to larger *( )p θ θ> values. The fact that the pressure distribution along the capsid

wall appears to be quite uniform, provides a possible explanation for the high mechanical

stability of viral capsids.

2.3.5 Incomplete Ejection

In general, the DNA within the capsid is highly compressed. As a result, the free energy of

the ejection ( totalF as a function of outL ) monotonically and quite rapidly decreases. This is

always the case at the early stages of the ejection process. However, the final stage of the

ejection is a moderate one and thus it is easily influenced by the details of the system.

Consequently, the decrease in totalF may proceed monotonically, all the way to outL L= , but

may also stop at some value of outL , *out outL L L= < , corresponding to a minimum of totalF ,

0capsidtotal solution

out in out

FF FL L L

∂∂ ∂= − + =

∂ ∂ ∂ (2.20)

The derivatives ( )capsid in capsidF L μ∂ ∂ = and ( )solution out solutionF L μ∂ ∂ = are the chemical potentials

(per unit length) of DNA within the capsid and solution, respectively. Thus, ejection proceeds

as long as the driving force, capsid solutionμ μ μΔ = − is positive, and stops if the internal and

external chemical potentials become equal at some *out outL L= . (Notice, that looking at the

loading process, the chemical potential difference is the minimum mechanical force that must

be supplied by the motor protein to load the DNA into the capsid). The ejection monotonic

behavior can be changed, for example, by having an osmotic pressure in solution.

Osmotic Pressure Effect - In principle, DNA ejection can be opposed even arrested, by

increasing the osmotic pressure in solution. Suppose that a neutral polymer (e.g., PEG) has

been added to the external solution, resulting in an osmotic pressure Π, see Figure 2.14. Then,

41

the ejection of DNA occurs against an external pressure in solution in the amount of 2

0

2sol outdV L π

γ⎛ ⎞Π = Π ⎜ ⎟⎝ ⎠

.

In other words, upon DNA ejection, solvent enters the capsid to fill up the released volume,

whereas the PEG is too large to enter the capsid and must remain in solution. This results in a

larger volume fraction of PEG polymer in solution and a corresponding free energy increase.

Adding this osmotic pressure term to the total free energy, Eq. (2.1) is replaced by

2

0( ) ( ) ( )2total out capsid out solution out outdF L F L L F L Lπ

γ⎛ ⎞= − + + Π⎜ ⎟⎝ ⎠

(2.21)

Figure 2.13 The effect of osmotic pressure in solution on the

total chain free energy, as a function of the ejected genome

length. A minimum in Ftot appears for Π ≥ 0.4 atm. Assuming

that DNA ejection stops ( or at least delayed) at Lout=L*out

corresponding to the minimum in Ftot . The inset shows L*out

as a function of the external osmotic pressure.

Figure 2.14 A schematic drawing of

ejection of DNA into a solution

which contains PEG polymers.

The osmotic pressure term in Eq. (2.21) opposes the injection, suggesting that for large

enough Π one should expect a barrier to injection at some intermediate outL . In Figure 2.13,

we show totalF as a function of outL for several values of Π. For Π values larger than a certain

threshold value *Π , a minimum in totalF (implying a barrier to injection) appears at

100outL ξ≈ , shifting toward lower values of outL as Π increases. The threshold osmotic

pressure is * 0.4 atmΠ ≈ , and can be rationalized as follows. From the totalF curve in Figure

42

2.13corresponding to Π=0, we note that during the second stage of DNA release, 0

total capsid solutionF F F≡ + decreases nearly linearly with outL , so that

0 3.6total out BF L k Tμ ξ∂ ∂ ≈ Δ ≈ ≈ constant, μΔ denoting the nearly constant chemical potential

difference during the second injection stage. Thus, a minimum in totalF is expected for

20 0.4 atmdμΠ ≥ Π ≈ Δ ≈ , consistent with the results shown in Figure 2.13.

2.4 Concluding Remarks

The basic result of our present work is a demonstration of the interplay between chain

bending and repulsion energies in determining the structural and energetic properties of

packaged DNA in phage capsids. This problem is related intimately to the more general one

of a self-repelling, semi-flexible chain confined to a volume whose dimensions are

comparable to the chain persistence length but small compared to its contour length. Our key

conclusion, based on the bending and repulsion energies of dsDNA, is that the bending

energy is dominant, in the sense that it ultimately prohibits the chain from filling in the core

of the capsid, forcing it instead to be crowded on itself in a spool-like structure. This situation

arises only after a large fraction (of order two-thirds to three-quarters) of the genome is

packaged. Up until this point, increasing lengths of the chain are accommodated by its

bending with smaller radii of curvature and thereby filling up progressively more of the

capsid volume. Beyond this point, however, additional length is accommodated largely by the

chain crowding onto its nearest neighbors, filling in a spool-like volume with decreasing

interhelical spacing. Because of the strong short-range repulsions acting between neighboring

chains, this latter process is associated with a dramatic increase in packaging stress. We show

in particular that the force required of the viral motor protein, to load the chain at the capsid

portal, increases from a few pN to tens of pN as the final 20% of the genome is packaged.

This is in good agreement with the recent experimental determination of motor packaging

force as a function of loading fraction [39]. Similarly, we calculate the angular distribution of

pressure acting on the inner wall of the capsid and demonstrate that its average is a strongly

increasing function of the fraction of chain loaded, rising steeply from a few atmospheres to

tens of atmospheres in the final 20% of packaging. We have also treated the inverse process

of DNA ejection, relevant to the first step of the viral infection cycle in which the phage

ejects its genome into a host bacterial cell. This process is driven initially by precisely the

stored stress established in the packaging step of phage replication, i.e., by the force

43

total outf F L= −∂ ∂ discussed at length above. As long as the chain free-energy totalF is a

monotonically decreasing function of ejected length outL , this force will remain repulsive and

drive the chain completely out of the capsid. In the presence of an attraction between the

chain and the inside of the viral capsid, however, the ejection force will vanish whereas some

of the chain remains inside. This is the case first treated phenomenologically by Gabashvili

et. al. [34].

More generally, we expect that the ejection of phage DNA into its host bacterial cell

will be incomplete because of the osmotic pressure in the cell. More explicitly, the high

concentration of cytoplasmic proteins gives rise to an effective force (work of insertion per

unit length) which resists the ejection force associated with stored packaging stress. Indeed,

we find that as soon as this osmotic pressure exceeds half an atmosphere (and realistic

estimates for macromolecular crowding in bacterial cells suggest that it does), at most, one-

third of the genome is ejected. Accordingly, it becomes important to investigate physical

mechanisms that make possible the delivery of the rest of the genome to the infected cell.

One scenario for pulling in the remaining DNA involves transcription of the genes that have

been delivered, i.e., translocation is driven by motor protein action of the host cell’s RNA

polymerase (see for example the case of T7 [41]). Another, alternative, scenario involves the

adsorption of DNA-binding proteins on the ejected (cytoplasmic) portion of the viral genome;

the adsorption here gives rise to an effective force (binding energy per unit length) which

pulls the rest of the chain into the cell.

In the last recent years, there is a growing interest in viruses among physicists and

chemists. Several papers were published on the ejection and packaging processes. Among

them, an article which addresses the kinetics of phage injection [42]. The kinetic model

presented consider the diffusion of DNA, the driving force due to DNA compression inside

the capsid, resisting forces associated with osmotic pressure and pulling and ratcheting forces

associated with DNA-binding proteins in the host cell cytoplasm. It is demonstrated that

stress in the capsid is the dominant factor in early ejection stages and binding particles take

over at later stages. The ability of binding molecules found in the bacterial cell to encourage

ejection was demonstrated experimentally using multivalent ions [43]. Experiments following

DNA ejection from a λ phage in the presence of PEG in solution, demonstrated the ability to

control the extent of ejection by varying the external osmotic pressure [44,45].

44

Chapter 3 Viral Budding

3.1 Introduction

In the previous chapter, we studied the infection mechanism of bacteriophages. In this

chapter, we turn to animal viruses. Most animal viruses enter their host cells via active cell

processes [47]. One common example is receptor-mediated endocytosis, in which the binding

of a viral spike protein to some specific receptor protein on the outer cell membrane triggers

the internalization of the virus inside an endosome. Lowering the endosomal pH causes

fusion of the viral membrane with the endosome membrane and the release of the viral

genome into the cytoplasm. The subsequent translation and replication of the viral genome by

the cellular machinery ultimately leads to the generation of many copies of viral proteins and

genome. As mentioned in Section 1.5, there are two possible pathways for viral assembly. In

the current chapter, we study Alpha viruses which assemble through the CAN pathway (see

Figure 1.10). That is, first their proteins and genome coassemble into nucleocapsids which,

however, still have to leave the cell and are not yet covered by a lipid membrane. These

remaining two tasks are solved simultaneously in a process termed budding [48]. In the

course of budding, the viral nucleocapsid becomes wrapped at a cellular membrane – often,

the plasma membrane and either leave the cell or at least enter the secretory pathway. In this

chapter, we address the budding mechanism of preassembled nucleocapsids1.

The scenario described above poses a critical difficulty: inasmuch as the presence of

spike proteins is crucial for the virus to be infective (no spikes, no trigger for endocytosis),

the budding mechanism must ensure that enough spikes are incorporated into the bilayer coat

during envelopment. Even though the viral genome will direct the cellular machinery to

synthesize the spike proteins and deposit them in the membrane at which budding will

ensure, this by itself does not imply that enough of them will actually end up in the viral coat

- unless they are severely overexpressed in the membrane, which appears not very

economical.

� 1 The results presented in this chapter were previously reported in [46].

45

Thirty years ago Garoff and Simons [49] proposed a solution to this puzzle which

rests on the simple idea that the spike proteins also mediate the adhesion between the

nucleocapsid and the lipid membrane. This automatically guarantees that after budding, the

mature virion contains spikes, because otherwise it would not have been able to bud in the

first place. Even though it was subsequently realized that this simple model does not hold for

all enveloped viruses (for a review, see [48]), it is by now clearly established as the

maturation route for Hepadnaviruses and Alphaviruses. The extensively studied model

system in the latter case is the Semiliki Forest virus (SFV). This is a tightly enveloped,

roughly spherical, animal virus of ~70 nm in diameter, containing one molecule of linear

positive-sense single-stranded RNA (~104 nucleotides), enclosed inside a capsid of

icosahedral symmetry (T=4) and ~40 nm diameter. The virus is covered with 80 spikes, each

consisting of a trimer of glycoproteins, which dock at specific binding sites of the capsid and

thereby also reflect the T=4 icosahedral symmetry. SFV buds at the plasma membrane (see

[50] for a general review on alphaviruses).

The intuitively appearing budding model outlined above poses a number of questions

which deserve both qualitative and quantitative understanding. For instance: The model

ensures that spikes will be present in budded virions, but why is it that actual virions are

basically fully covered with spikes, that is, why are no spikes missing? Is there a certain

minimum concentration of spikes in the membrane required before budding can commence?

What happens if several capsids compete for spikes? How are spikes drawn to the budding

site? And is there a way to adjust the production of spikes and capsids such as to maximize

the overall production of mature virions?

The formation of a stable bud requires that the bending energy should be

counterbalanced by the spike-mediated adhesion between the nucleocapsid and the lipid

membrane, which provides the driving force for viral budding. Based on this notion, we

develop a simple theoretical model for the budding scenario proposed by Garoff and Simons.

The model takes into account that two mechanisms oppose the enveloping of the

nucleocapsid by the lipid-spike membrane. First, wrapping the membrane around the capsid

involves an elastic bending energy penalty; and second, efficient capsid-membrane binding

requires accumulation of spike protein in these membrane regions. That is, spike proteins

must diffuse from the surrounding planar bilayer into the curved budding regions [51],

rendering the spike distribution nonuniform, which involves a demixing entropy penalty.

Another important factor which we take into account is the line energy [52,53] associated

46

pϕ

bϕ

with the saddle-like rim connecting the immature bud to the embedding planar membrane.

Whether or not budding occurs depends upon a delicate balance of all these energetic and

entropic contributions, which determine the spike populations in different membrane regions

and the size distribution of the budding virions. Our aim is to study this balance with a

statistical thermodynamic scheme that will enable us to address several of the questions put

forward above in qualitative and quantitative terms.

The outline of this chapter is as follows. Section 3.2 describes the theory. To simplify

the picture, we neglect in Section 3.2.1 the finite size of the buds and their size distribution,

which, however, are taken into account in Section 3.2.2. The results are described in Section

3.3. A discussion of our main findings is found in Section 3.4.

Figure 3.1 A schematic representation of the budding process. Naked nucleocapsids arrive at the cytoplasmic

leaflet of the cell membrane, where linker glycoproteins (i.e., spikes) help to anchor and envelope them by the

membrane. The spike concentration in the curved membrane around the partially wrapped buds (φb) is generally

different from that in the planar regions (φp).

3.2 Model Viral budding is a dynamical process, whereby nucleocapsids arrive at one side of the plasma

membrane and are released, enveloped by a membrane coat, at the other side. Electron

micrographs of virally infected cells generally reveal a population of bud sizes at different

stages of maturations, as illustrated schematically in Figure 3.1. The goal of our model is to

quantify the principal characteristics of this bud population. Underlying our model is the

assumption that the time required for viral bud maturation (many minutes usually) is long

enough to allow spike diffusion and equilibration between the curved (buds) and planar

membrane regions. Consequently, the distribution of bud sizes and spike densities in a

membrane containing given numbers of spike linker proteins (L) and adsorbed viral

nucleocapsids (N) can be treated using equilibrium statistical thermodynamics. In reality,

47

both L and N are time-dependent quantities. Our model does not describe the temporal

evolution of these (supposedly slowly varying) quantities but, rather, the momentary bud

population corresponding to given L and N.

Suppose N viral nucleocapsids have adsorbed onto a cell membrane embedding L

linker proteins (spikes). The capsids are wrapped to different extents by the adsorbing

membrane, resulting in a polydisperse 2D “solution” of buds, with the lipid-spike membrane

serving as the embedding solvent. Let Ma denote the total membrane area, where a is the

cross-sectional area per spike, at maximal membrane coverage. (Of course, even at full

coverage, the spikes are embedded in the lipid matrix). From the definition of a it follows that

the maximal number of membrane adhesion sites (equivalently, spikes) on the capsid’s

surface is K=4πR2/a, where R is the radius of the membrane-coated viral capsid. This limit is

achieved when the capsid is fully wrapped by a lipid membrane “saturated” with spikes. In

the numerical calculations presented in the next section we shall use K=80, as for SFV,

corresponding to a≈192 nm2 for R≈35 nm. Hereafter, we shall use a as our unit of area, and

a=R 4π/K (≈14 nm) as our unit of length. All energies will be measured in units of the

thermal energy kBT, with kB denoting Boltzmann’s constant and T the temperature.

Assuming that the membrane is tightly attached to the (spherical) capsids, the

membrane curvature in all buds is the same, except for the existence of a small circular rim at

the point where the membrane detaches from the capsid and where the curvature is not

spherical but rather toroidal. We shall use κ to denote the membrane bending energy in the

“bud phase”, and ε for the binding energy between a spike protein and the capsid. Upon

wrapping a nucleocapsid, spike glycoproteins, which mediate the interaction between the

nucleocapsid and the membrane, migrate into the curved membrane region enveloping the

capsid. Clearly then, the energy of the bud is lowered by the spike presence. Yet, this

segregation of spikes into the curved regions is entropically unfavorable. Furthermore, spike

diffusion into the budding domains is correlated with an increase in the degree of wrapping,

that is, with an increase of the overall curved area and hence also increases the total

membrane bending energy. The equilibrium densities of spikes in the planar and curved

regions are governed by the balance of these free energy contributions. So far, we discussed

the free energy of a single bud. However, buds have a finite size and do not form a

continuous phase and there is a distribution of bud sizes on the membrane. In Section 3.2.1,

we ignore these finite size effects in order to obtain a simplified picture. However, in Section

3.2.2, we take these effects into account, introducing additional terms to the free energy.

48

3.2.1 Macroscopic Phase Approximation To get a feeling for the relative contributions of the free energies introduced so far, we will

start by describing a crude approximation which enables us to solve the model analytically.

We refer to it by the macroscopic phase approximation. The key idea of this approximation is

to neglect the fact that the total curved membrane area is split-up between N buds and rather

think of it as one single phase which coexists with the planar membrane phase, as shown

schematically in Figure 3.2. Since we do not have individual capsids in this model, we cannot

obtain the bud size distribution but only the density of spikes inside the buds (and in the

planar membrane).

Figure 3.2 A schematic drawing of the “macroscopic phase approximation” a curved membrane with a

curvature that corresponds to the capsid radius coexists with a planar membrane. Spikes glycoproteins can

diffuse between both phases.

In other words, the total membrane area, M, (in units of a) is divided into two macroscopic

regions: a planar phase of area PM and a bud phase of total area bM with corresponding

spike densities, /p P PL Mϕ = and ( ) ( ) /b p P b bL L M M L Mϕ = − − = . Where pL and bL are the

number of spikes in the planar and curved (bud) regions respectively.

The free energy of such a system can be written as

p bF F F= + (3.1)

Where the tilde stand for the macroscopic phase approximation and PF and bF are the free

energies of the planar phase and the bud phase respectively, and are given by

[ ln (1 ) ln(1 )]p p p p p pF M ϕ ϕ ϕ ϕ= + − − (3.2)

[ ln (1 ) ln(1 )] ) b b b b b b b bF M L Mϕ ϕ ϕ ϕ ε κ= + − − − + (3.3)

The free energy in the planar phase involves only the configurational entropy of the pL spikes

embedded in the planar parts of the membrane, expressed in terms of a two-dimensional

lattice gas model. The free energy in the curved (bud) phase involves three free energy terms.

The first term accounts for the entropy associated with the rest of the spike linkers

( b pL L L= − ) which are distributed among the bud phase. The next two terms are energetics:

b b bL Mε ϕ ε− = − is the total spike-capsid binding energy. bMκ is the total membrane

pϕ bϕ

49

curvature energy in the bud phase. Note that κεϕε −≡ b~ , may be interpreted as the effective

adhesion energy per unit area in the bud phase. For a lipid bilayer characterized by a bending

modulus kc and a spontaneous curvature c0, the bending energy per unit area around a bud of

radius R is κ=(1/2)kc(2/R-c0)2/K [17]. The spontaneous curvature of cell membranes is

usually nonzero, because their two constituent leaflets are generally of different

compositions. Similarly, nonzero spontaneous curvature can also be induced by asymmetric

membrane proteins. The wedge-shaped spike glycoprotein themselves can give rise to a

nonzero spontaneous curvature thus obtaining a non-trivial dependence of the spontaneous

curvature on the spike density at the bud. In this work, we neglect the contribution of spike

glycoproteins to the spontaneous curvature as well as possible interactions between them. For

the simplest case of vanishing spontaneous curvature (c0=0) and a typical bending modulus

of kc=20kBT [54] we find κ=2π (using K=4πR2), which we will frequently use as a

characteristic value. However, one should keep in mind that for a given bending modulus kc,

the bending energy per unit area, κ, may actually be smaller (if c0>0) or larger (c0<0) than

the value implied by c0=0.

The equilibrium state of the system can be found by minimizing F with respect to bL and

bM . From 0bF L∂ ∂ = we obtain

ln ln1 1

p b

p b

ϕ ϕ ε μϕ ϕ

= − ≡− −

(3.4)

expressing the equality of the spike’s chemical potential (μ) in the planar and curved regions.

Recall that ln[ϕp/(1-ϕp)] is the chemical potential of a non-interacting lattice gas of density ϕp

[55]. Similarly, ( )ln 1b bϕ ϕ ε− −⎡ ⎤⎣ ⎦ is the chemical potential of a non-interacting lattice gas

of particles with lower (-ε) “ground state” energy.

Minimizing of F with respect to bM , i.e., 0bF M∂ ∂ = , we obtain

ln(1 ) ln(1 )p bϕ ϕ κ− − = − − − ≡ Π (3.5)

Recall that ( ) )1ln(~pLpp

pMF ϕ−−=∂∂−=Π is the familiar expression for the pressure of an

ideal lattice gas [55], in our case the 2D gas of spikes in the planar membrane. Similarly,

ln(1 )bϕ κ− − = Π + should be interpreted as the pressure in the budding region. It is larger

than Π (by κ) because of the bending energy penalty associated with increasing the area of

50

the bud phase. Note that Eq. (3.5) is analogous to Laplace’s equation for the pressure

difference across a curved surface [56] with κ playing the analog role to that of the surface

tension. Note finally that if we reinsert the two equilibrium conditions, Eqs. (3.4) and (3.5)

back into the free energy Eqs. (3.1)-(3.3), we regain the familiar thermodynamic relation

( ) ( )b p b pF L L M M L Mμ μ= + − Π + = − Π , between the Helmholtz (F) and Gibbs (G Lμ= )

free energies; in this case the free energies of a 2D system of area M containing L spikes at

pressure Π.

Equations (3.4) and (3.5), expressing the equality of chemical potentials and pressures

of the spike gas in the “p” and “b” phases, dictate the spike densities (ϕp and ϕb) in two

(hypothetical) macroscopic coexisting phases. Solving these equations we obtain

1 1 and 1 1b p

e ee e

κ κ

ε εϕ ϕ−

−

− −= =

− − (3.6)

with the tilde reminding us that these equations are only valid in the macroscopic phase

approximation. From Eq. (3.6) it follows that phase coexistence is only possible if 0ε κ≥ ≥ .

Physically, this is a consequence of the fact that for ε κ≤ the energy of the “p” phase is

lower than that of the “b” phase, ( ) 0b b b bL M Mε κ ϕ ε κ− + = − + ≥ , even if the buds are

densely covered by spikes; i.e., negative effective adhesion energy 0bε ϕ ε κ= − ≤ even for

1bϕ = . Under these circumstances there is no thermodynamic driving force for phase

separation (and hence spike density segregation). Thus, ε κ= marks a critical value for the

adhesion energy, below which budding cannot take place. Note that ε κ= implies

1b pϕ ϕ= = , whereas for ε κ> we must have 1 b pϕ ϕ> > .

One interesting and immediate prediction of Eq. (3.6), pertaining to the case where

coexistence is possible, i.e.,ε κ≥ , is that for most lipid membranes (where, typically, 3κ ≥ ),

the spike density in the curved membrane regions coating the buds is nearly saturated

( )1bϕ → . This agrees with experimental results showing that the virus is always fully

occupied with spikes. In other words, there is always a define stoichiometry between the

number of capsid proteins and the number of spike glycoproteins with no spike ever missing.

3.2.2 Bud Size Distribution So far, we study the partitioning of spikes between curved and planar membranes, ignoring

the fact that there is finite number of nucleocapsids of finite size. In order to calculate the bud

51

size distribution and the amount of fully wrapped nucleocapsids that leave the membrane, we

need to take these facts into account. The improved model system is shown schematically in

Figure 3.1.

Let nk denote the number of capsids wrapped around by a membrane section of area k, which

varies between k=0 and k=K (as already defined, 24K R aπ= is the maximal number of

membrane adhesion sites). The former value corresponds to a free capsid which has arrived at

the membrane and is ready to wrap (we may think of it as being loosely associated with the

membrane without involvement of spikes), while the latter value corresponds to a capsid

which is fully enveloped by the membrane. The bud size distribution { }kn should satisfy the

following conservation constraints

0

K

kk

n N=

=∑ (3.7)

0

K

k bk

kn M=

=∑ (3.8)

In addition, to the spike entropy, bending and interaction energies, we have now to take into

account the configurational entropy of the buds in the membrane plane and a line energy term

associated with the bud rims [see e.g., 52,53]. The origin of the line energy is mainly the

different curvature at the point where the membrane detaches from the capsid and the

different lipid composition found inside the bud as compared to the average composition in

the planar membrane. All these contributions are accounted for by the free energy functional

( ),{ } | , , ( )[ ln (1 ) ln(1 )

( ([ ln (1 ) ln(1 )]

b k k p p p p

k b b b b

b k

k

F L n L M N M kn

kn

L kn

n

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

ε κ

= − + − −

+ + − −

− +

+

∑∑

∑∑ ( ) [ln( / ) 1]k kk n n MΛ + −∑

(3.9)

with )/()( ∑−−= kbp knMLLϕ and ∑= kbb knL /ϕ .

