Osu 1275421702

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ON THE KRATKY-POROD MODEL FORSEMI-FLEXIBLE POLYMERS IN AN EXTERNAL

FORCE FIELD

DISSERTATION

Presented in Partial Fulllment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of the Ohio State University

By

Philip D. Kilanowski, M.S.

Graduate Program in Mathematics

The Ohio State University2010

Dissertation Committee:

Dr. Peter March, Advisor

Dr. Yuan LouDr. Saleh Tanveer

Dr. Harold Fisk

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c Copyright by

Philip D. Kilanowski

MMX

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ABSTRACT

We prove, by means of matrix-valued stochastic processes, the convergence, in a

suitable scaling limit, of the position vectors along a polymer in the discrete freely

rotating chain model to that of the continuous Kratky-Porod model, building on an

earlier result for the original model without a force, and showing that it holds when an

external force eld is added to the system. In doing so, we also prove that the process

of tangent vectors satises a stochastic differential equation, showing that it is the

sum of a spherical Brownian motion and a projective drift term, and we analyze this

equation to prove statements about the polymer in the regimes of high and low values

of the force parameter and persistence length. We augment these theoretical results

with a numerical Monte Carlo simulation of the polymer and formulate a conjectureto describe how the correlation function between tangent vectors changes with the

force.

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Dedicated to Mary.

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ACKNOWLEDGMENTS

It would be impossible to acknowledge everyone individually on one page, but here I

present the highlights:

First, I give thanks to God for giving me the talents and the perseverance to

accomplish this task– and for leading me to the beginning of my next adventure.

To my family, and to all the friends I have made in this time, for the support they

have given me (you know who you are).

To my professors who have instructed me and prepared me during this program,

especially to my advisor, Dr. Peter March, for his continued mentorship over a great

distance.

To the organizers of the Graduate Student Conference in Probability, for theopportunity to share this research with others in the eld.

To those who have gone before me and whom I have known as they completed

their doctoral work (especially my mother), for inspiring me to continue my work

until completion.

To my classmates for their help along the way, particularly Marko Samara, Ying

Wang, Dan Munther, Bill Mance, and Jim Adduci.

Finally, to Dr. Donald Knuth (Case ’60) for developing T EX, and to J. S. Bach

for writing the music the kept me going during the writing phase.

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VITA

June 9, 1982 . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Greenwich, CT

2000-2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Undergraduate,

Case Western Reserve University

2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. in Mathematics and Astronomy,


2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. in Mathematics,


2004-Present . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Teaching Associate,

The Ohio State University

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Applied Probability

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TABLE OF CONTENTS

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

CHAPTER PAGE

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Summary of the Kratky-Porod Model . . . . . . . . . . . . . . . . . . . 14

2.1 Construction of the Freely Rotating Chain . . . . . . . . 162.2 Convergence to the Kratky-Porod Model . . . . . . . . . 282.3 The Time-dependent Kratky-Porod Model . . . . . . . . 342.4 Physical Properties and Limiting Behavior . . . . . . . . 42

3 Introduction of the External Force . . . . . . . . . . . . . . . . . . . . . 56

3.1 Formulation of a Dynamical System . . . . . . . . . . . . 583.2 The Polymer in the Lowest-Energy Conguration . . . . . 68

4 Convergence of the Forced Polymer . . . . . . . . . . . . . . . . . . . . 72

4.1 Potential Energy and the Boltzmann-Gibbs Distribution . 754.2 Driving Process of the Forced Kramers Chain . . . . . . . 914.3 Convergence to the Forced Kratky-Porod Model . . . . . 1044.4 Characterization of the Tangent Vector Process . . . . . . 119

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5 Extreme Cases of the Forced Polymer . . . . . . . . . . . . . . . . . . . 138

5.1 Characterization of the K-Curve . . . . . . . . . . . . . . 1415.2 Results for the Extreme Cases . . . . . . . . . . . . . . . 152

6 Numerical Results for the Forced Polymer . . . . . . . . . . . . . . . . 158

6.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . 1596.2 Results of the Numerical Scheme . . . . . . . . . . . . . . 168

7 Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.1 Analysis of the Correlation Function . . . . . . . . . . . . 1887.2 Two Conjectures for the Correlation Function . . . . . . . 204

8 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

8.1 Alternative Models . . . . . . . . . . . . . . . . . . . . . 2138.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 218

APPENDICES

A MATLAB Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

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LIST OF TABLES

TABLE PAGE

7.1 Associated Legendre functions P ml (θ) for l ≤3 . . . . . . . . . . . . . 202

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LIST OF FIGURES

FIGURE PAGE

2.1 Construction of the discrete polymer from its segments Qk . . . . . . 17

2.2 Commutative diagram for H n−

1 and Θn

−1 . . . . . . . . . . . . . . . 20

2.3 Mean-square end-to-end length, radius of gyration, and their ratio vs.persistence length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1 Plots of the K-curve for various values of C and z 0 = −0.9 . . . . . . 146

6.1 Probability density curves for various values of µ . . . . . . . . . . . 165

6.2 Plot of RMS length for an initial angle of 45 degrees . . . . . . . . . 170

6.3 Plot of RMS length for an initial angle of 90 degrees . . . . . . . . . 171

6.4 Plot of RMS length for an initial angle of 135 degrees . . . . . . . . . 1726.5 Plot of radius of gyration for an initial angle of 45 degrees . . . . . . 173

6.6 Plot of radius of gyration for an initial angle of 90 degrees . . . . . . 174

6.7 Plot of radius of gyration for an initial angle of 135 degrees . . . . . 175

6.8 Plot of the ratio R2/R 2g for an initial angle of 45 degrees . . . . . . . 176

6.9 Plot of the ratio R2/R 2g for an initial angle of 90 degrees . . . . . . . 177

6.10 Plot of the ratio R2/R 2g for an initial angle of 135 degrees . . . . . . 178

6.11 Mean-square end-to-end length vs. log 2 ζ for z 0 = 0 and various valuesof p/L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

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6.12 Mean-square radius of gyration vs. log 2 ζ for z 0 = 0 and various values

of p/L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.13 Phase diagram for the forced Kratky-Porod model . . . . . . . . . . 184

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CHAPTER 1

INTRODUCTION

The Kratky-Porod model is a basic model of semi-exible polymer molecules in which

thermal effects, noticeable on small time and length scales, balance the elastic or me-

chanical effects. This model is particularly useful for representing polymers, which are

molecules composed of many copies of the same monomer unit or the same statistical

segment, that are immersed in a dilute solution. The actual model is a secondary

result of a 1949 paper by Austrian chemical physicists O. Kratky and G. Porod [1]

in which they describe the X-ray scattering of cellulose macromolecules in colloidal

suspension.

The foundations for this model date back to the period immediately following the

Second World War. In [2], H.A. Kramers in 1946 introduced a bead-rod model

called the freely rotating chain in which all segments have the same length, and the

bond angles between successive segments are constant, but the torsional angles are

independent and identically distributed. Later, Kirkwood and Riseman introduced

another bead-rod model, called the freely jointed chain, where all rods or segments

between beads, of a polymer have the same length and are independent, identicallydistributed random variables [3], [4]. Prince Rouse in 1953 formulated his own model,

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which involves Gaussian-distributed segments [5]. The Rouse model and Kirkwood-

Riseman chain describe exible polymers, in contrast to the Kramers chain, which

models semi-exible polymers. There are thousands of papers published in the chem-

ical physics literature since the introduction of these models. We make no attempt

to survey this vast body of work here, other than to refer to standard reference texts

such as [6], [7], [8], [9] and references therein.

The freely rotating chain (FRC) models a polymer as a collection of vectors,

RN = RN 0 , RN

1 , · · · , RN N (1.1)

where RN k is the location of the kth bead and

QN k = RN

k −RN k−1 (1.2)

is the kth rod or segment connecting adjacent beads. The model depends on a set of

parameters ( N,a,θ ), where N

∈N is the number of segments, a

∈

R + is the common

length of each segment, and θ ∈ (0, π) is the angle supplementary to the common

bond angle between any pair of adjoining segments, so that the steric angle between

segments is actually π −θ. Setting

Q N = QN 1 , QN

2 , · · · , QN N , (1.3)

we have,

QN k = a (1.4)

QN k ·QN

k+1 = a2 cos θ. (1.5)

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Throughout this dissertation, we choose a specic coordinate system for RN by speci-

fying the location and orientation of the polymer. We pin the polymer at one end and

set R0 = (0 , 0, 0) and R1 = (0 , 0, a), so that the rst segment Q1 = ae3 points along

the positive z -axis. If we describe each segment by its spherical coordinates centered

at the base of Qk , with the north pole in the direction of Qk , then the radius ( a) and

polar angle (θ) are determined for the next segment Qk+1 , but the azimuthal angle, or

torsional angle , φk , remains unspecied. These angles measure the extent to which a

given segment twists out of the plane determined by the previous two segments. andthey vary randomly with the thermal forces that the solution exerts on the polymer.

(Notice, that since Q1 is determined, the rst torsional angle φ1 gives the orientation

of Q2, and the indices of the torsional angles trail those of the immediately affected

segments by one.) If we know the set of torsional angles Φ = φ1, . . . , φN −1, then

we know the locations of all the segments and beads; since each segment Qk can

be considered as a rotation of the previous segment Qk

−1 through an angle θ, in a

direction determined by the angle φk−1. By convention, the value of φk = 0 results

in the segments Qk−1 and Qk+1 on the same side of Qk , that is, in the cis position,

while φk = π corresponds to the zig-zag, or trans position. Thus, the torsional angles

provide coordinates for the polymer, and the randomness inherent in the FRC and

its variants is fully reected in the randomness of the torsional angles.

Since the thermal forces acting on the polymer serve to randomize the torsional angles,

we consider these angles to be random variables. The FRC corresponds to the case

in which the φk are independent and identically distributed on the circle [0 , 2π). In

this case, if a segment Qk is known, then the endpoint of the next segment Qk+1

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is uniformly distributed on the base of a cone with vertex Rk , axis in the direction

of Qk , lateral length a and aperture angle θ. This model is the basis for the other

polymer models that we will discuss.

The Kratky-Porod polymer model , or wormlike chain, introduced in [1], is a similar

model that can be seen as a continuous analog of the freely rotating chain. Let Q(s)

be a standard Brownian motion process on the unit sphere S 2 and let ( L, p) be

auxiliary parameters. We interpret the time parameter s of Q(s) as the arc length

distance from the xed end s = 0 of the polymer, and we let L > 0 be the totalcontour length of the polymer and p > 0 be its persistence length. Letting R(s) be

the position of the segment a distance of arc length s along the polymer, we have

R(s) = s

0Q( −1/ 2

p σ)dσ. (1.6)

Recall that the persistence length p is a length scale that represents the exponential

decay of tangent-tangent correlations; that is, if T (s) denotes the unit tangent vector

to the polymer at an arc length distance of s from the xed end, then

E [T (0) ·T (s)]∼= e−s/ p . (1.7)

A folklore result from the chemical physics literature shows how the freely rotating

chain model and Kratky-Porod model are related under an appropriate scaling; more

precisely, that the Kratky-Porod model is the continuous limit of the FRC.

Theorem 1.1. Let RN be the freely rotating chain with parameters (N,a,θ ), and let

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R(s) be the Kratky-Porod model with parameters (L, p). Set

a = LN

(1.8)

θ = κ√ N

(1.9)

p = 2L

κ2 (1.10)

and interpolate the FRC model by dening RN (s) for s∈[ka, (k + 1) a), as follows:

RN (s) =sa −k RN

k + k + 1 − sa

RN k+1 . (1.11)

Then, as N → ∞, RN converges in distribution to R.

This theorem is proven rigorously in [10], by means of introducing a matrix-valued

stochastic processes in which the unit tangent vector to the polymer, Q(s), is aug-

mented by including two other basis vectors to form an orthonormal frame, Z (s).

Then, the orthogonal matrix Z produced from the frame vectors can be identied

with a transformation that rotates the frame of canonical basis vectors to the new

orthonormal frame. This technique allows us to represent the spherical Brownian

motion Q(s) as a random rotation of the initial value

Q(s) = Z (s)Q(0) (1.12)

and serves to linearize the problem in the following sense. Rather than work in the

Riemannian geometry of the sphere, we operate instead in the Lie group of orthog-

onal 3

×3 matrices and the associated Lie algebra of anti-symmetric matrices. This

treatment is a major part of the mathematical theory behind this physical model.

Another theoretical result of [10] concerns the limiting behavior of the Kratky-Porod

model when the persistence length tends to zero or innity.

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Theorem 1.2. The Kratky-Porod model exhibits the hard rod/random coil transition

in the following sense:

1. As p →0, R(s) →0 in probability, and −1/ 2 p R(s) converges in distribution to

a standard 3-dimensional Brownian motion W (s).

2. As p → ∞, R(s) → se3 in probability, and p(R(s) −se3) → s

0 Q(σ) dσ in

distribution.

The main result of this thesis concerns the inclusion of a constant force eld to thefreely rotating and Kratky-Porod models. The force f is rst added to the discrete

model via a potential energy function:

U (QN ) = −αN

k=1

QN k ·f, (1.13)

in which α is a constant that represents the fact that the strength of the electric

dipole moment of each segment is constant as N → ∞. (To show this, we must let

α = N .) This energy is dened as a function of the segment vectors that comprise the

polymer. We wish to use this energy to dene a probability measure, the Boltzmann-

Gibbs distribution with potential U , for the torsional angles Φ N = φN 1 , . . . , φN

N −1.

This is possible because both Q N and ΦN can be written as deterministic functions

of each other so that knowledge of one set of variables determines the other set. With

this in mind, we can write

U (QN ) = U (QN (ΦN )) = U (ΦN ). (1.14)

Letting P N 0 (dΦ) represent the uniform product measure on the set [0 , 2π)N −1, which

is the distribution of the set of angles in the original FRC model, and letting τ = kB T

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be the thermal energy of the system, we dene the Gibbs measure for the torsional

angles of the forced model as follows:

P N f (dΦ) = exp( −U (ΦN )/τ )P N

0 (dΦ). (1.15)

As with the freely rotating chain, the forced chain converges in distribution to a

continuous model, Rf (s), which is a forced version of the Kratky-Porod model.

The forced Kratky-Porod model is described in terms of a stochastic differential

equation for the tangent process T (s), that is, the derivative with respect to arclength of Rf (s), and a set of parameters ( L, p, ζ, f ), where ζ is a dimensionless

number that measures the relative magnitude of the force, and f is the unit vector

in the direction of the force vector f . More precisely, if

dT (s) = 1

pdQ(s) +

ζ p

(I −T (s)⊗T (s)) f ds (1.16)

with initial condition

T (0) = e3 (a.s. ) (1.17)

then

R f (s) = s

0T (s)dσ. (1.18)

Here is the rst major result of this dissertation.

Theorem 1.3. Let RN f (s) be the forced chain polymer model with parameters (N,a,θ,f ),

where f = f f , whose torsional angles follow the Boltzmann-Gibbs distribution

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(1.15). Let Rf (s) be the forced Kratky-Porod model with parameters (L, p, ζ, f ) and

suppose the parameters satisfy

a = L

N (1.19)

θ = κ√ N

(1.20)

p = 2L

κ2 (1.21)

Let τ = kB T and set

ζ = L f

τ . (1.22)

If RN (s) is the forced chain interpolated as in (1.11), then, as N → ∞, RN f converges

in distribution to Rf , and the process of tangent vectors T (s) to the polymer is the

solution to the stochastic differential equation (1.16).

The tangent vector therefore is an Itˆ o process consisting of a diffusion term, the

scaled Brownian motion on S 2

, and a quadratic drift term, which is proportional tothe orthogonal projection of the unit force vector f onto the plane perpendicular to

T (s). This means that T is a spherical Brownian motion with a bias towards the

direction of the force. While this differential equation cannot be solved explicitly for

T (s), the behavior of the polymer can be described fully in several limiting cases.

If we x the persistence length p and consider extreme values of ζ , we expect to

see two radically different behaviors. If ζ

→ 0, the Brownian motion term dQ(s)

dominates equation (1.16), and the polymer behaves exactly as in the Kratky-Porod

model; while if ζ → ∞, the drift term dominates, and the polymer becomes a rigid

rod pointing in the direction of the force, f .

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The more interesting cases, however, occur when ζ and p are proportional and

ζ → ∞. In this case, the Brownian motion can be ignored, but the polymer is

stiff enough that it does not point completely in the direction of the force. Rather, it

forms a deterministic curve in the plane, whose shape changes with the constant of

proportionality between the two large parameters.

Here is the second main result of this dissertation.

Theorem 1.4. The forced Kratky-Porod model has the following behavior:

1. Fix p and let ζ → 0. Then the polymer Rf (s) converges to the Kratky-Porod

model R(s).

2. Fix p and let ζ → ∞. Then the polymer converges in probability to a rigid rod

in the direction of the force: Rf (s) →sf .

3. Let C = ζ / p be constant, and let ζ → ∞. Then, R f (s) converges in probability

to the curve

X (s) = s

0x(σ)dσ (1.23)

where x is the solution of the ordinary differential equation x = V (x), x(0) =

e3, where

V (x) = C (I −x⊗x) f . (1.24)

This equation can be solved explicitly, leading to a exact formula for X (s).

4. Let C = ζ / p be constant as above, and let ζ →0. Then, in probability, Rf (s)

shrinks to a point.

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In addition, the dissertation contains several other results that arise from the research

on the addition of an external force to the Kratky-Porod model. Our original moti-

vation for doing so was to investigate how a polymer with these physical properties

acted when stretched by optical tweezers, a typical method of experimentation on

macromolecules, and to produce force-extension diagrams for the polymer, demon-

strating how the distance it stretches varies with the magnitude of the force applied,

as well as the inherent stiffness of the polymer. We considered the physics of applying

a force to the free end of the polymer, but found this model to be intractable. There-fore, we modied it so that the external force was applied uniformly at all segments,

as described previously.

Yet, we were able to obtain some information which we originally sought about the

physics of the forceed chain, namely, some numerical results regarding the root-mean-

square (RMS) end-to-end polymer length R and radius of gyration Rg, and how they

vary with the parameters p and ζ . These include force-extension diagrams and show

how increasing the force stretches the polymer and pulls it to align in the direction

of the force vector f .

We also examined a theoretical approach to describing this behavior, and formulated

a conjecture to explain it. Since the RMS polymer length can be written as a double

integral of the tangent-tangent correlation:

R2 =

L

0

L

0E [T (s)

·T (t)] dsdt, (1.25)

we can derive a formula for R if we know the correlation function exactly. As this

sems unlikely, we may try to approximate the correlation function as an exponential

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if we know the decay of correlation, or persistence length p(f ) of the polymer, which

now depends on the external force f . To distinguish this quantity from the intrinsic

persistence length of the polymer chain p, let us denote it as c for correlation length.

Then, we could approximate the integrand as

E [T (s) ·T (t)] = e−|t−s|/ c (f ) (1.26)

and derive an approximate expression for the force-extension diagram.

To investigate the forced persistence length, we consider an eigenvalue approach: the

stochastic differential equation (1.16) for the tangent vector process T (s) is related

to an elliptic operator L acting on smooth functions on the sphere S 2.

L= 1

p∆ S 2 +

ζ p

V (1.27)

where

V (x) = ( I −x⊗x) f . (1.28)

We show that L is an unbounded, self-adjoint operator in the space L2(m), where

the measure

m(dx) = e−f ·x/τ dx/ Z (1.29)

is the invariant measure, or weight function, on the sphere for the diffusion T (s), and

Z is the partition function that makes m a probability measure. Considering the

eigenvalues of the operator L leads us to make the following conjecture:

If λ1(f ) is the principal (rst non-zero) eigenvalue of L, then the persistence length

is given by

p(f ) = λ1(f )−1 (1.30)

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which yields an approximate formula for R2 that holds in a more general situation

for ζ and p, for polymers that are very short ( L 1) or very long (L 1).

With these results having been stated, the dissertation is organized according to the

following structure.

Chapter 2 presents the original forceless models, both the discrete freely rotating

chain and the continuous worm-like chain, by constructing them from the segments

and then taking the scaling limit as the number N of segments goes to innity. An

element of time dependence is also introduced, as the torsional angles are taken tobe independent one-dimensional Brownian motions. The main results from [10] are

stated concerning the convergence of the freely rotating chain to the Kratky-Porod

model. We also review some classical results on the root-mean-square end-to-end

length R and radius of gyration Rg.

Chapter 3 considers the freely rotating chain with a point force acting on the free

end as a model for optical tweezer experiments. In the chapter, we propose to replace

this force with a uniform eld, with which we work for the remainder of the disser-

tation. Some partial results are discussed, such as a description of the lowest energy

conguration.

Chapter 4 is concerned with the construction of the forced rotating chain by means

of the Boltzmann-Gibbs distribution. Under this probability measure, the segments

in the discrete forced model form a Markov chain, and this allows us to compute

the driving process Bf (s) of the polymer by its Doob-Meyer decomposition. The

main result of this chapter is that the discrete model converges in distribution to a

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continuous model, and the tangent vector traces a spherical Brownian motion with a

drift proportional to the orthogonal projection of the force vector.

Chapter 5 deals with several limiting cases and proves the behavior of the forced

Kratky-Porod model in various regimes, such as the rigid rod and random coil limits.

The main result is that as both the force parameter and stiffness increase together,

the polymer converges in probability to a deterministic curve that can be described

explicitly.

Chapter 6 gives the results of a numerical Monte Carlo simulation of the forced poly-mer model and provides a surface plots and force-extension diagrams, illustrating how

the RMS length of the polymer and the radius of gyration vary with the dimensionless

force parameter ζ and the persistence length p. The varying values of these physical

characteristics of the polymer divide its behavior into the regimes mentioned above.

Chapter 7 strives to explain these numerical data in a manner consistent with the

theoretical results and provides evidence to support conjectures regarding the rela-

tionship between the physical properties of the polymer and the parameters. By

interpolation of the limiting cases as well as eigenvalue analysis, we show how the

correlation length of the forced polymer varies with the magnitude of the force, and

how the added force affects the mean-square end-to-end length.

Chapter 8, nally, discusses several directions for future research can go, by altering

some of the constraints on the model or formulating different forms of potential

energy. A comparison to treatments of semi-exible polymers in the chemical physics

literature is also included.

