Hierarchical Modeling of Biomolecular Systems: From Microscopic to Macroscopic

Simulations

VAGELIS HARMANDARIS

Department of Applied MathematicsUniversity of Crete, and FORTH, Heraklion, Greece

Cell Biology and Physiology: PDE models, 05/10/12

Outline

Introduction: General Overview of Biomolecular systems. Characteristic Length-Time Scales.

Multi-scale Particle Approaches: Microscopic (atomistic), Mesoscopic (coarse-grained) simulations, Macroscopic PDEs.

Conclusions – Open Questions.

Applications:

Self-assembly of Peptides through Microscopic Simulations.

Elasticity of Biological Membranes through Mesoscopic Simulations.

INTRODUCTION - MOTIVATION

Systems biological macromolecules (cell membrane, DNA, lipids)

Applications

Nano-, bio-technology (biomaterials in nano-dimensions)

Biological processes

Radius of gyration ~ 1-10 nm (10-9 m)

Time – Length Scales Involved in Biomolecular Systems

Self-assembly of biomolecules ~ 10 μm (10-5 m)

Bond length ~ 1 Å (10-10 m)

Multi-compartment biological systems (e.g. cell) ~ 1 mm (10-3 m)

Bond vibrations: ~ 10-15 sec

Segmental relaxation: 10-9 - 10-12 sec

Maximum relaxation time of a biomacromolecule, τ1: ~ 1 sec (in Τ < Τm)

Angle rotations: ~ 10-13 sec

Dihedral rotations: ~ 10-11 sec

Time – Length Scales Involved in Polymer Composite Systems

Dynamics of multi-component system: ~days

THEORIES & COMPUTER SIMULATIONS:

-- probe microscopic structural features-- organization of the adsorbed groups-- dynamics at the interface-- study in the molecular level

Α) description in quantum level

Β) description in microscopic (atomistic) level

C) description in mesoscopic (coarse-grained) level

D) description in macroscopic - continuum level

Hierarchical Modeling of Molecular Materials

Main goal: Built rigorous “bridges” between different simulation levels.

Quantitative prediction of properties of complex biomolecular systems.

Molecular Dynamics (MD) [Alder and Wainwright, J. Chem. Phys., 27, 1208 (1957)]

Classical mechanics: solve classical equations of motion in phase space (r, p).

System of 3N PDEs (in microcanonical , NVE, ensemble):

Liouville operator:

The evolution of system from time t=0 to time t is given by : ( ) exp (0)t iLt

1

, t

N

i ii i i

iL H

r Fr p

K

it i

im

pr 1 2, ,..., N c

t i ii

U

r r r

p Fr

2

( )2

iNVE

i i

pH K V Vm

rHamiltonian (conserved quantity):

Microscopic – Atomistic Modeling: Molecular Dynamics Simulations

Molecular model: Information for the functions describing the molecular interactions between atoms.

bonded non bonded extU V V V R R R

Molecular Interaction Potential (Force Field): Atomistic Simulations

Important question: What is the potential energy function?

Assumption - The complex quantum many-body interaction can be:

1) Described by semi-empirical functions.

2) Decomposed into various components.

Vbonded: Interaction between atoms connected by one or a few (3-5) chemical bonds.

Vnon-bonded: Interaction between atoms belonging in different molecules or in the same molecule but many bonds (more than 3-5) apart.

Vext: External potential (force) acting on atoms.

1 2: , ,..., NU UR r r r

Potential parameters are obtained from more detailed simulations or fitting to experimental data.

21 ( )2 obend bendV k bending potential

21 ( )2 ostr strV k l l stretching potential

dihedral potential5

0cos ( )i

tors ii

V c

non-bonded potential12 6

4LJV r r

Van der Waals (LJ) Coulomb

ri j

qij

qqV

ε

bonded str bend torsV V V V r

Molecular Interaction Potential (Force Field): Atomistic Simulations

non bonded LJ q hybridV V V V r

MULTISCALE – HIERARCHICAL MODELING OF BIOMOLECULAR SYSTEMS

Limits of Atomistic Molecular Dynamics Simulations (with usual computer power):

-- Length scale: few (4-5) Å - (10 nm)

-- Time scale: few fs - (0.5 μs)

-- Molecular Length scale (concerning the global dynamics):up to ~ 10.000 – 100.000 atoms

Need: Study phenomena in broader range of time-length scales Study more complicated systems.