The first two terms in Eq. (3.9), represent the configurational entropy of the pL spikes

embedded in the planar part of the membrane and of the rest of the spikes ( bL in number)

which are distributed among the curved budding regions. Note that we do not a-priori assign

a particular number of spikes (say, *kl ) to a bud of size k. In fact, by allowing for all possible

distributions of the Lb spikes among all buds, we also account for all fluctuations around the

average lk. (The average spike density ,k b k bl k ϕ ϕ= ≡ is independent of k, because the

52

spikes chemical potential in all buds, bµ , must be the same everywhere in the membrane,

including in the various k-buds; see below.) The next three terms in Eq. (3.9) are energetic:

b b bL Mε ϕ ε− = − is the total spike-capsid binding energy. b kM knκ κ= ∑ is the total

membrane curvature energy in the budding regions. The third energetic term, the sum

)(knk Λ∑ , is the total line energy of the rim, with Λ(k) denoting the line energy of a k-bud

(see below). Finally, the last term in Eq. (3.9) accounts for the configurational entropy of the

polydisperse 2D bud mixture, treated here as a multicomponent ideal gas. More elaborate

models, taking into account excluded area effects and other interactions between buds are

possible, but not warranted here.

The equilibrium values of pϕ , bϕ and the equilibrium bud size distribution { }*kn are

determined by minimizing ( ,{ } | , , )b kF L n L M N with respect to Lb and { }kn (or another set of

K independent variables). The latter minimization should obey the conservation condition of

the total number of spikes, Eq.(3.7).

From 0bF L∂ ∂ = , we obtain Eq. (3.4) which express the equality of the spike’s chemical

potentials in the planar and curved regions.

Minimizing F with respect to all nk, subject to Eq. (3.7), we find

( ) ( ) ( )ln 1 ln 1 ( ) ln 0p b kk k n Mϕ κ ϕ λ⎡ ⎤− − + + − + Λ + − =⎣ ⎦ (3.10)

with λ denoting the Lagrange multiplier conjugate to Eq. (3.7). Hence, the normalized bud

size distribution is given by

* ( )

( )0

k kk

k K k kk

n epN e

αα

−Λ

−Λ=

= =∑

(3.11)

where we have used Eq. (3.7) to eliminate λ and defined

( )11

p p

b b

e e ε κκϕ ϕα

ϕ ϕ−−−⎛ ⎞ ⎛ ⎞

= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ (3.12)

To evaluate the pk’s (for known ε, κ, Λ(k), L, M, and N) we need pϕ and bϕ . From Eq. (3.4)

we obtain one equation relating these two variables. Another one is provided by the spike

conservation condition between the two phases (“lever rule”):

(1 )p k b kc kp c kpϕ ϕ φ− + =∑ ∑ (3.13)

53

In general, since Λ(k) is not a simple function of k, the evaluation of pϕ and bϕ , and hence of

*kp , is only possible numerically. In all the calculations presented in the next section, the line

energy associated with a k-bud will be modeled as being proportional to the length, L(k), of

its rim, with a constant line energy per unit length γ. Simple geometry then yields

( ) ( ) 2 4 1 k kk k RK K

γ γ π ⎛ ⎞Λ = = −⎜ ⎟⎝ ⎠

L (3.14)

where R is the radius of the capsid. Note that L(k) vanishes for k=0 and k=K, and is maximal

(2πR) when the membrane coats one capsid “hemisphere” (k=K/2).

The saddle-like curvature of the lipid-protein membrane at the bud’s rim is different from

both the simple spherical shape of the membrane around the bud, and the planar geometry of

the surrounding membrane. If the membrane is under nonzero lateral tension (which is the

case for all cell membranes [57]) this rim will contribute an additional bending energy [58].

Its dependence on k is not as simple as assumed in Eq. (3.14), but the general features of

large energies near the equator (k ≈ K/2) and small values at the poles (small or large degrees

of wrapping) are identical. Another contribution to γ may arise from the possibly different

lipid-protein compositions across the boundary separating the curved and planar membrane

regions. In addition to the difference in the density of spike proteins these two regions may

also differ in lipid composition and the content of other proteins. In fact, some studies suggest

that the chemical composition of various viral membranes, e.g., certain retroviruses, is

different from that of the host plasma membrane, resembling the composition of “lipid rafts”

[23]. It is less clear whether raft-like composition is also typical of Alphaviruses; yet, it has

been shown that increased concentrations of cholesterol (which is also abundant in membrane

rafts) are vital for their efficient budding [59].

Changes in curvature and composition at the bud rim are most likely coupled to each

other, because different lipid species involve different spontaneous curvatures. If this were

the boundary between ordinary phase separated (planar) domains of different compositions,

then γ would be on the order of 1kBT per molecular diameter; [see e.g., 60]. Most recently,

the coupling between curvature and composition has been clearly demonstrated in mixed

lipid vesicles, revealing line energies on the order of 1kBT per nm [61]. The origin of the line

energy in (say, binary) lipid membranes is the non-ideal mixing of the lipid species. In our

problem, assuming that the lipids in the planar (bud-free) membrane are randomly mixed, the

chemical contribution to γ should be smaller. (The difference in composition is enhanced by

54

the different curvatures.) In the absence of detailed information pertaining to the line energy

between the budding and planar membrane regions, we shall treat γ as a variable; ranging

between zero and 1kBT per unit length, a .

The final stage of the budding process, i.e., the pinching-off of the fully wrapped bud

and its release into the intercellular space, involves an energy barrier associated with the

fusion and scission of the lipid-protein membrane of the bud’s narrow neck. This process is

most likely mediated by special scission proteins, (e.g., TSG101 in the case of HIV [62,63]).

Our theoretical model is meant to account only for those stages of the budding process

preceding the final scission of the bud. That is, the process leading to the formation of a

nearly mature, almost fully wrapped (“narrow neck”) bud; assuming that its formation leads

to irreversible pinching-off of the viral particle. In our model calculations we shall assume

that this irreversible pinching-off is the fate of all buds for which k≥0.9K. (The value 0.9 is

quite arbitrary, but its precise value is immaterial for our purposes.) The concentration of

these buds, cw, would be proportional to the rate of budding, if this were a steady state

process.

3.3 Results

The numerical results presented in this section focus on the equilibrium densities of spikes in

both planar and budding membrane domains, the distribution of bud sizes and the

concentration of mature (fully wrapped) nucleocapsids, as a function of the average spike

density, L Mφ = and capsid density, c N M= for several choices of the adhesion energy,

ε , the membrane bending energy, κ and the line energy at the bud’s rim, γ .

3.3.1 Choice of Parameters Largely due to the lack of detailed information pertaining to all the relevant physical

constants and parameters in our model, (e.g., c, ε, and φ) our calculations are not intended to

mimic any particular system. Whenever possible, however, our choice of physical constants

was guided by data corresponding to Alphaviruses. Thus, in all calculations we have used

K=80 for the number of available spike binding sites per nucleocapsids [48]. In most

calculations ε, κ, and γ are treated as variables. Some calculations require specific values for

these material constants, which were chosen as follows: For the bending energy per unit

55

area, κ=2π (corresponding to kc=20 for lipid membranes of zero spontaneous curvature, but

to “softer” or “harder” membranes if c0 is positive or negative, respectively). The spike-

capsid adhesion energy is not known. However, following the suggestion that aromatic

residues in the capsid protein create a hydrophobic docking pocket for the side chain of the

spike glycoprotein [64], and assuming that the corresponding binding energy is comparable

to typical antibody-antigen interactions (for which the dissociation constant dK is on the

order of 10-10M), we have 20ε ≈ [2], which is the “typical adhesion energy” used in some of

the calculations. (Recall that both ε and κ are measured in units of Bk T ). For γ we have

examined several values in the range 0-1 (in units of kBT per unit length, a).

In some of the calculations below the capsid density, c, is treated as a variable. As a

specific representative value in many of the calculations we have used c=0.005. Note that this

is actually a rather large 2D capsid concentration, since c is the number of capsids per unit

membrane area, a, which is much smaller than the capsid’s surface area. More specifically,

the capsid’s surface area is 4πR2/a=K=80, so that its projection on the membrane plane is

πR2=20. Thus, just for comparison, the maximal value of c, corresponding to the hypothetical

limit where all capsids are unwrapped and densely packed in the membrane plane is 0.045,

(that is, (1/20)×0.91, 0.91 marking the maximal projected area fraction of spheres in 2D).

Another limit, also hypothetical but of interest for the choice of c, corresponds to the case

where all capsids attached to the membrane are fully enveloped by the lipid-spike coat and,

furthermore, densely packed against each other within the membrane plane. The total

membrane area per bud is now 4πR2+πR2/0.91, with the second term accounting for the

planar membrane area per bud. For K=80 this yields 0.01c ≈ . Thus, anticipating a

distribution of different bud sizes our “default” choice, c=0.005, amounts to a rather crowded

though not closely packed population of capsids at the cell surface.

No quantitative data are available for φ. Based on partial experimental information, φ

appears to vary in the range 0.01-0.1 [65]. From the rate of spike synthesis (∼105 spikes per

cell per minute), cell surface area ( 2 63000 15 10m aµ ≈ ×∼ ), and protein dwell time (∼15

minutes) one can estimate that 0.1φ ≈ , assuming that all spikes arrive at the plasma

membrane [24]). Some of the calculations presented below were carried as a function of φ for

its entire range, [0,1]. Others were performed for selected values of φ. All bud size

distributions were derived by solving for α using the expression for the optimal bud size

56

distribution (Eq. (3.11)), the chemical potential equality (Eq. (3.4)) and the conservation

condition (Eq. (3.13)).

3.3.2 Spike Partitioning Figure 3.3 shows the equilibrium densities of spikes in the budding (ϕb) and planar (ϕp)

membrane regions in the (ϕ,1/ε) plane, for a given bending rigidity κ=2π. (Notice that for a

given strength of the adhesion energy, 1/ε is proportional to the temperature T. For a given T

it is of course inversely proportional to the interaction strength. Phase diagrams are often

plotted in terms of this “effective temperature”). Similarly, Figure 3.4 shows the

corresponding phase diagram in the (ϕ,κ) plane for a given spike adhesion energy, ε=20. In

both figures, one set of data corresponds to the macroscopic phase approximation in which

the curved and planar membrane regions are treated as macroscopic phases, ignoring finite

(bud) size effects and line energy contributions. These results (shown by the solid curves in

Figure 3.3 and Figure 3.4) are ordinary phase diagrams, as obtained by solving the

coexistence conditions, Eqs. (3.4) and (3.5), which yield ( ),b pϕ ϕ , as given by Eq. (3.6).

Consistent with the discussion in the previous section, both figures reveal that spike phase

separation can only take place if ε is larger than κ (1 1 2ε π< in Figure 3.3). Otherwise, the

membrane bending energy overcomes the adhesion energy and prohibits budding. In the two-

phase region, coexisting densities are connected by horizontal tie lines, such as the light

dashed/dotted lines in the figures. As usual, the relative proportions of material in the two

phases is dictated by the lever rule, Eq. (3.13). As we have mentioned above, a noteworthy

prediction of the macroscopic phase approximation is that whenever phase separation takes

place, the spike density in the budding virions is essentially saturated, i.e., 1bϕ → ; in line

with the experimentally observed “stoichiometric” ratio between the number of available

adhesion sites on the capsid and the number of spike trimers in the virion [66].

Also shown in Figure 3.3 and Figure 3.4 are the coexisting spike densities when the

discreteness of the virions and hence the entropy of their polydisperse size distribution are

taken into account. For these calculations, which utilize Eqs. (3.4), and (3.11)–(3.13), we

must specify the capsid concentration c and the average spike density φ. In Figure 3.3 we

show the coexisting densities for c=0.005 and φ=0.01, 0.1 and 0.5. (φ=0.5 is rather

hypothetical and mainly used to emphasize the role of the buds’ finite size). The numerical

results shown here are all for γ=0, yet it should be noted that very similar coexisting densities

57

(ϕb,ϕp) are obtained for nonzero line energies. The value of γ is more important in its effect

on the size distribution of buds, as discussed below.

From Eq. (3.4) it follows that for every (positive) value of ε the spike density in the

budding regions, ϕb, should be higher than that in the planar ones, ϕp. Note, however, that

unlike in the coexistence of two macroscopic phases, whose densities are independent of the

average density, φ, and whose relative proportions are governed by the “lever rule”, in a

system containing discrete buds, ϕp and ϕb may depend on both φ and c.

In Figure 3.3, we see that the coexisting densities calculated for a system of finite-size

buds are very similar to those obtained in the macroscopic phase approximation: 1b bϕ ϕ≈ ≈

and 0p pϕ ϕ≈ ≈ . Deviation from this behavior occurs at two extremes:

1. For high values of φ and large ε , all the buds will be fully wrapped

( ), 1k bkkp K ϕ= =∑ . For a given number of capsids, it might happen that after all the

binding sites on the capsid get occupied, there are still available spikes, so that pϕ

will be different than 0pϕ ≈ . For instance, for c=0.005, K=80, 0.5φ = , we will get

using Eq. (3.13), ( ) ( )1 0.1667p cK cKϕ φ= − − = , consistent with the result in Figure

3.3.

2. When ε get smaller (1/ε increases), the planar membrane phase becomes much more

stable and pϕ increases to the point where pϕ φ> . Since pϕ can not exceed the

average density of spikes in the membrane, pϕ φ→ as 1/ε keeps increasing. In this

regime bϕ must also deviate from bϕ ( )b bϕ ϕ< , as follows from Eq. (3.4). Notice

that there is no critical point in a system of finite-size buds, and from Figure 3.3 it

follows that the spike density in the budding regions is still much larger than in the

planar membrane, even for ε κ< . This might seems surprising at first, since there is

no physical driving force for budding in this case. It should be stressed, however, that

this result does not contain any information regarding the number and extent of

membrane-wrapped capsids. This information can only be provided by the bud size

distributions, as discussed below. In fact, due to the configurational entropy of the

finite bud phase, the membrane indeed contains an exponentially small fraction of

small buds. However, as we shall see below, no bud maturation is possible.

58

0 0.2 0.4 0.6 0.8 1ϕ

0

0.05

0.1

0.15

0.2

1/ε

κ=2π

ϕp ϕ

b

Figure 3.3 Equilibrium spike densities in the budding (φb) and planar (φp) membrane regions. This (φ,1/ε)

diagram was calculated for κ=2π, and γ=0. The solid (black) curve is a phase diagram describing the coexisting

spike densities ( ),b pϕ ϕ when all budding regions are treated as one macroscopic phase, in equilibrium with a

planar membrane phase. The pairs of purple, red and orange curves are the coexisting spike densities calculated

for a system of discrete buds where φ =0.01, 0.1 and 0.5, respectively. In all cases c=0.005. The horizontal

dotted lines are representative tie lines, connecting pairs of coexisting spike densities.

Figure 3.4 Phase diagram in the (φ;κ) plane for ε=20 and γ=0. The solid curves describe the coexisting spike

densities ( ),b pϕ ϕ when all budding regions are treated as one macroscopic phase. The dashed curves describe

the results for a system of discrete buds of two-dimensional density c=0.005, in a membrane with spike

densityφ =0.1. Shown are a few tie lines (light dashed horizontal lines). The inset magnifies the behavior in the

low κ regime.

59

Figure 3.4 shows the phase diagram in the (φ,κ) plane, for ε=20. The results corresponding to

the macroscopic bud phase approximation are shown by the solid curves, which bound

(above and below) the two-phase region. Coexisting densities are connected by horizontal tie

lines. The dashed curves in this figure describe the results for finite size buds, embedded in a

membrane where c=0.005 and φ=0.1. The figure shows that in the limit κ→0, both bϕ and

pϕ must vanish, as follows from Eqs. (3.4) and (3.5); the first requiring b pϕ ϕ= for κ=0 and

the second showing that for nonzero ε this can only be fulfilled if both densities vanish. The

bottom right corner of the diagram (small κ and nonzero φ) is a one phase region where only

the curved, “budding phase” exists. Indeed, for low κ and nonzero φ the spikes-rich bud

phase is of much lower energy (chemical potential) as compared to the planar membrane.

As κ increases, the gap between the coexisting densities widens rapidly, with

1bϕ → (saturation) and 0pϕ ≈ , reflecting the strong preference for the budding phase

forε κ> . Again, a critical point, beyond which no phase separation can take place is reached

when κ ε= , and the critical density is φ=1. Above the coexistence line we again find only

the planar phase.

As we found for Figure 3.3, within the two-phase region the coexisting densities in a

system of discrete buds are generally similar to those obtained in the macroscopic (bud)

phase approximation. Differences appear when κ becomes comparable or larger than ε. As

already remarked with respect to Figure 3.3, once κ gets larger than ε, no buds are formed, as

will become apparent after discussing the bud size distributions. Differences between the

finite and macroscopic bud systems appear also in the low κ limit. In the finite-bud system,

when κ=0, buds bearing a finite spike density coexist with a planar, spike-free, membrane.

3.3.3 Bud Size Distribution In Figure 3.5 we show several distributions of bud sizes, corresponding to different choices

of γ, as obtained by solving Eqs. (3.4) and (3.11)–(3.13), with a line energy modeled

according to Eq. (3.14); in all cases for c=0.005, ε=20, κ=2π, and φ=0.1. For γ=0 the size

distribution is rather broad, with the probability of finding a bud of size k decreasing

monotonically with k (solid curve in Figure 3.5a). This is since when no line tension penalty

is involved, the size distribution of the buds is primarily determined by the last, “mixing

entropy”, term in Eq. (3.9), which favors a random distribution of the available spikes among

the various buds, and hence a broad (exponential) distribution of bud sizes. Consequently, the

60

fraction of nearly fully wrapped buds is small. For nonzero γ we expect an increase in the

populations of the two extreme bud sizes; the nearly fully wrapped ( k K≈ ) capsids on the

one hand, and totally naked nucleocapsids ( 0k ≈ ) on the other. This is, of course, a

consequence of the fact that the circumference of the bud is maximal at the “equator”

(k=K/2), and minimal near the “poles”, (k=0,K).

Indeed, as γ increases the size distribution becomes bimodal, i.e., in addition to the

maximum at k=0, a second maximum emerges at k=K, with a concomitant depletion of

intermediate size capsids. For 0.5γ ≥ , the size distribution is sharply bimodal, with peaks at

k=0 and k=K. In other words, budding becomes an “all-or-none” process, whereby

nucleocapsids arriving at the membrane either become fully wrapped by a membrane, or

remain naked; no partially wrapped capsids are stably attached to the membrane. This

scenario suggests the existence of a kinetic barrier which must be overcome to achieve full

wrapping. Using our estimation for the line energy, this barrier is on the order of 100 kT’s,

much larger than the energy scale of thermal fluctuations. Therefore, it might imply that

additional cellular mechanisms participate in the wrapping process in order to reduce this

barrier. For example, the bud’s rim can be enriched with cellular proteins whose spontaneous

curvature is comparable to the saddle-like curvature at the rim.

For the conditions corresponding to Figure 3.5 (namely, ε significantly larger than κ)

the spike density in the fully enveloped buds is essentially saturated, 1bϕ ≈ (whereas

1pϕ << ). Thus, for large γ (≥0.6 in Figure 3.5b), the number of fully wrapped capsids, Kn , is

dictated by the total number of spikes embedded in the membrane, L. Since each essentially

fully enveloped nucleocapsid engages K spikes, it follows that Kn L K≈ , and hence the

fraction Kp of these capsids in the system, is Kp c Kφ≈ . For large γ the high-k-peak is

essentially confined to k=K, thus Kp is almost the same as the fraction of essentially wrapped

capsids 0.9w kk K

p p≥

= ∑ . Consistent with the results in Figure 3.5b we find that, indeed,

0.1/(0.005 80) 0.25wp c Kφ = × =∼ . Pictorially then, when nucleocapsids arrive at a

membrane characterized by a large value of γ, they get fully enveloped by membrane coats,

recruiting spike proteins to ensure tight membrane-capsid binding. Once all spike proteins are

engaged in bud coats, newly arriving nucleocapsids necessarily remain naked. Similar

qualitative behavior is found for other values of c and φ.

61

Figure 3.5 The distribution of bud sizes in a system where φ =0.1, c=0.005, ε=20, and κ=2π. The solid, dashed

and dotted lines in a correspond to γ=0, 0.1 and 0.2. The solid curve in b is for γ=0.6. For larger values of γ, the

size distribution is strictly bimodal, with peaks at k=0 and k=K, as shown by the triangles for γ=1.

Figure 3.6 The concentration (a) and fraction (b) of essentially fully wrapped viral capsids ( 0.9k K≥ ) as a

function of the two-dimensional concentration of nucleocapsids at the membrane plane, for a system with

20ε = , 2κ π= , 0.1φ = , and 0γ = (solid curve), 0.3γ = (dotted curve), 0.5γ = (dashed curve), and 1γ = (dot-

dashed curve). c*= Kφ is the optimum (i.e., stoichiometric) value of the capsid concentration for efficient

budding (here c*=0.1/80=0.00125).

3.3.4 Mature Buds Figure 3.6 describes the fraction, wp and the concentration, w wc cp= of “mature virions”,

i.e., nearly or fully enveloped capsids (k≥0.9K), as a function of the bud density in the

membrane plane, c. The different curves correspond to different values of γ, all for φ=0.1,

62

κ=2π, and ε=20. We know already that for these values of ε and κ the spike density in bud

membranes is nearly saturated, 1bϕ ≈ . For small values of c, that is *c c< where *c Kφ=

(here c=0.1/80=0.00125) there should be enough spikes to fully envelope all the

nucleocapsids arriving at the membrane, so that wc c= , consistent with the low c behavior in

Figure 3.6. However, since the number of spikes in the membrane is not unlimited, as soon as

c increases beyond *c , the bud size distribution is bound to change, since the total curved

(budding) membrane area is distributed among a larger number of buds. For large γ, as noted

in analyzing Figure 3.5, { }kp is bimodal with peaks at k=0 and k=K, and hence an increase in

c beyond *c Kφ= hardly affects cw, and hence w wc c p= is inversely proportional to c. On

the other hand, when γ is small (e.g., γ=0 in Figure 3.6) both the absolute number and the

fraction of fully wrapped buds decreases with c, indicating that efficient viral budding

requires a nearly stoichiometric ratio of spikes to capsids, c Kφ = . If, as our model

assumes, c and φ are indeed slowly varying quantities, then cw could be interpreted as being

(proportional to) the momentary budding rate.

In Figure 3.7 we show the fraction of mature buds as a function of the average spike

concentration in the membrane. For the two curves describing the behavior of a system with

nonzero γ, pw increases linearly with φ, saturating at the threshold spike concentration

* cKφ φ= = , above which there are always enough spikes to fully wrap all nucleocapsids

arriving at the membrane. This is the behavior expected for a bimodal (k=0 or K) distribution

of bud sizes, as we found to be the case for these values of γ. The nonlinear increase of pw

with φ for γ=0 is a consequence of the highly polydisperse size distribution of buds in this

case; (see Figure 3.5).

Finally, in Figure 3.8 we show how pw depends on the spike-mediate adhesion energy.

These calculations confirm that budding cannot take place unless the adhesion energy

counterbalances the membrane bending energy penalty. For large γ, once ε exceeds the κ

threshold, the budding fraction increases rapidly, and saturates when all available spikes have

been consumed. For very small values of γ (here represented by γ=0), the threshold behavior

is more moderate, reflecting the broad distribution of bud sizes and the relatively small

fraction of fully wrapped buds.