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CHAPTER 2

SUMMARY OF THE KRATKY-POROD MODEL

The focus of this dissertation is to present a rigorous mathematical explanation of

the force-driven Kratky-Porod polymer model, or worm-like chain (WLC). However,

in order to understand this model fully, it is necessary to consider rst the model

with no external force, in which the only forces acting on the polymer are those

which constitute the Gaussian thermal noise due to interactions between the polymer

molecule and the smaller molecules comprising the solution. Kratky and Porod were

the rst to describe the WLC for macromolecules when they examined the X-ray

scattering of cellulose in a colloidal suspension, in [1], and this approach has since

been applied to other semi-exible polymers, such as actin [11], polyelectrolytes [12],

[13], [14], viruses [15], [16], [17], and DNA [18], [19]. A mathematical analysis of this

model is the basis of my doctoral work, and a full treatment of the problem appears in

[10]. The following is a summary of the work, so that the reader may gain familiarity

with the problem, and more importantly, with the notation used in the description

of the model with an added force.

We will begin by constructing the discrete Kramers chain from a nite set of N segments of equal length, which always meet at the same bond angle, but may rotate

freely about each previous segment. To model the free rotation, we let each angle

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of rotation be uniformly distributed on the circle [0 , 2π), and derive a sequence of

rotation matrices from these angles to show the orientation of the segments. When

we introduce a time dependence to the model, we let each angle follow the law of a

Brownian motion with uniform stationary distribution.

Next, in order to prove convergence of the discrete model to the continuous model,

we introduce a pair of matrix-valued stochastic processes that describe the model:

a driving process , BN , which is a weighted sum of innitesimal rotations, and an

orthonormal frame process Z N

, which shows the orientation of the orthonormal framealong the polymer. The matrix Z N is, in effect, an augmentation of the tangent vector

by two orthonormal vectors. By scaling the parameters of bond length and bond angle

and letting the number of segments go to innity while keeping the overall length L

and persistence length p constant, we show that the limiting processes B and Z

describe the position of the polymer in the WLC model.

This is the major theorem of [10], but other results are included here. For example,

the limiting two-parameter process B is a matrix containing entries that are Ornstein-

Uhlenbeck sheets, and the polymer behaves as a rigid rod or shrinks to a point in the

extreme cases of p going to innity or zero, respectively. Our analysis also includes

computations of the root-mean-square end-to-end length R of the polymer, and the

radius of gyration Rg. We will compare these values to numerically calculated results

from the forced Kramers chain.

In this chapter we describe, without a full proof, some of the major results of [10].

The purpose of this part of the dissertation is to familiarize the reader with the main

ideas and basic arguments of the paper which will be used in later chapters. Nota

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bene : The proofs in Chapters 4 and 5 are self-contained and do not depend on the

results described here.

2.1 Construction of the Freely Rotating Chain

Consider a chained polymer made up of N segments, each of length a, with each pair

of consecutive segments meeting at the same planar angle. For the sake of notational

simplicity, denote by θ the supplement of the bond angle, so that the consecutive

segments have an angle of π −θ between them. Denote the vectors comprising the

segments of the polymer, by Q1, Q2, . . . , Q N , and let the position of each bead on the

polymer, or bond between segments, be denoted by R0, R1, . . . , R N . Note that these

segments are not actual monomers, but collections of monomer molecules that act

together as a statistical unit, similar to Kuhn segments [18], [20].

Without loss of generality, we can specify the position and orientation of the polymer

in R 3 by choosing our coordinates appropriately. Fix one end of the polymer (thezeroth bead or xed end) at the origin, so that R0 = 0. It then follows that

Rk =k

j =1

Q j . (2.1)

Next, orient the polymer so that the rst segment always points straight up, in the

positive z -direction. That is,

R1 = Q1 = ae3. (2.2)This pinning down of one end of the polymer serves to specify boundary conditions

for the model. Additionally, since the rst segment is xed, the remaining segments

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Figure 2.1: Construction of the discrete polymer from its segments Qk

can be described by a recurrence relation; since the bond angle is constant, the k +1 st

segment can be expressed as the rotation, by a xed angle θ, of the kth segment. The

construction of the segments can be seen in gure 2.1.

However, the axis of the rotation has not yet been specied. Therefore, it is necessary

to take into account the torsional angles. We assume, in this model, that each segment

of the polymer is free to rotate about each previous one, and that all possible torsional

angles in [0, 2π) are accessible. (Other models, such as the rotational isomeric state,

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only allow three values for the angle [8].) This occurs when the polymer is placed in

a dilute solution, so that the constraints on which states are accessible vanish [20].

Each segment of the polymer, after the rst, requires a torsional angle to specify

its exact position. This is because if the segment Qk+1 is expressed in spherical

coordinates relative to the previous segment Qk , the radial coordinate (the bond

length a) and the polar angle (the supplement to the bond angle, θ) are already

known, so only the azimuthal, or torsional, angle remains. Denote the torsional angle

relative to segment Qk as φk ; since the rst segment Q1 requires no torsional angle tospecify its coordinates, it follows that the angle φk is used to give the position of Qk+1

for each of the N −1 torsional angles in the polymer, that is for k = 1, . . . , N −1.

For the freely rotating chain, let all the torsional angles be independent uniformly

distributed random variables on the circle, [0 , 2π).

Once the torsional angles are known, it is possible to determine how one segment is

rotated to form the next. Dene Θ k to be the transformation that maps Qk to Qk+1 ,

for each k = 1, . . . , N −1. The transformation Θ k is a rotation through an angle of

θ, so that the proper bond angle is obtained. Then, dene Z k to be the composition

of the rst k rotations, so that

Qk+1 = Θ kQk = Θ k · · ·Θ1Q1 = Z kQ1. (2.3)

However, each Θk is a rotation about a different axis, and thus the kth transfor-

mation depends on each of the rst k torsional angles. There is a way around this

complication, though, so that the each individual transformation depends only on one

torsional angle at a time. Since Z k = Θ kZ k−1, dene a transformation (a rotation)

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H k conjugate to Θ k :

H k = Z −1k−1ΘkZ k−1. (2.4)

That is, Z k = Z k−1H k , and the transformation Z k taking Q1 to Qk+1 can be built

from the H ’s in reverse order.

Notice that

H kQ1 = Z −1k−1ΘkZ k−1Q1 = Z −1

k−1Z kQ1 = Z −1k−1Qk+1 . (2.5)

Thus, while Θ k is a rotation acting on Qk , H k is a rotation acting on Q1, for any k.

In fact, if the previous rotations H 1, . . . , H k−1 are all known, it is possible to consider

Z k−1H k as a rotation taking Q1 to Qk+1 . Since each Θk is a rotation by the angle

θ, and H k is conjugate to Θ k by means of a composition Z k−1 of rotations, it follows

that each H k is also a rotation through an angle θ, only with respect to a different

basis from Θk . But all of the H k’s are rotations acting on the same vector, Q1, by

the same angle; only the angle of orientation, or the torsional angle, is different.

Since the torsional angles φ1, . . . , φN −1 are i.i.d., it follows that the transformationsH 1, . . . , H k−1 are all i.i.d. as well. This yields a great advantage in calculating the

expected behavior of the polymer, especially when the continuous limit is taken. In

particular, it becomes possible to use the Law of Large Numbers and Central Limit

Theorem.

Figure 2.1 shows how the i.i.d. rotation H k relates to the rotation Θ k mapping one

segment to the next.

We can now dene the rotations H k for k = 1, . . . , N in matrix form, so that each

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Figure 2.2: Commutative diagram for H n−1 and Θn−1

H k depends only on the single torsional angle φk . Recall that φk is a uniformly

distributed angle in the interval [0 , 2π). Dene a unit vector vk in the xy-plane by

vk = (cos φk , sin φk , 0) (2.6)

so that vk is uniformly distributed on the unit circle. This unit vector becomes an

axis of rotation; dene bk to be the innitesimal rotation about ˆ vk :

bk =

0 0 sinφk

0 0 −cos φk

−sin φk cosφk 0

. (2.7)

Subsequently, express H k in matrix form to be the rotation about ˆ vk by the angle θ:

H k = exp( θbk). (2.8)

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By expanding the exponential of the matrix into a power series, we nd that

H k =

1 −(1 −cos θ)sin2 φk (1 −cos θ)sin φk cosφk sin θ sin φk

(1 −cos θ)sin φk cosφk 1−(1 −cos θ)cos2 φk −sin θ cos φk

−sin θ sin φk sin θ cos φk cosθ

(2.9)

and thus

H kQ1 = a(sin θ sin φk , −sin θ cos φk , cos θ), (2.10)

which is a uniformly distributed angle on the circle whose polar angle, in spherical

coordinates, is equal to θ. Therefore, any vector which differs from Q1 by an angle of

θ is equally likely to be chosen as the vector H kQ1, which is the result of the rotation.

By changing the basis with respect to the rotation Z k−1, that is, conjugating the

matrix H k by Z k−1, we see that the vector Qk+1 is chosen uniformly at random from

the circle of points which differ from Qk by an angle of θ. The construction of the

freely rotating polymer chain is now in place.

We have expressed the matrix Z k as a composition of rotations, or equivalently, aproduct of matrices. Dene also Z 0 = I , the empty product. This product Z is

therefore the stochastic process that determines the position of the polymer; if the

angles φ1, . . . , φn−1 are all known, then the segment Qn can be calculated as

Qn = Z n−1Q1 = aH 1H 2 · · ·H n−1e3. (2.11)

Observe that the third column of Z n−1, that is, Z n−1e3, is the unit vector in the

direction of Qn . In fact, we will show later that all three columns of Z are unit

vectors which determine an orthonormal frame, namely the tangent vector and its

normal development, as the polymer is traversed.

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One more important observation follows: the innitesimal rotations bk are anti-

symmetric matrices, while the full rotations H k , and thus their product Z k , are

orthogonal matrices, so that bT k = −bk and H T

k = H −1k . This will simplify many

of our calculations later on.

The innitesimal rotation matrices, which are independent and identically distributed,

can also form a process, which we will call the driving process of the polymer (because

it will appear in the stochastic differential equation for Z ). Dene

B N n = 1√ N

n

k=1

bk(φk) (2.12)

and for consistency, for a polymer with N segments and n < N , dene

Z N n = H 1(φ1)H 2(φ2) · · ·H n (φn ). (2.13)

This two processes, which are discrete, will become the basis for the continuous

processes that describe the motion of orthogonal frames along the polymer, and they

are instrumental in proving the convergence of the discrete model to the continuous

model. But rst, let us turn our attention to the dynamics of the discrete model.

So far, the freely rotating polymer has been dened so that the N −1 torsional angles

are chosen once and for all, independently and uniformly at random, and a unique

conguration of the polymer is constructed from the values of the angles. Now, let

us add a time parameter to the model, so that the polymer can move, while still

maintaining the properties of a xed bond length and a xed bond angle.

Let t ∈ [0, ∞) be the time parameter, and (Ω , F , P ) be a probability space. Dene,

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for each k = 1, . . . , N −1, a random variable φk : [0, ∞) ×Ω →R dened by

φk(0, ω) = U ([0, 2π)) (2.14)

φk(t, ω) = φk(0, ω) + N (0, t). (2.15)

That is, each φk is a Brownian motion on the real line (modulo 2 π) in the time

variable t, with a uniform initial distribution. Moreover, it follows that the unit

vector vk dened as in (2.6) is a Brownian motion on the unit circle in the xy-plane,

and that the product of the rotation matrix, H kQ1, is a Brownian motion on the circleof all possible positions of that vector, that is, the circle with center (0 , 0, a cos θ) and

radius a sin θ.

If all the torsional angles up to a certain point are known, we can speak of the

distribution of the next angle, and thus of the next segment. Let

F k = σ(φ1, . . . , φk), (2.16)

that is, the σ-eld generated by the rst k torsional angles. Then, the conditional

expectation satises

E [Qk+1 |F k] = Qk cosθ (2.17)

and the distribution of Qk+1 , given F k , is that of a Brownian motion on the circle of

all points that are a distance a from the endpoint Rk of the previous segment of the

polymer, so that Qk

·Qk+1 = a2 cos θ, preserving the common bond angle.

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The driving process BN n and the rotation process Z N

n can now be redened so that

the time parameter is included:

B N n (t) =

1√ N

n

k=1

bk(φk(t)) (2.18)

Z N n (t) = H 1(φ1(t)) · · ·H n (φn (t)) . (2.19)

The most important observation about the moving discrete model is that the Brown-

ian motions φk(t) are all independent. That means that each segment of the polymer

is free to rotate about the axis dened by each previous segment, independently of

the motions of all other segments. Notice also that the processes are continuous in

time, since they are based on the continuous Brownian motions, but each Z N n and B N

n

is dened discretely with respect to the index of the segment of the polymer. As the

number of segments increases toward innity, these processes will approach a pair of

continuous processes Z and B , but they can no longer be indexed by the number of

the segment. We must therefore redene the position along the polymer in terms of

a continuous parameter, the arc length, rather than an integer index.

In order to scale the polymer model, so that the discrete freely rotating chain can

converge to a continuous polymer, it is best to consider the polymer at rest, that is,

when the torsional angles are stationary uniform distributions. We wish to let the

number of segments of the polymer increase to innity while keeping the length of the

polymer constant, so that the length of each segment goes to zero. Likewise, we wish

to decrease the supplementary bond angle towards zero, while keeping the curvature,

or stiffness, of the polymer constant as well. Let us therefore dene new parameters

which are intrinsic to the polymer model, regardless of the number of bonds:

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N , the number of segments, remains as before. We wish to take the limit of the

processes as N → ∞.

L is the total length of the polymer, and the bond length is given by a = L/N .

κ is a dimensionless number, called the curvature parameter , which is a measure of

the stiffness of the polymer. A value of κ = 0 means that the polymer is a

straight, rigid rod, while a very large value of κ means that the polymer is free

to coil in any direction. This parameter is related to the supplementary bond

angle: θ = κ/ √ N .

p is an intrinsic property of the polymer, called the persistence length , and it has

dimensions of length. It measures the decay of correlations between segments of

the polymer, so that two segments a distance of n segments apart have covari-

ance exp(−Cn/ p) where C is a constant of proportionality. The persistence

length p

is related to the other parameters L and κ by the following formula:

p = 2Lκ2 . (2.20)

Thus, p is a measure of the stiffness of the polymer, as opposed to κ, which

measures curvature. The regime in which p 1 corresponds to a exible

polymer, while if p 1, the polymer is a rigid rod. While our original for-

mulation of the problem used κ chiey, most of the chemical physics literature

speaks of p as an intrinsic parameter instead, as a measure of the stiffness of

the polymer at the microscopic level [21], [22]. Our treatment will follow this

convention and use p, except when it is more convenient to use κ, such as in

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the differential equations governing the processes Z and B. Notice also that

our use of κ differs from other sources, in that it represents curvature rather

than stiffness; for example, Chaudhuri [22] uses κ for (d−1) p/ 2 where d is the

number of dimensions, and the symbol t = L/ p for κ2/ 2 in our notation.

Since the number of segments of the polymer is now increasing toward innity, it

is no longer useful to index the processes BN and Z N by the number of segments

along which the polymer has been traversed. Rather, consider them as functions

of a continuous parameter, the arc length s. Let s ∈ [0, L] be such that the value

s = 0 corresponds to the end of the polymer that is xed at the origin, and s = L

corresponds to the other end, or the free end. Then, for a xed N (and a), the values

of s in the interval [( k −1)a,ka ] correspond to the segment Qk of the polymer, for

any k = 1, . . . , N . The processes BN and Z N , therefore, expressed as functions of

s, are locally constant, with jumps at each multiple of a, so that the processes are

continuous from the right with limits from the left, or cadlag . In short, the matricesB N and Z N , expressed as functions of the continuous parameter s, have the forms

B N (s) = 1√ N

[Ns/L ]

k=1

bk(φk) (2.21)

Z N (s) =[Ns/L ]

k=1

H k(φk) (2.22)

where in the product in (2.22), the rotation matrices H k are arranged in increasing

order, from left to right, and [ Ns/L ] is the number of segments required to reach an

arc length of s.

This covers the conversion of the notation from indexing the processes by an integer

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to dening them in terms of a continuous parameter. However, the processes are

still discrete, and the polymer that they describe is still a discrete chain with a nite

number of segments. Yet, it is now possible to keep the total length and stiffness

constant while letting the number of segments go to innity, thereby turning the

polymer into a continuous worm-like chain, equivalent to the Kratky-Porod model.

Thus, rather than describing the position of the polymer in terms of rotated segments,

we can give use the matrix-valued processes to do so, thus transferring the problem

from the regime of vector spaces to the Lie group O(3) of rotations, of which Z N

isa member, and the corresponding Lie algebra o(3), in which we nd BN .

Dene the unit tangent vector to the polymer T N (s) as

T N (s) = Q[Ns/L ]+1 /a = Z N [Ns/L ](Q1/a ) = Z N (s)e3 (2.23)

that is, as the third column of the matrix Z N (s), and since it is a unit vector, its

length is invariant as the scaling limit is taken.

From the tangent vector at a given arc length along the polymer, we can now compute

the position of the polymer at that same value of s. Recall that in the discrete case,

Rn =n

k=1

Qk , (2.24)

so if we dene RN (s) = R[Ns/L ], the above equation becomes

RN (s) =[Ns/L ]−1

k=0

Z N k ae3, (2.25)

or, alternatively, since Z N is a locally constant function of s,

RN (s) = s

0Z N (s)e3 ds, (2.26)

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so, naturally, the position is the integral of the tangent vector along the polymer

curve.

2.2 Convergence to the Kratky-Porod Model

We are now ready to take the limit of the processes BN , Z N , and RN as N → ∞.Dene ∀s∈[0, L],

B(s) = limN →∞

1

√ N

[Ns/L ]

k=1

bk . (2.27)

Thus, B(s) is a pointwise limit, although this denition does not indicate the mode of

convergence. However, since the process B(s) is continuous in s, the sequence ( B N )

converges weakly to B. However, we cannot the limits Z and R of the orthonormal

frame processes and the position vectors, respectively, as pointwise limits and keep

their relationships to B intact. In order to do so, we must show that BN and Z N

satisfy a difference equation, and that the limiting processes B and Z follow the

limiting differential equation. The key to proving weak convergence of the Z N is a

theorem of Kurtz and Protter [23]. Then, weak convergence of the RN , which are

taken from the integrals of the Z N , will follow from the limit theorems of measure

theory.

In the discrete model, the processes Z N and BN are closely related; the former is a

product of matrices which are the matrix exponentials of those innitesimal rotations

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whose sum is the latter. This can be expressed as a difference equation:

Z N n +1 −Z N

n = H 1 · · ·H n H n +1 −H 1 · · ·H n (2.28)

= H 1 · · ·H n (H n +1 −I ) (2.29)

= Z N n exp

κ√ N

bn +1 −I (2.30)

= Z N n

κ√ N

bn +1 + κ2

2N b2

n +1 + r(N ) (2.31)

= Z N n κ(B N

n +1 −B N n ) +

κ2

2 (B N

n +1 −B N n )2 + Z N

n r (N ) (2.32)

where r(N ) is the remainder term from the power series expansion:

r (N ) =∞

j =3

κ j

j !(B N

n +1 −B N n ) j (2.33)

and each term is the increment of the cubic (or higher order) variation in the process

B N . However, the quadratic variation of BN is nite, so when the increments are

summed and N → ∞, the remainder term r(N ) → 0. This is because r(N ) is the

increment of the cubic variation, and since it is of order O(N −3/ 2

), the sum of N suchincrements will converge to 0 in the limit.

Since the processes Z N and BN are locally constant with jumps at each multiple of

∆ s = L/N , the difference equation (2.32) is equivalent to a differential equation:

dZ N (s) = Z N (s) κdB N (s) + κ2

2 d[B N , B N ]s + Z N (s)r (N ) (2.34)

where [B N , B N ]s is the nite quadratic variation of the process BN in s. Since the

cubic variation of BN is zero, the right side of (2.34) is precisely the Stratonovich

differential (plus the remainder):

dZ N (s) = κZ N (s)∂B N (s) + Z N (s)r (N ). (2.35)

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The question remains: does the differential equation (2.35) hold if the discrete pro-

cesses Z N and BN are replaced by Z and B ? That is, does

dZ (s) = κZ (s)∂B (s) (2.36)

hold when N → ∞?In fact, it does, by the theorem of Kurtz and Protter [23] on the convergence of

stochastic differential equations. The notation in the original has been modied to

suit this situation, and some conditions of the original theorem, which are trivial in

this context since the integrand is the identity function and the initial condition I is

constant, have been omitted.

Theorem 2.1. [Kurtz and Protter, 1995] Suppose that (B N , Z N ) satisfy

Z N (s) = I + κ s

0Z N (s−) ∂B N (s), (2.37)

that (B N , Z N ) are relatively compact in the Skorohod topology, and that B N ⇒B and

that (B N )n≥1 is good. Then any limit point of the sequence (Z N )n≥1 satises

Z (s) = I + κ s

0Z (s) ∂B (s). (2.38)

In the same paper by Kurtz and Protter, a sequence of semimartingales is said to be

good if and only if the processes have uniformly controlled variation (UCV); that is,

if B N (s) has the Doob-Meyer decomposition

B N (s) = M N (s) + AN (s) (2.39)

for M N a martingale and AN an adapted process, then

supN

E N [M N , M N ]s + E N s

0dAN (s) < ∞. (2.40)

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In this case, the driving process BN for any N is a martingale (since its increments

have mean 0), whose quadratic variation is bounded, so the sequence is UCV, and

thus good. Likewise, by the denition of B in (2.27), the processes BN converge

weakly to B. The only condition remaining is the relative compactness of the pair

of sequences (Z N , B N ) in the Skorohod topology. This is proven with a theorem

by Ethier and Kurtz (1986) that gives necessary and sufficient conditions for relative

compactness in the Skorohod topology [24]. Essentially, the rst condition for relative

compactness is uniform boundedness (also called compact containment), and thesecond is equicontinuity in the Skorohod topology, both in a stochastic sense. In [10],

we show that these conditions hold for B N by comparing it to the limiting Gaussian

process and obtaining stochastic bounds, and for Z N by breaking it into martingale

and adapted parts, that is, the Doob-Meyer decomposition. This technique will be

employed extensively in later chapters.

Thus Theorem 2.1 applies to the processes BN (s) and Z N (s), and equation (2.38)

holds for the limiting processes Z (s) and B(s). Likewise, R(s), the stochastic integral

of Z (s)e3, is the limit of RN (s), so that the continuous process gives the position of

a point, corresponding to the given arc length, along the continuous polymer.

This yields the rst major result of [10], the convergence of the Kramers chain to the

Kratky-Porod model:

Theorem 2.2. Let

RN be the freely rotating chain with parameters (N,a,θ ), and let

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R(s) be the Kratky-Porod model with parameters (L, p). Set

a = L

N (2.41)

θ = κ√ N

(2.42)

p = 2L

κ2 (2.43)

and, interpolate the FRC model by dening RN (s) for s∈[ka, (k + 1) a), as follows:

RN

(s) =sa −k R

N k + k + 1 −

sa R

N k+1 . (2.44)

Then, as N → ∞, RN converges in distribution to R.