COARSE-GRAINED MESOSCOPIC MODELS

Integrate out some degrees of freedom as one moves from finer to coarser scales.

GENERAL PROCEDURE FOR DEVELOPING MESOSCOPIC PARTICLE MODELS DIRECTLY FROM THE CHEMISTRY

1. Choice of the proper mesoscopic description.

2. Microscopic (atomistic) simulations of short chains (oligomers) for short times.

-- number of atoms that correspond to a ‘super-atom’ (coarse grained bead)

3. Develop the effective mesoscopic force field using the atomistic data.

4. CG (MD or MC) simulations with the new CG model.

Re-introduction (back-mapping) of the atomistic detail if needed.

r

BONDED POTENTIAL Degrees of freedom: bond lengths (r), bond angles (θ), dihedral angles ()

PROCEDURE: From the microscopic simulations we calculate the distribution functions of the degrees of freedom in the mesoscopic representation, PCG(r,θ,).

PCG(r,θ, ) follow a Boltzmann distribution: ( , , ), , exp

CGCG U rP r

kT

Assumption:

( , ) ln , , ( , , )CG CGBU x T k T P x T x r

, ,CG CG CG CGP r P r P P

Finally:

( ) ( ) ( )CG CG CGtotal bonded non bondedU U U Q Q Q

DEVELOP THE EFFECTIVE MESOSCOPIC CG POLYMER FORCE FIELD

NONBONDED INTERACTION PARAMETERS: REVERSIBLE WORK

Reversible work method [McCoy and Curro, Macromolecules, 31, 9362 (1998)] By calculating the reversible work (potential of mean force) between the centers of mass of two isolated molecules as a function of distance:

exp ,( , ) lnCG ATnb UU T rq

,

,AT ATij

i j

U U r r

Average < > over all degrees of freedom Γ that are integrated out (here orientational ) keeping the two center-of-masses fixed at distance r.

1( , ).... exp , ,...CG

nb

ATNU T

N

U T d

Ze

qr r r

CG Hamiltonian – Renormalization Group Map:

( , ) ( , ) |CG AT

nbU T U TNP de e q r r q q

APPLICATION I: SELF – ASSEMBLY OF PEPTIDES THROUGH ATOMISTIC MOLECULAR SIMULATIONS

Experimental Motivation Diphenylalanine FF

Peptides can assemble into various structures (fibrillar, or spherical) depending on conditions such as solvent.

The diphenylalanine core motif of the Alzheimer’s disease b-amyloid

E. Gazit et al, 2003, 2005, 2007

Simulation Method and ModelAtomistic Molecular Dynamics (MD) NPT Simulations.P=1atm (Berendsen barostat)T=300K (velocity rescaling thermostat)

Periodic boundary conditions were used in all three dimensions.Gromos53a6 Atomistic Force Field was used

Di-alanine (AA) / Di-phenylalanine (FF) molecule in explicit solvent

21 2

2

( , ,..., )i Ni i

i

d Um

d t

r r r rFr

Simulated SystemsSystem Name N-peptide

(# molec.)N-solvent(# molec.)

#atoms c(g pep./cm3

solv.)

T(K)

1 AA in Water 16 3696 11328 0.0385 300

2 AA in Methanol

16 1632 5120 0.0385 300

3 FF in Water 16 6840 21112 0.0385 300

4 FF in Methanol

16 3024 9648 0.0385 300

5 FF in Water 16 25452 76948 0.0103 300

6 FF in Methanol

16 11648 35520 0.0103 300

7 RE FF in Water 16 6840 21112 0.0385 395-343

8 RE FF in Methanol

16 3024 9648 0.0385 385-332

Potential of Mean Force (PMF): Alanine

0.0 0.3 0.6 0.9 1.2 1.5-5

0

5

10

15

20 AA in Water AA in Methanol k

BT

r(rm)

V(r

)(kJ

/mol

)

Effect of solvent:

Slight attraction of Alanine in Water.