63

Figure 3.7 The two-dimensional fraction pw of essentially fully wrapped capsids ( 0.9k K≥ ) as a function of

the average spike density in the membrane plane, for a system with ε=20, κ=2π and c=0.005, and γ=0 (solid

curve), γ=0.3 (dotted curve), γ=0.5 (dashed curve), and γ=1 (dot-dashed curve). * cKφ = is the stoichiometric

value of the spike concentration for optimum budding (here * 80 0.005 0.4φ = × = ).

Figure 3.8 The two-dimensional fraction, pw, of essentially fully wrapped capsids ( 0.9k K≥ ) as a function of

the spike adhesion energy, ε, for a system with κ=2π, c=0.005, 0.1φ = and γ=0 (dotted curve), γ=0.5 (dashed

curve), and γ=1 (solid curve).

3.4 Concluding Remarks

We have cast the budding scenario of Alphaviruses into a statistical thermodynamic model,

which has enable us to address in both qualitative and quantitative terms a variety of

questions raised by this scenario. One of our first results is the fact that, for essentially all

biologically meaningful values of the membrane elastic constant and the spike binding

strength, the spike density on wrapped capsids is saturated. In other words, if budding takes

64

place, all binding sites on the capsid will be occupied. This is a nontrivial result of our

calculation in the sense that it is not a necessary consequence of the mechanism of spike-

assisted budding alone. The underlying reason for this is rather that entropic terms involved

with the spikes are basically outweighed by energetic ones for the interaction strengths

present in nature. Intermediate densities, which are entropically favorable, do not occur,

because a vast coexistence region spans almost the entire range between 0φ = and 1φ = .

The opposite side of the coexistence region, describing the planar membrane phase,

depends strongly on ε , and pϕ covers a wide density range (see Figure 3.3). In the

macroscopic phase approximation, pϕ , ε and κ are linked by a very simple equation, Eq.

(3.6), which provides a useful link between these important but difficult to measure

quantities. In fact, it turns out that under all interesting conditions the coexistence lines of the

macroscopic phase approximation describe the preferred densities in the planar regions and

on the capsids for any given average spike density quite well. For weak binding ε κ< , the

density pϕ in the planar region essentially (i.e., up to an exponentially small correction)

coincides with the average density, φ , as one would expect; thus dictating bϕ by Eq. (3.4)

(which generally implies b pϕ ϕ>> ; see Figure 3.3). Since, the bud entropy term favors the

existence of buds, some (small) buds should form even in the ε κ< regime. However, the

fraction ( )bM M of the membrane area occupied by these buds (as confirmed by

calculations not reported here) is negligibly small and bud maturation is obviously

impossible.

Once the adhesion strength get large enough such that the average density φ finally

exceeds the macroscopic coexistence density pϕ , the spike density pϕ in the planar region

departs from φ , joins into the macroscopic coexistence line pϕ , and thereby begins to

decrease. When this happens, the bud phase finally acquires a macroscopic number of spikes

and budding becomes possible. Upon further increasing ε , spikes are continuously shifted

from the p phase to the b phase. However, in the discrete case there exists one more

limitation which the macroscopic phase approximation does not know about – namely, that

the total number of capsids per area (and thus the amount of occupiable binding sites) is

finite. It may thus happen that all spikes have been transferred into the b phase before the p

phase is emptied. If this occurs, pϕ can no longer follow pϕ (which approaches 0 as

65

ε → ∞ ), and instead saturates at a finite density. This is evidently favored if the spike

concentration is high, but also if the capsid concentration is low.

The above scenario is also nicely reflected by our studies of the bud size distribution.

For instance, Figure 3.8 illustrates that wrapping will only commence once ε exceeds the

critical threshold κ , even though a bud phase existed before. Perhaps an even more

interesting insight from the analysis of the bud size distribution is that the line energy

associated with the bud rims, although acting as an additional penalty toward wrapping,

nevertheless promotes the production of more mature virions, as has been clearly

demonstrated by Figure 3.8. It has been pointed out previously [58,67] that a line energy

suppresses partially wrapped states and can therefore also shift the wrapping balance toward

full envelopment. The same effect is at work in the present situation, only with the subtle

additional feature that the bud size distribution comes along with an entropy, which is thereby

also reduced.

The line energy thereby helps to increase the efficiency of budding. Indeed, Figure 3.6

shows that the budding rate increases, and that the maximum in wc as a function of c, which is

most strongly pronounced for 0γ = is broadened. This again follows because capsids are not

wasted in partially wrapped states. However, one should not overlook that the fraction of

budding virions nevertheless starts to decrease beyond the stoichiometric point, which is

therefore the optimum point at which the virus should operate.

Lerner et al. [68] estimate that on reasonable experimental timescales (~10-20

minutes) efficient budding from membranes of zero spontaneous curvature may only take

place if the membrane bending modulus is rather small ( 7 Bk T≤ ). They suggest that budding

could possibly occur from precurved membrane regions, where 0 0c > , thereby reducing the

membrane bending energy barriers. Lerner and co-workers further suggest that the wedge-

shaped spike proteins could possibly be the origin of the nonzero spontaneous curvature. It

should be noted that such estimates of the membrane bending rigidity based on calculated

budding rates may be quite sensitive to the details of the kinetic model.

Simultaneously, a paper was published estimate mean-field equilibrium values for the

surface concentration of adsorbed colloids, and the average number of ligand-receptor bonds

per colloid, as a function of bulk colloid concentration. While the effects of curvature, line

energy and bud size distribution effects are neglected, they also find that there is an optimal

ratio between colloids and linkers for adhesion by the membrane, [69].

66

Chapter 4 Adsorption of Flexible Macromolecules on

Fluid Membranes

4.1 Introduction

In Chapter 2, we considered the packaging of dsDNA in bacterial viruses. In that case the

genome was found to be strongly stressed in its rigid protein shell, with an associated

pressure as large as fifty atmospheres, and a strong motor protein was needed to perform the

work of packaging. Contrariwise, in most ssRNA animal viruses the genome is significantly

less compressed and the nucleocapsids are believed to form spontaneously. In the previous

chapter we studied the budding of pre-assembled nucleocapsids at the plasma membrane. The

assembly stage, as already mentioned in the Introduction to this thesis, might occur either as a

preliminary step for budding (capsid assembled nucleocapsids – CAN, characterizing alpha

viruses) or simultaneously with budding (membrane assembled nucleocapsids – MAN, of

which the HIV-1 virus is a characteristic example).

Since the work presented in this chapter was motivated by these assembly processes,

we begin the introduction with a discussion of the characteristics of HIV-1 assembly and of

the different degrees of freedom and physical entities involved therein. Next, we describe the

model system we study in this chapter (that is, adsorption of a flexible macromolecule on a

fluid membrane) and its main statistical-thermodynamical properties1.

The assembly pathway of retroviruses is presented schematically in Figure 4.1. The

main entities participate in this process are the retroviral structural proteins (Gag

polyproteins), the RNA and the lipid membrane.

The Gag polyproteins consist of three major domains: the matrix domain (MA), the

capsid domain (CA), and the nucleocapsid domain (NC). The MC, CA and NC domains are

linked by short and flexible amino acid sequences. These flexible chains are cleaved in the

� 1 The results presented in this chapter were previously reported in [70].

67

mature infectious virus, after the bud pinches off from the membrane, with a resulting

dramatic structural change. The MA, which contains a cluster of basic residues and often a

myristoyl chain, is attracted electrostatically to the membrane and most probably is

responsible for the segregation of anionic lipids to the viral envelope. The CA is the

electrostatically neutral capsid domain, which in a mature virus constitutes the viral capsid.

Mutual interactions between CA domains of neighboring Gag precursors are believed to play

an important role in determining the curvature of the immature virion. The NC contains

special regions for specific sequences of the viral genomic RNA, as well as many residues

mediating nonspecific electrostatic binding of RNA. It is known that there is a well defined

stoichiometry between the number of NC proteins in the virion and the length of RNA

[71,72].

The immature viral bud consists of three concentric spherical shells, as illustrated

schematically in Figure 4.1. The outermost shell is a lipid bilayer envelope, the middle layer

consists of several thousands, tightly packed Gag polyproteins and the innermost shell is the

viral ssRNA. The tri-layer structure of the viral bud, with Gag-proteins sandwiched

electrostatically between the membrane and RNA, can be visualized experimentally using

cryoelectron microscopy, revealing three peaks in the radial density distribution,

Figure 4.1 The basic MA domain of

Gag adsorbs onto the inner leaflet of

the plasma membrane, attracting

anionic lipids. The genomic RNA

adsorbs to the basic NC domains.

These interactions, in addition to the

nonpolar attraction between the CA

domains, govern the assembly and

structure of the premature virion. The

figure displays an early stage in the

budding process, by the end of which

the membrane-Gag-RNA closes on

itself into a nearly complete spherical

shell. The pinching off from the

membrane is mediated biochemically.

68

corresponding to the MA-lipid interface, the shell of CA domains, and the NC-RNA interface

[72-74].

It appears that electrostatic and hydrophobic interactions are not the only ones to play

a role in this complex process. Several observations hint at additional significant factors. For

example, ssRNA is known to change its secondary structure upon interaction with NC

proteins [75]. Thus, the flexibility of the RNA as well as its secondary structure, namely its

ability to form base pairs, might affect the assembly. Another observation is that the lipid

composition of the viral envelope is different from that of the original cell membrane,

resembling the composition of lipid rafts — rich in anionic lipids, sphingolipids, and

cholesterol [23,76,77]. It is apparent that membrane fluidity and more specifically the ability

to segregate “favorite” lipids into the adsorption zone play a role in viral assembly.

Various other experimental observations like the ones presented above suggest that

retroviral assembly involves the coupling between several degrees of freedom: RNA

flexibility and its ability to form base pairs, the fluid nature of the membrane, the flexibility

of Gag-proteins embedded in the membrane and the electrostatic and hydrophobic

interactions between the various entities. This delicate interplay between the different free

energy contributions may explain why the observed size distribution of retroviral particles is

not monodisperse [72-74]. The wide distribution of retroviral particle sizes is quite different

from the crystal structures obtained for alpha viruses.

As a first step toward modeling this complex system, we study in this chapter the

interactions between flexible polyelectrolytes (such as ssRNA) and mixed fluid charged

membranes (such as a membrane decorated by Gag proteins). This study can be extended to a

wide range of biological systems including the one described above

Whenever a macromolecule — DNA, RNA, or protein — is adsorbed onto a fluid membrane,

the membrane can adjust its configuration to optimize the interaction by segregating those

specific lipids that best interact with the adsorbed molecule. This process leads to local

changes in lipid composition around the adsorbed molecules which, under certain conditions,

may evolve into larger-scale reorganization of membrane components, resulting in domain

formation (e.g., “lipid rafts”).

The ability of integral proteins to induce local and global changes in lipid composition

has been extensively documented experimentally [54], and analyzed theoretically, [78,79].

Similarly, experiments reveal that when charged macromolecules, such as certain kinds of

69

proteins or DNA, are adsorbed onto a mixed membrane containing a small amount of

oppositely charged lipids, the charged species migrate toward the adsorbed macromolecule

[80-82], tending to achieve local electrical neutrality. Although lowering the electrostatic free

energy of the system, the segregation of charged lipids induced by peripheral

macromolecules may involve a non-negligible entropic penalty. Several recent theoretical

studies have carefully analyzed the energetic-entropic balance associated with electrostatic

adsorption of rigid macromolecules onto fluid membranes [83-89].

The adsorption of flexible macromolecules onto such membranes is considerably

more complicated, since the macromolecule is capable of changing its own conformation in

order to enhance binding. Upon adsorption both the macromolecule and the lipid membrane

thus lose entropy, since they are no longer found in their most probable configuration. Still,

adsorption is favorable if they gain enough interaction energy.

In a recent paper, adsorption of polyelectrolytes on fluid membranes was shown to

increase the critical temperature of lipid phase separation at the membrane plane, as

compared to that of a bare lipid membrane [90]. Similar result was obtained for adsorption of

rigid bodies [91]. In this chapter, we do not include explicitly short range attractions between

membrane lipids. Lipid segregation occurs because of the “effective attraction” induced by

polymer adsorption and leads to a local change in the distribution of lipids at the interaction

zone, rather than a global phase separation.

Our main goal in this chapter is to describe the relative contributions of these different

degrees of freedom to the adsorption free energy and to the structural properties of the

macromolecule and membrane at the interaction zone. To demonstrate the important role of

lipid mobility, our results for the fluid (annealed) membrane are compared to those obtained

for a quenched membrane and for a uniform charged membrane of the same average lipid

composition. An additional achievement is the development of a simulation scheme that will

enable us to sample the equilibrium configurations and obtain free energies in a fast and

accurate way.

In Section 4.2, we describe the basic statistical-thermodynamic background

underlying the adsorption of a flexible macromolecule onto the various types of lipid

membranes mentioned above. In Section 4.3 we introduce the model system, and in Section

4.4 we provide a description of our extended version of the Rosenbluth Monte Carlo (MC)

simulation scheme [92,93]. A discussion of the simulation results for the adsorption of a

flexible macromolecule onto various types of lipid membranes appears in Section 4.5. We

70

show that the structural and energetic characteristics of a polyelectrolyte interacting with a

fluid membrane are qualitatively and quantitatively very different from those pertaining to

any specific quenched lipid membrane (Secs. 4.2.1 and 4.5.2). In the limit of vanishing

polymer concentration, however, the adsorption probabilities and the partition coefficients are

equal (Secs. 4.2.1 and 4.5.2). Differences between the two kinds of membranes appear at

nonzero concentrations of macromolecules, and will be accounted for using a simple cell

model (Secs. 4.2.2 and 4.5.2). Mixed membranes containing charged lipids are sometimes

modeled as uniformly charged surfaces or as quenched membranes of a random configuration

(thus taking the discrete nature of charges into account). We will show that in general,

uniformly charged surfaces adsorb more weakly than either a quenched or a fluid lipid

membrane. The adsorption properties of a fluid membrane can be derived using an ensemble

of quenched membranes, but only by using a weighted average over the polymer adsorption

properties of all quenched membranes in the ensemble (Sec. 4.2.1).

The strength of our simulation lies in its capacity to calculate the free energies of

complex systems with coupled degrees of freedom — systems for which most simple MC

steps will be rejected. It therefore gives us an appropriate tool for the study of complex

biological systems. We demonstrate the use of this simulation to study a particular biological

system (the MARCKS protein) in Appendix A.

4.2 Adsorption Thermodynamics

4.2.1 Single Polymer Adsorption A fluid membrane can change its lipid distribution upon polymer adsorption. A quenched

membrane, on the other hand, can not. For a quenched membrane, the polymer can “feel”

different lipid environments by adsorbing on different areas of the membrane, but the lipid

composition is fixed and can not anneal upon adsorption. A practical definition of a quenched

membrane is an ensemble of independent membrane cells, each characterized by a specific

quenched lipid configuration, m. The fraction of quenched membranes in configuration m,

( )qP m , equals to the Boltzmannic weight,

71

(0)

exp[ ( )]( )qf

U mP mq−

= (4.1)

where ( )U m is the inter lipid interaction energy of a membrane in configuration m and (0)fq is

the partition function per membrane cell of a bare fluid membrane

(0) exp[ ( )]fm

q U m= −∑ (4.2)

The partition function of a fluid membrane, occupied by an adsorbed macromolecule is given

by

(1)

,

(0) (1) (0) (1)

exp[ ( , )]

= exp[ ( )] exp[ ( | )]

( )

fm p

m p

f q m f m qm

q U m p

U m U p m

q P m q q q

= −

− −

= =

∑

∑ ∑

∑

(4.3)

Here, ( , ) ( ) ( | )U m p U m U p m= + is the potential energy corresponding to the membrane-

polymer configuration (m,p). The term ( | )U p m stands for the energy of a polymer in state p

interacting with a membrane in a given configuration m. It includes the self-energy of the

polymer (i.e., the sum of its intersegment potentials), and its interaction energy with a

membrane in state m. By ,p α= r we refer to the polymer chain conformation, α , and the

position, r of the polymer relative to the membrane plane. As above, ( )U m is the inter lipid

interaction energy.

The sum of Boltzmann factors,

(1) exp[ ( | )]mp

q U p m= −∑ (4.4)

introduced in the third equality in Eq. (4.3), is the partition function of a macromolecule

adsorbed onto a membrane of a specific lipid configuration m. The quantity (1)m q

q introduced

in the last equality of Eq. (4.3) may be interpreted as the average partition function per

membrane cell in a Boltzmann weighted ensemble of quenched membranes. From Eq. (4.3) it

follows that the partition function of a macromolecule interacting with a fluid membrane can

be expressed as a Boltzmann average of the partition functions corresponding to the ensemble

of quenched environments.

Two immediate conclusions can be made regarding the adsorption of a single

macromolecule on these two kinds of membranes:

72

I. Thermodynamic Averages - Let A be any thermodynamic property (e.g., interaction

energy, polymer radius of gyration, etc.). The average (over polymer configurations) of A for

a polymer adsorbed on a quenched membrane in configuration m is

(1)( ) ( , ) exp[ ( | )] mp

A m A m p U p m q= −∑ (4.5)

(Notice that in order to avoid confusion we denote differently the averages over the annealed

degrees of freedom and the averages over the ensemble of quenched membranesq)

The average of A for a polymer adsorbed on a fluid membrane is

,(1)

(1)

(1) (1)(0) ______

(1)(1)

( , ) exp[ ( , )]

exp[ ( )] ( , ) exp[ ( | )]

exp[ ( )] ( ) ( ) ( ) ( )

( )

m pf

f

m p

f

m mm f m

qmm q m

A m p U m pA

q

U m A m p U p m

q

U m q A m P m q A mqA m

P m qq

−=

− −=

−

= = ≡

∑

∑ ∑

∑ ∑∑

(4.6)

Note that all the quantities on the right hand side of this equation depend only on quenched

membrane properties. Eq. (4.6) thus offers a way of calculating f

A as a weighted average

of the polymer conformational averages ( )A m in the ensemble of quenched membranes. In

this biased average, here denoted by _______

( )qA m , the weight of the quenched membrane

configuration m, is the product of the fraction of such membranes ( ( )P m ) with the statistical

weight ( (1)mq∝ ) of all polymer conformations on this membrane. This formal relationship

between fluid and quenched membrane averaging may be given a physical meaning in the

limit of vanishing macromolecule concentration as will be explained below.

Note, however, that this average differs from the simple average ( ) ( ) ( )q

mA m P m A m=∑ .

II. Partition Coefficient - As will be shown in the next section, the equilibrium constant for

the adsorption process (equivalently, the adsorption probability or the partition coefficient

between lipid membrane and solution) in the limit of low polymer concentration can be

written as ( )(1) (0)b bK q q qθ ϕ= = where θ is the fraction of adsorbed sites and bϕ is the

bulk density of macromolecules. (1)q and (0)q are the partition functions for an occupied and

73

free membrane cell respectively and bq is the internal partition function of a polymer in the

bulk solution

exp[ ( )]bq Uα

α= −∑ (4.7)

The partition coefficient for both a fluid membrane and for an ensemble of quenched

membranes is identical, as can be seen in Eq. (4.8) below.

(1)(1) (1)

(0) (0)

mf q m

b f b b m q

qq qKq q q q q

= = = (4.8)

Notice that, (0) 1mq = since the quenched membrane has no degrees of freedom.

The physical intuition behind this result is that for the case of adsorption of a single

polymer (low density limit), the fluid membrane can rearrange its lipids upon polymer

adsorption, while the polymer may sample with different probabilities the different

environments of the quenched membrane. Therefore, in the low density limit, we obtain an

identical statistics for both cases. This intuition will not hold when some of the membrane

cells are already taken by macromolecules, as is the case in finite concentrations of polymer

in solution as explained in detail in the next section.

4.2.2 Finite Polymer Concentration To account for the adsorption behavior at nonzero surface concentrations we should

consider a many-cell membrane in equilibrium with a solution of macromolecules. In this

model, the membrane area A is divided into an array of /M A a= non-interacting cells, all

the same area, a , and the same lipid composition. The cell area is large enough to

comfortably accommodate one adsorbed macromolecules. The model thus, approximately,

accounts for excluded area effects but ignores other inter-macromolecule interactions.

Suppose the bulk solution is of volume V, and contains bN macromolecules of chemical

potentialμ . For simplicity we assume dilute solution behavior, in which

case ln lnb bqμ ϕ= − + , where /b bN Vϕ = is the bulk density of macromolecules, and bq is the

internal partition function of a polymer in the bulk solution as defined in Eq. (4.7). Note that

the summation here is over all possible conformations of the macromolecule, ensuring that its

center of mass (or one of its segments) is kept fixed in space. Note also that bq , like all

partition functions in our treatment, is a configurational partition function. The momentum

factors in the partition function cancel out identically in all relevant expressions [94].

74

Adsorption on a Fluid Membrane

The cells comprising a fluid membrane are identical. Treating the membrane as an open

system with respect to macromolecule exchange, the grand-canonical partition function of

the membrane is

(0) (1)( ) [ ]M Mf f f fq qξ γΞ = = + (4.9)

where / exp( )b bqγ ϕ µ= = is the absolute activity, and (0) (1)f f fq qξ γ= + is the two-state

(empty and occupied) partition function of a membrane cell. We denote by sN , the number of

macromolecules adsorbed on the membrane surface and by / /f s sN M N a Aθ = = , the

fraction of adsorbed cells (which is also the membrane area occupied by macromolecules,

i.e., the “surface coverage”). Using the relationship lnsN µ= ∂ Ξ ∂ , we get (1) /f f fqθ γ ξ= .

Therefore, we obtain a Langmuir-like adsorption equation

(1) (1)

(0) (0)1fFf f fb

bf f b f

q qe

q q qθ γ ϕ ϕθ

−∆= = =−

(4.10)

In the third equality we have introduced the dimensionless bulk concentration

/b b bN Vϕ ϕ ν ν= = , where ν is a volume per macromolecule defined in more detail below.

Thus, bϕ may be regarded as the volume fraction of polymers in solution.

Adsorption on a Quenched Membrane

The grand-canonical partition function of the quenched membrane is given by

(1)( ) [1 ]m mM Mq m m

m m

qξ γΞ = = +∏ ∏ (4.11)

where ( )m qM P m M= is the number of membrane cells with a 2D lipid distribution m. Using

Eq. (4.11), we can write

ln ln( ) ( )q m

m q q mqm m

P m P mξθ θµ µ

∂ Ξ ∂= = =

∂ ∂∑ ∑ (4.12)

Where mθ is the probability of finding an m-cell occupied by a macromolecule. Notice that the

fraction of occupied membrane cells is simply an average of mθ over all quenched membrane

configurations, m. From the equations above we get, (1) /m m mqθ γ ξ= , so that

(1)

(1)

1mFm m

m b bm b

qq eq

θ γ ϕ ϕθ

−∆= = =−

(4.13)

75

A Comparison between Fluid and Quenched Membranes

When averaging over all the quenched configurations, m, and using Eqs. (4.3), (4.10) and

(4.13), we find

(1)

(1)(0)( )

1 1 1f fm m b b

q m qmm m b b f fq

qP m q

q q qθθ θ ϕ ϕ

θ θ θ= = = =

− − −∑ (4.14)

Equivalently,

ln mf q

FF e−∆∆ = − (4.15)

Actually, with the definitions of fF∆ and mF∆ given in Eqs. (4.10) and (4.13), the last

equality follows directly from Eq. (4.3).

Both /(1 )m mθ θ− and exp( )mF−∆ are convex functions of their arguments. Using Jensen’s

inequality of convex functions [95], it thus follows from Eqs. (4.14) and (4.15) that

and m f m fq qF Fθ θ≤ −∆ ≤ −∆ (4.16)

for any probability distribution ( )qP m . In other words, on average, macromolecule

adsorption onto an ensemble of quenched membranes (whose lipid configurations appear

with probabilities ( )qP m ) is always weaker (lower and smaller - Fθ ∆ ) than adsorption onto

a fluid membrane of the same lipid composition.