The linear interpolation of RN (s) comes naturally, since the matrix processes B N and

Z N are locally constant. This gives us a description of the Kratky-Porod model in

terms of stochastic processes: B, Z , and R. This is a novel approach to the model;

several sources in condensed matter physics, such as [25], dene the model in terms

of energy by means of the Hamiltonian.

In the continuous wormlike chain, the stochastic differential equation (2.36) can also

be written in terms of the vectors that comprise the orthonormal frame, which is

represented by the matrix Z . Recall that the tangent vector process is given by T (s) =

Z (s)e3. Denote the rst two columns of Z (s) as M 1(s), and M 2(s), respectively. Also,

since B is an anti-symmetric matrix, label its non-zero entries as such:

B(s) =

0 0 β 1(s)

0 0 β 2(s)

−β 1(s) −β 2(s) 0

(2.45)

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Then, expanding the equation (2.36) in terms of the columns of the matrices yields

the following system of equations:

dT (s) = κdβ 1(s)M 1(s) + κdβ 2(s)M 2(s) (2.46)

dM 1(s) = −κdβ 1(s)T (s) (2.47)

dM 2(s) = −κdβ 2(s)T (s) (2.48)

The above equations describe the movement of the rotating Bishop frame (see [26]) as

the polymer is traversed, and how this orthonormal frame varies with the arc length

parameter s.

We close this section with one of the most important results in [10]: that the process

of tangent vectors along the polymer, with respect to the parameter of arc length, is

a spherical Brownian motion. To show this, remember that a Brownian motion on

any manifold is generated by the Laplace-Beltrami operator for that manifold; for

the sphere S 2, this operator is the Laplacian in spherical coordinates when the radial

coordinate is xed at 1.

The proof in [10] is done as follows: We rst let Φ : R 3 → R be a C2 function, and

we differentiate Φ by the Itˆo formula:

dΦ(T (s)) = ∇Φ(T (s)) ·dT (s) + 12

dT (s)T H Φ(T (s))dT (s). (2.49)

This formula can be evaluated with the fact that the driving process B(s) is Gaussian

(by the Central Limit Theorem) and is a martingale. Since the increments of B have

mean zero, so do the increments of T , and the rst term on the right side of (2.49) has

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mean zero. We can calculate the second term using the system of equations (2.46),

which says that

dT (s)T H ΦdT (s) = κ2 ds

2 (ΦM 1 ,M 1 + ΦM 2 ,M 2 ) (2.50)

when the Hessian of Φ is taken in terms of directional derivatives along the or-

thonormal Bishop frame, ( M 1|M 2|T ). This implies that the process T (s) satises the

conditions of Levy’s theorem, which states that a martingale whose generator is the

Laplace-Beltrami operator for a manifold is exactly the Brownian motion on that

manifold. Therefore, the process T (s) is a Brownian motion on the unit sphere S 2.

If we let Q(s) represent the spherical Brownian motion normalized so that

E [ Q(s + ∆ s) −Q(s) 2] = ∆ s, then to assure that T (s) has the proper variance as

shown in (2.50), the tangent process is scaled so that

T (s) = Q κ√ 2L

s (2.51)

or, in terms of the persistence length,

T (s) = Q( −1/ 2 p s). (2.52)

The analogous result for the Kratky-Porod model with an external force is one of the

main theorems of this dissertation.

2.3 The Time-dependent Kratky-Porod Model

Now that we have dened the continuous polymer in terms of arc length and shown

results about its behavior, we can re-introduce the time parameter and consider how

the behavior of the continuous polymer changes with time. Recall that we introduced

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the time parameter t as the independent variable of the torsional angles, each of

which is an independent Brownian motion on the real line, with a uniform initial

distribution. In fact, at t = 0, the time-dependent model is exactly the same as the

static case. We aim to determine how these discrete random variables, which depend

on time, converge to a continuous dynamic model.

From equations (2.18) and (2.19), we know that since the torsional angles φk deter-

mine the innitesimal rotations bk (and in turn the full rotations H k), we can use the

Central Limit Theorem to evaluate the limiting process B (s, t ), at a xed arc lengths, at any given time t. Thus

B(s, t ) = limN →∞

1√ N

[Ns/L ]

k=1

bk(t) (2.53)

Now, the innitesimal rotations are all independent and identically distributed; their

non-zero entries, sin φk(t) and cos φk(t) describe Brownian motion on the circle. For

any xed s

∈ [0, L], the number of terms in the sum in (2.53) increases to innity;

thus the sum in the same equation can be evaluated via the Central Limit Theorem.

Thus, it follows that the process B(s, t ) is Gaussian, and we can know its distribution

completely by knowing its mean and variance. This can be done by examining the

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mean and variance of a single innitesimal rotation.

E [bk(t)] =

0 0 E [sin φk(t)]

0 0 −E [cosφk(t)]

−E [sinφk(t)] E [cosφk(t)] 0

(2.54)

E [sinφk(t)] = E [sin φk(0)] = 0 (2.55)

E [cosφk(t) = E [cosφk(0)] = 0 (2.56)

E [bk(t)2] =−E [sin2 φk(t)] E [sinφk(t)cos φk(t)] 0

E [sin φk(t)cos φk(t)] −E [cos2 φk(t)] 0

0 0 −1

(2.57)

E [sin2 φk(t)] = E [sin2 φk(0)] = 12

(2.58)

E [cos2 φk(t)] = E [cos2 φk(0)] = 12

(2.59)

E [sinφk(t)cos φk(t)] = E [sin φk(0) cos φk(0)] = 0 (2.60)

E [bk(t)] =0 0 00 0 0

0 0 0

(2.61)

Var[ bk(t)] =−1

2 0 0

0 −12 0

0 0 −1

(2.62)

Thus, the mean of each innitesimal rotation is the zero matrix, and the variance isa diagonal matrix. We will call this matrix D throughout the dissertation.

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It follows that the mean of B N (s, t ) and (since the constituent matrices are indepen-

dent) the variance both satisfy

E [B N (s, t )] = 0 (2.63)

E [B N (s, t )2] = 1N

E [Ns/L ]

j =1

b j (t)[Ns/L ]

k=1

bk(t) (2.64)

= 1N

[Ns/L ]

j =1

[Ns/L ]

k=1

E [b j (t)bk(t)] (2.65)

= 1N

[Ns/L ]

j =1

[Ns/L ]

k=1

E [b j (t)]E [bk(t)] (2.66)

= 1N

[Ns/L ]

k=1

E [bk(t)2] (2.67)

= [Ns/L ]

N D (2.68)

By the Bounded Convergence Theorem, we have the following expressions for the

mean and variance of the limiting process, which is Gaussian:

E [B(s, t )] = 0 (2.69)

E [B(s, t )2] = s

LD (2.70)

and we now have a complete description of the limiting process B(s, t ) for any s.

In fact, as we will show soon, by considering the covariation, this driving process is

an Ornstein-Uhlenbeck sheet. But rst, we must examine the process of rotations

Z (s, t ),which gives the formula for the tangent vector (and thus the position) of the

polymer, at an arc length of s and time t. We could simply dene Z to be the limit of

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the Z N , but, it would be best to dene it as it relates to B , as we did in the previous

section.

Thus, we must rst derive a stochastic differential equation involving Z N and BN ,

showing how they relate with respect to a small change in time, then pass to the

limit, using the same Theorem 2.1 from Kurtz and Protter.

We begin the proof, presented in [10], by nding the change in Z N (s, t ) with respect

to time. Since for each N , Z N is continuous, we can simply differentiate the process,

in the manner of Stratonovich, so that we can use the ordinary rules of calculus:∂ ∂t

Z N (s, t ) = ∂ ∂t

[H 1(t)H 2(t) · · ·H [Ns/L ](t)] (2.71)

=[Ns/L ]

k=1

H 1(t) · · ·H k−1(t) ∂ ∂t

H k(t)H k+1 (t) · · ·H [Ns/L ](t) (2.72)

=[Ns/L ]

k=1

Z N (k −1)LN

, t ∂ ∂t

H k(t)Z N kLN

, t−1

Z N (s, t ) (2.73)

while

∂ ∂t

H k(t) = ∂ ∂t

exp κ√ N

bk(t) (2.74)

= κ√ N

∂ ∂t

bk(t) + κ2

2N bk(t),

∂ ∂t

bk(t) + ∂ ∂t

r (N ) H k(t). (2.75)

This is complicated by the fact that bk(t) does not commute with its time derivative.

However, we can notice that the rst term in (2.75) is related to the mixed partial

derivative of B N with respect to both s and t, and the second term (the commutator)

goes to zero when summed, by the Law of Large Numbers. This is proven in more

detail in [10].

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For simplicity of notation, we can move the differentiation into the arguments of B

and Z , for example;d∂

ds∂tB N (s, t ) = B N (ds,∂t ). (2.76)

Then, as N → ∞, the sequence ( Z N ) converges to the limit of the following stochastic

integro-differential equation:

Z (s,∂t ) = s

0κZ (σ, t )B(dσ,∂t )Z (σ, t )−1 Z (s, t ) (2.77)

This is a result of xing s and applying the Kurtz-Protter theorem (2.1) to the process

as it varies in t. We have now derived a pair of stochastic partial differential equations

that the process Z (s, t ), which determines the orthonormal frame of vectors, satises.

The question remains, however: what is the nature of the scalar processes β 1 and β 2,

which are components of the driving process B (s, t )?

Theorem 2.3. The processes β 1(s, t ) and β 2(s, t ) are independent Ornstein-Uhlenbeck

sheets; that is, they are Gaussian and satisfy

E [β i(s, t )β j (s , t )] = δ ijs∧s

2L e|t−t |/ 2. (2.78)

In [10], we prove this by rst dening the discrete versions of the component processes:

β N 1 (s, t ) =

1√ N

[Ns/L ]

k=1

sin φk(t) (2.79)

β N 2 (s, t ) = − 1√ N

[Ns/L ]

k=1cos φk(t) (2.80)

and calculating the means and variances of each increment. We rst nd that, by

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the independence of both the Brownian motions φk(t) and the increments of each

Brownian motion,

E [sinφ j (t)cos φk(t )] = 0 ∀ j, k, t, t . (2.81)

By summing these increments, we nd that

E [β N 1 (s, t )β N

2 (s , t )] = 0, (2.82)

so the two component processes are uncorrelated. We then nd the covariance of each

component process β i(s, t ) with itself by examining the increments, starting with β 1.

By independence of the Brownian motions, only the variance of each bk contributes

to the covariance of β 1:

E [β N 1 (s, t )β N

1 (s , t )] = 1N

[Ns/L ]∧[Ns /L ]

k=1

E [sin φk(t)sin φk(t )]. (2.83)

We must now calculate the covariance of the sine (or cosine) of a single torsional

angle in time. In order to do this, re-write the term in the sum in the above equationas a sum of independent increments, using the angle addition formula:

sin φk(t)sin φk(t ) = 1

2(cos(φk(t) −φk(t )) −cos(φk(t) + φk(t ))) (2.84)

= 1

2(cos(φk(t) −φk(t )) −cos(2φk(t ) + φk(t) −φk(t )))(2.85)

= 1

2 (cos(φk(t) −φk(t )) −cos(2φk(t )) cos(φk(t) −φk(t ))

+ sin(2 φk(t )) sin( φk(t)

−φk(t ))) . (2.86)

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Without loss of generality, assume that t > t ; then φk(t) −φk(t ) is equal in distri-

bution to φk(t −t ), which is normal with mean 0 and variance t −t . It follows that

every term but the rst on the right side of (2.91) has mean 0, and

E [sin φk(t)sin φk(t )] = 12

E [cosφk(t −t )]. (2.87)

To calculate this, we can expand the right side of the above equation in a power

series: Let

ξ = φk(t

−t ). (2.88)

Then ξ is normal with mean 0 and variance t −t . It follows that

E [cosξ ] =∞

j =0

(−1) j

(2 j )! E [ξ 2 j ] (2.89)

=∞

j =0

(−1) j

(2 j )! (2 j −1)!!(t −t ) j (2.90)

=∞

j =0

(−1) j

(2 j )!!(t −t ) j (2.91)

= ∞ j =0

(−1) j

2 j j ! (t −t ) j (2.92)

= e−(t−t ) / 2. (2.93)

Therefore,

E [β N 1 (s, t )β N

1 (s , t )] = 1N

[Ns/L ]∧[Ns /L ]

k=1

12

e−(t−t )/ 2 (2.94)

= [Ns/L ]

∧

[Ns /L ]

2N e−|t−t |/ 2, (2.95)

and by the Dominated Convergence Theorem,

E [β 1(s, t )β 1(s , t )] = s∧s

2L e−|t−t |/ 2, (2.96)

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as desired. By a similar argument,

E [cosφk(t)cos φk(t )] = 12

E [cosφk(t −t )], (2.97)

and equation (2.96) also holds if β 1 is replaced by β 2.

This completes the construction of the dynamic Kratky-Porod model of a semi-exible

polymer and the description of the two-parameter processes Z (s, t ) and B(s, t ). Next,

we will investigate some of the physical properties of this model, as well as its behavior

if the stiffness (or curvature) parameter, κ, is very small or very large, or respectively,if the persistence length p is very long or very short, rst for the static case, then

for the dynamic case.

2.4 Physical Properties and Limiting Behavior

When studying the characteristics of a polymer, it is necessary to take into an account

the physical quantities which measure the behavior of the polymer as a whole. Twoof these quantities are the root-mean-square end-to-end distance, R, and the radius

of gyration, Rg. The rst quantity measures the average length of the polymer, as

the square root of the average Euclidean distance between the endpoints, and is used

to model how the molecule moves laterally, while the second measures the effective

radius for rotational motion, as the average (RMS) distance from each bead to the

center of mass of the polymer. These length scales are usually presented as the squares

of the actual quantities, since these are easier to calculate. If the positions of all the

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beads, or bonds, of the polymer are all known, these quantities can be calculated

thus:

R2 = E [RN ·RN ] (2.98)

R2g =

1N + 1

N

k=0

E [ Rk −Rcm2] (2.99)

where

Rcm = 1N + 1

N

k=1

Rk (2.100)

is the position of the center of mass.

Notice that these formulas are dened only for discrete polymers. Thus, in order to

determine the RMS length and radius of gyration for the Kratky-Porod model, we

will nd a formula for the discrete case rst, for parameters N , L, and p, then take

the limit of the expressions as N goes to innity.

To nd the RMS length of the polymer, let us rst calculate its square, the mean-

square end-to-end length, R2. Assume that R0 = 0. We can simplify equation (2.98)

in this way:

R2 =LN

2 N

j =1

N

k=1

E [ Q j · Qk] (2.101)

by the linearity of both expected value and the dot product. Thus it is necessary

to compute the expected value of the dot product of unit vectors, Q j · Qk for any

integers j, k between 1 and N .

If j = k, then the unit vectors in the dot product are identical, and the value is 1.

If j and k differ by 1, then the two segments of the polymer are consecutive, and by

denition of the model, the angle between them is θ, or κ/ √ N . Thus

E [ Q j · Q j +1 ] = cos(κ/ √ N ) (2.102)

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We claim that each additional segment in between Q j and Qk contributes another

factor of cos θ to the expected value of the dot product.

Lemma 2.4. For a discrete polymer of N segments, for each j, k ∈ 1, . . . , N , the

expected value of the inner product between the two segments Q j and Qk satises

E [ Q j · Qk] = cos|k− j |(κ/ √ N ) (2.103)

Proof. We proceed by induction. We have shown the base case in (2.102). Suppose

that (2.103) holds, and (without loss of generality) k > j ; we wish to show thatE [ Q j · Qk+1 ] = cosk− j +1 (κ/ √ N ). For each j = 1 . . . , N −1, let F j (temporarily)

stand for the σ-eld generated by the rst j segments of the polymer. Also, for

simplicity of notation, let x = cos(κ/ √ N ). Then

E [ Q j · Qk+1 ] = E [E [ Q j · Qk+1 ]|F k] (2.104)

= E [ Q j ·E [ Qk+1 |F k]] (2.105)

= E [ Q j ·x Qk] (2.106)

= xE [ Q j · Qk] (2.107)

= xk+1 − j (2.108)

as desired.

From this, we can calculate R2 = E [ RN 2] and investigate its limit as N → ∞:

Theorem 2.5. The mean-square end-to-end length R2

of the wormlike chain polymer has the following formula:

R2 = 2L p −2 2 p 1 −e−L/ p . (2.109)

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Proof. We calculate R2 as a double geometric sum:

E [ RN 2] =

LN

2 N

j =1

N

k=1

E [ Q j · Qk] (2.110)

=LN

2 N

j =1

N

k=1

x| j−k| (2.111)

=LN

2 N

j =1

1 + 2N

j =2

j−1

k=1

x j−k (2.112)

=L

N

2

N + 2N

j =2

x −x j

1 −x (2.113)

=LN

2

N + 21 −x

(N −1)x +N

j =2

x j (2.114)

=LN

2

N + 2x1 −x

(N −1) − x −xN

1 −x (2.115)

=LN

2

N + 2(N −1)x

1 −x − 2x2

(1 −x)2 (1 −xN −1) . (2.116)

In terms of N , L, and κ, this equals

R2 = L2 1N

+ 2(N −1) cos(κ/ √ N )N 2(1 −cos(κ/ √ N )) −

2cos2(κ/ √ N )N 2(1 −cos(κ/ √ N ))2

1 −cosN −1(κ/ √ N ) .

(2.117)

We seek the limit of this expression as N → ∞. The key to this calculation is

evaluating the limit of cos N −1(κ/ √ N ), which is identical to the limit of cos N (κ/ √ N ).

limN

→∞

cosN κ√ N

= limN

→∞

1 − κ2

2N + O(N −2)

N

(2.118)

= limN →∞

1 − κ2

2N

N

(2.119)

= e−κ 2 / 2. (2.120)

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Therefore, as N → ∞, the mean-square polymer length R2 converges to

R2 = L2 4κ−2 −8κ−4(1 −e−κ 2 / 2) . (2.121)

By expressing R2 in terms of p for the limiting polymer, we obtain exactly equation

(2.109), as desired.

This result (2.109) is consistent with formulas for R2 in other papers. Chaudhuri,

for example, gives a generalized version of this formula for polymers of different

dimensions, which is equivalent to formula (2.109) in the case that d = 3 [22]. The key

calculation in the preceding proof, that is, the derivation of equation (2.120), provides

us with the link between the parameter p and the tangent-tangent correlation of the

polymer.

Lemma 2.6. Let T (s) denote the unit vector tangent to the polymer curve in the

worm-like chain model. Then the tangent-tangent correlation between any two vectors

an arc length distance of s and s from the xed end is give by

E [T (s) ·T (s )] = e−|s−s |/ p . (2.122)

Proof. Let s, s ∈[0, L], and without loss of generality let s < s . Observe that for s,

the tangent vector T (s) is the limit of the unit vectors in the direction of the discrete

segments, as the number of segments in the chain goes to innity:

T (s) = limN →∞

Q[Ns/L ] (2.123)

and similarly for T (s ). Thus, for each N , the vectors are [ Ns /L ]−[Ns/L ] segmentsapart. By Lemma 2.4, the tangent-tangent correlation satises

E [T (s) ·T (s )] = limN →∞

cos κ√ N

[Ns /L ]−[Ns/L ]

. (2.124)

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By the calculation similar to that in Theorem 2.5, this limit equals

limN →∞

cos κ√ N

[Ns /L ]−[Ns/L ]

= limN →∞

1 − κ2

2N

N (s −s)/L

(2.125)

= limN →∞

1 − κ2(s −s)/ 2L

N (s −s)/L

N (s −s)/L

(2.126)

= limN →∞

1 − (s −s)/ p

N (s −s)/L

N (s −s)/L

(2.127)

= e−(s −s)/ p (2.128)

and so the correlation satises equation (2.122). Therefore, the parameter p actsas a true persistence length, as it is the reciprocal of the coefficient of exponential

decay. A similar formula is also used by Toan and Thirumalai in [19] to explain

the tangent-tangent correlation of a discrete polymer. In their case, the number of

segments, multiplied by the common effective Kuhn length ( | j −k|a in our notation)

replaces the arc length distance |s −s |.

The calculation for the radius of gyration is somewhat more complicated, but we cancalculate it in a manner similar to that of R, as in Theorem 2.5.

Theorem 2.7. The mean-square radius of gyration R2g for the wormlike chain satis-

es the following formula:

R2g =

L p

3 − 2 p +

2 3 p

L − 2 4

p

L2 1 −e−L/ p . (2.129)

Proof. As before, we begin with the discrete model and take the limit as N → ∞.In lieu of equation (2.99), which uses the intermediate quantity of the center of mass

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vector, we can use an equivalent alternative formulation for the radius of gyration,

which considers the RMS length of the vectors joining every possible pair of beads:

R2g =

1(N + 1) 2

N

j =0

j

k=0

E [ R j −Rk2] (2.130)

But we know the quantity inside this sum, because it is precisely the mean-square

distance between the j th and kth beads, or equivalently, the mean-square length of

the subpolymer from R j to Rk , with k − j segments. Thus the sum can be evaluated

(using x as before):

R2g =

1(N + 1) 2

N

j =1

j−1

k=0

LN

2

j −k + 2( j −k −1)x

1 −x − 2x2

(1 −x)2 (1 −x j−k−1)

(2.131)

= L2

N 2(N + 1) 2

N

j =1

j−1

k=0

j −k + 2x1 −x

j −k −1 − x1 −x

(1 −x j−k−1)

(2.132)

= L2

N 2(N + 1) 2

N

j =1

j ( j + 1)

2 +

2x

1 −x

j ( j −1)

2 − x

1 −x j

− 1 −x j

1 −x(2.133)

= L2

2N 2(N + 1) 2

N

j =1

j ( j + 1) + 2x1 −x

j ( j −1) − 4x2

(1 −x)2 j − 1 −x j

1 −x

(2.134)

= L2

2N 2(N + 1) 2N 3 + 3 N 2 + 2 N

3 +

2x1 −x

N 3 −N 3

−

4x2

(1 −x)2

N (N + 1)

2 −

N

1 −x +

x −xN

(1 −x)2 . (2.135)

Taking the limit as N → ∞, we obtain

R2g = L2 2

3κ2 − 4κ4 +

16κ6 −

32κ8 (1 −e−κ 2 / 2) (2.136)

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and if we express R2g so that it depends on p rather than κ, we obtain equation

(2.129).

These two quantities have a relationship between them that depends on the poly-

mer model being used. For example, in the freely jointed chain or Rouse model, in

which the segments are independent Gaussian vectors, the RMS length and radius of

gyration satisfy [6]R2

R2g

= 6 , (2.137)

while with the freely rotating chain or Kratky-Porod model, the same ratio depends

on the persistence length p.