No attraction in Methanol.

Potential of Mean Force (PMF): Diphenylalanine

0.0 0.5 1.0 1.5-5

0

5

10

15

20

V

(r)(

kJ/m

ol)

r(rm)

FF in Water FF in Methanol k

BT

Attraction is apparent only in Water.

Phenyl groups are responsible for strong attraction between FF molecules.

STATIC PROPERTIES : LOCAL STRUCTURE

radial distribution function gn(r): describe how the density of surrounding matter varies as a distance from a reference point.

pair radial distribution function g(r)=g2(r): gives the joint probability to find 2 particles at distance r. Easy to be calculated in experiments (like X-ray diffraction) and simulations.

1 11 2

.... exp ,...!( , )( )!

NN n Nn

nN

U dr rV Ng r rN N n Z

2, 1

1( )N

iji j

g r rN

r

choose a reference atom and look for its neighbors:

Strong tendency for self assembly of FF in water in contrast to its behavior in methanol.

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5 FF-FF in CH3OH

G(r

)

G(r

)

r(nm)

0.0 0.5 1.0 1.5 2.0 2.5 3.002468

1012

FF-FF in H2O

Structure – Self Assembly of Peptides

Self Assembly of Peptides: Experimental Data

Vials A: Peptide is dissolved in water, vials labelled as B: Peptide is dissolved in methanol.

Self-assembly of Peptides in water.

Self Assembly of Peptides: More Experimental Data

Peptide in water

SEM Pictures (A. Mitraki, Dr. E. Kasotakis, E. Georgilis, Department of Material Science, University of Crete)

Peptide in methanol

Self Assembly of Peptides: More Experimental Data

Peptide in water

SEM Pictures (A. Mitraki, Dr. E. Kasotakis, E. Georgilis, Department of Material Science, University of Crete)

Peptide in methanol

Dynamics of PeptidesDynamics can be directly quantified through mean square displacements of molecules

10 100 1000 10000 100000

1

10

100

FF in Methanol FF in Water

<r2 >/

N (n

m2 )

t(ps)

22 ( ) (0)cm cmr t R t R

Dynamics of Peptides

Systems D (cm2/sec) stdevAA in Water 1.1567 +/- 0.4352

FF in Water 0.5370 +/- 0.2897

AA in Methanol 2.3904 +/- 0.5372

FF in Methanol 0.8252 +/- 0.2190

Slower Dynamics in Water

Phenyl groups retard motion

2( ) (0)lim

6cm cm

t

R t RD

t

Temperature Dependence at the same concentration: c= 0.0385gr/cm3

0.0 0.5 1.0 1.5 2.0 2.5 3.00

4

8

12 T=295K T=316.39 T=342.74K

g(r)

r(nm)

FF in Water

Temperature increase reduces structure in water.

Aggregates do not exist at any temperature in methanol.

FF in Methanol

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

1.2

T=285K T=311.84 T=331.12K

g(r)

r(nm)

Temperature Dependence at the same concentration: c= 0.0385gr/cm3

280 290 300 310 320 330 340 350

4

6

8

10

12

14

16

FF in H2O FF in CH3OH

Mea

n nu

mbe

r of F

F m

olec

ules

in a

n ag

greg

ate

T(K)

CM - radius of 2nm

Number of FF in the aggregates decreases with temperature for water solutions.

-- An amphiphilic - lipid membrane: one water-loving (hydrophilic) and one fat-loving (hydrophobic) group.

-- Works as a selective filter which controls transfer of ions, molecules, large particles (viruses, bacteria, ..) between extracellular and cytoplasm.