From Eq. (4.14) it follows that the equality m fqθ θ= is obtained only in the limit of

vanishing surface coverage, i.e., when 0bϕ → . ( m fqF F∆ → ∆ requires that all mF∆ are

negligibly small, as can be seen from Eq. (4.15).)

Note also that in the limit 0bϕ → , we obtain (see Eq. (4.13)) (1)m mqθ ∝ . In a Boltzmann

weighted ensemble of quenched membranes, the probability of finding a macromolecule

bound to a membrane of lipid configuration m, is ( ) q mP m θ and hence is proportional

to (1)( )q mP m q . If measured over an ensemble of quenched membranes, the average of a

physical observable ( , )A m p would then be given by

(1) (1)( ) ( ) ( ) / ( )q m q mqm m

A m P m q A m P m q=∑ ∑ , which, as noted in Eq. (4.6), is equal to f

A .

Thus, in the 0θ → limit, both the average surface concentration, and the “θ -weighted”

average of ( , )A m p over the quenched membrane ensemble approaches the corresponding

quantities in the fluid membrane.

76

A physical interpretation of the last conclusion can be given in terms of the cell model,

as follows. When 0θ → each of the adsorbed macromolecule can freely and independently

explore all membrane environments m either by lateral diffusion on the membrane surface or

by desorption from one local membrane region and adsorption into another, thereby sampling

all possible lipid-polymer configurations m,p. Furthermore, since none of the cells is blocked,

all m and hence all m,p are sampled according to their Boltzmann weights, just like the states

sampled by a macromolecule on a fluid membrane. The difference is, of course, that a

macromolecule adsorbed on a fluid membrane need not migrate from one cell to another in

order to sample the entire configuration space. Consequently, fθ θ= is the same for all cells

of the fluid membrane, whereas a wide distribution of 'smθ characterizes the ensemble of

quenched membranes with an average that converges to the fluid membrane value

m fqθ θ→ . (Note that kinetic barriers may interfere with this behavior)

Upon increasing the concentration of macromolecules in solution, the more strongly

adsorbing cells (m-values) of the quenched membrane will be occupied first. Once these

favorable local environments are populated, further adsorption is necessarily suppressed.

This implies fqm θθ ≤ , because in the fluid membrane every cell can independently anneal

its lipid distribution, thereby enhancing adsorption.

Our conclusions regarding the relationship between macromolecule adsorption on

quenched vs. fluid membranes agree with previous works pertaining to polymer statistics in

random media. Cates and Ball [96] have studied the behavior of a single long polymer chain

in a random medium and concluded that as long as the environment is infinite the quenched

and annealed averaging will yield the same statistical chain properties. Our fluid and

quenched membranes are analogous to the annealed and quenched random potentials in the

treatment above. Inequalities valid for the multi-chain adsorption, have been obtained by

Andelman and Joanny [97,98] for neutral chains adsorbing on annealed and quenched flat

surfaces. The main conclusion there is that the density of polymers on an annealed surface

(membrane) is always higher than in the quenched case.

4.2.3 Adsorbed State Definition In the discussion found in the previous sections, we use a two state model. We treat the

membrane cell as either occupied or empty and the polymer molecule as either adsorbed or

free. However, the partition function of an adsorbed polymer is a continuous function of the

77

distance of the polymer from the membrane plane. How then can we distinguish between an

adsorbed polymer and a free polymer in solution?

A consistent definition of an adsorbed polymer is: a polymer which is found within

the adsorbed layer, that is, whose distance from the membrane, z, is smaller than a certain

cutoff distance λ; beyond that distance, which is comparable to the range of membrane-

polymer interactions, the macromolecule is not affected by the membrane. The distance z

from the membrane can be measured in terms of any chain segment (or the center of mass).

In the simulations described in the next section, we find it convenient to measure this distance

in terms of 1z , the normal displacement of the first chain segment (see Figure 4.2). For

1z λ≤ , the macromolecule is considered adsorbed, and otherwise as free in solution.

The cutoff distance can be determined in one of three alternatives, which are all equivalent

(i) The distance where the polymer properties are effectively identical to its bulk

properties, i.e., (0)1 1( ) ( )f f f bq z q z q qλ= ≅ = ∞ = .

(ii) The distance where the polymer segment density is equal to its bulk density, i.e.,

( ) ( ) bz zρ λ ρ ρ= = = ∞ = .

(iii) The distance on which the surface excess is converged. The surface excess is the

increase in polymer density near the surface in comparison to its density in the

bulk.

All three definitions are equivalent but use different thermodynamic quantities. In the

following we will elaborate on each one of them.

The partition function profile

The partition function of an adsorbed polymer on a fluid membrane, whose first segment is

anchored at distance 1z from the membrane, is given by

1 1,

( ) exp[ ( , | )]fm

q z U m zα

α= −∑ (4.17)

For large enough 1z ( 1z λ> ), there is no interaction between the polymer and the membrane

and thus we have 1( , | ) ( ) ( )U m z U m Uα α= + , and hence (0)1( ) ( )f f f bq z q q qλ> = ∞ = .

With 1,p z α≡ , we find from Eq. (4.3) that the partition function of a macromolecule

adsorbed on a fluid membrane is given by

(1) (1)1 1 1 1

,0 0

ˆexp[ ( , | )] ( )f f fm

q a U m z dz a q z dz vqλ λ

α

α= − = =∑∫ ∫ (4.18)

78

with v aλ≡ and

(1)1 1 1

,0 0

1 1ˆ ( ) exp[ ( , | )]f fm

q q z dz U m z dzλ λ

α

αλ λ

= = −∑∫ ∫ (4.19)

The factor a in Eq. (4.18) results from the fact that the partition function per molecule must

be proportional to the cell’s area. The volume element, v aλ≡ , may now be interpreted as

the volume of a membrane cell. Thus, (1)ˆ fq represents the (average) partition function per unit

volume of an adsorbed macromolecule.

For a quenched membrane, the partition function as a function of 1z for a given membrane

configuration, m, is defined as

1 1( ) exp[ ( | , )]mq z U m zα

α= −∑ (4.20)

Here, the cutoff distance is different for every quenched membrane cell, ( ) mmλ λ≡ . For a

specific membrane configuration, m, we can write that for 1 mz λ≥ , 1( ) ( )m m m bq z q qλ> = ∞ = .

The partition function of a macromolecule adsorbed on a quenched membrane with

configuration m is given by

(1)1 1 1 1

0 0

exp[ ( | , )] ( )m m

m mq a U m z dz a q z dzλ λ

α

α= − =∑∫ ∫ (4.21)

We may now rewrite Eq. (4.10) and (4.13) in the form

11(0)

0

( )11

fFf fb b

f f b

q zdz e

q q

λθϕ ϕ

θ λ−∆⎛ ⎞

= =⎜ ⎟⎜ ⎟− ⎝ ⎠∫ (4.22)

And

11

0

( )11

m

mFm mb b

m b

q z dz eq

λθ ϕ ϕθ λ

−∆⎛ ⎞

= =⎜ ⎟⎜ ⎟− ⎝ ⎠∫ (4.23)

In Eq. (4.23), we keep using the same concentration units, b baϕ λϕ= , to enable

straightforward comparison with the fluid membrane.

The segment density profile

For chain molecules composed of L segments, the segment density in the bulk solution is

( ) b bz Lρ ρ ϕ= ∞ = = . Near the membrane surface, the segment density, ( )zρ , is different

from bρ and is given by

1 1 1( ) ( ) ( | )z dz z n z zρ ϕ= ∫ (4.24)

79

where 1( )zϕ is the density of macromolecules whose first segment is at 1z , and 1( | )n z z dz is

the average number of chain segments between z and z dz+ due to chains originating at 1z .

For small values of θ we can write 1 1( ) ( ) / ( )bz q z qϕ ϕ= ∞ with 1( ) / ( )q z q ∞ derived from our

single-chain simulations. Approximate density profiles for nonzero surface concentrations

can be derived by expressing ( )zρ as the product of the probability (θ ) to find the cell

occupied and the normalized density profile corresponding to one adsorbed molecule.

We then find for z λ≤ ,

1 11 1 1 1(1)

0 0

( ) ( )( ) ( | ) (1 ) ( | )( )b

q z q zz n z z dz n z z dzq q

λ λ

ρ θ ϕ θ= = −∞∫ ∫ (4.25)

The second equality was observed using Eqs. (4.10) and (4.13). Notice that this expression

converges for low polymer concentrations to the one given in the text for small θ values.

For z λ> we must require ( ) bzρ ρ= . Eq (4.24) and (4.25) apply to the fluid membrane, as

well as to any quenched membrane in state m.

The surface excess

The surface excess is defined as

( ) ( )0

( ) b s bz dzρ ρ λ ρ ρ∞

Γ = − = −∫ (4.26)

where in the second equality, ( )0

1 ( )s z dzλ

ρ λ ρ≡ ∫ is the average (three dimensional) density

of chain segments within the surface layer. Note that the upper limit in the integral defining

Γ , can be replaced by λ (or any larger value). The second equality may also be regarded as

the definition of the surface layer thickness λ.

In practice, 1 10 0

( ) (1/ ) ( )z dz L z dzλ λ

ϕ ρ≈∫ ∫ , so that the integral over ( )zρ can be replaced by the

integral over ( )zϕ . Namely, we can choose λ either as the smallest value of 1z beyond

which the ratio 1 1( ) ( ) ( ) bq z q zϕ ϕ∞ = is practically one or according to Eq. (4.26), and the

two definitions are indistinguishable. The equality of the two integrals follows from the fact

that for practically all 1z within λ, all chain segments will be found inside the surface layer.

For chains originating near λ, say at 1z λ δ= − (δ λ ), some conformations will cross the

z λ= surface, contributing less than L segments to the surface layer density. By symmetry,

80

however, chains originating at 1z λ δ= + will compensate for the loss of segments from

the 1z λ δ= − chains. The (near) equivalence of chains originating at 1z λ δ= ± follows from

the fact that these chains are hardly affected by the membrane.

4.3 The Model System Our model system consists of a single polyelectrolyte interacting with a finite size membrane,

large compared to the size of the polymer and the range of intermolecular potentials. In most

of our simulations, the polyelectrolyte chains are composed of L=20 spherical segments of

diameter d, interacting with a (considerably larger) 2D membrane cell consisting of

2500M = lipid headgroups. The lipid membrane is modeled as a perfectly flat and

impenetrable 2D hexagonal lattice, with lipid headgroups occupying all its lattice sites. The

lattice constant is set equal to d. The membrane may thus be regarded as an hexagonal array

of closely packed disks of diameter d, as illustrated schematically in Figure 4.2. Using a

typical lipid headgroup area of 65Å2, we find d=8.66Å. We simulate three-component

membranes, composed of electrically neutral ( 0z = ), monovalent ( 1z = − ), and tetravalent

( 4z = − ) headgroups. These may be regarded as representing, respectively, the phosphatidyl-

choline (PC), phosphatidyl-serine (PS), and phosphatidylinositol 4,5 bisphosphate (PIP2)

lipids mentioned in the introduction to the thesis. The average membrane concentration of

PIP2 is ~1% (versus the 10-30% abundance of monovalent acidic lipids, primarily PS), yet it

tends to localize in viral envelopes and membrane rafts, as well as in the binding zones of

various proteins involved in signal transduction pathways. Among these proteins is

MARCKS, which is studied in Appendix A.

The lipid charges are treated as point charges residing at the grid points of the

hexagonal lattice, and the electrostatic repulsion between them is modeled in the Debye-

Huckel (DH) approximation. Explicitly, the interaction potential between lipids of

valences 1z and 2z at distance r apart, and in units of Bk T , is

1 2exp( )( )DH B

ru r z z lrκ−

= (4.27)

where 2 /B Bl e k Tε= is the Bjerrum length, and 1κ − is the Debye screening length; e denoting

the elementary charge and ε is the dielectric constant. In all calculations we use Bl =7.14Å,

appropriate for water ( 78)ε = at room temperature, and 1 10κ − = Å, which corresponds to

typical physiological conditions (monovalent ionic strength of about 0.1M). Note that 1κ − is

81

comparable to the other relevant length scale in our system, namely, the distance (d=8.66Å)

between adjacent lipid charges, as well as between adjacent polymer charges.

In the simulations each polymer bead carries a unit positive charge ( 1z = + ), localized

at its center. While the polymer bond length d is fixed, there are no other restrictions on bond

angles, except for those implied by electrostatic and spatial (excluded volume) repulsion

between non-bonded segments. For the electrostatic interaction between polymer charges we

again use DH potentials. The spatial repulsion is modeled using the (shifted and truncated)

Lennard-Jones potential:

12 6 1/6

1/64 [( / ) ( / ) ] for 20 for 2

( )LJr r r

ru r ε σ σ ε σ

σ− + ≤

≥⎧

= ⎨⎩

(4.28)

Note that only the short-range repulsion of the 6:12 Lennard-Jones potential is retained.

Setting 1/ 62 dσ = and 0.1 Bk Tε = ensures the onset of steep repulsion as soon as r falls

below d [99].

The electrostatic attraction between the oppositely charged polymer and membrane is

also modeled using screened DH potentials. In addition, the membrane surface is treated as

an impenetrable wall to the polymer, implying a minimal distance of d/2 between polymer

and lipid charges. At this distance the electrostatic attraction between a polymer ( 1z = + )

segment and a monovalent ( 1z = − ) lipid headgroup is 1.07 Bk T . For comparison, the

electrostatic repulsion between neighboring monovalent lipids or adjacent polymer beads,

taking the distance of closest approach to be r d= , is 0.35 Bk T . Since the distances between

charges in the system are either comparable to or larger than the Debye length, i.e., 1r d κ −≥ ≈ , screening by counterions is expected to be effective. Under physiological

conditions, when 1κ − is small (of the order of few Angstroms), the long-range character of

the electrostatic interactions is screened and DH potentials offer a reasonable approximation.

These potentials are commonly employed in simulation and theoretical studies of

polyelectrolyte-surface interactions, see e.g., [100,101]. Henceforth, we shall measure all

distances in units of d . Recall also that energies are measured in units of Bk T .

4.4 Simulation Method The Rosenbluth MC method, [92] or its ‘configurational-bias’ variant, provides an efficient

means for simulating polymer statistics [93]. In this approach, chain conformations are

generated, segment after segment, with preference for conformations of large statistical

82

weight. Based on these ideas we present below our extension of the Rosenbluth scheme for

modeling polyelectrolyte adsorption on fluid, as well as quenched and uniform membranes.

In order to get a configuration of a polymer on a quenched/uniform membrane, the polymer

is grown segment after segment in the membrane “field”. When growing the polymer on a

fluid membrane, both the polymer and the membrane are grown simultaneously as one

dimensional and two dimensional chains respectively. This simulation can be extended quite

easily to include various kinds of objects other than polymer and lipid membrane and various

interaction potentials.

Figure 4.2 A schematic drawing of the simulation model. A 20-segment long chain of spherical segments, each

carrying a single point charge in its center, interacts with a mixed membrane composed of neutral, singly

charged and tetra-valent anionic lipids, which occupy the sites of a 2D hexagonal lattice. Lipid charges are

concentrated in the centers of the corresponding discs. The lipids can diffuse (exchange positions) within the

membrane plane. The polymer chain is flexible, but subjected to electrostatic and short range spatial repulsion

between its constituent segments. The diameters, d, of polymer segments and lipid disks are equal.

4.4.1 Quenched membrane Consider first a polymer interacting with a membrane of quenched lipid configuration

m. The simulation begins by placing the first chain segment at distance 1z above the center

of the membrane cell, where its interaction energy with membrane lipids is 1( , )u z m , see

Figure 4.2. We then sample k random directions (and hence positions, 2r ) for segment 2 and

select one, say 22jr , with probability 2

2 1 2exp[ ( | , )] /ju z m w− r , where

22 1 2 1( | , ) ( | , )ju z m u j z m≡r is the interaction energy of segment 2 with segment 1 and the

membrane, and 2

2

2 2 11exp[ ( | , )]

kj

jw u z m

=

= −∑ r is a local partition function. This procedure is

continued until all segments of the chain are generated. Repeated applications of this scheme

83

[for the given ( 1,m z )] yield an ensemble of conformations 2 1{ ,..., | , }L z mα = r r with

probabilities

3 2 12 1 2 1 1

1 1

( | , , )( | , ) ( | ... , , )

12 3( | , ) ( | , )

11 2 1

( | , )

( | , )

L LU z mU z m U z m

RL

U z m U z m

LL

e e eP z mw w w

ke ew w w k W z m

α α

α

α

−−− −

− −

−

= × ⋅⋅⋅×

= =⋅⋅ ⋅

r rr r r r

(4.29)

As above, 1 1 1 12( | , ) ( , ) ( | ,..., , )L

l llU z m u z m u z mα −=

= +∑ r r is the sum of the polymer self

energy and its interaction with the membrane.

The “partition function”

1 1( | , ) ( / )L

llW z m w kα

==∏ (4.30)

(with 1 1exp[ ( , )]w k u z m≡ − ), is the complete “Rosenbluth factor” of the polymer-membrane

configuration 1( ; , )z mα . Note that W becomes independent of k in the limit k →∞ . In our

calculations we generally use k=50. Note also that some of the k vectors pointing from

segment l to l+1 may cross the membrane interface, especially if segment l is near the

surface. Their probability, and likewise their contribution to 1lw + (and hence to W) is zero,

reflecting the loss of entropy associated with the presence of the hard membrane wall.

Since every possible conformation α is sampled with probability proportional to

exp[ ( )] / ( )U Wα α− , proper Boltzmann averaging requires weighting each α by its

Rosenbluth factor ( )W α ; i.e., the average (over α, for the given 1,z m ) of any structural or

energetic polymer property A is given by 1 1 1 1( , ) ( ; , ) ( ; , ) / ( ; , )A z m W z m A z m W z m

α αα α α=∑ ∑ (4.31)

Note also that the partition function corresponding to all polymer conformations originating

at 1z is

1 11 1 1( ) ( , ) ( | , ) / 1L L

mq z k W z m k W z mα α

α− −= = ∑ ∑ (4.32)

where it should be stressed that the sum runs over all theα generated by the Rosenbluth

scheme

(and thus, 1 11 1 1( , ) ( | , ) ( | , )L L

Rk W z m k W z m P z mα

α α− −= ∑ and using Eq. (4.29), we get

84

11

( | , )( | , )1 1

1 1 111

( , ) ( | , ) ( )( | , )

U z mU z mL L

mL

ek W z m k W z m e q zk W z m

αα

α α

αα

−−− −

−

⎛ ⎞= = =⎜ ⎟⋅⎝ ⎠

∑ ∑ )

For 1z λ> we have 1( ) ( )m m bq z q q= ∞ ≡ .

Averaging 1( , )A z m over all 1z λ≤ we obtain the average of A (over all conformations) for

molecules adsorbed on a quenched membrane of lipid configuration m,

1 1 1 1 10 0( ) ( ) ( , ) / ( )m m

m mA m q z A z m dz q z dzλ λ

= ∫ ∫ (4.33)

Similarly,

(1) 11 10

ˆ (1/ ) ( ) ( )m

m

Lmq q z dz k W m

λλ −= =∫ (4.34)

is the partition function of the adsorbed polymer, (see Eq. (4.23)).

4.4.2 Fluid Membrane From Eqs. (4.3) and (4.6) we know that the thermodynamic and structural properties of a

fluid membrane can be modeled based on simulating an ensemble of quenched membranes.

However, this procedure is rather indirect and often impractical. Alternatively, adsorption on

the fluid membrane could be simulated by combining the Rosenbluth and Metropolis

methods. That is, after generating a polymer in conformation 1( ; )p zα= for a given lipid

configuration m, the membrane is allowed to relax to a new configuration m’ through a series

of Metropolis moves. Another polymer conformation p’ can then be generated for m’, letting

the membrane relax to m", and so on. The problem here is that the relaxed membrane is no

longer the one which served to generate the last polymer conformation. A “retracing”

procedure [93] can be used to improve this scheme, but not fully eliminate its

inconsistencies. We have, therefore, adopted an alternative simulation method for the fluid

membrane whereby, in the spirit of the Rosenbluth sampling scheme, we generate

simultaneously both polymer conformations p and membrane configurations m, as follows.

Any joint polymer-membrane configuration p,m is fully specified by the coordinates of ( 1) ( 4)K L M M− −= + + particles; that is, L polymer segments, ( 1)M − monovalent lipids and

( 4)M − tetravalent lipids. ( ( 0 ) ( 1) ( 4 )M M M M− −= − − neutral lipids occupy all other membrane

sites.) We now generate a joint (p,m) configuration by randomly adding either a polymer

segment or a charged lipid, until all particles have been placed. More explicitly, suppose the

new configuration is already partly grown, consisting of a polymer chain of length l, and a

85

partially charged membrane containing ( 1)m − and ( 4)m − anionic lipids. One of the remaining

( ( 1) ( 4)K l m m− −− − − ) particles is now randomly selected and added to the system. If this is a

polymer segment it is added as the ( 1l + )-th segment of the chain. As before, this segment is

placed in one of k possible positions, with probability ( 1) ( 4)1exp[ ( 1; , , )] / lu l l m m w− −+− + ;

( 1) ( 4)( 1; , , )u l l m m− −+ is the interaction potential of the added particle with all those already

placed, and 1lw + is defined as usual. If the new particle is, say, a monovalent lipid it is placed

with probability ( 1)( 1) ( 1) ( 4)

1exp[ ( 1; , , )] /

mu m l m m w −

− − −+

− + in one of mk randomly chosen

membrane sites, where ( 1) ( 1) ( 4)( 1; , , )u m l m m− − −+ is the interaction energy of this lipid with the

rest of the system, and ( 1) 1mw − +

is the sum of the Boltzmann factors corresponding to the mk

membrane sites. (In the simulations we usually sample mk =1000 sites, some of which are

possibly occupied already and thus do not contribute to w.) This procedure is repeated until

all chain segments and all charged lipids are placed, resulting in a statistical distribution of

p,m configurations, whose probabilities are

( 1) ( 4)[ 1] [ ]

exp[ ( , )]( , )( , )R L M M

m

U p mP p mk k W p m

− −− +

−=

⋅ (4.35)

where

( 1) ( 4)

1 0

( , ) ( / ) ( / )L M M

l i ml i

W p m w k w k− −+

= =

= ×∏ ∏ (4.36)

is the (generalized) Rosenbluth factor of configuration p,m.

As for the quenched membrane, we generally sample many polymer-membrane

configurations corresponding to various 1z values and only then average over this variable.

The averaging procedure is analogous, i.e.., the average of A for a given 1z is 1 1 1 1, ,

( ) ( , | ) ( , | ) / ( , | )m m

A z W m z A m z W m zα α

α α α=∑ ∑ (4.37)

The partition function, introduced in Eq. (4.17), corresponding to all system configurations

(membrane and polymer) where the polymer first segment is found on 1z is

( 1) ( 4) ( 1) ( 4)[ 1] [ ] [ 1] [ ]1 1 1, ,

( ) ( ) ( , | ) / 1 L M M L M Mf m m m m

q z k k W z k k W m zα α

α− − − −− + − += = ∑ ∑ (4.38)

where the sum runs over all the configurations ( , )mα generated by the Rosenbluth scheme

(and therefore are generated according to the probability given in Eq. (4.35)).