Figure 2.4 shows how the mean-square end-to-end length R2, mean-square radius of

gyration R2g, and their ratio all depend on p for a polymer of xed length L. Notice

that they all are sigmoidal in shape, and increase monotonically with p. Also, we

have non-dimensionalized the length scales by dividing each quantity by the overall

contour length L.

While it is impractical to describe a complete picture of the Kratky-Porod polymer

model for all values of the parameters L and p, we can consider some extreme cases.

In our treatment of the problem, we hold the total arc length L constant, since if

we use the formulas for R2 and R2g that are functions of L and κ, we see that they

scale linearly with L. Thus, by varying κ alone, we will change the shape of the

polymer, as exhibited in the relationship between ¯R and Rg. Since, as mentioned

before, the chemical physics papers use p as the intrinsic parameter for the stiffness

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−10 −8 − 6 −4 − 2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

log2(lp /L)

R b a r 2

/ L 2

−10 −8 − 6 −4 − 2 0 2 4 6 8 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

log2(lp /L)

R g 2

/ L 2

−10 −8 − 6 −4 − 2 0 2 4 6 8 106

7

8

9

10

11

12

13

log2(lp /L)

R b a r 2

/ R g 2

Figure 2.3: Mean-square end-to-end length, radius of gyration, and their ratio vs.persistence length

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of the molecules, we will use the formulas (2.109) and (2.129) that express the mean-

square end-to-end length and mean-square radius of gyration as functions of p and

L, and then examine these quantities, as well as the position vector R(s), for extreme

values of p for a xed length L. As we let p → 0 or p → ∞, we see that the

Kratky-Porod model reduces to one of two simpler limiting cases.

First, examine the case of a high persistence length: if p 1, then by formula

(2.122), the tangent-tangent correlation is nearly 1, and the polymer will not deviate

much from its initial direction (which we take to be along the positive z -axis). As p → ∞, the polymer will approach a rigid rod, pointing in the positive z -direction.

To show this, we see that the RMS length approaches the total arc length L:

limp →∞

R2 = limp →∞

2L p −2 2 p 1 −e−L/ p (2.138)

= limp →∞

2L p −2 2 p

L p −

L2

2 2 p

+ L3

6 3 p

+ O( −4 p ) (2.139)

= limp →∞2L p −2L p + L2

− L3

3 p + O( −2

p ) (2.140)

= limp →∞

L2 1 − L3 p

(2.141)

= L2 (2.142)

so the expected length of the polymer is the entire arc length, which implies that

the whole polymer must lie in a straight line. Notice that equation (2.141) gives an

asymptotic form of R2 for high values of p. We can also nd the limit of the radius

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of gyration for the p 1 case:

limp →∞

R2g = lim

p →∞L p

3 − 2 p +

2 3 p

L − 2 4

p

L2 1 −e−L/ p (2.143)

= limp →∞

L p

3 − 2 p +

2 3 p

L − 2 4

p

L2L p −

L2

2 2 p

+ L3

6 3 p −

L4

24 4 p

+ L5

120 5 p

O( −6 p )

(2.144)

= limp →∞

L2

12 − L3

60 p(2.145)

= L2

12 (2.146)

which is exactly the square of the radius of gyration of a straight rod; this result

is known from basic physics. Notice also that equation (2.145) gives an asymptotic

formula for Rg for the high p limit. The ratio between the RMS length and radius

of gyration thus satises

limp →∞

R2

R2g

= 12 (2.147)

which is markedly higher than in the Rouse model. However, we will show that inthe low p limit, the same ratio approaches that of the Rouse model.

If p 1, then only the lowest-order terms in p will contribute:

R2 ≈ 2L p (2.148)

R2g ≈ L p/ 3 (2.149)

and both quantities show that the polymer shrinks to a point:

limp →0

R = 0 (2.150)

limp →0

Rg = 0 (2.151)

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Although both the RMS length and the radius of gyration of the polymer approach

0 in the low p limit, they can still be compared:

limp →0

R2

R2g

= limp →0

2L p + O( 2 p)

L p/ 3 + O( 2 p)

(2.152)

= 6 (2.153)

which is the same is the ratio in the Rouse model. This makes sense, because the

Rouse model is the limit of the freely jointed chain, and a low persistence length

allows for each segment to point in any direction, not only at a xed bond angle.To complete the discussion of the limiting cases, let us turn our attention to the

behavior of the unit tangent vector T (s) to the polymer, as well as the position

vector R(s), in each case. By the system of equations (2.46), the rate of change

of the tangent vector with respect to arc length, ∂T ∂s , is directly proportional to the

curvature parameter κ. This means that if κ 1, equivalently if p 1, the unit

tangent vector will remain nearly constant. This is consistent with the statement

made before that in the high p limit, the polymer approaches a straight rod, as the

tangent vector always points in the same direction. However, in the low p limit,

the rate of change of the spherical Brownian motion increases without bound, so

that the distribution of the tangent vectors on the unit sphere approaches a uniform

distribution. The following theorem, stated and proven in [10], summarizes these

results for T (s) and R(s) in both extreme cases.

Theorem 2.8. The Kratky-Porod model exhibits the hard rod/random coil transition

in the following sense:

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1. As p →0, R(s) →0 in probability, and −1/ 2 p R(s) converges in distribution to

a standard 3-dimensional Brownian motion W (s).

2. As p → ∞, R(s) → se3 in probability, and p(R(s) −se3) → s

0 Q(σ) dσ in

distribution.

The second part of case (2) is a simple calculation: let R(s) = p(R(s) −se3).

Then, it follows that

R(s) = s

01/ 2 p (T (σ) −T (0)) dσ (2.154)

= s

0Q(σ) dσ. (2.155)

Thus the rate of convergence of R(s) is of order 12 , and the scaled process is the

integral of the spherical Brownian motion. This differs from the Rouse model, in

which the scaled limit is the Brownian motion itself, rather than its integral.

This summarizes the extreme cases for the time-independent Kratky-Porod model. If

we re-introduce the time parameter, we nd that Theorem 2.8 has a dynamic analog.

The correlation of the segments, E [ Q j · Qk], is the same regardless of time, and so

the limits of R(s, t ) in the low and high p cases are unchanged. However, the scaled

processes are now Gaussian with a second parameter:

limp →0

−1/ 2 p R(s) = W (s, t ) (2.156)

limp →∞ p(R(s) −se3) =

s

0 Q(σ, t ) dσ (2.157)

where W (s, t ) is a 3-dimensional Brownian sheet, and Q(s, t ) is a Brownian sheet on

the sphere.

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Thus, we have a picture of the Kratky-Porod polymer model, in both the static and

dynamic cases, described as the limit of the freely rotating Kramers chain. This

description, including its theorems, is written in [10], and it serves as a foundation

for this thesis, which concerns a polymer with the same physical characteristics as

the Kratky-Porod model, but inuenced by a constant external force. The next step

is to introduce this force into the model, and determine its effect on the behavior of

the polymer.

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CHAPTER 3

INTRODUCTION OF THE EXTERNAL FORCE

In the model thus described, the polymer has an initial random conguration and is

then set in motion. The only entity driving the motion of the polymer, that is, the

only force acting on it in this model, is the thermal noise, which is expressed by an

Ornstein-Uhlenbeck sheet B(s, t ). In the discrete case, this is modeled by an indepen-

dent Brownian motion at each bond, acting on each torsional angle. This formulation

works well for an isolated polymer in a dilute solution, but a more realistic model

would place the molecule under an external force. This force can act on the polymer

in one of various ways, as noted by Marko and Siggia [18]. For example, the polymer

could be stretched or compressed by molecular tweezers in a hydrodynamic ow, or

it can have a charged particle placed at one end and be placed in an electric eld.

These physical situations will alter the distributions of the torsional angles, which

will now exhibit some dependence between bonds, and will thus change the overall

behavior of the polymer, namely, in the orientation, shape, and size, as expressed by

the root-mean-square length and radius of gyration. To begin this consideration of

the force-driven polymer, let us begin by letting a constant force f ∈ R3

act at thefree end of the molecule.

Our original idea was to keep one end of the polymer xed, as before: R(0) =

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0, T (0) = e3, and to add a constant force at the free end, R(L). Then, we would

modify our stochastic processes to include the force, and compute the mean-square

end-to-end length and radius of gyration of the polymer in this situation, to produce

force-extension diagrams such as those in [18]. The motiviation for this idea was to

simulate and describe mathematically three dynamical situations for a forced semi-

exible polymer described by Hallatschek, Frey, and Kroy: Pulling , in which both

ends of the polymer are acted on by opposing forces which stretch the molecule;

Towing , in which a force tugs on one end of a stretched polymer to move it throughthe uid; and Release , in which the force lets the molecule go, and it shrinks or curls

back to its original state; although we would modify the situation to include a xed

end [27], [28]. Additionally, Chaudhuri describes the situations in which both ends,

one end, or neither end of the force polymer is xed [22].

However, these situations proved to be mathematically intractable, because of the

way the external force is propagated along the polymer through each bond in the

discrete model. While Marko and Siggia state that the tension in the wormlike chain

is uniform in this case [18], our analysis of the discrete model, by balancing the

components of force at each bead, gives a much more complicated formula for the

amount of the external force acting at each segment of the polymer. Therefore, we

decided to re-formulate our problem by placing the polymer in a uniform force eld,

such as the electric eld described in [18], and suppressing the time dependence, so

that the polymer is static, yet may be found any of its possible states. Then, the force

acting at each segment is constant in magnitude and direction, and the equation of

the balance of forces is simplied. This force eld induces a potential energy gradient

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for each segment, which in turn determines the distribution function for each torsional

angle. We close this chapter by describing the most likely state in which each segment

is found and then analyze the distribution function in the following chapter.

3.1 Formulation of a Dynamical System

To explain the physics of the polymer with an added force, let us turn our attention

to the discrete model. Let N be the number of segments in the polymer; thus we

must describe the motion of N vectors, or 3N coordinates. This may seem daunting,

because there are only N bonds at which we can balance the forces, yielding N

equations. However, if we convert the problem to spherical coordinates, we notice

that the radial coordinate r is always the same, since the bonds all have the same

length, and the polar angle θ is also the same in each bond. Only the azimuthal

angle φ is variable. Recall that each torsional angle φi is dened as the angle that

the random axis of rotation, which is a unit vector in the xy-plane, makes with thex-axis, so that the rotation matrices H i can be identically distributed. Notice also

that since the position of the rst segment of the discrete polymer is xed at both

ends, that is,

Q1 = ae3, (3.1)

it does not have a torsional angle. The number of angles (and thus the number

of moving bonds) is therefore N −1, and the indices of the torsional angles trailthe indices of the segments by one. For example, the rst torsional angle φ1 most

immediately determines the position of the second segment Q2. Each torsional angle

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can be in the range [0 , 2π), which means that each segment is free to rotate about the

axis determined by the previous segment. This gives rise to an equation of motion,

balancing all forces and torques acting on each bond, and so the result is a system of

N −1 differential equations in N −1 unknown coordinates.

In the model already discussed, in which the only force is the thermal noise, the

system of equations can be expressed as, for each i = 1, . . . , N −1,

φi(t) = c W i(t) (3.2)

where c is a constant depending on the temperature and viscosity of the uid, and

W i(t)for i = 1, . . . , N −1 is an ensemble of independent Brownian motions on the

real line. If an external force f i were to be applied to each bond, equation (3.2) would

read:

φi(t) = f i + c W i(t) (3.3)

in general. If a net force of f is applied to the polymer at the free end, or anywhereelse along the length, we must determine how it affects the motions of the torsion

angles.

Notice that no torsional angle, save for the last one φN −1, causes only one segment to

move. If there is a change in the angle φi , then the subpolymer comprising all posterior

segments Q j for j > i moves, as a single rigid body, around the axis along the segment

Qi . This is because the end R0 of the polymer is xed, and it can be seen by working

with a plastic polymer model and adjusting the torsional angles manually. Therefore,

we must consider how a force acting on one bead can propagate through the polymer

to an anterior bead. This can be done in one of two ways, so that the system of

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differential equations can be derived: by considering the ensemble of torsional angles

as a Lagrangian system and examining how the total kinetic energy changes, and by

balancing the forces and torques that act on each bead of the polymer.

If we consider the molecule as a Lagrangian system, then we represent the polymer

by the collection of its torsional angles:

Φ = φ1, φ2, . . . , φn−1. (3.4)

This system is based on the following physical principles: for any given conguration

Φ of the angles, the kinetic energy T (Φ) can be calculated. From T , we can nd the

momentum of each angle directly as a partial derivative of T :

pi = ∂ ∂ φi

T (Φ), (3.5)

and the derivative of angular momentum is torque:

τ i = ddt

∂

∂ ˙φi

T (Φ) . (3.6)

The net change in work on the polymer is equal to the sum of the torques multiplied

by the change in the angles (a total derivative):

δW =N −1

i=1

τ iδφi , (3.7)

but the work is also the sum of dot products of the applied forces with the change in

position of each bead:

δW =N

−1

i=1f i+1 ·δR i+1 . (3.8)

The indices are one higher in this equation, because the position of the rst bead,

R1, is unchanging.

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We can equate the two expressions for the change in work in equations (3.7) and (3.8)

to derive a relation for the torque and the applied force. By expressing δRk as a total

derivative with respect to all the φi’s for i < k , we obtain

τ i =N

j = i+1

f j · ∂R j

∂φ i. (3.9)

Thus, if we know an expression for the torque, we can derive a differential equation

for each torsional angle. Notice that the term R j appears in the equation for φi , for

any j > i , and that each R j depends on all torsional angles φi for i < j , by the

construction of the position vectors from the torsional angles in chapter 2. Therefore,

the equations governing the angles are by no means independent, even though they

were in the case with only thermal forces.

We can determine the torque on each angle from the kinetic energy of the system (the

whole polymer) using equations (3.5) and (3.6). Let m be the mass of each bead on

the polymer. Then, we can calculate the angular momentum at each torsional angle

directly, in terms of the rotational inertia I i at each bond:

pi = I i φi (3.10)

= m φi

N

j = i+1

ρ2i,j (3.11)

where ρ j,i is the distance from the j th bead to the i th axis of rotation.

The time derivative of the angle φi depends on the unit vector φ in the direction

along which the angle increases. This vector is perpendicular to both the segmentbeing rotated, Qi+1 , and the axis of rotation, Qi , so we have the relation

φi = Qi ×Qi+1

Qi ×Qi+1. (3.12)

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Note that the entire posterior polymer (the part after R i) must be considered, since

all segments after the ith rotate around it, because of the way the distance ρ j,i is

dened:

ρ j,i = projQ⊥

i(R j −R i) (3.13)

= (R j −R i) − (R j −R i) · Qi Qi . (3.14)

By factoring out the rotation matrix Z i , which preserves the lengths of vectors, we

nd a somewhat simpler computation of ρ j,i :

ρ j,i = a (I −e3⊗e3) j

k= i+1

Z −1i Z k e3 . (3.15)

This gives a picture of how to dene the moving polymer as a Lagrangian system.

However, since the motions of the torsional angles are no longer independent as they

were in the model without an external force, the differential equations describing the

motions of the segments are more difficult to express, because the force acting on thefree end is propagated through the polymer, and it affects the motions of all torsional

angles in the system. Namely, the equation for motion about the axis Qi contains

the rates of change of all the angles φ j for j ≥ i, because these angles appear in

the distances ρ j,i , whose time derivative must be taken in order to solve for the two

expressions of torque that must be balanced.

However, there is another equivalent method of deriving a system of stochastic partial

differential equations to describe the system, one that does not involve taking the time

derivative of several complicated expressions ( ρ j,i ) that depend on several torsional

angles. Rather than treat the physics of the entire molecule at once, as we did with

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the Lagrangian system, let us consider each bond individually, and balance the forces

and torques acting on each bead to derive an equation of motion for that bead. As

before, we will place an external force f at the free end RN of the polymer, and

impose a thermal uctuation W i at each bead Ri+1 ; the indices are chosen so that

W i immediately affects φi .

Thus, the following forces acting on each bead must be balanced:

The external force f , which is applied only to the last bead RN ;

The tension in the rods Qk and Qk+1 , which meet at the bead Rk ;

The thermal force c W k−1, which is the increment in a Brownian motion, so that the

entire set of W k’s are independent of each other;

A corrective force which holds the bond angle at Rk in place at a xed value of π−θ.

At the xed end, this force keeps the rst segment in place, so all work acting

at that point is lost as heat.

A frictional force f r k , which, for the moving polymer, is proportional to the trans-

lational velocity of the bead, and for the static polymer, holds the segment in

place.

This gives a system of several equations (one at each bead Rk) in several unknowns,

namely, the tension and corrective forces. Through the balance of these forces, the

external force f is propagated from the free end through the segments of the polymer.

The segments are set in motion by the torque that the net force at each bead exerts

about the axis determined by the previous segment. Let us hold the polymer in

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kinematic equilibrium; then the sum of all forces, as well as the sum of all torques,

acting at each segment, must be zero. Since the motion of each bead is rotational,

about a specied axis, only one component of the net force exerts any torque on

the bead ( Rk) about that axis ( Qk−1). This is the component of the force that is

perpendicular to Qk−1. However, we cannot express these components in terms of a

xed basis (such as x, y, and z ), since the axes are themselves variable. We need

another orthonormal basis of vectors that moves with the polymer. For this, we use

the Frenet-Serret frame of tangential, normal, and binormal vectors, to separate f k ,the applied force acting directly at the point Rk , into these three components:

Tangential ((f k ·tk−1)tk−1), which runs along the rod Qk . This component tends to

compress or stretch the rod, and is balanced by an equal and opposite internal

force, namely, the tension in the rod.

Normal ((f k ·nk−1)nk−1), which runs perpendicular to Qk in the same plane as Qk−1.

This component tends to exert a torque on the bead at Rk about Rk−1, changing

the bond angle, and is balanced by an equal and opposite torque that preserves

the angle; this torque is exerted by the corrective force.

Binormal ((f k ·bk−1)bk−1), which runs in the plane perpendicular to both Qk−1 and

Qk , i.e. along the direction of φk−1. This is the only component that does work

on the bead, and it tends to produce a torque on Rk about the axis of Qk−1.

The opposing force that maintains static equilibrium is the friction, f r k−1; in

the dynamic case, the friction is proportional to the angular velocity φk−1.

It is the binormal component that exerts a torque on the bead, while the others must

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be balanced by the tension in the previous segment, and the corrective force that

exerts a torque to hold the bond angle in place. To examine how the external force

f applied at the free end of the polymer propagates through the segments, which

gives us an equation of motion for each torsional angle, let us make an inductive

argument. Consider how, if N > 2, the force at the free end RN is propagated

through the polymer. At any torsional angle φk on the polymer, the resultant force is

f k+1 , plus the sum of all posterior (larger index) forces, minus their components that

have already caused torques on any posterior torsion angles. The balancing frictionalforce contains the effects of the bead at Rk+1 and all posterior beads, as they all

move together about the axis at Qk . All of these factors contribute to the equation

of motion of the polymer. We thus make the following inductive claim:

Lemma 3.1. For each k = 1, . . . , N −1,

N −1

j = k+1

ρ j,k f j

−

j−1

i= k+1

f j,i φi

· φk = b

N

j = k+1

ρ2 j,k φk (3.16)

or, equivalently,N −1

j = k+1

ρ j,k f j,k = bN

j = k+1

ρ2 j,k φk (3.17)

[Here, f j is the total force applied to the bead at position R j , and f j,i is dened above

to be the component of the force f j , after projections, that does work on the angle φi .]

Proof. Assume that equation (3.16) is true for a polymer with N segments. Consider

a polymer with N + 1 segments; then f N +1 = f ext + c W N and for all smaller j ,

f j = c W j−1. Let k ∈ 1, . . . , N ; we seek to balance the torques acting directly on

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the angle φk . By the inductive hypothesis, the net torque acting at bead Rk+1 , that

is, immediately affecting the angle φk , is

τ k =N

j = k+1

ρ j,k f j,k + τ N +1 ,k (3.18)

where τ N +1 ,k is the torque acting about axis Qk as a result of the force f N +1 . That

is, the net torque at φk is the sum of the torque due to the subpolymer from Rk+1

to RN , which is assumed from the inductive hypothesis, plus the additional torque

τ N +1 ,k . It remains to calculate this torque due to the force applied at the last bead.By the inductive hypothesis again, the force f N +1 will project onto a component

binormal to the polymer at the segment Qk+1 , by means of a calculation similar to

the Gram-Schmidt process. However, the number of projected vectors that must be

subtracted from f N +1 increases by one. To use the inductive hypothesis here, notice

that the amount of the original force f N +1 that remains at angle φk+1 is

f N +1 −N

i= k+2f N +1 ,i (3.19)

and the magnitude of its component binormal to Qk+2 is the dot product of the above

expression with the unit vector φk+1 ; this is equal to the tension in the rod Qk+2 .

f N +1 ,k+1 = f N +1 −N

i= k+2

f N +1 ,i · φk+1 φk+1 . (3.20)

This is true by considering the subpolymer consisting of the segments from Q2 to

QN +1 ; it has N segments, and thus the inductive hypothesis can be applied to nd

the effective torque due to f N +1 at φk+1 . By subtracting this binormal component

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from the force (3.19), we obtain the component, or amount, of the force which has

not done any work yet on the polymer, namely,

f N +1 −N

i= k+1

f N +1 ,i

and we have a recursive formula for the components of the force vector. When this

remaining force is projected onto the unit vector φk , it becomes the quantity f N +1 ,k ,

which is the component of the force f N +1 that does work on the angle φk . This proves

the inductive statement, and we have the system of differential equations (3.16) forthe polymer.

However, this model is very cumbersome; it can be modeled numerically, but again,

it is nearly impossible to solve analytically except in the simplest cases corresponding

to the smallest values of N . The N = 3 case, for example, corresponds to the chaotic

double pendulum, in which both the sine of each torsional angle and its second

derivative appear in each coupled differential equation.What we can do, however, is simplify both this model, and the previous case, in

which the polymer was considered as a Lagrangian system, to consider the case in

which the polymer is in the lowest possible energy state. This will be the most likely

conguration in which we would expect to nd the polymer, based on the magnitude

and direction of the external force, as well as the other parameters in the model, such

as the persistence length p or the total length L. By imposing these assumptions,

we can describe a model for the polymer which is both physically tractable and

mathematically soluble, though not soluble in the uid, which accurately describes

the stochastic behavior of the polymer molecule.