CELL MEMBRANE Formation of a membrane: Self-aggregation of amphiphilic molecules

-- Molecules try to reduce contacts with water. They form various structures:

• micelles

• bilayer membranes

• closed bilayers (vesicles)

• …... etc

MULTI-SCALE MODELING OF BIOLOGICAL MEMBRANES

SIMULATIONS OF BIOMEMBRANES

Motivation to Study Biomembranes:

• “Biophysical” reasons: -- 2D systems with novel physical properties,-- their composition involves many components, self-organization of multi-component systems, -- specified membrane function can be studied on the molecular level,-- possible role of universal physical properties,-- ………………. etc

• “Biotechnical” reasons: -- drug delivery (directly connected with the vesicles),-- biosensors (combinations of membranes + electronics), -- ………………. etc

Atomistic ------------------> Mesoscopic ------------------> Macroscopic (MC, MD, …) (CG, DPD, Triangulated surfaces, …) (continuum)

COARSE-GRAINED LIPID MODEL (SOLVENT FREE MODEL):[I.R. Cooke, M. Deserno, K. Kremer, J. Chem. Phys. 2005]

Interactions:• Bonded Interactions: FENE bonds (h-t1, t1-t2), harmonic bending angle (h-t1-t2)• Excluded volume potential: (Repulsive, WCA potential (fix size of the lipid)

• Attractive (t – t):

: hydrophilic group, “head” particle : hydrophobic group, “tail” particles : no solvent (water) particles

h

t1

t2

2

,

( ) cos , 2

0 ,

c

catt c c c

c

c c

r r

r rV r r r r w

w

r r w

Lipid model: Real Lipid molecule:

Integrated with a DPD (pairwise) thermostat using ESPResSO package

PARAMETERIZING CG PHENOMENOLOGICAL MODEL

-- length unit: σ -- energy unit: ε

-- wc : model parameter that control the ¨hydrophobic effect¨.

Phase Diagram: Select wc so as to simulate a stable liquid phase.

gel like

fluid

unstable

Application 1: Studying The Curvature Elasticity Of Biomembranes Through Numerical Simulations

OUR GOAL: Study the curvature elasticity (predict the elastic constants) through simulation methods

[V. Harmandaris, M. Deserno, J. Chem. Phys. 125, 204905 (2006)]

Definitions: two principal radius R1 and R2

Mean curvature:

Gaussian curvature:

Fluid Membranes: Free Energy (Continuous Approach)

1 21/ 1/ / 2K R R

1 21/(GK R R

Bending Elasticity Theory: [Helfrich, 1973]

-- κ: bending rigidity-- κG: Gaussian bending rigidity

Assumptions: fluidity of the membrane, 2D representation, insolubility (constant number of lipids)

Membrane shape can be calculated by minimizing F under constant area A and volume V [Seifert, 1997; Lipowsky 1999; …]

222 G GE dA K dAK

Question: how can someone calculate κ, κG from simulations?

-- Main idea: impose a deformation on the membrane and measure the force required to hold it in the deformed state.

STUDYING THE CURVATURE ELASTICITY – AN ALTERNATIVE WAY: CALCULATION OF ELASTIC CONSTANTS FROM DEFORMED VESICLES

[V. Harmandaris, M. Deserno, J. Chem. Phys. 125, 204905 (2006)]

Simple Method: Stretch a Membrane ! (a well-controlled bending deformation is created by the periodic boundary conditions).

Cylinder with fixed area:(one principal curvature radius R).

Helfrich theory:

STUDYING THE CURVATURE ELASTICITY: CALCULATION OF ELASTIC CONSTANTS FROM DEFORMED VESICLES.

2...zz A

EFL R

2

12

E AR

Tensile force:

2A RL

Bending rigidity:2zF R

R

Lw

Coarse-graining MD simulations: (5000 lipids, kBT = 1.1 ε, radius R = 6 – 24 σ)

z zz x yF L L

The smaller the radius R, the higher the bending of the cylinder

Tensile Force (due to the deformation), Fz

-- Stress tensor, τ, can be calculated directly in the simulation (using the Virial theorem).