86

Averaging over 1z ,the average of the thermodynamic quantity, A for an adsorbed polymer

will get the form

1 1 1 1 10 0( ) ( ) / ( )f ff

A q z A z dz q z dzλ λ

= ∫ ∫ (4.39)

Similarly, the total partition function is given by

( 1) ( 4)(1) [ 1] [ ]

1 10ˆ (1/ ) ( )

f

L M Mf m f

q q z dz k k Wλ

λ− −− += =∫ (4.40)

4.4.3 Free Energies of Adsorption Quenched Membrane

Following Eq. (4.23), the free energy of adsorption on a quenched membrane in configuration

m is given by

( ) 1 10

ln 1 ( )m

m b mF q q z dzλ

λ⎡ ⎤

∆ = − ⎢ ⎥⎢ ⎥⎣ ⎦

∫ (4.41)

where bq is the partition function of a free polymer in solution as introduced in Eq. (4.7). We

extract bq from the simulation using a similar scheme to the one described in Section 4.4.1,

but without the presence of a membrane, i.e., ( , ) ( )U m Uα α≡ or alternatively, 1z = ∞ .

Therefore, in analogy to Eq. (4.32), bq is given by

1 1 ( ) 1L Lb b

q k W k Wα α

α− − ⎛ ⎞= = ⎜ ⎟⎝ ⎠∑ ∑ (4.42)

where again, the conformationsα were generated by the Rosenbluth scheme.

Using Eqs. (4.41), (4.32) and (4.42), the free energy of adsorption is given by

( )1

1 10

1 110

( , )1 1ln ln ( , )

m

m

L

m L bb

k W z m dzF W z m W dz

k W

λ

λ

λ λ

−

−

⎛ ⎞⎜ ⎟ ⎛ ⎞⎜ ⎟∆ = − ⋅ = − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠

∫∫ (4.43)

Notice that we use the same value for k in all the simulations, and hence the value of the free

energy does not depend in our choice of k, as long as we sample enough configurations.

Fluid Membrane

Following Eq. (4.22), the free energy of adsorption on a fluid membrane is

87

( )(0)1 1

0

ln 1 ( )f b f fF q q q z dzλ

λ⎡ ⎤

∆ = − ⎢ ⎥⎣ ⎦

∫ (4.44)

where (0)fq is the partition function of an empty membrane as introduced in Eq. (4.2). In order

to extract (0)fq from the simulation, we use a similar scheme to the one described in section

2.4.2, but without the presence of a polymer, that is, we add charged lipids until all lipids

have been placed. More specifically, say we add a monovalent lipid, we add it to one of mk

randomly chosen membrane sites with probability ( 1)( 1) ( 1) ( 4)

1exp[ ( 1; , )] /

mu m m m w −

− − −+

− +

where ( 1) ( 1) ( 4)( 1; , )u m m m− − −+ is the interaction energy of this lipid with the rest of the

system (that includes only the membrane) and ( 1) 1mw − +

is the sum of the Boltzmann factors

corresponding to the mk membrane sites. Therefore, in analogy to Eq. (4.38), (0)fq is given

by

( 1) ( 4) ( 1) ( 4)(0)(0) [ ] [ ] ( ) / 1 M M M M

f m mf m mq k W k W m

− − − −+ += = ∑ ∑ (4.45)

where the conformations m were generated by the Rosenbluth scheme.

Using Eqs. (4.38), (4.42), (4.45) and (4.44), the free energy of adsorption is given by

( )( 1) ( 4)

( 1) ( 4)

11 1

(0)01 1(0)1

0

1 ( )1ln ln ( )

L M Mm

f b fL M Mmb f

k k W z dzF W z W W dz

k W k W

λ

λλλ

− −

− −

− +

− +

⎛ ⎞⎜ ⎟ ⎛ ⎞⎜ ⎟∆ = − = − ⎜ ⎟⎜ ⎟⋅ ⎝ ⎠⎜ ⎟⎝ ⎠

∫∫

(4.46)

Notice that again, in order to enable comparison between the different partition functions, we

use the same values for mk and for k in all the simulations we perform.

4.5 Results From the simulations we have derived the basic thermodynamic characteristics of

macromolecules interacting with fluid, quenched, and uniformly charged membranes. In

parallel, for every system considered we have calculated a variety of structural properties,

such as the 2D distribution of charged lipids in the membrane plane, or the density profile of

chain segments along, as well as perpendicular to, the membrane normal. Two membrane

compositions were analyzed in detail:

88

i) PC:PS:PIP2 =98:1:1 membrane, i.e., a membrane containing 98% neutral

( 0z = , or “PC”) lipids, 1% monovalent ( 1z = − , PS) lipids and 1% tetravalent

( 4z = − , PIP2) lipids.

ii) PC:PS:PIP2 =89:10:1.

Note that the average charge, per lipid, corresponding to these membranes (hereafter also

referred to as the “weakly charged” and the “strongly charged” membranes) is

0.01( 1) 0.01( 4) 0.05wz = − + − = − and 0.1( 1) 0.01( 4) 0.14sz = − + − = − , respectively. For both

compositions, simulations were performed for fluid, quenched and uniformly charged

membranes. In the uniformly charged membrane all lipids carry the same partial charge, z .

For the sake of comparison we have also performed a limited number of simulations

for a stiff (rod like) polymer, as well as for a weakly charged ( 1/ 2)z = + polymer. For the

3D case of a polymer in solution we have also carried out, for comparative reasons, one set

of simulations for an electrically neutral polymer.

The number of chain-membrane conformations generated for (each 1z value of) a

polymer adsorbed on a fluid membrane is of the order of 610 . The number of chain

conformations generated for (each 1z value of) a given quenched membrane m is about 310 ,

and the number of membrane configurations is 410 . The increments in chain origin positions

are 1 1z∆ = . (Recall that distances are measured in units of d.)

A pictorial illustration of the polymer-membrane configurations generated by our

simulations is given in Figure 4.3. The figure shows top and side views of two (rather

arbitrary) simulation snapshots of a polyelectrolyte interacting with a fluid membrane of

composition PC:PS:PIP2=98:1:1. Only part of the membrane is shown, yet it is apparent that

the local concentration of charged lipids in the vicinity of the polymer exceeds significantly

the membrane average.

We begin by describing the structural properties of both the macromolecule and the

lipid distribution which occur upon adsorption on fluid membranes. Next, we compare the

adsorption thermodynamics between fluid, quenched and uniform membranes, referring to

the thermodynamic background described in Section 4.2.

4.5.1 Structural Properties

The structural and thermodynamic properties of the adsorbed macromolecules are intimately

related to each other. For instance, the density profile of chain termini, 1( )zϕ , enters the

89

calculation of partition functions and free energies. In this subsection we present information

pertaining to the configurational statistics of the adsorbed polymer and the concomitant

changes in the 2D distribution of membrane lipids. Since changes in lipid distribution can

only occur in fluid membranes the discussion in this subsection involves only fluid

membranes.

Figure 4.3 Side and top views of two, rather arbitrary, simulation snapshots (left and right), of a polyelectrolyte

interacting with a weakly charged fluid membrane (1% PIP2, and 1% PS). For visual clarity only a section of the

membrane is shown, and polymer segments and lipid headgroups are depicted as small spheres, (recall,

however, that short range repulsions keep these segments at distance d≥ ). PIP2 and PS lipids are represented

by blue and purple spheres, respectively. Note the localization of the charged lipids in the vicinity of the

polymer.

Polymer dimensions

In Table 4.1, we present the results of our simulations for some of the basic conformational

characteristics of the 20-segment polyelectrolyte chain, when adsorbed onto the weakly and

strongly charged membranes. For the sake of comparison we also list the corresponding

values of the charged polyelectrolyte, as well as for the corresponding neutral chain, in an

isotropic bulk solution.

The 3D polymer radius of gyration, 1/ 23 2 2 2D

gR x y z= + + gives information about the

effective size of the polymer in three dimensions, while the 2D radius of gyration 1/ 22 2 2( , )D

gR x y x y= + in a plane parallel to the membrane surface, and the 1D radius of

gyration 1/ 21 2( )D

gR z z= along the membrane normal give information about the degree of

90

extension of the polymer on the surface and into solution respectively. In solution, 3DgR is

significantly larger for the charged polyelectrolyte than for the neutral polymer, owing to

electrostatic repulsion between chain segments, [4,5]. We find 3DgR =2.97 vs. 2.50 for the

neutral polymer; the corresponding end-to-end distances are 3DeR =7.92 and 6.36,

respectively. The ratio 3 3/D De gR R is close to the theoretical value, 6 , for an ideal chain [6].

From Table 4.1, it is apparent that the polymer’s 3D radius of gyration, ( 3DgR ), does

not change much upon adsorption. Yet, 2 ( , )DgR x y and 1 ( )D

gR z are quite different from the

corresponding bulk values. As expected, upon adsorption, the polymer flattens parallel to the

membrane plane (see also Figure 4.3), resulting in larger 2 ( , )DgR x y and smaller 1 ( )D

gR z with

enhanced anisotropy on the strongly adsorbing membrane. We also note a substantial

increase in the persistence length, pξ , upon adsorption, reflecting the stretching of the

polymer chain along the membrane plane.

In Table 4.1 we also list the width of the chain density profile along the membrane normal,

22( )z z zσ = − , where 0 0

( ) / ( )k kz z z dz z dzλ λ

ρ ρ= ∫ ∫ is the k-th moment of the segment

density ( ) ( , , )z dxdy x y zρ ρ= ∫∫ . Here ( )zρ is the same quantity defined in Eq.(4.24), and

1 1 1( , , ) ( ) ( , , | )x y z dz z n x y z zρ ϕ= ∫ is the segment density at x,y,z, where 1( , , | )n x y z z dxdydz

is the number of segments in dxdydz around , ,x y z , due to chains originating at 1z . Note that

in calculating 1( , , | )n x y z z , and hence ( , , )x y zρ , we average over many chain

conformations, ensuring that their centers of mass reside on one z-axis. Figure 4.4 shows the

radial distribution of chain segments relative to the membrane normal, ( )rρ which is defined

by

2 2 2

0 0 0

( ) ( , , ) ( )r dz dxdy x y z x y rλ

ρ ρ δ∞ ∞

= + −∫ ∫ ∫ (4.47)

with 2 2r x y= + denoting the distance from the z-axis and ( )xδ being the delta function.

It is not difficult to show that if the centers of mass of all chain conformations are

superimposed onto one z-axis then

1/ 2

2 2

0 0( , ) ( )2 / ( )2D

gR x y r r rdr r rdrρ π ρ π∞ ∞⎡ ⎤= ⎢ ⎥⎣ ⎦∫ ∫ (4.48)

91

0 2 4 6 8 10r

0

0.02

0.04

0.06

ρ(r)

Equivalently, ( )rρ is the projection of the segment density distribution on the membrane

plane. The radial distribution in Figure 4.4 is shown for macromolecules adsorbed on the

weakly and the strongly charged membranes, indicating a radial span of ~5 segment

diameters in both cases. As noted already in Table 4.1, and as follows by comparing Figure

4.4 and Figure 4.10, the lateral dimensions of the adsorbed macromolecule are about twice

larger than its extension along the membrane normal.

3DgR 2 ( , )D

gR x y 1 ( )DgR z ( )zσ pξ

Solution 2.97 (2.50) 2.42 (2.04) 1.72 (1.45) 1.72 (1.45) 4.70 (3.15)

0.14fz = − 2.91 2.80 0.79 2.13 8.23

0.05fz = − 2.91 2.63 1.25 4.00 5.37

Table 4.1 Macromolecule conformational properties: Conformational properties of the polymer in solution and

when adsorbed on the weakly ( 0.05fz = − ) and strongly ( 0.14fz = − ) charged membranes. The numbers in

parenthesis are for an electrically neutral polymer. 3DgR is the 3D radius of gyration of the polymer, 2 ( , )D

gR x y is

the 2D radius of gyration in a plane parallel to the membrane surface, 1 ( )DgR z is the z component of the radius of

gyration (measured, as usual, with respect to the center of mass), ( )zσ is the width of the segment density

distribution along z, and pξ is the persistence length.

Figure 4.4 The integrated 2D density, ( )rρ , of chain segments as a function of the radial distance from the

membrane normal. The solid and dashed curves are for the strongly and weakly charged fluid membranes,

respectively.

92

0 2 4 6 8 10r

1

3

5

0 2 4 6 8 10r

1

1.5

2

(a) (b)

PS

PIP2

PS

ψ(r)ψ

0

ψ(r)ψ

0

Lipid Redistribution

One important characteristic of the 2D lipid distribution is the “enrichment factor” ( )i irψ ψ .

This is the ratio between the local concentration of lipid species i at distance r from the

(projection on the membrane plane) of the polymer’s center of mass, and the average (or

“bulk”) concentration of this lipid in the membrane. The enrichment factor thus measures the

change in local lipid composition following macromolecule adsorption.

In Figure 4.5 we show the enrichment factor for two ternary membranes, PC:PS:PIP2

=89:10:1, and 98:1:1 (left); and two binary membranes, PC:PS=90:10 and 99:1 (right).

Comparing Figure 4.4 and Figure 4.5 we find that the range of the lipid region enriched with

charged lipids, namely, ~5 lipid diameters, correlates closely with the lateral dimensions of

the adsorbed polymer.

Figure 4.5 The enrichment factor of charged lipids associated with macromolecule adsorption on a ternary lipid

mixture of PC/PS/PIP2 (a) , and a binary mixture PC/PS (b), as a function of the radial distance from the

polymer’s center of mass. The bulk molar fraction of PIP2 in (a) is in all cases 0.01. Solid curves and dashed

curves, in both figures, correspond to PS molar fractions of 0.1 and 0.01, respectively.

Another view of the lipid density profile is shown in Figure 4.6a, which displays the

distribution of tetravalent lipids around the projection onto the membrane plane of the

polymer’s center of mass. Figure 4.6b shows, for comparison, the results corresponding to a

stiff, rod like, polymer of the same length and charge. For both cases shown the lipid

composition is PC:PS:PIP2 =98:1:1, but it should be noted that the PIP2 distribution in the

PC:PS:PIP2 =89:10:1 membrane is very similar. Since the range of the lipid region enriched

with charged lipids correlates closely with the lateral dimensions of the adsorbed polymer,

the lipid distribution gives us information on the polymer shape. The lipid enrichment serves

as a stamp of the polymer projection shape on the membrane.

93

Figure 4.5a and Figure 4.6a reveal a rather dramatic enrichment of the interaction

zone by the tetravalent lipids, and essentially no change in the local concentration of the

monovalent lipids. This phenomenon has been discussed and analyzed both theoretically and

experimentally [82,84,88]. Qualitatively, its origin involves two basic physical principles.

The first is that the electrostatic interaction free energy between a charged macromolecule

and a charged surface is minimal at “isoelectricity”, i.e., when the net amounts of negative

(in our case lipid) and positive (in our case polymer) charges are equal [51,102,103]. Thus,

when a highly charged polymer is brought into contact with a weakly charged fluid

membrane, oppositely charged lipids tend to migrate towards the polymer, attempting to

achieve the desired charge matching. In the case of a flexible polyelectrolyte on a mixed

membrane this tendency is partly opposed by the entropic penalties associated with the loss

of polymer flexibility and lipid mixing freedom. The second physical fact is that importing

one tetravalent lipid into the interaction zone involves a much lower entropy loss as

compared to that of bringing four monovalent lipids to get the same amount of charge.

Figure 4.6 Contour maps of PIP2 density in the membrane plane. The area per square of the grid corresponds to

one lipid molecule. (The square grid is used here just for display, the simulations were carried out using 2D

hexagonal lattice.) The figure on the left, (a), is for a flexible, 20-segment, chain interacting with a fluid

membrane of average composition PC:PS:PIP2 =98:1:1. The figure on the right, (b), is for a rod-like polymer of

the same length and charge. The rod is placed at distance 1z = from the membrane. The polymer’s first

segment is fixed at 1 1z = . The numbers labeling the color code indicate the local mole fractions of PIP2 . The

average membrane concentration is 0.01.

94

Lipid Demixing

Figure 4.5b shows the simulation results for polymer adsorption on a binary membrane

containing only neutral and monovalent lipids. PS enrichment in these membranes is very

poor and the adsorption free energy is small for 10% PS and positive for 1% PS. The addition

of 1% tetravalent lipids changes the structural and energetical properties of the adsorption

dramatically, since these lipids are localized to the interaction zone, thereby enhancing the

polymer-membrane interaction significantly. As argued previously, the dramatic enrichment

of tetravalent lipids is a result of the reduced entropic penalty associated with their

localization as compared to that of monovalent lipids (since more monovalent lipids are

needed to achieve electroneutrality). To account for this qualitative argument quantitatively,

we can estimate the entropic loss upon lipid segregation in the different cases.

The entropy change upon transferring one lipid molecule of type i from a region

where its molar fraction is iψ into a region of local mole fraction ( )i rψ is, (for small iψ )

ln[ ( ) / ]i i iS rψ ψ∆ = − (4.49)

A crude estimate of the average lipid charge within the interaction zone can be obtained by

calculating the amount of charge required to neutralize the charge of the adsorbed polymer.

Our simulations of a fluid membrane containing 1% PIP2 and 10% PS reveal that, on average,

most of the 20 polymer charges reside within a rather thin surface layer (see Figure 4.10 and

Table 4.1). The radius of the lipid interaction zone is about 5 headgroup diameters,

corresponding to a membrane patch containing about 80 lipids. The simulations show that the

total lipid charge within this patch is roughly -20. About 8 charges are provided by the

monovalent lipids (corresponding to their average fraction in the membrane) and the

remaining 12/4=3 tetravalent lipids, implying an average enrichment factor of 3, see Figure

4.5. The entropic cost of bringing the three tetravalent lipids into the interaction region is thus

3ln(0.03/ 0.01) 3.3− ≈ − (which is nearly one half of the total entropy loss in adsorption; see

Table 4.2). In the absence of tetravalent lipids, effective charge neutralization would require

the import of 12 additional monovalent lipids into the interaction region. In this case the

entropic penalty would be intolerably high, 12ln(20 / 8) 11− ≈ − , comparable to the gain in

electrostatic energy. In the case of the weakly charged (1% PIP2 and 1% PS) membrane, complete charge

neutralization would require the recruitment of 5 tetravalent lipids, implying a substantially

higher entropic penalty as compared to the strongly charged membrane. Here we found that,

on average, only 12 lipid charges have accumulated in the interaction zone and that a similar

95

number of polymer charges reside within the narrow surface layer. In other words, in this

case the system settles on less then complete neutralization of all polymer charges, thereby

retaining more lipid translational freedom and polymer flexibility. Finally, we note that a

membrane containing a large amount of monovalent lipids to begin with, need not relocate

lipids upon polymer adsorption. For our case, using the same, rather crude, estimates as

above, we conclude that a membrane containing about 20-25% PS need not recruit additional

molecules into the interaction region. These qualitative conclusions appear consistent with

experiments, measuring the interaction between the MARCKS effector domain (and similar

peptides) and mixed PC:PS:PIP2 membranes [82,88].

4.5.2 Adsorption Thermodynamics

Potential of Mean Force

Figure 4.7 shows how 1( )F z∆ , the “differential” adsorption free energy, and 1( )E z∆ , the

differential adsorption energy, vary with the distance ( 1z ) of the chain origin from the surface

of the weakly charged (PC:PS:PIP2 =98:1:1) membrane. Figure 4.8 shows the same quantities

for the strongly charged (PC:PS:PIP2 =89:10:1) membrane. 1( )F z∆ is the free energy change,

or, the potential of mean force, associated with bringing the first segment of the

macromolecule from the bulk solution to distance 1z from the membrane. 1( )E z∆ and

1 1 1( ) ( ) ( )T S z E z F z∆ = ∆ −∆ are the energetic and entropic components of this free energy

difference. More explicitly, for the fluid and uniformly charged membranes

1 1( ) ( , | ) ( ) ( )f f b fE z U m z U U mα α∆ = − − and 1 1( ) ( | ) ( )u b

E z U z Uα α∆ = − ,

respectively. For the quenched membrane we show here the average energy change

corresponding to the (Boltzmann weighted) ensemble of quenched membranes,

1 1 1( ) ( ) ( ) ( )[ ( | , ) ( ) ( )]m qq bm mE z P m E z P m U m z U U mα α∆ ≡ ∆ = − −∑ ∑ (4.50)

The differential adsorption free energy onto the fluid membrane is given by

(0)1 1( ) ln[ ( ) / ]f f f bF z q z q q∆ = − , with a similar definition of 1( )uF z∆ . The corresponding free

energy change for the quenched membrane is defined here as

1 1( ) ( ) ln[ ( ) / ]q m bq mF z P m q z q∆ ≡ −∑ . It should be noted that the net (or “integral”)

96

adsorption energy of the quenched membrane is not a simple integral of 1( )q

E z∆ .

Similarly, the net free energy change of all membranes is not a direct integral of 1( )F z∆ .

For example, for a fluid membrane,

( )(0)1 1 1 1

0

ln ( ) ln exp ( ) ( )f f f b f fF q z dz q q F z F zλ

λ⎡ ⎤

∆ = − = − −∆ ≠ ∆⎢ ⎥⎣ ⎦∫

For a quenched membrane, ( ) 1 10

ln 1 ( )m

m b mF q q z dzλ

λ⎡ ⎤

∆ = − ⎢ ⎥⎢ ⎥⎣ ⎦

∫ and thus,

1 1( ) ( ) ( ) ln[ ( ) / ]q m m m q m bq q mm

P m F F F z P m q z q∆ ≡ ∆ ≠ ∆ ≡ −∑ ∑

The main contribution to the adsorption energy is the interaction energy between the polymer

and the membrane. That is, although the polymer and the fluid membrane change their

configuration upon adsorption, the change in their self energy is negligible. Thus, it is

expected that the interaction between the polyelectrolyte and all three types of oppositely

charged membrane is attractive, i.e., 1( ) 0E z∆ < for all 1z . This is indeed the case as shown in

both Figure 4.7 and Figure 4.8. However, this attractive interaction may not suffice to ensure

adsorption and 1( )F z∆ might even be positive. Notice also (as demonstrated in Figure 4.7)

that when a polymer approaches the surface from the bulk, 1( )E z∆ always monotonically

decreases, while 1( )F z∆ might show non-monotonic behavior. That is, decreases first but

starts to increase in small distances from the surface. The differences in behavior between the

adsorption energy and free energy are caused by the entropy losses,

1 1 1( ) ( ) ( )T S z E z F z∆ = ∆ −∆ , associated with polyelectrolyte adsorption. In the quenched and

uniform membranes, these entropy losses reflect the lower conformational entropy of the

adsorbed molecule compared to that of a polymer in solution (as experienced for any flexible

molecule near a rigid wall). The entropy loss is even higher in the case of the fluid membrane

because of the additional loss of lipid mixing entropy. As a result of these entropy losses, the

value of the free energy is lower than the interaction energy in both Figure 4.7 and Figure 4.8

and even get positive, i.e., 1( ) 0F z∆ > for the case of a ( 1z = + ) polymer interacting with the

uniformly ( 0.05wz = − ) charged membrane, (see Figure 4.7). The weakly charged quenched

membrane is on average, non-adsorbing as well. Only the fluid membrane appears attractive

owing to its ability to recruit charged lipids into the interaction zone. However, even this

membrane is repulsive when the polymer charge is reduced to 1/ 2z = + . Figure 4.8 reveals

97

2 4 6 8 10z

1

-8

-6

-4

-2

0

∆E

2 4 6 8 10z

1

-2

-1

0

1

∆F(a) (b)

2 4 6 8 10z

1

-16

-12

-8

-4

0

∆E

2 4 6 8 10z

1

-5

-4

-3

-2

-1

0

∆F(a) (b)

that upon increasing the membrane charge (to 0.14sz = − per lipid), all membranes become

attractive. The strongest binding is to the fluid membrane and the weakest corresponds to the

uniformly charged one.

Figure 4.7 The differential energy of adsorption (a), and free energy of adsorption (b), of a flexible

macromolecule adsorbing on a membrane of lipid composition PC:PS:PIP2=98:1:1. 1z is the distance of the first

polymer segment from the membrane plane. The solid, dashed and dotted curves correspond to the fluid,

quenched and uniformly charged membranes, respectively. The dotted-dashed curve in (b) is for a weakly

charged ( 1/ 2)z = + polymer interacting with a fluid membrane. The free energy change corresponding to this

polymer is not shown because it very nearly overlaps the dotted curve in (a).