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3.2 The Polymer in the Lowest-Energy Conguration

Let us consider a nite polymer chain with N segments, in which rotation of each

bead about any segment is still free, and let the interaction of the surrounding uid

with the polymer be such that each torsional angle φk experiences an independent

Brownian motion W k . So far, this is equivalent to the original model described in

Chapter 2, without any external forces. In the models that we have considered so far,

we have imposed an external force f at the free end of the polymer only; physically,

this occurs when a charged particle is placed at the last bead, RN , and the polymer is

placed in an electric eld, among other situations. This caused great complication in

solving the equations of motion, because the force at the free end propagated through

the entire polymer, and it was necessary to consider which component of the force

vector exerted how much torque on each torsional angle. (Additionally, the thermal

forces propagated through the polymer as well, but now we are assuming that each

angle has an independent Brownian motion acting on it.) To simplify the problem, letthe external force be a constant force eld, that affects each torsional angle equally.

That is, the force vector due to the external force is the same at every bead of the

polymer. If f ext is the vector of the external eld, then the equation of motion is

simpler:

ρk+1 ,k ((f ext + c W k) · φk)φk = bρ2k+1 ,k φk (3.21)

or, since the distance from one bead to the previous axis is a sin θ,

((f ext + c W k) · φk)φk = absin θ φk . (3.22)

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Recall that the unit vector φk satises the relation

φk = Qk ×Qk+1

Qk ×Qk+1 (3.23)

and thus it depends on the current values of all torsional angles from φ1 to φk , since

they are all needed to determine the segment Qk+1 .

We wish to consider a model that can be solved analytically. To do this, let us

eliminate the time parameter, and consider the static polymer in the most likely state.

This means that the rate of change of the torsional angles, φk for any k = 1, . . . , N

−1,

equals 0, and the above equation becomes

((f ext + c W k) · φk)φk = 0, (3.24)

in which case the direction of φk must assimilate to the value of the increment of the

Brownian motion, W k , making the torsional angle a random variable on [0 , 2π).

It is thus necessary to determine this most likely state.

Recall, that in the lowest-energy state,

f · φk φk = 0 (3.25)

(since we are holding time constant, there is no W k term). This equation will hold

true if the vectors f and φk are orthogonal. By equation (3.23), this is equivalent to

saying that

f ·(Qk ×Qk+1 ) = 0 (3.26)

and the scalar triple product is zero exactly when all three vectors lie in the same

plane. By an inductive argument, we can show that the entire polymer must lie in a

plane to be in static equilibrium.

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Lemma 3.2. To achieve minimum energy, all segments of the discrete polymer must

be coplanar with the force vector f and the initial segment Q1; that is, for all k =

2, . . . , N ,

f ·(Q1 ×Qk) = 0 (3.27)

Proof. Since the vectors f and Q1 are xed, Q2 must be chosen to be in the plane

determined by f and Q1, so that f · φ1 = 0. Assume that f , Q1, . . . , Q k all lie in

the same plane. Then in order that f · φk = 0, the vector Qk+1 must be in the same

plane as f and Qk , meaning that all k + 2 segment vectors are in the same plane.

Therefore, in the state of lowest energy, the entire polymer is coplanar, and it sits in

the plane dened by the force vector f and the initial direction vector Q1.

Therefore, the distribution functions of the torsional angles will be biased towards

the state in which the entire polymer lies in a plane. Investigating exactly how

these distributions vary with the magnitude and direction of the force and with other

parameters such as the persistence length p is the major theme of this dissertation

and will be the central idea of the description of the forced polymer model that

follows.

Once the distributions of the individual torsional angles are known, we can analyze

the entire polymer in a manner very similar to the methods of Chapter 2: dene

the process of rotating Bishop frames of orthonormal vectors, Z N , and the driving

process, BN

; take the scaling limit of these processes as N goes to innity and thetotal length and curvature of the polymer remain constant; then, describe the shape,

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size, and orientation of the polymer in the limit, using physical quantities such as root-

mean-square end-to-end length and radius of gyration to determine how the forced

polymer model differs from the unforced one. These simplications to the physical

model of imposing a uniform force eld and considering the stochastic behavior of

the polymer in static equilibrium make for a self-contained problem that is the focus

of this dissertation.

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CHAPTER 4

CONVERGENCE OF THE FORCED POLYMER

So far, we have described the two models of semi-exible polymers without the ex-

ternal force, and shown that in the scaling limit, the Kramers chain converges to the

Kratky-Porod model. Also, we have introduced a force eld to act on the polymer

and formulated a way to include it in the model. Now that we have added the force

as a vector that acts equally at all points of the polymer, we wish to show the same

results for the polymer in the eld, namely, that the force-driven Kramers chain con-

verges to the force-driven Kratky-Porod model. We also aim to derive a stochastic

differential equation for the tangent vector, which is no longer simply a scaled Brown-

ian motion as in the unforced model, but now contains a drift term, which introduces

a bias in the direction of the force. This is the equation that the tangent vector in

the Kratky-Porod model satises, and we show that it is the limiting equation for

the forced model.

These two results, the convergence of the discrete forced model to the continuous

one, and the differential equation for the tangent vector T (s), comprise one theorem

about the forced Kratky-Porod model that is the main focus of this chapter, and infact, of this dissertation. The proof of this theorem requires several lemmas, which

we will prove according to the following scheme:

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1. The collection Q of segments and the set Φ of torsional angles, which describe

the discrete Kramers chain, may each be written as a function of the other, as

a change of coordinates.

2. The integral of a function F (Φ) with respect to Lebesgue measure of the tor-

sional angles may be expressed as an integral of an equivalent function G(Q)

with respect to the transition probabilities of the segments.

3. The segments of the discrete polymer, under the original Lebesgue product

measure, form a Markov chain.

4. The transition functions for the segments, given by the Boltzmann-Gibbs dis-

tribution, form a probability measure.

5. The segments of the discrete polymer, under the Boltzmann-Gibbs measure,

form a Markov chain.

6. The driving process BN (s), for the discrete polymer in a force eld, has a Doob-

Meyer decomposition whose nite variation part includes the orthonormal frame

process Z N (s), thus yielding a projective drift term.

7. The martingale part M N (s) of B N (s) converges weakly to a Gaussian matrix-

valued process.

8. The sequence of ordered pairs of processes (B N , Z N ) is relatively compact inthe Skorohod topology.

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9. The bounded variation part AN (s) of B N (s), along with the orthonormal frame

process Z N (s), converges weakly along subsequences.

10. The sequence (B N ) is of uniformly controlled variation, and therefore good in

the sense of Kurtz and Protter [23].

11. The sequence of discrete processes Z N of orthonormal frames converges weakly

to the continuous process Z , dened as the solution to a limiting stochastic

differential equation.

12. The martingale part M (s) of the driving process B(s) has non-zero entries that

are scaled Brownian motions.

13. The driving process B(s) of the continuous forced polymer satises a differential

equation involving both a drift and a diffusion.

14. The tangent vector process T (s) satises a differential equation in terms of

Gaussian matrices.

15. The diffusion term in the equation above is a scaled spherical Brownian motion.

By following these steps, we reach the main result of this chapter, about the conver-

gence of the discrete forced model to the forced wormlike chain and the stochastic

differential equation governing the resulting tangent vector process.

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4.1 Potential Energy and the Boltzmann-Gibbs Distribution

Before proving these steps, let us rst consider the fact that the external force eld

induces a potential gradient on the polymer. According to Lemma 3.2, the lowest

potential energy state for each segment of the discrete polymer corresponds to the

case in which the segment points as closely as possible to the direction of the force.

Therefore, the energy due to each segment must attain its minimum at the most

likely position of each segment; thus, dene the energy due to each segment to be a

negative constant multiplied by the dot product of the segment with the vector f :

uk = −α(Qk ·f ). (4.1)

This inner product is +1 when the vectors align and −1 when they point in opposite

directions. If we let α > 0, then the negative sign means that the potential energy

for the segment is maximized when the vectors point as far apart as possible, and

minimized when they are aligned as best as possible. Then, take the sum of all termsfor the energy over all segments of the polymer to obtain the total potential energy,

since the inner product is linear and the cumulative sum of the Qk is RN :

U = −α(RN ·f ). (4.2)

This denition, in which the energy for each segment is proportional to the inner

product of the force and each segment vector, and the total potential energy is pro-

portional to the dot product of f with the position of the free end, is consistent with

other sources [19], [21], [29], [30]. In these cases, the energy due to the force is a

linear term added to the Hamiltonian.

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The potential energy is proportional to the magnitude of the force f , which makes

physical sense. If there is only a small deviation from the equilibrium position, then

the angular deviation for one segment, φk −φ0k , is also small. The potential energy

for that segment is then approximately equal to

−α(Qk ·f ) ≈ −α f a(1 −(φk −φ0k)2) (4.3)

which parallels the formula for the potential energy for a torsion pendulum, another

physical model in which a restoring force brings an angle towards a favored rest state,

as well as the harmonic potential well considered in [22].

The most important property, however, of the potential energy formula in (4.2) is

its additivity. This is what facilitates the calculations of the conditional probability

distributions for the torsion angles, and in turn, the position of the polymer. In these

calculations, we will show that the energy of a single segment must be O(1), even as

the length of the segment goes to zero. To maintain these properties and to give a

physically consistent explanation of the mathematics, suppose that f is the force due

to an external electric eld, and it induces a potential energy because each segment

has a dipole moment proportional to the length. As N → ∞, the length a = L/N

of each segment goes to zero, so to keep the proper scaling for the energy, the dipole

moment must be kept constant, and this is done by increasing the charge. Thus, we

let α = N , and the potential energy for one segment Qk is

U k = −Nf ·Qk = −L f f · Qk . (4.4)

Motivated to dene a potential energy for the polymer, we can use this quantity

to determine at last the distribution of the torsional angles. The key to nding

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the density function is the Boltzmann-Gibbs distribution , a concept from statistical

mechanics that describes the density of particles that follow a certain potential energy.

If the potential energy U , depending on the conguration of polymer segments, is

known for a system, and each state of torsional angles Φ corresponds to a value of U ,

then the Boltzmann-Gibbs probability measure P for each state is determined thus:

P (Φ∈dΦ) = e−U (Φ) /τ dΦ

Z (4.5)

where τ = kB T is the thermal energy, and

Z is the partition function:

Z = T N − 1e−U (Φ) /τ dΦ (4.6)

while the integral above is over the N −1-dimensional torus, the product space of

the interval [0 , 2π) with itself N −1 times. Since the formula for U is additive, the

probability density is multiplicative.

However, in formula (4.5), the differential is product measure on the N −1-tuple of

angles,

dΦ = dφ1 ×dφ2 ×· · ·×dφN −1 (4.7)

while the exponential factor depends on the segments of the polymer (the exponent

is the normalized inner product of the force vector and the position vector of the

free end of the polymer, which can be broken up into the segment vectors). Thus,

if we wish to use the multiplicativity of the density function, it helps to convert the

product measure of torsional angles into the product of transition probabilities of the

segments.

This can be done by showing that the sequence of segments in the discrete polymer

is a Markov chain; then it becomes possible to calculate conditional expectations

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of functions of the torsional angles, which are necessary to nd the Doob-Meyer

decomposition of the driving process of the polymer, by iterated integration. If the

measure in the integral is left as the product measure of the torsional angles, then

the integral cannot be evaluated segment-wise. By using the Markov property of the

segments, the calculations are facilitated, as the positions of the segments can be

evaluated as single integrals. But before we prove this statement, let us dene some

notation:

Let F k represent the σ-eld generated by the rst k torsional angles; that is, F k =

σφ1, . . . , φk. Notice that since Q1 is deterministic, the segment Qk+1 , which

is a function of the rst k torsional angles, lies within F k , and subsequently

Rk+1 ∈F k as well.

Let Z N represent the partition function, dened as the integral of the density func-

tion over all states, for an N -segment polymer, and let

Z k denote the partition

function for the rst k segments only.

Let P 0(Φ∈dΦ) represent the probability density of the collection of torsional angles

for the polymer in which no external force is applied (that is, the model in

Chapter 2), and let P f (Φ∈dΦ) be the density of the angles under the force f .

Recall that P 0 is simply normalized Lebesgue measure on the N −1-dimensional

torus.

Finally, dene the transition probabilities: Let π0N (x,dy) represent the probability

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that in an N -segment polymer, one segment is in the set dy given that the

previous segment is x, in the model with no added force; that is:

π0N (q k , dq k+1 ) = P 0(Qk+1 ∈dq k+1 |Qk = q k) (4.8)

for a xed vector q k and set dq k+1 . Similarly, let πf N (x,dy) be the transition

probability for the polymer with the force.

Therefore, we have the transition probabilities for the conditional distributions of

each segment with respect to the location of the previous segment. However, the

Boltzmann-Gibbs distribution is given with respect to product measure on the tor-

sional angles. Yet, the density function depends only on the endpoint of the polymer

(since the force eld is uniform) and can be determined directly from the positions of

all the segments. In order to switch between the segments and the torsional angles,

we wish to use a formula that can express one set of variables in terms of the other.

This amounts to a change in the coordinates (similar to the ones we used in theLagrangian system in Chapter 3), and we can convert from one system to the other.

Lemma 4.1. [Change of coordinates] For each N ∈ N , if Q = Q1, . . . , Q N is the collection of segments that comprise a discrete N -unit polymer, and Φ =

φ1, . . . , φN −1 is the set of torsional angles of the same polymer, then the two col-

lections of variables can each be expressed as a function of the other.

Proof. The means by which Q can be expressed as a function of Φ is simple, and

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has already been discussed; the function can be expressed as a series of intermediate

matrix-valued functions:

Q1 = ae3 (4.9)

Qk = Z k−1Q1 (4.10)

for k > 1, where the matrices Z k follow this recursive relationship:

Z 0 = I (4.11)

Z k = Z k−1H k(φk) (4.12)

where the rotation matrix H k , depending only on the angle φk , is given as in equation

(2.9). This expresses the segments of the polymer, Q, as an explicit function of the

torsional angles Φ. For example, if k = 1, . . . , N is xed, then the segment Qk is

given by the formula

Qk = a k

j =1

H j (φ j ) e3 (4.13)

where the matrices are multiplied in the order such that the indices j increase from

left to right.

Next, let us nd a way to dene each torsional angle φk from the collection Q of

segments. It may seem that, since the angle φk affects the position of every segment

Q j for j > k , that all of those segments would appear in the formula for φk . However,

if we recall from the discussion of the dynamical model in Chapter 3 that the cross

product of two consecutive segments gives a vector in the direction in which the

nearest torsional angle increases, we can nd a formula that depends on only two

segments. The formula in question is (3.23), and it yields the unit vector φk in the

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direction in which φk increases from only the two segment vectors Qk and Qk+1 by

taking the cross product of the two segment vectors and normalizing the resulting

vector. We must then nd a way in which to isolate the torsional angle φk from that

unit vector. Recall from the theory of spherical coordinates that the unit vector φ,

which is the vector in which the azimuthal angle φ increases, has the rectangular

coordinates

φ =−sin φ

cos φ

0

. (4.14)

Now, this is dened in the standard orientation for spherical coordinates, in which φ

is the azimuthal angle in the xy-plane from the positive x-axis, and it represents the

angle of rotation about the north pole vector e3. Recall that, in our construction of

the discrete model in Chapter 2, the torsional angle φk represented the angle that the

axis of rotation vk , which determined the rotation matrices bk (so that for any vector

w, bkw = vk ×w), made with the positive x-axis. These axes of rotation vk were

independent with identical uniform distributions on the unit circle in the original

unforced model, and will show a bias towards the force vector in the forced model. In

turn, the axes gave rise to i.i.d. rotation matrices H k , and to determine the action of

each rotation form the segment Qk to the following segment Qk+1 , it is necessary to

conjugate the matrix H k by Z k−1, thus pulling back the segment Qk so that it lines

up with the north pole vector, or the initial direction vector e3. Therefore, to ndthe unit vector in the direction in which the segment Qk+1 rotates counter-clockwise

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(call it γ k), it is necessary to apply the inverse rotation Z −1k−1 to the cross product:

γ k = Z −1

k−1(Qk ×Qk+1 )a2 sin θ

. (4.15)

This is the unit vector that corresponds to the direction of increase of the azimuthal

coordinate of the segment Qk+1 when Qk is taken to be the axis of rotation; however,

because of the denition of the angle φk and the axis of rotation vk , it actually leads

the unit vector φk , that is, the direction of increase of the angle φk , by 90 degrees.

Thus, rather than follow the formula above which relates ˆφk and φk , we can determine

φk from the vector γ k :

γ k =

cos φk

sin φk

0

. (4.16)

Therefore, with γ k as dened in equation (4.16), we can isolate the torsional angle φk

as follows:

φk = atan2( γ k ·e2, γ k ·e1), (4.17)

that is, we take the two-argument arctangent of the y and x components of the vector

γ k . Now, in equation (4.16), we see that the unit vector γ k depends explicitly on the

segments Qk and Qk+1 , as well as the matrix Z −1k−1. But since Qk = Z k−1Q1, and Q1

is a constant, it is apparent that γ k depends only on the two segments Qk and Qk+1 .

Likewise, the torsional angle φk can be written as a function of only those same two

segments.

Therefore, each element in each collection Qof segments or Φ of torsional angles can

be written as a function of a subset of elements of the other set, as desired.

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Thus, we have established a relationship between the segments and the torsional an-

gles in the discrete polymer model, and we can pass from one set of random variables

to the other when discussing the probability distributions of each, thus simplifying

the calculations of integrals with respect to the Boltzmann-Gibbs distribution, which

can be evaluated in terms of either product measure on the torsional angles, or tran-

sition probabilities of the segments. We now wish to show that under the transition

probabilities πf N , the segments Qk form a Markov chain. To do this, we will rst

show that they are a Markov chain under π0N , then express π

f N in terms of π

0N and

the partition functions, then show that the Markov property carries over to the case

with the force, in which the distribution of the torsional angles (and of the segments)

is biased due to the potential energy. But to show that the sequence of segments

is a Markov chain, we need an intermediate lemma rst, to show that we can make

a change of variable between the torsional angles and the segments of the polymer.

This will allow us to evaluate an integral with respect to the transition probability of

the segments by switching to a simple normalized Lebesgue measure on angles.

Lemma 4.2. [Change of variables] For all N ∈N , for any collection

Φ = φ1, . . . , φN −1 of random variables on the set T = R / 2πZ , for any set of de-

terministic functions Q= Q1, Q2(φ1), . . . , Q N (φ1, . . . , φN −1), if F (Φ) is a bounded,

measurable function that can be expressed as G(Q), which has the following multi-

plicative property:

G(Q) = h(Q1, . . . , Q N −1)g(QN ), (4.18)

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then

T N − 1F (Φ)dφ1 · · ·dφN −1 = T N − 1

G(Q)π0N (q 1, dq 2) · · ·π0

N (q N −1, dq N ). (4.19)

Proof. We prove the assertion by induction. The base case is N = 1, which is trivial,

since there are no torsional angles; the function F is constant, and the only segment

Q1 is deterministic. Now, assume that equation (4.19) is true for N segments; we must

prove it for N + 1 segments. To do this, we must re-scale the transition probabilities

π0N +1 , so that they give the probability that a segment of length a = L/ (N + 1) is

located on a subset of the sphere S of that radius. By the rules of construction of

the polymer, the segment must lie on a circle C determined by

C = y ∈S 2 : x ·y = a2 cos θ (4.20)

where x is the previous segment and θ = κ/ √ N + 1. Therefore, given a segment

QN = q N , the transition probability measure π0N +1 (q N , dQN +1 ) lives on the circle C

dened above.

Since the function F (Φ) is bounded (and thus also is G(Q)), we can use Fubini’s

Theorem to interchange the order of integration. Then, we can evaluate the iterated

integrals, one at a time. We must, therefore, show that

T N − 1 T F (Φ)dφN dφN −1 · · ·dφ1 =

(S 2 )N − 1 S 2 G(Q)π0N +1 (q N , dQN +1 ) π

0N +1 (q N −1, dq N ) · · ·π

0N +1 (q 1, dq 2).

(4.21)

By the inductive hypothesis, it suffices to show that the inner integrals on each side

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are equal; by the multiplicative property of G, the function h can pass through the

inner integral, so we must show that

C g(QN +1 )dφN = S

g(QN +1 )π0N +1 (q N , dQN +1 ). (4.22)

Recall that dφN is one-dimensional Lebesgue measure λ1 on the circle [0, 2π), divided

by 2π so that the integral over the whole circle is 1, while π0N +1 is dened on the

sphere of radius a so that for any subset A of the sphere, and for C the circle dened

in (4.20) for x = q n ,

π0N +1 (q n , A) = λ1(A ∩C )

2π . (4.23)

Notice that π0N +1 (q n , A) = 0 if A ∩C =∅, showing that the measure lives on C . With

this, we can prove that the inner integrals are equal. First, consider the case in which

the function g is an indicator function, g = 1A for a subset A of the sphere of radius

a. Then to express g in terms of φN , g(φN ) must be the restriction of 1 A to the circle

C , i.e. g = 1A∪C . It follows that

C g(φN )dφN = C

1A∩C (φN )dφN = λ1(A ∩C )

2π (4.24)

and

S g(QN +1 )π0

N +1 (q N , dQN +1 ) = S 1A(QN +1 )π0

N +1 (q N , dQN +1 ) = λ1(A ∩C )

2π . (4.25)

Thus, equation (4.22) holds. Next, consider the case in which g is a simple function;

by linearity of the integral, (4.22) is true in this case as well. Finally, consider the

general case in the statement of the theorem, in which g is any bounded measurable

function on S ; then, by the Dominated Convergence Theorem, (4.22) holds in this

general case. Thus, the lemma is proven.

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Now, we can use this lemma to prove that the segments are a Markov chain when they

are uniformly distributed on a circle, which depends only on the value of the previous

segment. (In the following analysis, the domains of integration will be suppressed

when it is clear that they are products of either the full circle T = [0, 2π) or the

entire sphere S 2.)

Lemma 4.3. For an N -segment polymer, under normalized Lebesgue product measure

on the angles Φ = φ1, . . . , φN −1, the functions Q = Q1, . . . , Q N (dened as in

Chapter 2) form a Markov chain, and the induced measure is the transition probability

π0N .