, ,1

i ii

r FV

BENDING RIGIDITY[V. Harmandaris and M. Deserno, J. Chem. Phys., 125, 204905, 2006]

Result fromThermal fluctuations

Helfrich theory holds even for very small curvatures !!

2z eqF R

Application 2: Interaction between Proteins and Biological Membranes

Biological problem: how do membrane proteins aggregate? Do they need direct interactions? What is the role of the curvature-mediated interactions?

CG simulations

Modeling: needs simulations in the range of length ~ 100nm and times ~ 1ms.

Experimentally: very difficult to isolate curvature-mediated and direct (e.g. specific binding) interactions.

[Gottwein et al., J. Virol., 77, 9474 (2003)]

CG modeling of proteins and biomembranes:

[B. Reynolds, G. Illya, V. Harmandaris, M. Müller, K. Kremer and M. Deserno, Nature, 447, 461 (2007)]

CG lipids

CG proteins

No specific interactions: proteins are partially attracted to lipid bilayer but not between each other.

Interaction between Proteins (Colloids) and Biological Membranes

CG colloids

Evolution in time of the aggregation process:[ System: 46080 lipids and 36 big caps. (~ 106 atoms). Time: ~ 4 ms]

Curvature-mediated interactions: aggregation due to less curvature energy.

222 G GE dA K dAK

Interaction between Proteins (Colloids) and Biological Membranes

Colloidal spheres (model of viral capsids or nanoparticles)

Attraction and cooperative budding: clustering in form of pairs

[Gottwein et al., J. Virol., 77, 9474 (2003)]

Interaction between Proteins (Colloids) and Biological Membranes

Pair attraction: put two capsids on a membrane, calculate the constraint force needed to fix them at distance d.

Possible mechanism for attraction: capsids tilt towards each other thus reducing local curvature.

Interaction between Proteins (Colloids) and Biological Membranes

Summary - Conclusions

Microscopic (atomistic) Molecular Dynamics can give valuable information about the structure and the dynamics of small systems at the atomic resolution

Effect of solvent (water or organic) is very strong on the self-assembly of short peptides, like Di-alanine (AA) and Di-phenylalanine (FF).

Stronger attraction between FF molecules because of phenyl groups.

Slower Dynamics in Water. Phenyl groups retard motion.

Modeling of realistic multi-component biomolecular system requires multi-scale simulation approaches.

Mesoscopic (coarse-grained) simulations of biomembranes allows the study of more complicated systems as well as of continuum approaches

Interaction between colloids/proteins can lead to the rupture of membrane.

continuum elasticity is valid even for very small distances.

Current Work – Open Questions

Systematic Coarse-Graining in order to study much larger systems (thousands of peptide molecules).

Need for efficient numerical schemes to describe complex many-body terms

Study more complex systems: Boc-FF, FMoc-FF and porphyrines in water Bioconjugated hybrids: 8-mer peptide NSGAITIG (Asn-Ser-Gly-Ala-Ile-Thr-Ile-Gly) and polyethylene-oxide (PEO) and/or poly(N-isopropylacrylamide) (PNIPAM).

Length scales: from ~ 1 Å (10-10 m) up to 100 nm (10-7 m)

Time scales: from ~ 1 fs (10-15 sec) up to about 1 ms (10-3 sec)

ACKNOWLEDGMENTS

Modeling of PeptidesDr. T. Rissanou [Applied Math, University of Crete, Greece]

Prof. A. Mitraki, Dr. E. Kasotakis, E. Georgilis [Department of Material Science, University of Crete, Greece]

Funding:DFG [SPP 1369 “Interphases and Interfaces ”, Germany]ACMAC UOC [Greece]MPIP [Germany]

Biological MembranesProf. K. Kremer [Max Planck Institute for Polymer Research, Mainz]Prof. M. Deserno [Carnegie Mellon]Dr. I. Cooke [Department of Zoology, Cambridge]Dr. B. Reynolds [MPIP]