Figure 4.8 The differential energy of adsorption (a), and free energy of adsorption (b), of a flexible

macromolecule interacting with a lipid membrane of composition PC:PS:PIP2=89:10:1. The solid, dashed and

dotted curves correspond to the fluid, quenched and uniformly charged membranes, respectively.

98

Low Density Limit

According to Eq. (4.8) and as discussed in Section 4.2.1, the partition coefficient

(equivalently, the adsorption probability) is identical for a fluid membrane and for an

ensemble of quenched membranes. In a similar way, we can define the ‘differential partition

coefficient’

1 11( ) 1 ( )1 1

1 (0)0

( )( )( ) ( )( )( )

f mmF z f q F z

qf b b b

q zq zq z zK z e eq q q q

θ

ϕϕ

−∆ −∆

→

= = = = = =∞

(4.51)

Again we expect it to be identical for both fluid and ensemble of quenched membranes. The

last equality in Eq. (4.51) is a reminder that, in the limit of low surface coverage,

1( ) ( )q z q ∞ is equal to the ratio between the density of chain molecules (more precisely, chain

termini) at distance 1z from the membrane and the corresponding density in the bulk

solution. Figure 4.9 shows how the differential partition coefficient varies with 1z for our

three model membranes. It should be emphasized that the partition functions corresponding

to the quenched and fluid membranes have been obtained using the two different MC

simulation schemes described in Section 4.4. Apart from the small numerical noise, we

indeed find that the partition functions corresponding to the fluid and (the ensemble of)

quenched membranes are essentially identical, reassuring that the different simulation

methods indeed yield identical results. Comparing Figure 4.9a and Figure 4.9b, the ratio

( )1 1 0( ) / ( ) /b bq z q z

θϕ ϕ

→= reveals, as expected, the stronger attraction of the polyelectrolyte

to the strongly charged membrane. Similar behavior is shown by the average segment density

profiles, ( )zρ , as defined in Eq. (4.24) and shown in Figure 4.10. Again we see that for

0θ → the density profiles corresponding to the fluid and quenched membranes are the same.

Note, however, that although the partition coefficients are identical, the free energies (and the

differential free energies) are different, 1 1( ) ( )f m qF z F z∆ ≠ ∆ . To achieve this equality, all the

free energies involved must be small enough to allow exponent expansion, that is, 11 ( )( )

1 11 ( ) 1 ( )mf qF zF zf m q

F z e e F z− ∆−∆− ∆ ≅ = ≅ − ∆ .

Adsorption Free Energies

The adsorption free energies and related thermodynamic functions were calculated using Eqs.

4.22, 4.23, 4.25, 4.32 and 4.38. The obtained values depend on the cutoff distanceλ .

99

0 2 4 6 8 10z

0

1

2

3

4

0 2 4 6 8 10z

0

50

100

150(a) (b)

ρ(z)ρ

b

ρ(z)ρ

b

2 4 6 8 10z

1

0

1

2

3

4

2 4 6 8 10z

1

0

20

40

60

80

100(a) (b)

q(z1)

q( ) q( )

q(z1)

To determineλ , we defined it operatively as the distance beyond which ( ) / 1.1bzρ ρ ≤ for

attractive membranes ( 0F∆ < ), or larger than 0.9 for repulsive ones. (This criterion closely

satisfies the equality in Eq. 4.26).

Figure 4.9 The partition function ratio, 1( ) / ( )q z q ∞ ( )( )1 0( ) / bz

θϕ ϕ

→= for a macromolecule interacting

with a weakly charged membranes of composition PC:PS:PIP2 =98:1:1 (a), and a strongly charged membranes

where PC:PS:PIP2 =89:10:1 (b). Solid, dashed and dotted curves correspond to the fluid, quenched and

uniformly charged membranes, respectively.

Figure 4.10 Segment density profiles along the membrane normal, ( )zρ , relative to the segment density in the

bulk solution ( )bρ ρ= ∞ . In (a) the membrane composition is PC:PS:PIP2 =98:1:1, and in (b) PC:PS:PIP2

=89:10:1. The solid, dashed and dotted curves correspond to the fluid, quenched and uniformly charged

membranes, respectively. Two curves are shown for each type of membrane; the upper curve corresponds to the

low density limit 0bϕ → (and hence 0θ → ), and the lower one is for 0.034bϕ = .

100

-6 -4 -2 0 2∆F

0.2

0.4

0.6

P(∆F)

-15 -10 -5 0∆E

0.1

0.2

0.3

0.4

P(∆E)

(a) (b)

Given the λ values we have calculated, the integral adsorption energies, free energies, and

surface concentrations θ for the fluid, quenched and uniform membranes. The adsorption

energies were calculated using Eq. 4.33 and 4.39 (substituting E∆ for A). The adsorption

free energies were extracted from the simulation using Eqs. 4.43 and 4.46. The numerical

values of F∆ and E∆ for the fluid, quenched and uniform membranes are listed in Table 4.2

below.

Figure 4.11 Probability distributions of adsorption free energies ( )P F∆ (a), and adsorption energies ( )P E∆ (b),

for a Boltzmann weighted ensemble of quenched membranes. Solid and dashed curves correspond to

membranes with PC:PS:PIP2 =89:10:1 and PC:PS:PIP2 =98:1:1, respectively. In all cases the volume fraction of

macromolecules in the bulk solution is 0.034bϕ = . See Table 2.2 for more details.

As already mentioned in Section 4.2.2, although the partition coefficient is identical

for both fluid and quenched membranes, there is a major difference between these

membranes. Since any fluid membrane cell can change its lipid distribution so as to optimize

the interaction with the polymer, the adsorption energy for every fluid membrane cell is

identical. On the contrary, a polymer which adsorbs on a quenched membrane “feels” a

different environment depending on the membrane configuration m, and therefore will gain a

different free energy mF∆ upon adsorption. The distribution of adsorption energies,

adsorption free energies and surface coverage for an ensemble of quenched membrane is

quite wide as described in Figure 4.11 and Figure 4.12.

As shown in Figure 4.11, it is possible that for some quenched membrane states m,

m fF F∆ ≤ ∆ (and hence, m fθ θ≥ ). This may appear surprising in view of the fact that the fluid

membrane has more degrees of freedom (in other words, the free energy, (1)lnf fF q= − ,

101

0 0.05 0.1 0.15 0.2θ

0

10

20

30

P(θ)

0 0.2 0.4 0.6 0.8 1θ

0

1

2

3

4

P(θ)

(a) (b)

<θm

>q= 0.05 <θ

m>

q= 0.28

θf = 0.5

θu = 0.09

θf = 0.06

θu = 0.01

involves summation over both m and p and therefore is more negative than (1)lnm mF q= − that

involve summation only over the polymer degrees of freedom). After adsorption, however,

the distribution of lipids in the fluid membrane changes as discussed previously and it is no

longer the Boltzmann distribution found previously to adsorption, implying a loss of lipid-

mixing entropy. No loss of lipid entropy is involved upon adsorption onto a quenched

membrane, which explains why certain quenched states can be more attractive to

macromolecule adsorption than the fluid membrane. Note, that for the weakly charged

membrane, while 0mE∆ < for all m values, the bimodal distribution of mF∆ reflects two

distinct classes of quenched environments, corresponding to attractive ( 0mF∆ < ) and

repulsive ( 0mF∆ > ) membranes, (see Figure 4.11).

Figure 4.12 The distribution of surface concentrations, θ, for an ensemble of quenched membranes of

composition PC:PS:PIP2 =98:1:1 (a) and PC:PS:PIP2 =89:10:1 (b). Also listed are the average surface

coverages of the fluid, quenched and uniformly charged membrane. In (a), the solid curve is the overall

distribution of θ values, whereas the dashed and dash-dotted curves correspond to the distributions of θ values

for membranes with 0mF∆ < and 0mF∆ > , respectively.

Upon increasing the polymer concentration in solution, the more strongly adsorbing cells on

the quenched membrane will be occupied first. Once these are taken, further adsorption is

suppressed. For a fluid membrane every cell can anneal independently its lipid distribution

and thus we obtain that the average surface coverage is always lower on a quenched

membrane, m fqθ θ≤ . The distributions, ( )P θ , of surface concentrations for the ensembles

of weakly and strongly charged quenched membrane are shown in Figure 4.12. Also

mentioned there (and in Table 4.2) are the average values of θ for the fluid and uniform

102

membranes, confirming that adsorption onto the fluid membrane is, indeed, the strongest of

all in accordance with the free energy values which appear in Table 4.2.

An additional demonstration of this behavior is shown in Figure 4.10. In the limit of

low polymer density, 0bϕ → , we get that the density profiles corresponding to the fluid and

quenched membrane coincide, however for a higher polymer concentration (bulk density of

0.034bϕ = ), the profiles are different with a larger segment density in the case of a fluid

membrane. The values obtained for the surface coverage with the same bulk density support

this conclusion. Using Eq. 4.22 we get 0.5fθ = for the strongly charged fluid membrane,

whereas Eq. 4.23 implies a much smaller surface density for the quenched membrane,

28.0≅qmθ . Additional values are given in Table 4.2. Notice that for a repulsive

membrane such as the weakly charged uniform membrane, the ratio suu ϕλθ ~≈

( 5.2 ,01.0 ≅= uu λθ ), which may be interpreted as the 3D density of macromolecules very

near the membrane, is significantly smaller than 0.034bϕ = .

PC:PS:PIP2 =89:10:1 ( 0.14fz = − ) PC:PS:PIP2 =98:1:1 ( 0.05fz = − )

Fluid Quenched Uniform Fluid Quenched Uniform

E∆ -12.5 -7.4 -3.1 -5.0 -2.4 -0.7

F∆ -3.5 -1.4 -1.0 -0.7 0 1.3

θ 0.5 0.28 0.09 0.06 0.05 0.01

Table 4.2 Adsorption properties : Adsorption energies and free energies for the PC:PS:PIP2 =89:10:1

( 0.14fz = − ) and PC:PS:PIP2 =98:1:1 ( 0.05fz = − ) membranes. For the quenched lipid membrane we list

m qE∆ and m q

F∆ The surface concentrations, θ , in the bottom row ( m qθ for the quenched membrane) are

for a bulk concentration of macromolecules 0.034bϕ = .

4.6 Concluding Remarks Our major objective in this work has been to study the role of lipid mobility and composition

in the non-specific electrostatic adsorption of charged flexible macromolecules. Based on

computer simulations and qualitative theoretical considerations we have shown that a fluid

membrane, enabling lipid lateral diffusion, is substantially more effective in mediating

macromolecule binding than a quenched or a uniform membrane carrying the same average

charge. We also found that multivalent lipids, even if in small amounts, can substantially

103

enhance the electrostatic adsorption of flexible macromolecules. The crucial role of these

lipids in mediating membrane binding is a direct consequence of the fact that, in the fluid

lipid membrane, their localization in the macromolecule’s adsorption zone provides efficient

electrostatic binding at a minimal cost of lipid “demixing” entropy.

Previous theoretical studies suggest that a fluid membrane containing relatively small

(yet biologically relevant) amounts, say 10-20%, of monovalent lipids may effectively bind

rigid charged macromolecules (e.g., folded globular proteins) [86,104]. In such cases the

electrostatic binding free energy outbalances the lipid entropy loss. On the other hand, in the

case of a flexible macromolecule, binding involves the additional loss of conformational

entropy. Our calculations indeed suggest that in this case, 10% of monovalent lipids hardly

suffice to mediate polymer binding and the presence of multivalent lipids in the membrane,

whose localization in the interaction zone involves just a small entropy loss, appears critical.

Our conclusions regarding the ability of a medium size macromolecule to sequester

multivalent lipids upon membrane binding appear consistent with recent experimental

observations.

From the more technical-theoretical aspect, we have presented an extended version of

the Rosenbluth Monte Carlo sampling scheme, enabling the simultaneous generation of

polymer and membrane configurations. In addition, we have shown that, in principle, the

statistical aspects of polymer adsorption on a fluid membrane can be obtained by biased

superposition of simulation data of an ensemble of quenched membranes. An approximate

cell model has been presented in order to account for the different adsorption probabilities on

fluid and quenched membranes. In the limit of vanishing macromolecule concentrations, the

average adsorption probabilities become equal.

Notwithstanding the inherent approximations of our model (e.g., the use of DH

potentials) our results suggest that the electrostatic binding free energies of flexible

macromolecules onto lipid membrane are generally small and depend on a subtle interplay of

several factors. These include lipid mobility and composition on the one hand, and

macromolecule charge, shape and flexibility on the other hand. Finally, as noted in the

Introduction, in Appendix A, we extend this work by studying the adsorption of MARCKS

protein onto a fluid lipid membrane.

104

Chapter 5 Summary In the life cycle of any virus - bacterial, plant and animal, a crucial step involves the assembly

of new virions from many copies of viral genome and capsid proteins that have been

synthesized by the host cell. At the end of this assembly process, the viral genome is enclosed

in a rigid protein shell which is referred to as a “capsid”.

The mechanism of viral entry into host cells differs as one goes from bacterial to animal

viruses. In the first case, the genome is usually injected through the bacterial membrane, with

the capsid left outside the cell, while in the latter the complete virus enters the cell via

receptor-mediated endocytosis. These different entry mechanisms are intimately related to the

viral assembly process characteristics as well as to the physical properties of the participating

macromolecules.

In the case of bacterial viruses (as discussed in Chapter 2), the spontaneous ejection is

powered by the large amount of pressure stored inside the viral capsid. This pressure is

associated with the confinement of viral dsDNA into a volume whose dimensions are

comparable to its persistence length, yet hundreds of times smaller than its total contour

length. This confinement is associated with the strong repulsion between neighboring

portions of DNA and the large bending energy of DNA bent into radii of curvature that are

comparable to its persistence length. Since the pressure built up in the capsid can be as high

as tens of atmospheres, a viral-encoded motor protein, which can exert forces as large as tens

of pN’s , is needed to perform the work of packaging. As a consequence, when the capsid is

“opened” (by binding its tail to a receptor in the outer membrane of the bacterial cell) a force

on the order of tens of pNs initiates a spontaneous delivery of the genome into the host cell.

As we showed in Chapter 2, this high amount of stress is reduced to only a few atmospheres

after a relatively small fraction (about one-third) of the genome has been ejected. This

implies that subsequent ejection is easily influenced by environmental details. For example,

ejection might be arrested by an attraction to the capsid wall or the presence of osmotic

pressure in the solution. It could become favorable again if binding proteins or motor proteins

are translated in the host cell.

As opposed to bacterial viruses, the nucleocapsids of animal viruses are believed to

form spontaneously. No build-up of pressure in these nucleocapsids is required; moreover, it

105

can interfere with the assembly process. The entry mechanism of animal viruses such as

alpha viruses (as discussed in Chapter 3), requires the presence of spike glycoproteins

embedded in a lipid membrane coat. These spikes are transmembrane proteins whose

“inside” ends bind to the capsid, while their “outside” ends are available for binding to host

cell receptors and thereby enable cell entry via endocytosis. To ensure the engagement of

spike glycoproteins in the virion, the spike-capsid binding energy is the driving force for

budding. Only when this binding energy overcomes the bending energy of the membrane can

budding occur. Moreover, we find that, as known from crystallography, there is always a

defined stoichiometry between the number of capsid proteins and the number of spike

glycoproteins with no spike ever missing. The line energy associated with the bud rims,

although acting as an additional penalty toward wrapping, nevertheless helps to increase the

efficiency of budding since no capsids are wasted in partially wrapped states. One of the

sources of this line energy is the difference in lipid and protein compositions between the bud

and the cell membranes (i.e., across the boundary). For certain viruses such as retroviruses, it

was demonstrated that the chemical composition of various viral membranes is different from

that of the host plasma membrane, resembling the composition of lipid rafts.

The assembly and budding of retroviruses such as HIV-1 is even more complex, as

the assembly of Gag polyproteins with viral ssRNA is occurs simultaneously with budding at

the plasma membrane. The coupling between the different degrees of freedom (e.g., RNA

flexibility, lipid mobility, and the flexibility of Gag polyproteins) and their relative

importance is essential for understanding the physical principles underlying some of the

special trademarks of HIV-like assembly. Such features include a polydisperse size

distribution, raft-like viral membrane composition, a defined stoichiometry between RNA

length and the number of NC proteins incorporated into the virion, and drastic structural

change upon maturation.

As a first step toward this goal, in Chapter 4 we study the interactions between

flexible polyelectrolytes and fluid membranes. We show that adsorption onto a fluid

membrane is significantly stronger than adsorption onto quenched or uniformly charged

membranes. This is due to the localization of polyvalent lipids in the polymer region, which

enhances adsorption dramatically. The presence of polyvalent lipids is especially crucial to

the adsorption of flexible macromolecules, which lose their conformational entropy upon

adsorption. We demonstrate a way to derive the properties of adsorption on a fluid membrane

by correctly weighting the results obtained for an ensemble of quenched membranes. One

106

major achievement of this work is the development of an extended version of the Rosenbluth

Monte Carlo sampling scheme, enabling the simultaneous generation of multiple entities

(such as polymer and membrane) with coupled degrees of freedom. This simulation method

can be used to study specific complex biological systems, as demonstrated in Appendix A for

the “electrostatic switch mechanism” underlying the behavior of the MARCKS protein.

We can get one step closer to modeling retroviral assembly by successfully modeling

the assembly of ssRNA with structural capsid proteins. ssRNA is a special polymer, in the

sense that it is able to form base pairs. It therefore can gain stable secondary structures, which

makes its behavior more complex than a simple semi-flexible polymer. Following works such

as that of Hyeon and Thirumalai, ssRNA can be modeled as a polymer chain of nucleotides

where each nucleotide is modeled by three beads corresponding to the sugar, base and

phosphate groups. By introducing appropriate potentials, the ability of ssRNA to create base

pairs via hydrogen bonds and stucking interactions can be taken into account [105,106].

To conclude, in this work we have tried to relate the biological mechanism of viral

infection to underlying physical principles and the properties of the macromolecules

involved. In the process of this research, we developed a simulation method that enables the

study of complex systems with coupled degrees of freedom. We used this simulation to study

the general problem of interactions between flexible polymers and fluid membranes, which is

relevant to a number of biological systems.

107

Appendix A

MARCKS Protein –

The “Electrostatic-Switch Mechanism”

A.1 Introduction A delicate balance between the energetic and entropic contributions to the adsorption free

energy is exhibited in various biological processes, [107,108]. One important example is the

“electrostatic switch” mechanism underlying the operation of the myristoylated alanine-rich

C kinase substrate (MARCKS), and several other proteins, [109,110]. This natively unfolded

protein binds electrostatically to anionic lipids in the inner leaflet of the plasma membrane

through its relatively small (25 residues) but strongly charged ‘effector’ domain which

comprises 13 basic residues. The effector domain also contains five phenyl groups which

appear to insert into the membrane’s hydrophobic region and enhance binding. A 150 residue

long flexible polypeptide chain separates the effector domain from the myristoylated N-

terminus. The myristoyl chain inserts into the hydrophilic core of the lipid bilayer, helping to

anchor MARCKS to the membrane. A comparably long flexible chain connects the effector

domain with the C-terminus, (see Figure A.1).

Experiments reveal that the basic protein domain binds preferentially to the

multivalent lipid PIP2 (phosphatidylinositol 4,5 bisphosphate) introduced already in Chapter

4; approximately three PIP2 molecules per adsorbed protein, on average [82,111,112]. The

PIP2 charge here is z = -4, (but generally varying between -3 and -5), suggesting that the few

multivalent lipids provide full electrostatic neutralization of the 13 effector charges [113].

This is especially significant considering that the PIP2 concentration in the membrane is just

~1%, whereas the concentration of monovalent acidic lipids (primarily phosphatidylserine,

PS) is typically 10-30%. In fact, it was suggested that the binding of MARCKS to PIP2

lipids is an important part of its biological function. PIP2 lipids act at several levels to

regulate cell structure and metabolism, [2]. For example, phospholipase C hydrolyzes PIP2

lipids in response to hormonal signals, thereby releases two products. Both of them act as

intracellular messengers. PIP2 lipids may also bind to actin binding proteins, thereby

108

regulating cytoskeleton-membrane attachment [114]. By binding to PIP2 lipids, MARCKS

controls their accessibility for interaction with other cellular proteins, [115-117]. Upon

lowering the net charge of the effector domain from +13 to +7, MARCKS detaches from the

membrane, making the PIP2 lipids available for interactions and therefore initiating a series

of signal transduction events. The change in the effector domain charge is achieved, for

instance, through phosphorylation of three serine residues, in the basic domain, by protein

kinase C (PKC). The reversible binding of MARCKS to membranes is referred to as the

“electrostatic-switch” mechanism, [109].

Using the simulation method described in Chapter 4, we study the interplay between

the different entropic (polymer conformational freedom and lipid mobility) and energetic

contributions (hydrophobic and electrostatic interactions), and the way this interplay is

reflected in the “myristoyl-electrostatic-switch” mechanism.

Figure A.1 A schematic representation of the MARCKS protein. The red and green circles represent the

charged and neutral amino acids respectively. The purple hexagons represent the phenyl groups which are

inserted into the lipid bilayer. The blue and yellow lipids represent the tetravalent and monovalent lipids

respectively.

A.2 Model As mentioned in the introduction, MARCKS interacts with the membrane both

hydrophobically (through its myristoyl chain and Phenyl groups) and electrostatically

(through its charged basic domain). Upon detachment from the membrane, the MARCKS

protein gains both translational and configurational entropy. Hence, MARCKS

configurational entropy (mostly its long flexible chains’ entropy) is the main driving force

for desorption, while the electrostatic and hydrophobic interactions with the membrane are

the main driving force for adsorption.

109

Upon adsorption, the basic domain get an extended configuration on the membrane, while the

chains which do not interact strongly with the membrane (have low negative charge density) ,

are assumed to act as neutral chains near a surface. Thus, the C-terminal chain has one anchor

to the surface (the basic domain) and we referred to it as the “tail”. The N-terminal chain has

two anchors (both the basic domain and the myristoyl chain, see Figure A.1), hence we

referred to it as the “loop”. In accordance with this picture, we model the MARCKS as

composed of three independent domains, as shown in Figure A.2. The MARCKS free energy

is written as the sum of the domains’ free energies

ED loop tailF F F F∆ = ∆ + ∆ + ∆ (A.1)

where EDF∆ , loopF∆ and tailF∆ are the free energies of adsorption for the basic/effector

domain (BD/ED), loop and tail respectively. For every domain, we ran a separate simulation.

Similarly to the simulation described in Chapter 4, the protein is modeled by a chain

of spherical beads of diameter d where each bead represents a single amino acid. The lipid

membrane is modeled as a perfectly flat 2D hexagonal lattice, with lipid headgroups

occupying all its lattice sites. The bond length, d is fixed and equals the lattice constant

d=8.66Å (comparable to the effective diameter of an amino acid group). (See Chapter 4 for

additional details).

Since we neglect the interaction energy of the tail and loop with the membrane (other than

via the myristoyl chain), we can write,

tail tailF T S∆ ≅ − ∆ (A.2)

loop loop myrF T S F∆ ≅ − ∆ + ∆ (A.3)

Figure A.2 A schematic

illustration of the model used for

the MARCKS protein. The line of

circles shown beneath the amino

acid sequence corresponds to the

sequence of beads taken in the

simulation for the basic domain

(top: non-phosphorylated protein,

bottom: phosphorylated protein).

Phenyl groups are represented by

beads which are denoted by Ph.