Proof. Let G(Qk) be a bounded, measurable function of the kth segment in the poly-

mer. Assume that the values of the previous segments are specied: Q j = q j for

j < k . Equivalently, this means that the previous torsional angles are also specied:

φ j = ψ j for j < k −1. Then we wish to show that

G(Qk)dφk−1dψk−2 · · ·dψ1 = G(Qk)π0N (q k−1, dQk)dψk−2 · · ·dψ1. (4.26)

But this follows from the preceding lemma (4.2). The integral on the left side can be

re-parametrized so that the integration is in terms of the transition probabilities of the

segments. Then, the integrand depends only on the kth segment, so the integration

with respect to the last transition probability can be evaluated. Finally, the remaining

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integrations can be changed back into the product measure on the angles. That is:

G(Qk)dφk−1dψk−2 · · ·dψ1 = G(Qk)π0N (q k−1, dQk) · · ·π0

N (q 1, dq 2) (4.27)

= G(Qk)π0N (q k−1, dQk) · · ·π0

N (q 1, dq 2) (4.28)

= G(Qk)π0N (q k−1, dQk) dψk−2 · · ·dψ1 (4.29)

as desired. Since this holds for any k = 2, 3, . . . , N , the sequence of segments has the

Markov property.

However, we wish to show that the segments retain the Markov property when the

external force is added. To show this, we must rst express the transition probabilities

that take the force into account in terms of the transition probabilities that only use

the uniform distribution.

We can do this by means of the Boltzmann-Gibbs distribution. Consider each segment

as its own polymer; then recall that the potential energy associated with that segment

is

U k = −Nf ·Qk . (4.30)

Assume that the temperature is constant along the entire length of the polymer.

Then, τ = kB T is the thermal energy of the system.

Therefore, the transition probability for the N th segment under this force is

πf N (q N −1, dQN ) =

eNf ·Q N /τ π0N (q N

−1, dQN )

Z N / Z N −1 (4.31)

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in which the denominator is the ratio of partition functions, which are iterated in-

tegrals in which the transition probabilities of all previous segments are multiplied

together:

Z N

Z N −1= eNf ·R N /τ dφN −1dφN −2 · · ·dφ1

eNf ·R N − 1 /τ dφN −2 · · ·dφ1. (4.32)

We must show that the transition probability is, in fact, a probability measure; that

is, its integral over the sphere S of radius a = L/N is 1.

Lemma 4.4. The transition probability density πf N , which includes the dependence

on the external force, is a probability measure; that is,

S eNf ·Q N /τ π0

N (q N −1, dQn ) = Z N

Z N −1. (4.33)

Proof. To prove this, let us examine the ratio of consecutive partition functions, as

expressed in (4.33). By the rst lemma, the integrals with respect to the torsional

angles can be re-written in terms of transition functions:

Z N

Z N −1=

eNf ·R N /τ π0

N (q

N −1, dQ

N )π0

N (q

N −2, dq

N −1)

· · ·π0

N (q

1, dq

2)

eNf ·R N − 1 /τ π0N (q N −2, dq N −1) · · ·π0

N (q 1, dq 2) . (4.34)

Now, the expressions in both the numerator and denominator on the right side above

are iterated integrals, and the integrand can be decomposed into the product of

functions of each individual segment (since it is multiplicative). Thus, by isolating

the innermost integral in the numerator, we obtain

Z N

Z N −1

=

eNf ·R N − 1 /τ

eNf ·Q N /τ π0

N (q N −1, dQN ) π0N (q N −2, dq N −1) · · ·π0

N (q 1, dq 2)

eNf

·R N − 1 /τ

π0N (q N −2, dq N −1) · · ·π

0N (q 1, dq 2)

.

(4.35)

By the Markov property (Lemma 4.3), the inner integral can be evaluated indepen-

dently of the other iterations, and so the integral in the numerator can be expressed

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as the product of the inner integral, and the integral over the rst N −1 segments.

But this latter integral is exactly the denominator. Therefore,

Z N

Z N −1= eNf ·Q N /τ π0

N (q N −1, dQN ), (4.36)

as desired. Thus, the transition function π f N does integrate to 1 over the sphere, and

it is a true probability density.

Notice that since the Boltzmann factor exp( −U/τ ) and partition functions are always

positive, the transition probability πf N is absolutely continuous with respect to the

original measure π0N . Therefore, the potential energy term for the segment is a Radon-

Nikodym derivative.

With these lemmas in place, we can show that the Markov property extends to the

case in which the external force is applied to the polymer.

Lemma 4.5. For an N -segment polymer, under the Boltzmann-Gibbs measure on the

angles Φ = φ1, . . . , φN −1 corresponding to the potential energy due to a constant

force eld f , the functions Q = Q1, . . . , Q N (dened as in Chapter 2) form a

Markov chain, and the induced measure is the transition probability πf N .

Proof. This proof will mainly follow the proof of the Markov property for the unforced

polymer, except the Boltzmann-Gibbs distribution replaces the uniform distribution

of torsional angles, and the transition probabilities have the superscript f instead of

0. Let G(QN ) be a bounded, measurable function of the last segment of the polymer

(for a medial segment Qk , consider it to be the free end of a k-segment subpolymer).

Assume, as before, that all previous segments (and all previous torsional angles) are

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specied; that is, if j < N , then Q j = q j and φ j−1 = ψ j−1. Thus, we wish to show

that

G(QN )eNf ·R N /τ dφN −1dψN −2 · · ·dψ1 =

G(QN )π f N (q N −1, dQN )eNf ·R N − 1 /τ dψN −2 · · ·dψ1. (4.37)

By the Lemma 4.2, the left side of the above equation can be expressed in terms of

the transition probabilities:

G(QN )eNf ·R N /τ dφN −1dψN −2 · · ·dψ1 =

G(QN )eNf ·R N /τ π0N (q N −1, dQN ) · · ·π0

N (q 1, dq 2). (4.38)

But the exponential in the integral can be decomposed as the product of exp( Nf ·q k /τ )

for each segment q k , and the product of each of these exponentials with the transition

probability π0N is precisely π f

N . We can then express the left side of (4.37) as

G(QN )eNf ·R N /τ dφN −1dψN −2 · · ·dψ1 = G(QN )π f N (q N −1, dQN ) · · ·π f

N (q 1, dq 2).

(4.39)

But the right side of the above equation is an iterated integral, and the integrand

depends only on the innermost transition probability. By isolating this integral and

changing the other transition probabilities back to the Boltzmann-Gibbs measure on

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the angles, we nd that

G(QN ) πf N (q N −1, dQN ) · · ·π f

N (q 1, dq 2)

= G(QN )π f N (q N −1, dQN ) π f

N (q N −2, dq N −1) · · ·π f N (q 1, dq 2)

(4.40)

= G(QN )π f N (q N −1, dQN ) eNf ·R N − 1 /τ dφN −2 · · ·dφ1 (4.41)

which completes the proof.

4.2 Driving Process of the Forced Kramers Chain

Now that we have established that the segments of the forced Kramers chain form

a Markov chain, the next step is to use this property to characterize the driving

process, B N (s), in terms of drift and diffusion. This will lead to the two terms in the

differential equation (1.16) for the tangent vector in the Kratky-Porod model with

the added force.

Recall, from Chapter 2, that B N is given by the formula

B N (s) = 1√ N

[Ns/L ]

k=1

bk(φk) (4.42)

in which the matrices bk are innitesimal rotations, which comprise the segments of

the polymer in the following manner:

Qk =k−1

j =1

exp κ√ N

b j (φ j ) Q1 (4.43)

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where the product is written so that j increases from left to right, and Q1 = ae3. The

formula for each innitesimal rotation is

bk(φk) =

0 0 sinφk

0 0 −cos φk

−sin φk cosφk 0

. (4.44)

In the original model without a force (as in Chapter 2), the torsional angles φk all had

a uniform distribution on the circle; therefore, the expected values of the sines and

cosines of the angles were all zero. This meant that the mean of each innitesimalrotation, E [bk(φk)], was 0, and so the mean of the driving process E [B N (s)] was 0 for

all s∈[0, L]. In fact, since each increment bk of the driving process had mean zero,

their scaled cumulative sum B N (s), or B N k with a discrete index, was a martingale.

But now, the distribution of each torsional angle is biased towards a favored value

φ0k , which would make the segment Qk+1 line up as closely to the vector of the force

eld f as possible. Thus, the mean of the driving process will no longer be 0, but

instead will show a bias, or a drift, in the direction of the applied external force.

Additionally, the driving process BN (s) is no longer a martingale; however, it can

be expressed as a semimartingale. To show this, we can write BN in the form of its

Doob-Meyer decomposition. This formulation of B N will greatly aid us in our proof

of the convergence of the discrete forced polymer model to the continuous model.

Lemma 4.6. For an N -segment forced Kramers chain, the driving process BN (s)

can be expressed as

B N (s) = M N (s) + ζ κ p

[Ns/L ]

j =1

(Z N j )−1 f e T

3 −e3 f T Z N j

LN

(4.45)

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where M N (s) is a martingale and ζ is a dimensionless parameter.

Proof. Let us rst re-write BN in terms of a discrete index k, rather than a continuous

parameter s, because the Doob-Meyer decomposition requires a discrete index. Thus

B N k is given by

B N k =

1√ N

k

j =1

b j (φ j ) (4.46)

and we wish to decompose it as the sum

BN k = M

N k + A

N k (4.47)

where M N is a martingale, just like the driving process in the unforced model, and

AN k ∈ F k−1; that is, it is an adapted process. This decomposition can be found by

direct calculation. Consider the conditional expectation of each increment of BN k ,

with respect to the previous σ-eld:

E [B N k −B N

k−1|F k−1] = E [M N k −M N

k−1|F k−1] + E [AN k −AN

k−1|F k−1]. (4.48)

We choose the decomposition so that M N is a martingale, and AN is adapted. Thus,

the preceding equation simplies to

E [B N k −B N

k−1|F k−1] = AN k −AN

k−1. (4.49)

Therefore, the AN k can be calculated recursively, according to the following formula:

AN 0 = 0 (4.50)

AN k = AN

k−1 + 1√ N E [bk(φk)|F k−1] (4.51)

= 1√ N

k

j =1

E [b j (φ j )|F j−1]. (4.52)

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Thus, AN k ∈F k−1 as desired. To nd M N

k , simply subtract AN k from B N

k :

M N k = BN

k − 1√ N

k

j =1

E [b j (φ j )|F j−1] (4.53)

= 1√ N

k

j =1

(b j (φ j ) −E [b j (φ j )|F j−1]) (4.54)

and from the preceding equation, it is clear that M N k has mean 0, as do all of its

increments, since each increment M N k −M N

k−1 is equal to the corresponding increment

of BN

with its mean subtracted away. Thus M N

is a martingale, as desired.Therefore, to calculate the mean of BN

k , it suffices to know the mean of AN k . This

can be found by computing the conditional expectation of each innitesimal rotation

with respect to the preceding σ-eld. This calculation is best done entry-wise within

the matrix; for example, to nd E [bk(φk)|F k−1], we can calculate E [sinφk|F k−1] and

E [cosφk|F k−1]. To nd these conditional expectations, recall the denition of condi-

tional probability for random variables X and Y with joint density function f :

P (Y ∈dy|X = x) f (x, y) dxdy

f (x, y) dy dx (4.55)

that is, the ratio of the joint density of X and Y to the marginal density of X .

It would seem, then, that in the situation described above, X should stand for the

previous σ-eld, F k−1 (or the segment Qk , by the Markov property), and Y should

be the bounded measurable function (sine or cosine) of the angle φk .

However, the joint density function comes from the Boltzmann-Gibbs distribution,

which depends on the segments of the polymer. We can use the Markov property of

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the transition probabilities of the segments to make our calculations, which will yield

the following formula:

P (Y ∈dy|F k−1) = πf (Qk , dQk+1 ) · · ·π f (Q1, dQ2)/ Z N

πf (Qk , dQk+1 ) π f (Qk−1, dQk) · · ·π f (Q1, dQ2)/ Z N . (4.56)

This simplies to

P (Y ∈dy|F k−1) = πf (Qk , dQk+1 )

πf (Qk , dQk+1 ) (4.57)

and by using the formula for the transition probability and Lemma 4.2, we obtain

P (Y ∈dy|F k−1) = e−U k /τ dφk

e−U k /τ dφk(4.58)

so that the denominator has the appearance of a partition function for the segment,

Qk+1 . Here, Y can be any function of the torsional angles that govern this subpolymer,

but are independent of the prior subpolymer Rk , namely, the angles φk through φN −1.

However, we are interested only in trigonometric functions of φk . As an example, let

us consider in depth the top-right entry in E [bk(φk)|F k−1], that is, the conditionalexpectation of sin φk . (The conditional expectation of cos φk will turn out to be very

similar.) This is given by

E [sin φk|F k−1] = sin φk P (φk ∈dφk) (4.59)

= sin φk πf N (Qk , dQk+1 )

πf N (Qk , dQk+1 )

(4.60)

= sin φk eNf ·Q k +1 /τ π0

N (q k , dQk+1 )

eNf ·Q k +1 /τ π0N (q k , dQk+1 ) (4.61)

= sin φk eNf ·Q k +1 /τ dφk

eNf ·Q k +1 /τ dφk. (4.62)

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The last equation follows by Lemma 4.2. Thus, the conditional expectation can be

expressed as a ratio of integrals of functions of φk , in terms of φk . Recall that the

polymer segment in the exponent satises the relation

Qk+1 = Z k−1H k(φk)Q1 (4.63)

so the segment is a function of φk as well, and the result of the integration will be an

expression in terms of the previous k −1 torsional angles (namely, it will be in the

σ-eld F k−1).

This integral can be evaluated as follows: First consider the inner product f ·Qk+1

which appears in the exponent of the integrand. Then, introduce the notation

f k = ( xk , yk , z k) = Z −1k f (4.64)

to stand for a unit vector in the direction of the external force, under the same

rotation that sends Qk+1 back to the positive z -axis. Notice that f k ∈F k . The inner

product can then be expressed as

f ·Qk+1 = f ·(Z kQ1) (4.65)

= af ·(Z ke3) (4.66)

= a f f ·(Z k−1H ke3) (4.67)

= a f Z −1k−1 f ·(H ke3) (4.68)

= a f f k−1 ·(sin θ sin φk , −sin θ cos φk , cos θ) (4.69)

= a f [sinθ(xk−1 sin φk −yk−1 cosφk) + z k−1 cosθ]. (4.70)Thus, the lower integral in equation (4.62) equals

S eNf ·Q k +1 /τ dφk = eζz k − 1 cos θ T

exp(ζ sin θψk)dφk (4.71)

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where, for simplicity, we have made the substitutions

ζ = L f τ

(4.72)

and

ψk(φk) = xk−1 sin φk −yk−1 cosφk . (4.73)

Notice that ζ is a dimensionless parameter that measures the magnitude of the force

when contour length and temperature are xed.

Now, the integrand is an exponential that includes trigonometric functions, which is

impossible to integrate analytically. However, since the multiplier sin θ in the expo-

nent is small, on the order O(N −1/ 2), and we will eventually take the limit as N → ∞of the discrete model to obtain the continuous model, we can expand the integrand

in an asymptotic series, which will allow us to integrate the exponential. Recall that

the discrete process BN (s) is a sum of [Ns/L ] innitesimal rotations, with a factor

of N −1/ 2. Therefore, any terms in the asymptotic expansion for E [bk(φk)|F k−1] that

are smaller than O(N −1/ 2) can be ignored, as they will vanish in the limit. Let r j (N )

now stand for the remainder, consisting of all terms of order O(N − j ) or smaller, and

let µ = ζ sin θ, which is a dimensionless coefficient of order O(N −1/ 2). Notice that

µ ≤ ζκ√ N

. (4.74)

Then the integrand can be expressed as

eµψ k

= 1 + µψk + r1(N ). (4.75)By Taylor’s theorem, r 1(N ) has the form

r 1(N ) = µ2ψ2

k

2 ec (4.76)

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where c∈(0, µψ). Since |ψk| ≤2, this term is bounded above as follows:

|r 1(N )| ≤ 2e2µκ2ζ 2

N . (4.77)

Since the integral is over the whole circle T = [0, 2π), only the terms which have even

powers in both sine and cosine will have non-zero integrals. Therefore

Teµψ k dφk = T

1 + r1(N )dφk (4.78)

and the lower integral in (4.62) equals

TeNf ·Q k +1 /τ dφk = eL f zk − 1 cos θ/τ T

(1 + r1(N )) dφk (4.79)

= 2π exp(ζ z k−1 cosθ) + r1(N ). (4.80)

Notice that the integral, when evaluated, is in the σ-eld F k−1.

Now that we have derived a method for computing these integrals, let us use it to

calculate the conditional expectation,

E [sinφk|F k−1] = sin φk eNf ·Q k +1 /τ dφk

eNf ·Q k +1 /τ dφk, (4.81)

which is necessary for the Doob-Meyer decomposition of BN (s). The denominator

has been calculated, and we have shown that it equals 2 π exp(ζz k−1 cosθ) plus a rst-

order remainder term. The numerator can be calculated in a similar way, but the

integral involves the term sin φk ; that is, it can also be calculated as an asymptotic

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expansion. Notice that the multiplier exp( ζz k−1 cosθ) has been factored out of the

integral already:

Tsin φk exp(µψk)dφk = T

sin φk(1 + µψk + r1(N )) (4.82)

= Tµxk−1 sin2 φk + r1(N ) (4.83)

= (2 π)µxk−1

2 + r1(N ). (4.84)

The conditional expectation thus has the same factor, 2 π exp(ζ z k−1 cosθ), that ap-

pears in both the numerator and the denominator. Therefore, the desired conditional

expectation has the following simple formula:

E [sin φk|F k−1] = 2π(µxk−1/ 2 + r1(N ))

2π(1 + r1(N )) (4.85)

and this differs from µxk−1/ 2 by a small amount:

µxk−1

2 − µxk−1/ 2 + r1(N )

1 + r1(N )=

µxk−1/ 2(1 + r1(N )) −(µxk−1/ 2 + r1(N ))1 + r1(N )

(4.86)

= |r 1(N )||µxk−1/ 2 −1||1 + r1(N )|

(4.87)

which is itself of order O(N −1); call it r1(N ). When this conditional expectation is

summed, and N goes to innity, the remainder term will vanish:

κ√ N

N

k=1

µxk−1

2 + r1(N ) =

κ√ N

N

k=1

ζ 2

sin κ√ N

xk−1 + r1(N ) (4.88)

= ζκ2

2N

N

k=1

xk−1 + κ√ Nr 1(N ) (4.89)

= L

N ζ p

N

k=1

xk−1 + κ√ Nr 1(N ). (4.90)

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The remainder κ√ Nr 1(N ) is of order O(N −1/ 2), and so it vanishes in the limit, by

the following argument:

|κ√ Nr 1(N )| = κ√ N |1 −µxk−1/ 2||1 + r1(N )| |

r 1(N )| (4.91)

≤ κ√ N 1 + µxk−1/ 21 −r 1(N ) |r 1(N )| (4.92)

≤ 2κ√ N |r 1(N )| (4.93)

≤ 4e2µκ2ζ 2

√ N =

4e2ζ sin( κ/ √ N )κ2ζ 2

√ N . (4.94)

Thus for any > 0, if

N > 16e4ζ κ4ζ 4

2 , (4.95)

the remainder term is smaller than .

Similar formulas hold for the conditional expectations of the other trigonometric

functions of the torsional angles, and these can be used to calculate the Doob-Meyer

decomposition of the discrete driving process B N (s). The key is to examine integrals

of the type

Tg(φk) exp(µψk) dφk (4.96)

in which g(φk) is periodic on the interval T . We have already determined the integral

for the case in which g(φk) = sin φk . This gives the conditional expectation for

two of the non-zero entries in bk(φk). The other two entries require computation of

E [cosφk|F k−1], so that g(φk) = cos φk . As we did before, let us express the integrand

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in (4.96) in an asymptotic series:

cos φk exp(µψk) = cos ψk(1 + µφk + r1(N )) (4.97)

= cos φk + µ(xk−1 cosφk sin φk −yk−1 cos2 φk) + cos φkr 1(N ).

(4.98)

Recall that since the integral is over all of T = [0, 2π), only the terms that have even

powers in both sine and cosine will be non-zero. Thus, only the cos 2 term above will

have a sizable effect on the value of the integral, and we can conclude that

Tcos φk exp(µφk) dφk = T −µyk−1 cos2 φk + r1(N )cos φk dφk . (4.99)

But this integral can be calculated, and it equals

Tcos φk exp(µφk) dφk = −(2π)

µyk−1

2 + r1(N ) (4.100)

(following the error term analysis from the sine integral) and so the desired conditional

expectation, neglecting terms that are to small to affect the summation when AN k is

calculated, is

E [cosφk|F k−1] = −µyk−1

2 . (4.101)

Notice that this differs from the conditional expectation of sin φk by the fact that

this expression has −yk−1 where the latter has xk−1. From these two conditional

expectations, it is now possible to derive the formula for the conditional expectation

of each innitesimal rotation bk(φk) with respect to the previous σ-eld, as such:

E [bk(φk)|F k−1] = µ2

0 0 xk−1

0 0 yk−1

−xk−1 −yk−1 0

+ 1(N ) (4.102)

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From this conditional expectation, we can calculate the adapted process AN k that

appears in the Doob-Meyer decomposition of BN k , and which has the same mean as

B N k . Recall that

AN k =

1√ N

k

j =1

E [b j (φ j )|F j−1] (4.108)

and dene

AN k =

1√ N

k

j =1

b j . (4.109)

Notice that the difference between the above sums is small:

AN k − AN

k = 1√ N

k

j =11(N ) =

k√ N

1(N ) (4.110)

whose norm is of order at most N −1/ 2. Thus, we can substitute AN k for AN

k in our

calculations, since they will be equal in the limit.

By summing all the conditional expectations of the innitesimal rotations, and di-

viding by √ N , we obtain the following formula for the adapted part of the driving

process:AN

k = µ2√ N

k

j =1

[Z −1 j−1 f , e 3] (4.111)

= ζ sin κ√ N

12√ N

k

j =1

[Z −1 j−1 f , e 3] (4.112)

= ζκ2N

k

j =1

[Z −1 j−1 f , e 3] + r2(N ) (4.113)

= ζ κ p

LN

k

j =1[Z −

1 j−1 f , e 3] + r2(N ) (4.114)

using the rst-order approximation for sin( κ/ √ N ). This remainder will also vanish

in the limit. Thus, returning to our denition of BN (s) in terms of a continuous

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parameter (by letting k = [Ns/L ]), we obtain the desired formula for the Doob-

Meyer decomposition.

Once we know the structure of B N (s), we can use it to dene the process of orthonor-

mal frames, Z N (s), according to a stochastic differential equation:

dZ N (s) = Z N (s) κdB N (s) + 12

d[κB,κB ]s + r(N ) (4.115)

(recall here that r (N ) is the increment of the cubic and higher order variations of B ,

which vanish in the limit). This can also be written

∂Z N (s) = κZ N (s)∂B N (s) + r(N ) (4.116)

or in integral form,

Z N (s) = I + κ s

0Z N (σ)∂B N (σ) (4.117)

by the rule relating the Itˆ o and Stratonovich derivatives of stochastic processes, since

our initial condition is Z (0) = I .