110

where myrF∆ is the free energy of myristoyl chain insertion into the bilayer. This energy is

estimated experimentally by 8 kcal/mol= 13.5 Bk T− − , [109]. This estimation is based on

measurements of the partition coefficient of a short peptide which corresponds for the 15

first groups of Src protein into a neutral lipid membrane. The free energy obtained in this

experiment accounts for the hydrophobic interaction of the myristoyl chain with the

membrane as well as for the loss of entropy as a result of the reduction in the number of

peptide conformations near the interface. The reduction in configurational space is taken into

account explicitly in the simulation. This entropy penalty can be estimated by

0.5ln15 1.35 Bk T= , [118], or using the simulation by 1.68 Bk T . Therefore, in order to avoid

recounting, we use 13.5 1.68 15.18myr BF k T∆ = − − = − .

The MARCKS-ED chain consists of several types of beads. Basic amino acid groups

are represented by positively charged beads, hydrophobic amino acid groups (Leu, Phe or

Ala) are represented by neutral beads interacting with the membrane via the potentials

described in Section A.2.3. Polar amino acid groups are represented by neutral beads which

interact with the membrane only via excluded volume interactions. As will be explained next,

the charged and polar amino acid groups are not allowed to penetrate the membrane, while

the hydrophobic amino acid groups can. Upon phosphorylation, each of the beads which

represent the phosphorylated Serine groups, gain two negative charges. (For the amino acid

sequence of MARCKS-ED, see Figure A.2 and for the chemical structure of the side chains,

see Figure 1.3).

Upon adsorption, MARCKS-ED (MARCKS-effector-domain) acquires an extended

conformation on the membrane with the Phenylalanine groups penetrating into the lipid

bilayer, [119], see Figure A.1. Replacing the aromatic Phenylalanine groups by Alanines was

shown to reduce the adsorption strength of MARCKS-ED and its ability to sequester PIP2

lipids, [82]. Since MARCKS-ED penetrates the membrane interface, one should take into

account both the dielectric characteristics of the membrane and the hydrophobic interactions

with the Phenylalanine groups. Therefore, the interaction potentials used to study the

adsorption of the effector domain, are more complicated than the ones described in Chapter 4.

In addition to excluded volume and modified electrostatic interactions, we also consider the

hydrophobic interactions of the protein with the lipid bilayer. The next three subsections

(A.2.1-A.2.3) describe the potentials used to model the MARCKS effector domain.

111

A.2.1 Excluded Volume Interactions The excluded volume interactions between polymer segments are modeled using Lennard-

Jones-like potential:

12 6 1/6

1/64 [( / ) ( / ) ] for 20 for 2

( )LJr r r

ru r ε σ σ ε σ

σ− + ≤

≥⎧

= ⎨⎩

(A.4)

Setting 1/ 62 dσ = and 0.1 Bk Tε = ensures the onset of steep repulsion as soon as r falls

below d, [99].

The membrane surface is treated as an impenetrable wall to the polar polymer segments,

implying a minimal distance of d/2 between polymer and lipid charges. The hydrophobic

residues (Phe, Ala and Leu), however, are allowed to penetrate the membrane interface down

to z=0, as described in section A.2.3.

A.2.2 Electrostatic Interactions The charged amino acids are treated as spherical beads which carry a unit positive point

charge localized at their center. Similarly, the charged lipids are treated as disks which bear

point charges at their center. As already mentioned, when membrane bound, the MARCKS-

ED is localized at the membrane interface with the charged groups extended toward the

aqueous solution while the Phenylalanines are inserted into the membrane hydrocarbon,

[119]. Since the MARCKS-ED and the lipid hydrocarbons are intimately associated, the low

dielectric properties of the membrane should be taken into consideration. Recently, Netz

formulated the Debye-Huckel (DH) theory in the presence of a dielectric interface [120]. In

most cases, the interaction could not be solved in close form. However, it can be solved

explicitly for the case of a dielectric substrate with 0ε = , which is a fairly accurate

approximation for a substrate with a low dielectric constant such as a lipid bilayer, [120]. In

this case, the interaction potential between charged groups of valences q and q′ , at distance

r apart in units of Bk T is given by

2 4

2( , , )

4

r r zz

DH B Be eu r z z qq l qq l

r r zz

κ κ ′− − +

′ ′ ′= +′+

(A.5)

where z and z′ are the distances of the charges from the membrane interface, 2 /B Bl e k Tε=

is the Bjerrum length, and 1κ − is the Debye screening length. To account for the dielectric

boundary at the membrane interface, we use Eq. (A.5) as our interaction potential between

112

charged species (either charged lipids or charged polymer segments). In all calculations we

use Bl =7.14Å, appropriate for water at room temperature, and 1 10κ − = Å which corresponds

to typical physiological conditions (monovalent ionic strength of about 0.1M). Notice that at

the interface ( 0)z z′= = , the interaction becomes twice as large as the interaction with no

dielectric boundary.

The Born self-energy term, which represents the charging energy in the presence of a

low dielectric interface (a substrate with 0ε ≅ ) relative to its value in aqueous solution, is

given by

2 2

( )2 2

zselfDH B

q eu z lz

κ−

= (A.6)

where z is the distance from the interface. As a consequence, charges are repelled from the

low dielectric interface.

A.2.3 Hydrophobic Interactions To account for the interaction of hydrophobic amino acids with the membrane and their

ability to penetrate it, we use a square well potential.

{ for 0 /2 0 for /2 ( ) h

hr dr du r ε− ≤ ≤>= (A.7)

The width of the well was taken as the size of a phenyl group (i.e., 4Å / 2d≅ ). Wimley and

White, [121] determined experimentally the partitioning of short peptides into a lipid bilayer

interface. We calibrated the potential depth by using our simulation to simulate their

experiments and reproduce the measured free energies of adsorption. For the hydrophobic

amino acids of interest to us we find, , 2.4h Leu Bk Tε = , , 0.7h Ala Bk Tε = and , 3.5h Phe Bk Tε = .

A.3 Results and Discussion Using the simulation, we study several aspects of the “electrostatic switch” mechanism. Since

both the basic amino acids and the Phenylalanine groups are known to play a central role in

the MARCKS-membrane interaction, we study in addition to the non-phosphorylated and

phosphorylated isomers of MARCKS, the mutated protein whose Phenylalanines were

replaced by Alanine groups (MARCKS-FA). In order to study the influence of PIP2 lipids on

the adsorption properties, we carried out the simulation for three kinds of membranes: PC:PS

(=90:10), which contains PS lipids in relevant physiological concentration but with no PIP2

113

lipids, PC:PIP2(=99:1), which contains 1% PIP2 lipids but with no PS and a membrane which

contains both charged lipids, PC:PS:PIP2(=89:10:1).

Figure A.3 shows typical snapshots of the MARCKS-ED taken from the simulation. These

snapshots resemble the schematic drawing presented in Figure A.1, where the MARCKS-ED

conformation is fairly extended with the phenyl groups inserted into the hydrophobic core.

Also apparent is the sequestration of PIP2 lipids into the polymer vicinity, both with and

without monovalent charged lipids present in the membrane. (A typical snapshot of the intact

MARCKS are shown in Figure A.7).

Figure A.3 Typical snapshots of the MARCKS effector domain (MARCKS-ED) taken from the simulation for

PC:PIP2 membrane (a,b and c) and for PC:PS:PIP2 membrane (d and e). The red and green spheres represent

positively charged and neutral amino acids respectively. Phenyl groups are represented by purple spheres. PIP2,

PS and PC lipids are represented by blue, yellow and white spheres respectively. Notice the insertion of the

phenyl groups into the lipid membrane and the localization of PIP2 lipids into the polymer vicinity.

( )a ( )b

( )c

( )e( )d

114

Figure A.4 The segment distribution of an adsorbed protein. The solid and dashed curves correspond to the

non-phosphorylated and phophorylated isomers. On left: MARCKS protein, On right: MARCKS-FA protein. It

is apparent that the MARCKS is much more extended in the x-y plane than the MARCKS-FA on all the

different membranes. The distributions are shown for PC:PIP2 membrane, but similar results are obtained for

the rest of the membrane.

To demonstrate the role of the phenyl groups in the extended configuration of MARCKS

protein, we show in Figure A.4 the average segment distribution of an adsorbed protein along

the membrane normal. We calculate the distribution as 1 1 1 1 10 0

( ) ( ) ( | ) ( )P z q z n z z dz N q z dzλ λ

= ∫ ∫

where 10

( | )N n z z dz∞

= ∫ is the number of polymer segments. The smeared distribution of

MARCKS-FA in comparison to that of MARCKS is a result of the flattening of the

MARCKS on the membrane surface as the phenyl groups are inserted into the bilayer.

A.3.1 Lipid Distribution

Figure A.6 shows the enrichment factor 0( )i rψ ψ for PS and PIP2 lipids as a function of the

radial distance from the protein center of mass. As defined in Chapter 4, the enrichment

factor is the ratio between the local concentration of lipid species i and its average

composition at the membrane. Therefore, it measures the change in lipid profile upon

polymer adsorption. Similarly to the results presented in Chapter 4, Figure A.6 demonstrates

the localization of PIP2 lipids into the interaction zone with essentially no change in the PS

0 2 4 6 8z

0 2 4 6 8z

0 2 4 6 80

0.20.40.60.8

10 2 4 6 80 2 4 6 8

10 2 4 6 8

P(z)

PC:PIP2

PC:PIP2

1 MARCKS-FA MARCKS

115

1.5

00 3 6 9 12 15r

0

0.3

0.6

0.9

1.2

1.5

ρ(r)

PC:PS:PIP2

lipid distribution. Also apparent is the correlation between the lateral dimensions of the

adsorbed protein and the lateral distribution of charged lipids in the membrane. The lateral

dimensions of the adsorbed protein (approximately six lipid diameters, see Figure A.5)

correlates closely with the region enriched with charged lipids.

Integrating over the interaction zone, we find that the number of PIP2 lipids

sequestered by the polymer is ~4 PIP2 lipids per protein for the non-phosphorylated

MARCKS and ~3 for non-phosphorylated MARCKS-FA, comparable to the values known

from experiments, [111,122]. Note, however, that this is the number of PIP2 lipids localized

beneath (alternatively bound to) an adsorbed protein. In other words, this is the result of a

single molecule experiment, where the number of PIP2 lipids bound to a single protein

adsorbed on the membrane is detected. As will become clear next, the concentration of

adsorbed MARCKS is much larger than the concentration of adsorbed MARCKS-FA. Thus,

the similar number of bound PIP2 lipids per an adsorbed protein for both MARCKS and

MARCKS-FA, does not imply that the number of “free” PIP2 lipids at the membrane is

similar in both cases. By “free” PIP2 lipids, we refer to the lipids which are not associated

with an adsorbed protein, and hence are available for enzymatic reactions such as hydrolysis.

Figure A.5 The two dimensional density, ρ(r), of chain segments as a function of the radial distance from the

polymer center of mass. The solid and dashed curves correspond to the non-phosphorylated and phosphorylated

isomers respectively. The results are shown only for the adsorption of MARCKS-ED on the PC:PS:PIP2

membrane, but similar results are obtained for the additional membranes.

116

0 3 6 9 12 15r

0

2

4

6

8

10

0 3 6 9 12 15r

0 3 6 9 12 15r

0

2

4

6

8

10

0 3 6 9 12 15r

ψ(r) PC:PIP

2

ψ0

PC:PS

ψ0

MARCKS-FA

ψ(r)

PC:PS:PIP2

PC:PS:PIP2

PC:PIP2

PC:PS

MARCKS

Figure A.6 The enrichment factor of charged lipids as a function of the radial distance from the polymer’s

center of mass. Results are shown for the adsorption of MARCKS (a) and MARKCS-FA (b) on PC:PS:PIP2

membrane and of MARCKS (c) and MARCKS-FA (d) on PC:PS and PC:PIP2 membranes. The blue and purple

curves correspond to PIP2 and PS enrichment factors, respectively. The solid and dashed curves correspond to

the non-phosphorylated and phophorylated isomers, respectively.

A.3.2 Adsorption Free Energies

The localization of PIP2 lipids into the interaction zone is demonstrated in the previous

section. But, does the adsorption depend on the presence of PIP2 lipids in the membrane?

To answer this question, let us examine the adsorption behavior on the different membranes

introduced at the beginning of Section A.3.

As explained in Section A.2, we calculate separately the contributions of the loop, tail

and effector domains to the adsorption free energy. Let us start by describing the change in

free energy caused by the protein chains (i.e., tail and loop, see Figure A.2). Since the protein

chains are assumed to be approximately neutral in our model, the free energy change of the

chains upon adsorption is identical for any given membrane.

( )a ( )b

( )c ( )d

117

Figure A.7 A typical snapshot of

the intact MARCKS protein,

demonstrating the length of the

“tail” and “loop”. Here, green and

red spheres represent neutral and

positively charged amino acids,

respectively. The myristoyl chain

is represented by a yellow sphere.

PIP2, PS and PC lipids are

represented by blue, purple and

white spheres, respectively.

Assuming that the myristoyl chain is always anchored to the membrane for an adsorbed

protein (because of its large binding energy), we find that the difference in entropy upon

adsorption is 15.2tail loop BS k+Δ = − . In principle, the position of the first chain segment should

be dictated by the position of the adjacent segment which belongs to the basic domain.

However, since the value obtained for the entropic change using different segment positions

(in the appropriate range) is differs by ~0.5kBT, we use its average value. The low sensitivity

to the position of the first chain segment, may also serve as a justification for performing

separate simulations for the different MARCKS domains. The large entropic penalty (so as

the small change upon changing the first chain segment position) results from the length of

the chains. The confinement of such large chains near a surface, results in a huge reduction in

their configurational space due to their exclusion from the lower half of the plane. The

number of restricted configurations scales with the number of segments as: 1/ 2N for a tail and

as 3/ 2N for a loop [118]. Thus, the entropy loss dependence on N is (1/ 2) ln N and

(3 / 2) ln N for tail and loop respectively. To get a feeling for the large reduction in

configurational entropy associated with the chains confinement, we show in Figure A.7 a

typical snapshot of the intact MARCKS demonstrating the large length of its chains.

Using Eqs. (A.2) and (A.3) we get, 0.02 0tail loop myr tail loop BF F T S k T+ +Δ = Δ − Δ = ≅ . That

is, the binding energy of the myristoyl counterbalances the loss of configurational entropy of

118

the chains. Consequently, the adsorption free energy of intact MARCKS approximately

equals the adsorption free energy of MARCKS-ED.

Following the thermodynamic analysis presented in Chapter 4, we calculate the

adsorption free energies of MARCKS-ED as well as the energy and entropy differences

associated with it, for several cases as listed in Table A.1. Specifically, we use Eqs. (4.37)-

(4.40),(4.42),(4.44)-(4.46).

For the purpose of comparison with experimental results, we convert our free energies into

molar partition coefficients, [ ] [ ][ ]aK LP L P= where [LP], [L] and [P], are the molar

concentrations of the lipid-protein complex (the adsorbed protein), lipids and protein in

solution, respectively. The definitions of s molar concentrations in terms of the volume, V ,

the membrane area, A, and the number of adsorbed proteins ( LPN ) and free proteins ( PN ) are

( )[ ] ; [ ] ; [ ]LP P LLP N V P N V L A a V≡ ≡ ≡ (A.8)

where 65La = Å is the cross sectional area of a lipid head group.

The ratio between the number of adsorbed and free proteins equals the ratio of the respective

partition functions, i.e.,

(1)

(0)

ˆ fLP

P b m

qN AN q q V

λ= (A.9)

Remember that λ is the width of the adsorbed layer (as defined in Section 4.2.3), (0)fq is the

partition function of a bare fluid membrane (as defined in Eq. (4.2)), bq is the partition

function per unit volume of a protein in the bulk (as defined in Eq. (4.7) and (1)ˆ fq is the

average partition function per unit volume of an adsorbed macromolecule, (as defined in Eqs.

(4.18)-(4.19)).

Using Eqs. (A.8), (A.9) and (4.22), we can write

(1)

(0)

ˆ[ ][ ][ ]

f Fa L L

b f

qLPK a a eL P q q

λ λ −Δ= = = (A.10)

Using Eq. (A.10), we convert between the free energies obtained from the simulation and the

molar partition coefficients. The values obtained for aK are listed in Table A.1.

Experimental molar partition coefficients obtained for MARCKS-ED adsorption on

PC:PS:PIP2 (93:6:1), PC:PIP2(99:1) and PC:PS(10:1) membranes are 5 -15 10 M× , 6 -11 10 M×∼ and 3 -16 10 M× , respectively [111,123]. These values were observed using

different experimental techniques and hence it is not straight forward to compare to these

119

values and between them. This is since the experimental technique used might influence the

measured partition coefficient both by its different operative definition of an adsorbed protein

and by influencing the binding strength. For example, the partition coefficient for a

PC:PIP2(99:1) membrane, was obtained using acrylodan-labeled MARCKS-ED. This

hydrophobic peptide is known to increase binding to membranes by 50-100 fold, relative to

that of an unlabeled peptide [123].

PC:PS:PIP2 = 89:10:1 PC: PIP2 =99: 1 PC:PS =90:10

Non-phos phos Non-phos phos Non-phos phos

FΔ -12.6 -6.3 -7.8 -3.7 -4.6 -1.9

EΔ -43.1 -30.5 -41.2 -24.5 -21.1 -11.4

T SΔ -30.5 -24.2 -33.4 -20.8 -16.5 -9.5 MARCKS

1[ ]aK M − 62 10× 33 10× 42 10× 22 10× 24 10× 13 10×

FΔ -6.5 -3.1 -2.0 -1.2 -1.5 -0.61

EΔ -25.4 -14.9 -16.1 -9.8 -4.4 -2.1

T SΔ -18.9 -11.7 -14.0 -8.6 -2.9 -1.5 MARCKS-FA

1[ ]aK M − 34 10× 21 10× 14 10× 11 10× 8 4

Table A.1 Molar partition coefficients, adsorption free energies and the relative energetic and entropic

contributions to it, for the PC:PS:PIP2=89:10:1, PC:PS=90:10 and PC:PIP2=99:1 membranes. The energies are

given for MARCKS and MARCKS-FA, both phosphorylated and non-phosphorylated

A fairly good agreement between experiments and our simulation results is obtained for the

differences between free energies of the various cases studied. For example, from Table A.1,

we see that addition of 1% PIP2 lipids, increases MARCKS binding 410 -fold. This was also

found experimentally, [123].

The contribution of a single phenyl group to the adsorption free energy can not be

considered as a constant. Wimley and White [121] measure it for a phenyl group which was

incorporated into a short peptide of hydrophobic amino acids and adsorbed on a neutral

membrane. The value obtained was 1.3 kcal/mol, [121]. However, the presence of polar

amino acids and especially charged ones which do not penetrate into the hydrocarbon layer,

makes the phenyl group interaction with the membrane significantly weaker. The

composition of the membrane also changes the effective interaction energy even though the

phenyl group is not charged. From an analysis of the different free energy contributions, we

120

see that its contribution to the free energy has both hydrophobic and electrostatic

components. The strengthening of electrostatic interactions results from phenylalanine

groups’ ability to localize MARCKS at the membrane interface, close to charged lipid

headgroups. Indeed, it was found that the ratio between the partitioning of MARCKS and of

MARCKS-FA to PC:PS(10:1) membranes is ~6-10 corresponding to an effective free energy

contribution of 0.22-0.27 kcal/mol per phenylalanine group [123,124], different than the

value of 1.3 kcal/mol found previously, [121]. For a different membrane, PC:PIP2(99:1), the

ratio between the partition coefficients obtained is 300, corresponding to an effective free

energy of 0.7 kcal/mol per phenylalanine group [125]. From our simulations, we get for these

cases, effective free energy contributions of 0.46 kcal/mol and of 0.74 kcal/mol per

phenylalanine group, respectively.

A.3.3 Adsorption Isotherm – “Electrostatic-Switch Mechanism”

It was found, experimentally, that MARCKS-ED inhibits PIP2 hydrolysis by phospholipase

C. In order to account for this observation, notice that only membranes which contain PIP2

lipids are essentially adsorbing, as implied by Table A.1. As demonstrated in Figure A.6, any

adsorbed protein sequesters 3-4 PIP2 lipids. Therefore, we can think of the membrane as

divided into adsorption sites, each of them associated with ~3 PIP2 lipids. Whenever a protein

occupies an adsorption site, it binds the 3 PIP2 lipids present and thus prevents them from

being hydrolyzed. Figure A.8 shows an adsorption isotherms for all the cases studied in this

work. That is, the fraction of occupied adsorption sites as a function of polymer concentration

in solution. Using Eq. (4.10) we get,

1

F

F

ee

ϕυθϕυ

−Δ

−Δ=+

(A.11)

Where θ is the fraction of occupied adsorption sites (alternatively, the fraction of bound

PIP2’s), [ ]Pϕ ≡ is the molar concentration of the MARCKS protein and the volume of an

adsorbed protein is estimated by 210 8.66 (5.5 8.66)aυ λ π= ≅ × × × Å3 -1370 M= .

As apparent from Figure A.8, at physiological concentrations (~1μm), θ is different from

zero only for PIP2-containing membranes (i.e., PC:PS:PIP2 and PC:PIP2). At these

concentrations, MARCKS occupies essentially all membrane sites for PC:PS:PIP2 membrane

(and thus completely inhibits hydrolysis) and about 50% of the sites for PC:PIP2 membrane.

For comparison, the values obtained experimentally are >90% inhibition with 100nM

MARCKS-ED for PC:PS:PIP2(66:33:1) membrane and ~50% inhibition with 0.3-0.5μm

121

1e-08 1e-06 0.0001 0.01ϕ [M]

0

0.2

0.4

0.6

0.8

1

θ

MARCKS-ED for PC:PS:PIP2(83:17:0.15), [82,125]. Upon phosphorylation, θ is reduced to

essentially zero for PC:PIP2 membrane. Similarly, for PC:PS:PIP2 membrane, θ reduces to

~10% for 1μm protein and to essentially zero for 0.1μm MARCKS. Thereby, making the

PIP2 lipids available for hydrolysis and follows the “electrostatic-switch” mechanism. Notice

also that replacing the Phenylalanines by alanines, results in an effect which is similar to

phosphorylation of the three serine groups. Thus, with no aromatic residues (i.e., for

MARCKS-FA), the concentration of adsorbed proteins dramatically decreases, and

consequently the amount of bound PIP2 lipids.

Figure A.8 The Adsorption isotherms for MARCKS proteins. The fraction of adsorbed sites as a function of

the MARCKS concentration in solution is plotted for PC:PS:PIP2 (black), PC:PIP2 (blue) and PC:PS (purple)

membranes. The solid, dashed, dashed-dotted and dotted curves correspond to the MARCKS, MARCKS-FA,

phosphorylated-MARCKS and phosphorylated-MARCKS-FA respectively. At biological concentrations

( 1 mμ≈ ), only the non-phosphorylated MARCKS on a PC:PS:PIP2 membrane is significantly adsorbed.

A.3.4 Concluding Remarks

Our main findings are the following.

It is its large partition coefficient, that makes MARCKS inhibits hydrolysis much

more efficiently than MARCKS-FA. Both have similar ability to bind PIP2 lipids (~3-4 per

adsorbed protein).

The phenyl groups induce MARCKS extended conformation and contribute

significantly to its adsorption free energy. The free energy contribution per phenyl group is

not a constant and it depends on membrane composition and the amino acid sequence.

122

Upon phosphorylation, the partition coefficient decreases in up to three orders of

magnitude. Consequently, the concentration of adsorbed proteins (and of sequestered PIP2’s)

decreases significantly.

123

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phospholipid bicelles. Biophys. J. 85, 2442-2448.

120. Netz, R. R. (1999). Debye-Huckel theory for interfacial geometries. Phys. Rev. E. 60,

3174-3182.

121. Wimley, W. C., and White, S. H. (1996). Experimentally determined hydrophobicity

scale for proteins at membrane interfaces. Nature Struct. Biol. 3, 842-848.

122. Rauch, M. E., Ferguson, C. G., Prestwich, G. D., and Cafiso, D. S. (2002).

Myristoylated alanine-rich C kinase substrate (MARCKS) sequesters spin-labeled

phosphatidylinositol 4,5-bisphosphate in lipid bilayers. J. Biol. Chem. 277, 14068-

14076.

134

123. Arbuzova, A., Wang, L., Wang, J., Hangyás-Mihályné, G., Murray, D., Honig, D., and

McLaughlin, S. (2000). Membrane binding of peptides containing both basic and

aromatic residues. Experimental studies with peptides corresponding to the scaffolding

region of caveolin and the effector region of MARCKS. Biochemistry. 39, 10330-

10339.

124. Victor, K., Jacob, J., and Cafiso, D. S. (1999). Interactions controlling the membrane

binding of basic protein domains: Phenylalanine and the attachment of the

myristoylated alanine-rich C-kinase substrate protein to interfaces. Biochemistry 38,

12527-12536.

125. Wang, J. Y., Gambhir, A., Hangyás-Mihályné, G., Murray, D., Golebiewska, U., and

McLaughlin, S. (2002). Lateral sequestration of phosphatidylinositol 4,5-bisphosphate

by the basic effector domain of myristoylated alanine-rich C kinase substrate is due to

nonspecific electrostatic interactions. J. Biol. Chem. 277, 34401-34412.

VI

59 ......................................התפלגות דרגות ההיעטפות של הנוקלאוקפסידים 3.3.3

61 ...................................................................................שחרור הויריונים 3.3.4

63 ...................................................................................................הערות לסיכום 3.4

66 ............................................ספיחה של מאקרומולקולות גמישות על ממברנות נוזליות 4

66 .............................................................................................................הקדמה 4.1

70 .......................................................................תרמודינמיקה של תהליך הספיחה 4.2

70 ..........................................................................ספיחה של פולימר יחיד 4.2.1

73 ..................................................תלות הספיחה בריכוז הפולימר בתמיסה 4.2.2

76 ..............................................................................הגדרת המצב הספוח 4.2.3

80 ...................................................................................................מערכת המודל 4.3

81 ..............................................................................................שיטת הסימולציה 4.4

82 ...................................................................................ממברנות קפואות 4.4.1

84 ...................................................................................ממברנות נוזליות 4.4.2

86 ...................................................................אנרגיות חופשיות של ספיחה 4.4.3

87 .............................................................................................................תוצאות 4.5

88 .......................................................................................תכונות מבניות 4.5.1

95 ........................................................................תרמודינמיקה של ספיחה 4.5.2

102 ...................................................................................................הערות לסיכום 4.6

104............................................................................................................דבר-סוף 5

107 ..........................................."מנגנון המתג האלקטרוסטטי "-MARCKS-חלבון ה - Aנספח

A.1 107 .............................................................................................................הקדמה

A.2 108 ...................................................................................................מודלמערכת ה

A.2.1 111 ....................................................................."נפח אסור"אינטראקציות

A.2.2 111 ...............................................................לקטרוסטטיותאינטראקציות א

A.2.3 112 ....................................................................אינטראקציות הידרופוביות

A.3 112 .....................................................................................................תוצאות ודיון

A.3.1 114 ...............................................................................ליפידיםההתפלגות

A.3.2 116 ...................................................................אנרגיות חופשיות של ספיחה

A.3.3 120..............................."מנגנון המתג האלקטרוסטטי "-איזותרמת ספיחה

A.3.4 121 .......................................................................................הערות לסיכום

123 .................................................................................................................ביבליוגרפיה

V

תוכן עניינים

1 ............................................................................................................ הקדמה 1

1 ................................................................................................................מבוא 1.1

2 ............................................................................................פולימרים ביולוגיים 1.2

4 .......................................................................גודל אופייני של פולימרים 1.2.1

5 .............................................................................גמישות של פולימרים 1.2.2

7 ..........................................................................אלסטיות של פולימרים 1.2.3

11 .........................................פולימרים טעוניםשל ) Condensation(דחיסה 1.2.4

11 .............................................................................................ממברנות ליפידיות 1.3

14 ..................................................אינטראקציות בין פולימרים ביולוגיים ומשטחים 1.4

15 ..............................................................................................מערכות ויראליות 1.5

17 ....................................................................................................מבנה העבודה 1.6

18 ................................................................ים' מבקטריופאגDNAאריזה והזרקה של 2

18 .............................................................................................................הקדמה 2.1

22 ............................................................................................................תיאוריה 2.2

22 .............................................................האנרגיה חופשיתמודל ומערכת ה 2.2.1

DNA.......................................... 25פוטנציאל האינטראקציה בין מולקולות 2.2.2

26 .................................... בתמיסהDNAיווצרות של טורוס מודל אנליטי לה 2.2.3

27 ........................................................................................שיטת הפתרון 2.2.4

28 .....................................................................................................תוצאות ודיון 2.3

DNA...................................................................................... 28-אריזת ה 2.3.1

32 .......................................................האנרגיות בתהליך ההזרקהכוחות וה 2.3.2

34 ......................................................................מקורב" שני מצבים"ל מוד 2.3.3

36 ............................................................................ הנבנה בקופסיתלחץה 2.3.4

40 ...............................................................................בלתי שלמההזרקה 2.3.5

42 ...................................................................................................הערות לסיכום 2.4

44 ....................................................................................................הנצה ויראלית 3

44 .............................................................................................................הקדמה 3.1

46 .............................................................................................. התיאורטימודלה 3.2

48 .........................................................."קירוב הפאזות המאקרוסקופיות" 3.2.1

50 ......................................ידיםות דרגות ההיעטפות של הנוקלאוקפסהתפלג 3.2.2

54 .............................................................................................................תוצאות 3.3

54 .................................................................................בחירת הפרמטרים 3.3.1

spikes(........................................................ 56 (החלבונייםהחודים צפיפות 3.3.2

IV

4הסימולציה המתוארת בפרק . מורכבת הרבה יותרלהופכת את המכניקה הסטטיסטית של הפולימר

תפקיד חשוב קיים לה – המוביליות של הליפידים .אוצרת בתוכה את היכולת לתאר פולימרים מסוג זה

אנרגיה באופן עקיף באמצעות הוספת איבר ל 3 נלקחת בחשבון בפרק - A ובנספח 4ק המודגם בפר

אחת הסיבות לתשלום אנרגטי זה . וממברנת התאbud-בין ממברנת המצומד לשטח המגע ה, החופשית

, כמו גם האלסטיות של הממברנה,spikes-המוביליות של ה. ממברנותהינו השוני בהרכב בין שני סוגי ה

המוביליות של הליפידים ותרומתה האנטרופית והאנרגטית , ובנספח4בפרק . מודל זה במפורשנלקחות ב

סטטיסטיים של תהליך - כולו דן בהיבטים תרמודינמיים4 פרק .נלקחת במפורש, לאנרגיה החופשית

לאו , ניתנות להרחבה לכל סוג אינטראקציה4המסקנות המתקבלות מפרק , לעיל כפי שצוין .הספיחה

. אלקטרוסטטיתדוקא

הוירוס מדביק תאים לבין התכונות הפיסיקליות של המולקולות בו בין המנגנון הביולוגי קשרה

,ההדבקה הויראלית של חיידקים נעשית בצורה ספונטניתו מאחר - אלויולוגיות השותפות בתהליכיםהב

הגנום מתוך המעטפת יש צורך בכוח שיניע את יציאת , על תא החיידק)רצפטור(קולטן לאחר קישור ל

הגנום נדחס בכוח גדול אל תוך המעטפת החלבונית , לשם כך. החלבונית לפני שהיא תפורק על ידי המדיום

.(Motor protein) באנזים שיבצע את תהליך האריזהצורךועל כן ישנו , הליך ההתארגנות של הוירוסבת

קולטנים המצויים על ממברנת קישור לההדבקה על ידי וירוסים אנימליים נעשית על ידי , לעומת זאת

במקרה זה לחץ גבוה בתוך . איחוי של הממברנה הויראלית עם הממברנה התאיתעל ידי והתא המודבק

על מנת .ההתארגנות הויראלית הינה ספונטנית, עבור וירוסים אנימליים.המעטפת הויראלית הינו מיותר

המספקים את הם spikes-ה, בוירוס הנוצר)טניםההכרחיים לקישור לקול (spikesלחייב את נוכחות ה

. הכוח המניע לתהליך ההנצהוה אתומהאשר , אנרגית המשיכה בין הממברנה לנוקלאוקפסיד הויראלי

העוברים אינטראקציה ,חלבונים וליפידיםשל התרכזות . בוירוס השלםמובטח כי הם ישולבובצורה זו

buds- שטח המגע בין הורמים למחיר אנרגטי עלג, הנצהיד הויראלי באזור הקפסנוקלאומועדפת עם ה

מביאההיא הזו אנרגיה ,אולם. ההרכב השונה מההרכב הממוצע על פני הממברנה התאיתעקבוהממברנה

אינה מועדפת במקרה זהקפסידיםנוקלאו חלקית של מאחר והיעטפות, ר וירוסים יעיל יותרלשחרו

. אנרגטית

כי גם סביר להניח אולם, לא נכנס למסגרת הדוקטורטHIVסים כדוגמת מנגנון ההבשלה של רטרווירו

ניתן יהיה לראות את החותם של התכונות הפיסיקליות של המולקולות המשתתפות בתהליך ,זהבמקרה

.על מנגנון הכניסה הנבחר ותכונות הוירוס המתקבל

III

אחד הממצאים . נוזליות פולימרים על ממברנותתספיח בבעיה הכללית של החופשיות והמבנים המעורבים

לאנרגיות הספיחה והן לתכונות המרכזיים הינו שהמוביליות של הליפידים תורמת משמעותית הן

, קפואהבהשוואה לממברנה, חוזק האינטראקציה עם ממברנה נוזלית. המבניות של הפולימר והממברנה

על האינטראקציה . מתרכזים תחתיו, עם הפולימר אשר להם אינטראקציה חזקה,נובע מכך שהליפידים

להיות חזקה מספיק על מנת להתגבר על המחיר האנטרופי של ריכוז כמות משמעותית של ליפידים באזור

ניתן , אף על פי שתכונות הממברנה הנוזלית שונות מהותית מאלה של ממברנה קפואה. מסוים בממברנה

המתאימים לממברנה נוזלית מתוך אוסף בולצמני של לחשב את הגדלים התרמודינמיים והמבניים

snapshotsשל באופן הפרופורציוני לפונקצית החלוקה קל כל אחת מהן ְשאך יש לָמ, של ממברנות קפואות

באופן מקורב את התלות של תכונות הספיחה מוצאיםאנו , באמצעות מודל סריגי .פולימר הספוח עליה

פונקציות החלוקה עבור ספיחה של , נמוכותכי בגבול של צפיפויות ומראים , בריכוז הפולימר בתמיסה

נעשית עבור ספיחה של פולימר , 4הסימולציה בפרק .מתלכדותפולימר יחיד על ממברנה נוזלית וקפואה

טעונים ליפידיםPC( ,1%כדוגמת ( ליפידים ניאטרליים טעון חיובית על ממברנה המכילה תערובת של

.)PSכדוגמת (ערכיים - חד טעונים שלילית ליפידים10% או 1%- ו) PIP2כדוגמת (ערכיים - ארבעשלילית

זאת משום שישנה עדיפות , ערכיים-ערכיים ולא החד-הליפידים המתרכזים מתחת לפולימר הינם הארבע

" ריתוק"עדיפות הנובעת מכך שמתקבל רווח אלקטרוסטטי זהה מ. אנטרופית לריכוז ליפידים מסוג זה

- הימצאותם של ליפידים ארבע. יחידערכי -ערכיים או ליפיד ארבע- ליפידים חד ארבעהמר של לאזור הפולי

אשר ספיחתם מלווה באיבוד אנטרופיה קונפורמציונית , ערכיים הכרחית לספיחה של פולימרים גמישים

אינטראקציות אלקטרוסטטיות האינטראקציות בין הפולימר והליפידים הטעונים בממברנה הינן . רבה

. ניתן להכליל את התוצאות עבור כל אינטראקציות שהן, עם זאת. היקל-לקחות בקירוב דביהנ

את בו אנו חוקרים , Aמודגם בנספח , השימוש בסימולציה ללימוד מערכת ביולוגית מסוימת

-התפקיד הביולוגי של חלבון ה. מברנותמ מMARCKS-מנגנון הספיחה וההתנתקות של חלבון ה

MARCKSמסוג לליפידים תו להיקשר מיוחס ליכול PIP2, כמות הובכך לווסת את -PIP2 החופשיים

קישור של , פקודם בתהליכים שונים בתא כגון פתיחת תעלות סידןליפידים אלו ידועים בת. בממברנה

. לממברנה וכדומה(cytoskeleton) התאיהשלד

. הפרק החמישי מהווה סיכום קצר של המוצג בעבודה

האחד נוגע למכניקה הסטטיסטית של פולימרים . עוברים כחוט השני לאורך העבודהיים מרכזרעיונותשני

ממברנות ויחסי הגומלין ביניהם ואילו השני הוא הקשר בין המנגנון הביולוגי בו הוירוס מדביק , גמישים

) 1פרק (ההקדמה . תאים לבין התכונות הפיסיקליות של המולקולות הביולוגיות השותפות בתהליכים אלו

.שנימתמקד ברעיון ה) 5פרק (הסיכום בעוד ש, ראשוןרעיון המתמקדת ב

מכניקה הפרק הראשון עוסק ב- פולימרים גמישים וממברנות נוזליות של מכניקה הסטטיסטיתה

במעטפת חלבונית אשר גודלה הוא מסדר (Semi-flexible)גמיש למחצה פולימר הסטטיסטית של כליאת

הגנום של הוירוסים . ך הפולימרקטן הרבה יותר מאורם זאת עאך , persistence length- הגודל של

והאנטרופיה יותרועל כן הינו גמיש, גדילי-גדילי ולא דו-הינו חד, 3 המוצגים בפרק ,האנימליים

, גדילי- חדRNAהעובדה כי זהו , כמו כן. הקונפורמציונית המעורבת בתהליכים הופכת משמעותית

עובדה זו . יצור מבנים שניוניים על ידי זיווג בסיסיםליכולתו מאחר וב, פת למערכתמוסיפה דרגת חופש נוס

II

מצויDNA- כאשר כל ה,)spool( מצורה של סליל ופסיתבק המצוי DNA- האריזתשל הדרגתי מבני

אנו , למעשה. DNAלאחר ששוחררה כמות מספקת של ) הדומה לזה הנוצר בתמיסה(לטורוס , ופסיתבק

, שהיה קיים בקופסיתהמלווה בשחרור מרבית הלחץ, פטימליגדילי או-במרחק ביןאריזה כי מוצאים

איזון עדין בין התרומות השונות , מכאן והלאה. מתקבלת כבר לאחר שחרור של כשליש מכמות הגנום

שינויים קלים במערכת , כךמשום . א ליציאת שארית הגנום מתוך הקופסיתמביהוא ה , החופשיתהלאנרגי

הנגרם למשל כתוצאה (הימצאות לחץ אוסמוטי בתוך תא החיידק כגון , סיונית או התיאורטיתיהנ

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.סיוניותיהינם בהתאמה לתוצאות נ, המתקבלים מן המודלהגודל של הכוחות הדרושים לאריזה

להתארגנות אפשריים נגנוניםשני מקיימים . ם אנימליים בוירוסי עוסקפרק השלישי של העבודהה

בדרך כלל (ראשית נארז הגנום הויראלי : )Alpha viruses-אופייני לה ( הראשוןנגנוןמב. של וירוסים אלו

RNA ד בשלב הבא לממברנה הנוד, אוקפסיד נוקלחלבוני המבנה של הוירוס ליצירתיחד עם ) גדילי- חד

המצויים על ממברנת התא לממברנה באמצעות חלבונים ויראלייםאוקפסיד נקשרהנוקל. התאית

.אותו אנו ממדלים בפרק זה, נעטף בה ועוזב את התא בתהליך הנקרא הנצה, (spikes) הקרויים חודים

בכמותו) bud-גודל ה(קפסיד אונוקלדרגת ההיעטפות של הה ב תלויbudהאנרגיה החופשית עבור יצירה של

המתווכת על ידי (קפסיד לממברנה אונוקלוכוללת את אנרגית הקישור בין ה, bud-בהמצויים spikes-ה

spikes( ,אנרגית אזור המגע ו ,האנרגיה האלסטית של הממברנה)line energy (בין ממברנת ה-bud

ומכילה buds- היא פונקציונל של התפלגות גדלי ה האנרגיה החופשית של המערכת כולה.לממברנת התא

. buds- והתא והתפלגות אוכלוסיית הbuds- בין ממברנות הspikes- התפלגות העבוריים איברים אנטרופ

מתוך מינימיזציה של האנרגיה יםמתקבל, buds-כמו גם התפלגות גדלי ה, בשיווי משקלspikes-ריכוזי ה

אחת התוצאות המרכזיות של המודל הינה העובדה ששחרור ויראלי מתבצע . החופשית ביחס לגדלים אלו

סף הדרושה על מנת לכופף את ה גבוהה מאנרגית לממברנהspikes-הרק כאשר אנרגית הקישור של

, spikesכל וירוס מכיל מספר זהה של כי , אנו מוצאים, במקרה זה. אוקפסידנוקלהממברנה מסביב ל

ולקית אינאוקפסיד עטוף חנוקל גורמת לכך שאזור המגעאנרגית .המתאים לאיכלוס כל אתרי הקישור

בריכוזים נמוכים של .נעטף לחלוטין ומשוחרר מן התא, קפסיד שנעטף בממברנהאונוקלולמעשה כל , יציב

,מעל לריכוז מסוים. קפסידים המגיעים לממברנה נעטפים ומשוחרריםנוקליאוכל ה, קפסידיםנוקליאו

מגיעים קפסידים החדשים האונוקלה, קפסידיםאונוקל לspikesאשר מקביל ליחס סטויכיומטרי בין

, קפסידיםאונוקל לspikesישנו יחס אופטימלי בין , כלומר. לממברנה לא יכולים להיעטף והנצילות יורדת

. ליעילות מכסימליתהמביא

כדוגמת , בוירוסים מורכבים יותר וירוסים אנימליים מתרחשעבור התארגנות השני נגנוןהמ

מתבצע בד התחברות המרכיבים השוניםתהליך , במקרה זה). HIV-בה נכלל ה(רטרווירוסים משפחת ה

דרגות החופש של המולקולות נוצר צימוד בין , בצורה זו. על גבי הממברנה התאיתההנצה בבד עם

הספוחים על מוביליות הליפידים וחלבוני המבנה הויראליים, גמישות הגנום והממברנה: השונות

הפרק , על מנת לגשת לבעיה זו. רכב יותרהופך את פיתרון הבעיה למו, צימוד זה. וכיוצא בזההממברנה

פיתחנו , לחקור את הנושאכדי. נוזליותן בין פולימרים וממברנות טעונותיחסי הגומלי עוסק בהרביעי

המאפשרת לחשב בצורה יעילה את האנרגיות )Rosenbluthהרחבה לשיטת ( שיטת סימולציה ,ככלי

I

תקציר

, בכלל את העקרונות הפיסיקליים המצויים בבסיס תהליך ההדבקה הנגיפיחקור להינה מטרת העבודה

מאחר והתארגנות ויראלית . בפרט החלבונים והגנום הויראליים ליצירת וירוס שלםהתארגנותבשלב ו

דוגמת מעטפת כ(שטחים ומ)כדוגמת חומצות גרעין (במהותה מערבת אינטראקציה בין פולימרים טעונים

ובתפקידן פולימרים גמישים ומשטחיםהעבודה עוסקת באינטראקציות בין , )חלבונית או ממברנלית

.בהתארגנות הוירוס

פולימרים טעונים וממברנות ביולוגיות בדגש על תכונותיהן הפיסיקליות של מפורטות, בהקדמה

ת ומתואר, כמו כן). ראליים וי DNA אוRNAכדוגמת (מתמקד המחקרהפולימרים הביולוגיים בהם

. באות לידי ביטוי בהתארגנות של וירוסים תכונות אלהאופן בובקצרה מספר דוגמאות ל בפרק

כאשר . משטחיםספיחה עלהגמישות של הפולימרים הביולוגיים מאפשרת להם לשנות מבנה כתוצאה מ

זאת משום שממברנה הינה . ההממברנה אף היא יכולה להגיב לספיח, הפולימר נספח על ממברנה ביולוגית

ליפידים של המימדית -הדויכולת התנועה . חלבונים" יםמסומ" בו ,מימדי-דו, ליפידילמעשה נוזל

להתרכז באזור בעלי אנרגית אינטראקציה מועדפת עם הפולימר מאפשרת לאלו, חלבוניםהו

מההרכב הממוצע שונהרכב אזור ובו הנדידה זו של ליפידים גורמת ליצירת . ולחזק אותה האינטראקציה

לאבד מבלייה אינטראקצההאלסטיות של הממברנה מאפשרת לה לשנות עקמומיות כתוצאה מ. בממברנה

נעטף , הקפסיד הויראלי נקשר לממברנה בהם ,תהליכים כדוגמת הנצהל יכולת זו הכרחית. את שלמותה

.בה וניתק עמה מן התא

).םי'בקטריופאג( חיידקים על ידי וירוסים שלתהליך ההדבקה הפרק השני של העבודה עוסק ב

בדרך ( דרכה מוזרק הגנום הויראלי ,הנגיף" זנב" תעלה בפתיחתהקישור של נגיפים אלה לחיידק גורם ל

.נשאר מחוץ לתא, הגנום מצויהקרום החלבוני בתוכו כאשר זאת .החיידקאל תוך ) גדילי- דו DNAכלל

האנרגיה . )קופסית ( מקפסיד ויראליDNA של והטעינה מודל תיאורטי לתהליך ההזרקה מוצגבפרק

כלומר כפונקציה (השחרור דרגת או לחילופין, ת נכתבת כפונקציה של דרגת הטעינההחופשית של המערכ

). כפונקציה של הכמות ששוחררה לתמיסהאו לחילופין, ופסית עדיין מחוץ לק המצויה DNA-של כמות ה

החופשית בתמיסה ושל DNA-ל האנרגיות החופשיות של שרשרת הכסכום ש האנרגיה החופשית מבוטאת

אנרגית ו DNA של אנרגית כיפוףכאשר כל אחד מהאיברים הוא סכום של , ופסית בקה הכלואזו

האנרגיה עבור שרשרת כמו גם ,האריזהרת צו, בכל שלב בתהליך השחרור. DNAבין גדילי אינטראקציה

DNA גדילי המאפיינים - ולמרחק הביןונקציונל האנרגיה ביחס לצורה של פ מינימיזציה על ידימתקבלות

נעשה שימוש , לקבלת התוצאות הנומריות המוצגות בפרק. ופסית בתמיסה ובקDNA-את אריזת ה

התוצאות המתקבלות מן המודל . ואורך הגנוםופסיתעבור מימדי הק λ' פאגבקטריו האופיינים לבערכים

ה בשחרור של הלחץ הרב שהיה אצור באריזה ועל כן גנום מלוֶו של התהליך השחרורשמעידות על כך

אינטראקציות הדחייה הינו תוצאה של DNA- בה המלאופסית בקהלחץ הגבוה. מתרחש בצורה ספונטנית

- ה" כליאת"ן ושל האנרגיה האלסטית של הצפיפות המטען הגבוהה על פני בשל DNA-בין שרשרות ה

DNAשל הגודל מסדר בתוך קופסית שגודלה הוא-persistence length .שחרור ה-DNA מלווה בשינוי

אבינועם בן שאולפרופסור שלזו נעשתה בהדרכתו עבודה

של מקרומולקולות גמישות אינטראקציות

עם משטחים והאופן בו הן משחקות תפקיד

במערכות ויראליות

קטור לפילוסופיה תואר דולשם קבלתיבור ח מאת

צלילשלי

האוניברסיטה העברית בירושליםלסינטוגש ה

דצמבר 2006