4.3 Convergence to the Forced Kratky-Porod Model

We can now dene the limiting process B (s) by simply taking the pointwise limit of

B N (s) for each xed s as N → ∞. We will show, in fact, that it is a Brownian motion

with a non-linear drift. However, the analogous limiting rotation process, Z (s), can

only be dened as the solution to the limiting differential equation, which is the limit

of the above discrete equation, if we wish for this relationship between the B and Z

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processes to continue as the number of the segments of the polymer goes to innity.

The limiting stochastic differential equation is

dZ (s) = κZ (s)∂B (s), (4.118)

and it is often written in integral form:

Z (s) = I + κ s

0Z (σ)∂B (σ). (4.119)

We wish to show that the processes Z N converge weakly to Z . This is done, as in

Chapter 2, by means of the theorem (2.1) of Kurtz and Protter, which yields the

convergence in distribution of processes that are solutions of stochastic differential

equations under certain conditions. With the next few lemmas, we will prove that the

conditions of the theorem are satised. The rst step is to prove that the martingale

parts M N converge weakly to a Gaussian limit M , by means of a martingale version

of the Central Limit Theorem.

Lemma 4.7. The martingale part M N (s) of BN (s) converges in distribution to a

Gaussian matrix-valued process.

Proof. Our goal is to show that the martingale part, M N , when the limit as N → ∞is taken, is identical to the Gaussian process that governed the motion of the tangent

vector along the polymer in the unforced model. Recall that this process was matrix-

valued, and its non-zero entries were independent Ornstein-Uhlenbeck sheets. If the

dependence on time is suppressed, then the entries are Brownian motions in the arc

length parameter s. To do this, it is necessary to show that the discrete processes

satises conditions for a version of the Central Limit Theorem, so that as N → ∞, the

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discrete processes can converge weakly to a continuous Gaussian process. Since the

innitesimal rotation matrices, properly scaled and with the mean subtracted away,

are the increments of the martingale part M N , we must examine the conditional mean

and variance of each of these matrices with respect to the previous σ-eld. The mean

of the matrices has been calculated in equation (4.107). Now, let us nd the second

moment:

E [bk(φk)2|F k−1] = E

0 0 sinφk

0 0 −cos φk

−sin φk cosφk 0

2

F k−1 (4.120)

= E −sin2 φk sin φk cosφk 0

sin φk cosφk −cos2 φk 0

0 0 −1F k−1 .(4.121)

Therefore, we must compute the conditional expectations of the functions sin 2 φk ,

cos2 φk , and sin φk cosφk . As before, we will expand the integrand in an asymptotic

series and discard all terms that are smaller than O(N −1/ 2).

To begin, consider

E [sin2 φk|F k−1] = T sin2 φk exp(µψk)dφk

T exp(µψk)dφk. (4.122)

Again, we seek the terms in the expansion which, when multiplied by sin 2 φk , have

even powers in both sine and cosine. This yields

E [sin2 φk|F k−1] = 12π T

sin2 φk + 12

µ2(x2k−1 sin4 φk + y2

k−1 sin2 φk cos2 φk) + r2(N ) dφk

(4.123)

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where r2(N ) is a remainder term of order N 2. This integrates to

E [sin2 φk|F k−1] = 12

+ µ2 316

x2k−1 + 1

16y2

k−1 + r2(N ). (4.124)

In a similar way, we nd that

E [cos2 φk|F k−1] = 12π T

cos2 φk + 12

µ2(y2k−1 cos4 φk + x2

k−1 sin2 φk cos2 φk) + r2(N ) dφk

(4.125)

which integrates to

E [cos2

φk|F k−1] =

1

2 + µ2 1

16x2

k−1 +

3

16y2

k−1 + r2(N ). (4.126)The other conditional expectation, that of sin φk cosφk , is different, as we must use

the sin φk cosφk term that appears in ψ2k , but it is no more complicated:

E [sinφk cosφk|F k−1] = 12π T

[µ2xk−1yk−1 sin2 φk cos2 φk + r2(N )]dφk . (4.127)

Evaluating the integral yields the formula

E [sinφk cosφk|F k−1] = 18

µ2xk−1yk−1 + r2(N ). (4.128)

Let 2(N ) denote the matrix consisting of the second-order error terms that appear

in the entries above. Therefore, we have the following formula for the conditional

second moment of the matrix bk(φk):

E [b2k(φk)|F k−1] =

−12 0 0

0 −12 0

0 0 −1

+ 2(N ) (4.129)

+µ2

16

−3x2k−1 −y2

k−1 2xk−1yk−1 0

2xk−1yk−1 −x2k−1 −3y2

k−1 0

0 0 −4(x2k−1 + y2

k−1)

.

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As before, let us call the rst matrix on the right-hand side of the above equation D,

since it is a diagonal matrix. If we subtract off the square of the conditional mean as

in (4.102), we obtain the conditional variance of bk . First, compute the square of the

mean:

b2k = −

µ2

4

x2k−1 xk−1yk−1 0

xk−1yk−1 y2k−1 0

0 0 x2k−1 + y2

k−1

. (4.130)

Therefore, the variance is

Var[ bk(φk)|F k−1] = D + µ2

16

x2k−1 −y2

k−1 6xk−1yk−1 0

6xk−1yk−1 y2k−1 −x2

k−1 0

0 0 0

+ 2(N ). (4.131)

The difference between the conditional variance and the matrix D is small:

Var[ bk(φk)|F k−1]−D ≤ 3µ2

8 + r2(N ) (4.132)

≤ 3ζ 2L4N p

(4.133)

so the conditional variance has the form D + 1(N ), where 1 is a matrix whose norm

is of order O(N −1), which will aid in proving that the conditions for the Central Limit

Theorem are satised.

To show that the martingale version of the CLT holds, for each bk / √ N , we can

subtract off its conditional expectation, with respect to the σ-eld

F k

−1, and make

that the element in our triangular array. For notational simplicity, let

mk = bk −E [bk|F k−1]. (4.134)

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Then, each entry in the triangular array will be mk / √ N , and we can use an appropri-

ate form of the Central Limit Theorem to prove that M N (s), the martingale part of

B N (s), converges in distribution to a Gaussian process (the convergence is considered

one entry at a time).

There are several forms of the Central Limit Theorem for triangular arrays, and even

some that apply to conditional expectations. For example (from [31] p. 333):

For each N ≥ 1, let S N =

kN i=1 X N,i , F N,j , 1 ≤ j ≤ kN < ∞ be an L2

stochastic sequence withkN

j =1

Var[ X N,j |F N,j −1]−u2N → 0 in prob. for some u2

N ∈ F N, 1

(4.135)kN

j =1

E [X 2N,j 1 X N,j > ] → 0 in prob. (4.136)

kN

j =1

E [X N,j |F N,j −1] → 0 in prob. (4.137)

kN

j =1

Var[ X N,j |F N,j −1] → σ2. (4.138)

Then S N → N (0, σ2).

To show how this theorem applies, let us rst consider what each of the symbols means

in the context of our problem. X N,k , an element in the triangular array, represents

mN k / √ N . The row sum S N is identical to M N (s), and the number of terms in each

row, kN , is equal to [Ns/L ].

Now, let us check the conditions of the theorem. The Lindeberg-Feller condition

(4.136) holds because for any xed , if N > −2, then X N,j = N −1/ 2 is always

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less than . If we let both u2N = D/N for all N and σ2 = D(s/L ), then conditions

(4.135) and (4.138) are both satised, since the higher order terms in µ go to 0

in both conditions. To see this, recall that Var[ X N,j |F N,j −1] = 1N Var[ b j |F j−1] =

(D + 1(N ))/N , which is the same as uN + 1(N )/N , the sum of uN and a term of

order N −2. Likewise, for condition (4.138), each term is ( D + 1(N ))/N , which, when

summed, is D(s/L ) + 1(N ). The only remaining condition is (4.137), but this is

trivial, because each increment already has mean 0 by its construction. Therefore,

the theorem applies to the processes, and the limit M (s) is a matrix-valued Gaussianprocess with mean 0 and variance D(s/L ).

Now that we have weak convergence of the martingale part of the driving process B ,

we must show the same for the entire process, so that we can use Theorem 2.1. The

next step is to prove relative compactness of both the B N and the Z N .

Lemma 4.8. The sequence of pairs of processes (BN

(s), Z N

(s))N ∈

N is relatively com-pact in the Skorohod topology.

Proof. First, consider for each N the Doob-Meyer decomposition of B N into the sum

of a martingale M N and an adapted process AN . Recall that

AN (s) = 1√ N

[Ns/L ]

j =1

E [b j |F j−1] (4.139)

M N (s) = 1

√ N

[Ns/L ]

j =1b j −E [b j |F j−1]. (4.140)

Let us rst prove relative compactness for the sequence of processes ( B N ): this con-

sists of proving both a uniform stochastic boundedness condition and a stochastic

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equicontinuity condition. First, notice that by Lemma 4.7, the sequence of martin-

gale parts ( M N ) is already relatively compact, since it has a weak limit M . In order

to place a stochastic bound on B N and its increments, we can use the bound that we

already have for M N and nd one for AN as well. We know that

AN (s) = ζ κ p

LN

[Ns/L ]

j =1

[(Z N j )−1 f , e 3]. (4.141)

The expression in the sum is the matrix ( Z N j )T f e T

3 −e3 f T Z N j , which has norm at

most 2, since the vectors whose outer product makes the matrix are unit vectors.

Therefore, by the triangle inequality,

AN (s) ≤ 2ζsκ p

, (4.142)

and since s∈[0, L], the adapted part has a uniform bound of κ f L/τ . In a similar

fashion, we can bound the increment in AN between two points s and s on the curve

(here, as before, we take s < s ):

AN (s) −AN (s ) = ζ κ p

LN

[Ns /L ]

j =[ Ns/L ]+1

[(Z N j )−1 f , e 3] (4.143)

≤ 2ζ κ p |s −s |. (4.144)

and thus the aggregate size of the jumps in the adapted process is bounded above by

a constant multiple of the increment in the arc length. Therefore, both conditions of

relative compactness are satised for the adapted part of B N , and in turn, for all of

B N itself.

Next, we must show that the discrete processes of rotations Z N (s) are also relatively

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compact. In terms of the Doob-Meyer decomposition, we can express Z N by this

stochastic differential equation:

dZ N (s) = κZ N (s)dM N (s) + κZ N (s)dAN (s) + κ2

2 d[M, M ]s . (4.145)

Recall that since the adapted part AN has nite variation, the quadratic variation of

B N is precisely that of M N . The stochastic uniform boundedness of the Z N is trivial,

since all rotations have norm 1. Thus, we must show equicontinuity of each of the

three terms on the right side of equation (4.145). The key to this is a theorem of Ethier and Kurtz [24] which states that a family ( Z N ) is stochastically equicontinuous

if and only if ∀η > 0, L > 0, ∃δ > 0 such that

supN

P (w (Z N ,δ ,L) ≥η) ≤η. (4.146)

The quantity w is a modulus of continuity, and Ethier and Kurtz dene it thus:

w (Z N

,δ ,L) = inf s i maxi sups,s ∈[s i − 1 ,s 1 ) Z N

(s) −Z N

(s) . (4.147)

That is, w is the inmum over all partitions 0 = s0, s1, . . . , s N = L, of the maximum

over intervals in the partition, of the supremum over pairs of points s, s within the

interval, of the distance between the values of the process at the points s and s .

Let us rst nd a bound for the modulus of continuity for the two terms involving

M N . Denote by Y N the part of Z N that depends on M N only; that is,

dY N (s) = κY N (s)dM N (s) + κ2

2 Y N (s)d[M N , M N ]s . (4.148)

Suppose that s and s are two points in the same subinterval of the partition taken

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in the modulus of continuity w , and that s < s . Thus, we seek to put a stochastic

bound on the quantity

Y N (s) −Y N (s ) = κ s

sY N (σ)dM N (σ) +

κ2

2 s

sY N (σ)d[M N , M N ]σ . (4.149)

These integrals are non-anticipating, and the factor Y N (σ) that appears in each

integral is a rotation that does not affect the size of the increment. Since the processes

are discrete in s, we can rewrite the above equation in terms of a sum:

Y N (s)−Y N (s ) = κ[Ns /L ]

k=[ Ns/L ]+1

Y N k (M N

k+1 −M N k ) +

κ2

2

[Ns /L ]

k=[ Ns/L ]+1

Y N k (M N

k+1 −M N k )2 ,

(4.150)

and the rst term is the incremental product of processes, ( Y ·M ) from s to s .

Since Y is adapted to M in the product, and Y ≤ 1, it follows that Y ·M ≤M (s ) −M (s) . Thus, the matrix Y N

k does not change the norm of each increment,

and it can be removed from the norm to obtain

Y N (s) −Y N (s ) ≤κ[Ns /L ]

k=[ Ns/L ]+1

M N k+1 −M N

k + κ2

2

[Ns /L ]

k=[ Ns/L ]+1

(M N k+1 −M N

k )2 .

(4.151)

Notice that the rst sum is telescoping, and the second sum gives the quadratic

variation of M N , which in the Gaussian limit, is directly proportional to the arc length

distance |s −s |. Now, let us assume that if N > N 0, then M N (s) −M N (s ) ≤2 M (s) −M (s ) ; that is, the variation in the discrete process is less than twice thatof its Gaussian limit. Then it follows that

Y N (s) −Y N (s ) ≤2κ M (s) −M (s ) + κ2

2L |s −s |. (4.152)

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It is evident that the rst term on the right side is a random variable, which must be

stochastically bounded, while the second term is deterministic, and goes to zero just as

quickly as the increment in arc length does. To place a bound on the entire expression,

we can use the same method as before, except now that we nd a stochastic bound

for the increment of B. If we let χ stand for a standard normally distributed random

variable, then

P (2κ M (s) −M (s ) > a

|s −s |/ 2L) = P (|χ | > a ) (4.153)

and given any η > 0, there exists a(η) > 0 that makes the above probability less than

η/ 2, and if we choose s and s so that |s −s |(2L/κ 2) < η/ 2, we conclude that

P ( M N (s) −M N (s ) > η ) < η (4.154)

as desired.

Thus, we have established a stochastic bound for Z N (s) −Z N (s ) if Z N depends only

on the martingale part M N . It suffices now to show that the increment Z N dependingon the adapted part is also stochastically bounded, namely:

κ[Ns /L ]

j =[ Ns/L ]+1

Z N j (AN

j +1 −AN j ) =

ζL pN

[Ns /L ]

j =[ Ns/L ]+1

Z N j [(Z N

j )−1 f , e 3] (4.155)

is bounded above by a constant multiple of |s −s |. But this is true, because we

can use the triangle inequality on the increments in the sum: Z N j = 1 always,

and

[(Z N

j )−1 f , e

3]

≤2, since it is the difference of tensor products of unit vectors.

Therefore,

κ[Ns /L ]

j =[ Ns/L ]+1

Z N j (AN

j +1 −AN j ) =

ζ p |s −s | (4.156)

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and the relative compactness of Z N (s) for the forced model follows.

Next, we must show that BN converges to B weakly, which is the second condition

for the Kurtz-Protter theorem. Since we have shown the weak convergence of the

martingale part M N (s) to a Gaussian limit (call it M (s)), we must show the same

for the adapted part AN (s). In doing so, we will rst show convergence of the AN

and Z N together along subsequences. Since the processes AN , M N , and Z N satisfy

a stochastic differential equation, and the M N converge weakly to a limit M , we will

show that weak convergence of AN and Z N will follow, using the next three lemmas.

First, notice that existence of a subsequence along which the AN k converge follows

from relative compactness of the AN (as part of B N ) from Lemma 4.8.

Lemma 4.9. Let (N k) be the indices of a subsequence along which (B N k , Z N k ) con-

verges weakly to a limit (B, Z ). Then along the same subsequence, the processes

AN k (s) converge weakly to a limit A, satisfying the equation

A(s) = ζ κ p

s

0[Z (σ)−1 f , e 3]dσ. (4.157)

Proof. First, let ( B N k ) be a subsequence which converges to a limit B in distribution.

Then, since for all k, BN k = M N k + AN k , and M N k⇒ M by Lemma 4.7 above, it

follows that the subsequence ( AN k ) converges weakly to a limiting process; call it A.

Recall that if ∆ s = L/N , then AN k and Z N k satisfy

AN k (s) = ζ κ p

[Ns/L ]

j =1

[(Z N k j )T f , e 3]∆ s (4.158)

AN k (s) = ζ κ p

s

0[Z N k (σ)T f , e 3]dσ. (4.159)

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Since the processes AN k and Z N k are cadlag , the discrete sum can be expressed as

the integral of a piecewise constant function. Also, we have replaced Z −1 with Z T ,

since Z is an orthogonal matrix. This will simplify the next step of the proof.

We now claim that for all k, AN k is a continuous bounded transformation of Z N k .

The boundedness criterion is immediate from equation (4.141), so AN ≤ 2ζsκ p

Z N ,

and continuity is straightforward as well:

A(Z ) −A(Z ) = ζ κ p

s

0

[(Z (σ) −Z (σ))T f , e 3]dσ (4.160)

≤ 2ζs

κ pZ −Z . (4.161)

Therefore, the continuity theorem of weak convergence (Theorem 2.3 of Chapter 2 of

[32]) holds, and so equation (4.157) is true for the subsequential limits A and Z .

Thus, we see that the martingale part M N of B N converges weakly to a limit M , and

the bounded variation part AN retains its relationship with Z N along subsequences.

There now remains one condition to be proven to use Theorem 2.1 for subsequences:

that the sequence of processes B N is good , in the language of [23]. In the same paper,

Kurtz and Protter show that a sequence of stochastic processes is good if and only if

it is of uniformly controlled variation. This is the condition that we will prove.

Lemma 4.10. The sequence of processes (B N )N ∈N is of uniformly controlled varia-

tion (UCV), that is, for all N ∈N ,

supN

E N [M N , M N ]s + E N s

0dAN (s) < ∞. (4.162)

Proof. Note that each of these expressions above, that is, both the quadratic variation

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of the martingale part and the total variation of the adapted part, can be evaluated.

First,

E N [M N , M N ]s = 1N

[Ns/L ]

j =1

E (b j −E [b j |F j−1])2 (4.163)

= 1N

[Ns/L ]

j =1

Var[ b j ] (4.164)

= 1N

[Ns/L ]

j =1

D + 1(N ) (4.165)

= sL

(D + 1(N )) < ∞. (4.166)

Next,

E N s

0dAN (s) = E N 1

√ N

[Ns/L ]

j =1

E [b j |F j−1] (4.167)

= 1√ N

[Ns/L ]

j =1

E µ2

[(Z N j−1)−1 f , e 3] (4.168)

= µ2√ N

[Ns/L ]

j =1

E [ [(Z N j−1)−1 f , e 3] ] (4.169)

≤ Lζ Nκ p

[Ns/L ]

j =1

2 (4.170)

≤ 2sζ

κ p< ∞. (4.171)

Since these bounds hold for all natural numbers N , the sequence ( B N ) is UCV, as

desired.

Therefore, the Kurtz-Protter theorem holds for subsequences. With these lemmas in

place, we are now ready to show that convergence of the BN and Z N holds along

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the entire sequence, namely, that all subsequential limits are the same, and that the

limiting processes B and Z satisfy the limiting differential equation.

Lemma 4.11. The sequence of processes of orthonormal frames along the polymer

in the forced Kramers chain model, (Z N )N ∈N , converges weakly to the process of

orthonormal frames along the polymer in the forced Kratky-Porod model, Z .

Proof. Let us rst describe the differential equation for BN and Z N in terms of the

Doob-Meyer decomposition:

∂Z N (s) = κZ N (s)∂M N (s) + κZ N (s)∂AN (s). (4.172)

Let (A1, Z 1) and (A2, Z 2) be two subsequential limits of the sequence ( AN , Z N ). Then,

by Lemmas 4.7 and 4.9, and by the theorem in [23], these two subsequential limits

satisfy

∂Z 1(s) = κZ 1(s)∂M (s) + κZ 1(s)∂A1(s) (4.173)∂Z 2(s) = κZ 2(s)∂M (s) + κZ 2(s)∂A2(s) (4.174)

dA1(s) = ζ κ p

[Z 1(s)T f , e 3]ds (4.175)

dA2(s) = ζ κ p

[Z 2(s)T f , e 3]ds (4.176)

and so (A1, Z 1) and (A2, Z 2) are solutions of the same coupled system of stochastic

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differential equations, driven by the same process M . But the solutions to stochastic

differential equations are unique. Therefore, ( A1, Z 1) = ( A2, Z 2), and it follows that

AN ⇒ A (4.177)

B N ⇒ B (4.178)

Z N ⇒ Z (4.179)

and the limiting processes B and Z satisfy the SDE

∂Z (s) = κZ (s)∂B (s), (4.180)

as desired.

4.4 Characterization of the Tangent Vector Process

This last lemma (4.11) gives us the convergence of the processes of orthonormal

frames (and by projection, of tangent vectors along the molecule) which will greatlyaid us in proving the convergence of the discrete forced model to the continuous forced

model. As before, the tangent vector to the polymer satises a stochastic differential

equation. However, since the driving process B(s) is no longer a martingale, but

a semimartingale, and its adapted part leads to a drift, the equation governing the

tangent vector will include a drift term.

To derive this equation, we must rst show that the martingale part M (s) of B(s)

denes a Brownian motion. We have already established, in Lemma 4.7, that it is a

Gaussian matrix-valued process. Now, we will prove a statement about the entries of

the matrix.

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Lemma 4.12. The martingale part M (s) of the driving process B(s) has non-zero

entries that are scaled independent one-dimensional Brownian motions.

Proof. The key to this proof is to show that the process M is continuous in s, and

that its entries have the same mean and variance as a Brownian motion. Then,

Levy’s theorem tells us that the resulting process in each entry of the matrix M is a

Brownian motion.

Let us begin by considering the distance between two points of the discrete process

M N (s), and then take the limit as N → ∞. Let N ∈ N , and ∆ s = L/N as before.

Then, the difference between M N (s + ∆ s) and M N (s) is bounded as follows:

M N (s + ∆ s) −M N (s) =1N

(b[Ns/L ]+1 −E [b[Ns/L ]+1 |F [Ns/L ]]) (4.181)

≤ 1N

b[Ns/L ]+1 + κ f

2τ [Z N (s)−1 f , e 3] ∆ s(4.182)

≤ 1√ N

+ ζ κ p

∆ s. (4.183)

If ∆ s is xed and N → ∞, we have

M (s + ∆ s) −M (s) ≤ ζ κ p

∆ s. (4.184)

Thus, we have the desired continuity property, which can be seen as follows: let > 0.

Then if we let δ = ( κ p/ζ ) , if |s −s | < δ , then M (s) −M (s ) < . Now, we can

nd the covariance of the process M (s) and show that it matches that of a Brownian

motion. Recall that B(s), as well as both of its components in the Doob-Meyer

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decomposition, is an anti-symmetric matrix whose non-zero entries are in the third

row or third column, but not both. Thus M (s) can be expressed as follows:

M (s) =

0 0 β 1(s)

0 0 β 2(s)

−β 1(s) −β 2(s) 0

(4.185)

where the processes β i(s) for i = 1, 2 are continuous Gaussian scalar-valued processes,

as shown before. We wish to show now that the β i(s) are independent Brownian

motions; that is, for some positive constant C ,

E [β i(s)β j (s )] = Cδ ij (s∧s ). (4.186)

To prove this claim, we will consider each case of the values of i and j individually,

and without loss of generality, let s < s . If i = j = 1, then we will calculate the

auto-correlation of β 1, which has the following form:

β 1(s) = limN →∞

1√ N

[Ns/L ]

k=1

sin φk − µ2

xk−1 . (4.187)

Recall that µ = O(N −1/ 2), so the second term in the sum is smaller than the rst by

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a factor of O(N −1/ 2). Thus we can compute the auto-correlation function:

E [β 1(s)β 1(s )] = limN →∞

1N

E [Ns/L ]

j =1

sin φ j − µ2

x j−1

[Ns /L ]

k=1

sin φk − µ2

xk−1

(4.188)

= limN →∞

1N

[Ns/L ]

j =1

[Ns /L ]

k=1

E sin φ j − µ2

x j−1 sin φk − µ2

xk−1

(4.189)

= limN →∞

1

N

[Ns/L ]

j =1

[Ns /L ]

k=1

E sin φ j sin φk

− µ

2(x j

−1 sin φk + xk

−1 sin φ j )

+µ2

4 x j−1xk−1 . (4.190)

If j = k, then the summand simplies to

E sin2 φk −µxk−1 sin φk + µ2

4 x2

k−1

and the number of such terms is [ Ns/L ], since s < s . Thus, since there is a factor

of N −1 outside the sum, only the terms inside which are O(1) will have any effect on

the total as N → ∞. Namely, only the E [sin2 φk] will affect the sum, since it is equal

to 12 + r 1(N ), and the other terms contain negative powers of N , since µ = O(N −1/ 2).

Thus we have

limN →∞

1N

[Ns/L ]

k=1

E sin2 φk −µxk−1 sin φk + µ2

4 x2

k−1 = s2L

(4.191)

and this is true because the difference δ between N −1 times the sum and s/ 2L is

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By a similar argument, the above result holds for E [β 2(s)β 2(s )] as well; we can

apply the same reasoning to the proof above, while replacing β 1 with β 2, sin φk with

−cos φk , and xk−1 with yk−1. Thus it remains only to show that E [β 1(s)β 2(s )] = 0.

By a similar calculation to the one above, we nd that

E [β 1(s)β 2(s )] = limN →∞

1N

[Ns/L ]

j =1

[Ns /L ]

k=1

E sin φ j − µ2

x j−1 cos φk + µ2

yk−1 .

(4.200)

In this case, if j = k, the summand is

E [sin φk cosφk + r1(N )].

However, we have shown, in equation (4.128), that this expected value is of order

O(N −1) (call it η/N ). Thus, every term in the sum of [ Ns/L ] terms is too small to

be counted in the sum when the limit as N → ∞ is taken; that is,

limN →∞1N

[Ns/L ]

j =1

ηN = limN →∞

ηsLN = 0. (4.201)

If, however, j = k, the summand can be written as

−E [(sin φ j −E [sinφ j |F j−1]) (cos φk −E [cosφk|F k−1])]

and by taking the nested conditional expectation of this expression with respect to

the larger of the two σ-elds F j−1 and F k−1, we obtain, by a similar argument as

before, that each term in the sum for j = k is exactly 0.Therefore,

E [β 1(s)β 2(s )] = 0, (4.202)

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with w1 and w2 satisfying

E [wi(s)w j (s )] = δ ij (s∧s ), (4.206)

namely, that the wi are independent one-dimensional Brownian motions. Since the

matrix W (s) is standardized, that is, its non-zero entries are standard normally dis-

tributed random processes in s (namely N (0, s)), we will refer to it in future calcu-

lations.

Therefore, we have shown in this section that the discrete processes which govern

the orientation of the polymer, namely, the driving process BN and the process of

rotation matrices Z N , converge in distribution to limiting processes B and Z , and

the process B satises the equation

B(s) = 1√ 2L

W (s) + ζ κ p

s

0[Z (σ)−1 f , e 3]dσ. (4.207)

Now that we have expressed B(s) in terms of the independent standard Brownian

motions w1(s) and w2(s), let us do the same for the tangent vector process T (s),

which will lead to the desired differential equation.

Lemma 4.14. The tangent vector process T (s) for the forced Kratky-Porod model

satises the following differential equation in terms of the matrices Z (s) and W (s):

dT (s) = ζ

p (I −T (s)⊗T (s)) fds + 1

p Z (s)∂W (s)e3. (4.208)

Proof. Previously, we dened the Bishop frame to be the unit vector T tangent to the

curve representing the polymer, and M 1, M 2to be its normal development, that is,

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a pair of orthonormal vectors which span the plane normal to T at any given point

on the polymer. The three vectors are mutually orthogonal and have unit length,

and together they comprise the matrix-vaued process Z (s). Together, these vectors

satisfy the stochastic differential equation

∂ ∂s

(M 1(s)|M 2(s)|T (s)) = κ(M 1(s)|M 2(s)|T (s)) ∂ ∂s

0 0 β 1(s)

0 0 β 2(s)

−β 1(s) −β 2(s) 0

(4.209)with initial conditions

T (0) = e3 (4.210)

M 1(0) = e1 (4.211)

M 2(0) = e2. (4.212)

In fact, the initial direction of M 2 does not need to be specied, because for any s, itsatises the relation

M 2(s) = T (s) ×M 1(s) (4.213)

by the orthogonality of the vectors. This shows how the orthonormal frame of vectors

moves along the length of the polymer through parallel transport, and the equation

is identical to the stochastic differential equation for Z (s) in Stratonovich form.

This differential equation for Z (s) transforms into one for T (s) by simply multiplying

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each matrix that appears in the equation by the initial direction vector T (0), or e3

as we dene it:

dT (s) = κZ (s)∂B (s)e3 (4.214)

= κZ (s)(∂A(s) + ∂M (s))e3 (4.215)

= ζ

pZ (s)[Z (s)−1 f , e 3]e3ds +

κ√ 2L

Z (s)∂W (s)e3. (4.216)

Thus, the Doob-Meyer decomposition expresses the differential of the driving process

as the sum of a drift term and a diffusion term (involving the Brownian motion W (s)).

The matrix expression involving the outer product commutator in the drift term can

be simplied:

Z (s)[Z (s)−1 f , e 3]e3 = Z (s)(Z (s)T f e T 3 −e3 f T Z (s))e3 (4.217)

= f −T (s) f T T (s) (4.218)

= f −T (s)(T (s)T f )T (4.219)

= ( I −T (s)⊗T (s)) f . (4.220)

Therefore, the drift term is proportional to the projection of the unit vector in the

direction of the force, f , onto the plane perpendicular to the current tangent vector.

This plane is equal to the normal development, and this yields the following simplied

differential equation for T (s):

dT (s) = ζ

p(I −T (s)⊗T (s)) fds +

κ√ 2L

Z (s)∂W (s)e3. (4.221)

But recall that the persistence length is related to the curvature parameter κ: p =

2L/κ 2. Thus, the coefficient on the Brownian motion term is −1/ 2 p , and the equation

(4.221) becomes (4.208), as desired.

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This equation, however, is not self-contained, since Z (s) appears on the right side,

and it contains the normal development vectors M 1 and M 2 in addition to T . Also,

we wish to express the tangent vector in terms of the spherical Brownian motion

Q(s). However, by an argument of differential geometry, we can show that the part

of equation (4.208) containing the diffusion term is generated by the Laplace-Beltrami

operator on the sphere, thereby equaling a spherical Brownian motion in distribution,

according to the version of Levy’s theorem for Riemannian manifolds.

Lemma 4.15. The diffusion term in equation (1.16) is equal in distribution to a

scaled spherical Brownian motion.

Proof. As with the proof of the previous case (with no force), we seek to use the Itˆo

formula to show that the differential equation for T (s) yields the Laplace-Beltrami

operator, which generates the spherical Brownian motion, with drift as the remainder.

Let Φ : S 2 → R be a C 2 function on the unit sphere that extends to a function

F : R 3 → R which is invariant in the distance from the origin; that is, in spherical

coordinates, F (r,θ,φ ) depends only on θ and φ. Then if we compose Φ with the

vector-valued process T (s) and apply the Itˆo formula, we obtain

dΦ(T (s)) = ∇Φ(T (s)) ·dT (s) + 12

dT (s)T H Φ(T (s))dT (s). (4.222)

The rst term on the right side of (4.222) is known, since we have a formula (4.208)

for the differential of the tangent vector T (s). The second term, however, appearsto be more complicated, as it involves the Hessian matrix of Φ, being multiplied on

both sides by the differential of T (s). But this can be re-written as the sum of each

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second-order derivative of Φ, multiplied by each component of the quadratic variation

matrix of the vector T :

dT (s)T H Φ(T (s))dT (s) =3

i=1

3

j =1

Φi,j (T (s))d[T i , T j ]s . (4.223)

This simplies the second term on the right side of (4.222), because not every com-

ponent of the quadratic variation of T is non-zero. First, recall that in equation

(4.208), the differential of T (s) has two terms, one multiplied by the scalar ds, and

the other by the matrix dW (s). Since the arc length s is of bounded variation, the

only non-zero quadratic variation arises from the square of the dW (s) term. Recall

that the non-zero entries of W (s) are independent Brownian motions, so that the

quadratic variation of each pair of entries satises this equation:

dwi(s)dw j (s) = δ ij ds. (4.224)

Thus, the differential of the quadratic variation of T (s) is given by

d[T, T ]s = 1

p Z (s)dW (s)e3eT 3 dW (s)T Z (s)T (4.225)

= 1

pZ (s)diag(1 , 1, 0)Z (s)−1ds (4.226)

= 1

p(M 1(s)⊗M 1(s) + M 2(s)⊗M 2(s))ds. (4.227)

This is true because the matrices in the calculation have special properties: Z (s)

is orthogonal, and W (s) is antisymmetric, so the product on the right side can be

simplied:

dW (s)e3eT 3 dW (s)T = −

−(dw1)2 −dw1dw2 0

−dw1dw2 −(dw2)2 0

0 0 0

(4.228)

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and by the independence of the Brownian motions, this matrix is equal to ds diag(1, 1, 0).

The last equation in the array follows from the fact that the rst two columns of Z (s)

are M 1(s) and M 2(s), respectively. Thus

Z (s)diag(1 , 0, 0)Z (s)T = M 1(s)M 1(s)T (4.229)

Z (s)diag(0 , 1, 0)Z (s)T = M 2(s)M 2(s)T (4.230)

and the sum of the two has an equivalent formulation in terms of the tangent vector:

d[T, T ]s = ( I −T (s)⊗T (s)) ds. (4.231)

Notice that this expression also appears in the ds (drift) term of the differential of

T in equation (4.208). However, this matrix must be multiplied component-wise by

the Hessian matrix of second-order partial derivatives of the function Φ( T (s)). These

partial derivatives must be directional derivatives with respect to an orthonormal

basis.

Rather than take the canonical basis e1, e2, e3 as the set of vectors with respect to

which we calculate the partial derivatives, a better choice is the Bishop frame, because

the matrix of the derivative of the quadratic variation of T , d[T, T ]s , is expressed in

terms of the Bishop frame. The two orthonormal bases are related to each other in

this way:

(M 1(s)|M 2(s)|T (s)) = Z (s)(e1|e2|e3) (4.232)

and the Hessian, in which the directional derivatives are taken with regard to the

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Bishop frame, gives a simple result when multiplied entry-wise by the quadratic

variation increment, and the entries are summed:3

i=1

3

j =1

Φi,j (T (s))d[T i , T j ]s = ds

p

∂ 2Φ(T (s))(∂M 1(s))2 +

∂ 2Φ(T (s))(∂M 2(s))2 . (4.233)

In a shorter form, we can write the sum on the right side as Φ M 1 M 1 + ΦM 2 M 2 . This

gives us a form for the differential of Φ in which every occurrence of the differential

dT (s) has been evaluated:

dΦ(T (s)) = ∇Φ(T (s)) · ζ κ p(I −T (s)⊗T (s)) fds

+ 1

pZ (s)∂W (s)e3 +

1 p

(ΦM 1 M 1 + ΦM 2 M 2 )ds. (4.234)

The rst term, which contains the rst-order derivatives of Φ( T (s)), represents the

drift, while the second derivative term represents the diffusion. In order for the

tangent vector process to be a Brownian motion with a drift on the sphere, then,

the diffusion must be the Laplace-Beltrami operator on the sphere. To show this, we

must describe the Laplace-Beltrami operator in terms of the spherical Riemannian

metric tensor g.

In spherical coordinates ( θ, φ) with r held constant at 1, this metric tensor has the

form

g =1 0

0 sin2 θ. (4.235)

Thus the distance ds between two points on the sphere, whose polar angles (co-latitudes) differ by dθ and whose azimuthal angles differ by dφ satises

ds2 = dθ2 + sin 2 θ dφ2. (4.236)

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Once we know the metric tensor, we can use it to calculate the Laplacian on the

Riemannian manifold of our function Φ:

∆ S 2 Φ = 1√ det g

∂ j (gij det g ∂ 1Φ). (4.237)

The upper indices on g indicate the ( i, j ) entry of g−1, which is diagonal with entries

equal to the multiplicative inverses of the diagonal elements of g, since g itself is a

diagonal matrix. Also, det g = sin 2 θ. This shows that the Laplace-Beltrami operator

is∆ S 2 Φ =

1sin θ

∂ θ(sin θ ∂ θΦ)) + 1sin2 θ

∂ 2φΦ, (4.238)

and this is identical to the Laplacian of F (T ) in spherical coordinates, in a neighbor-

hood of the sphere S 2, because F |S 2 = Φ and F is constant on rays emanating from

the origin (so the radius is always 1, and the radial derivative is 0). Moreover, the

Laplace-Beltrami operator on Φ is equivalent to the usual three-dimensional Lapla-

cian of F , whether it is expressed in spherical or Cartesian coordinates. But sincethe Laplacian is invariant under rotations, the quantity ∆ F is the same whether we

take the derivatives with respect to the canonical basis e1, e2, e3 or as directional

derivatives with respect to the rotating Bishop frame, M 1, M 2, T , which is also

orthonormal. Namely,

F xx + F yy + F zz = F M 1 M 1 + F M 2 M 2 + F T T (4.239)

and recall that F is equal to Φ on the sphere, and it is constant in the direction of T .

Therefore, ∆ S 2 Φ = ΦM 1 M 1 + Φ M 2 M 2 , and the diffusion term is equal to the generator

of a Brownian motion on S 2. Notice that since the coefficient in front of the spherical

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Laplacian in (4.234) is 1/ p, it follows that the speed of the Brownian motion is −1/ 2 p ,

which is also a multiple of κ as in the original model.

With these facts in place, we are now ready to state and prove the rst major result of

this dissertation, namely, the convergence of the discrete force-driven polymer model

to the continuous model, and its corresponding stochastic differential equation that

describes the tangent vector to the polymer.

Theorem 4.16. Let RN

f (s) be the forced chain polymer model with parameters (N,a,θ,f ),

where f = f f , whose torsional angles follow the Boltzmann-Gibbs distribution

(1.15). Let Rf (s) be the forced Kratky-Porod model with parameters (L, p, ζ, f ) and

suppose the parameters satisfy

a = L

N (4.240)

θ = κ√ N

(4.241)

p = 2Lκ2 . (4.242)

Let τ = kB T , and let the dipole moment of each segment remain constant for all N .

Set

ζ = L f

τ . (4.243)

If RN (s) is the forced chain interpolated as in (2.44), then, as N

→ ∞, RN

f converges

in distribution to Rf , where

R f (s) = s

0T (σ)dσ (4.244)

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and the unit tangent vector T (s) to the polymer is the solution of the stochastic

differential equation

dT (s) = 1

pdQ(s) +

ζ p

(I −T (s)⊗T (s)) f ds (4.245)

with initial condition

T (0) = e3 (a.s. ). (4.246)

Proof. To show the convergence of RN f (s) weakly to RN (s), it is best to consider each

vector-valued process in terms of its corresponding matrix-valued process, Z N (s) and

Z (s), respectively, since we have shown in Lemma 4.11 that Z N converges to Z in

distribution. Recall that the position vectors depend on the matrix-valued processes

as follows:

RN f (s) =

[Ns/L ]

k=1

Z N k e3 ∆ s (4.247)

R f (s) = s

0 Z (σ)e3 dσ. (4.248)

Since the process Z N is discrete and piecewise continuous, we can express the sum

as an integral:

RN f (s) =

s

0Z N (σ)e3 dσ. (4.249)

Now, x s∈[0, L], and let > 0. We wish to show that if N is sufficiently large, then

RN

f (s) −R f (s) < . (4.250)

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To show this, examine the difference of the integrals:

RN f (s) −R f (s) =

s

0(Z N (σ) −Z (σ))e3 dσ (4.251)

≤ s

0Z N (σ) −Z (σ) e3 dσ (4.252)

≤ s sup0≤σ≤s

Z N (σ) −Z (σ) . (4.253)

We know, from Lemma 4.11, that Z N ⇒ Z . By the topological denition of weak

convergence, for any continuous bounded linear functional F , it follows that F (Z N ) →F (Z ). This is a consequence of the aforementioned continuity theorem (2.3 in Chapter

2 of [32]). Now, the integral F (X ) = s

0 X (s)e3 ds is such a functional, since

F = maxF (X )

X = s (4.254)

so it is bounded, and it is continuous because it is an integral of a bounded function.

Thus, it follows that F (Z N )⇒F (Z ); that is, RN f ⇒Rf .

The equation 4.245 for the tangent vector process T (s) follows directly from Lemmas

4.14 and 4.15. By Levy’s theorem, the tangent vector process T (s) is a scaled Brow-

nian motion Q(s) on the sphere, plus a drift as given in equation (4.245). This drift

is proportional to the projection of the force vector f onto the normal development

of T (s) for any given s, as was to be proven.

As an aside, notice that the differential equation (4.245) for T (s) is expressed as the

sum of a diffusion term and a drift term. This the form with which we will work in

the dissertation, because it shows the two sources of change in the tangent vector.

However, this form is not standard with regard to other texts on stochastic differential

equations, because the diffusion term dQ(s) is not a martingale. To express equation

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(4.245) in standard form, we must subtract the quadratic variation of the spherical

Brownian motion and add it to the drift. Recall that if x = T (0) is the starting

position of the tangent vector, then

dT (s) = dZ (s)x (4.255)

= κZ (s)∂B (s)x (4.256)

= κZ (s)∂M (s)x + κZ (s)∂A(s)x (4.257)

where the second term above is exactly the drift, ζ

pV (T (s)) ds. Recall also, by the

It o formula, that

dQ(s) = κZ (s)dM (s)x − κ2

2 T (s) ds (4.258)

since the quadratic variation of M (s) is D ds , and De3 = −e3. The rst term is

a martingale; let us denote it by Q(s). Then the standard form of the stochastic

differential equation for T is

dT (s) = 1

p d ¯Q(s) −

1 p T (s) ds +

ζ p (I −T (s)⊗T (s))

ˆf ds, (4.259)

where, by analyzing the Bishop frame, the martingale Q satises

d Q(s) = M 1(s)dβ 1(s) + M 2(s)dβ 2(s). (4.260)

Therefore, when the external force is added to the system, the discrete forced Kramers

chain converges in distribution to the continuous Kratky-Porod model. The spatial

differential equation (4.245) for the tangent vector can be used to describe the position

and orientation of the molecule in this situation, but it cannot be solved explicitly in

all cases. The next chapter concerns several extreme cases, determined by the values

of the parameters ζ and p, in which an exact solution can be found.

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CHAPTER 5

EXTREME CASES OF THE FORCED POLYMER

In the previous chapter, we described the forced Kratky-Porod model for semi-exible

polymers by rst introducing the external force to the discrete model (the Kramers

chain) and investigating how it changed the distribution of the tangent vectors along

the polymer, then showing that it converges to the continuous model in the scaling

limit. This also yielded a stochastic differential equation, expressed as the sum of

diffusion and drift terms, that the tangent vector obeys:

dT (s) = 1

p

dQ(s) + ζ

p(I −T (s)⊗T (s)) f ds (5.1)

with initial condition T (0) = e3. This can also be expressed in integral form, using

the scaling property of the spherical Brownian motion:

T (s) = e3 + Q s

p+

ζ p

s

0(I −T (σ)⊗T (σ)) f dσ. (5.2)

However, this equation cannot be solved explicitly for T , due to the quadratic drift

term. Thus, we cannot formulate an exact expression for the position vector R(s)

of the polymer in this model, except in some limiting cases. In our analysis of theunforced model, we discussed the behavior of the Kratky-Porod model for extreme

values of the persistence length, and we found that as p → ∞, the molecule becomes

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a rigid rod pointing in the initial direction, while as p → 0, the polymer shrinks to

a point.

Now that the force is included in the system, we must also examine the extreme cases

for the dimensionless force parameter, ζ , as well as p. Namely, what happens to a

polymer of xed persistence length as ζ →0 and as ζ → ∞? And what can we expect

to see when the two parameters tend to zero or innity together? Investigation of

these cases will enable us to map out a phase diagram (seen at the end of chapter

6) for the polymer, in which we can divide the space of all possible parameter values(ζ and p) into regimes, in which we expect to observe the polymer in certain states.

Some of these are the rigid rod, the compact coil, and a deterministic bent polymer

which we call the K-curve . This curve is planar, and so the polymer is reduced to a

two-dimensional vector-valued function of the parameters C = ζ / p and z 0, the cosine

of the angle between the initial direction T (0) and the unit force vector f . Several

chemical physicists have examined the relationship between force and extension of the

wormlike chain in two dimensions [30], [33], often for the case in which the molecule

is adsorbed to or embedded in a surface [34]. However, none have considered the case

in which both force and persistence length simultaneously go to innity. Yet, some

of the sources give results for small and large forces for xed p: the extension goes

to zero proportionally to the force as ζ → 0, and to the entire polymer length as

ζ → ∞, forming a rigid rod [30]. This type of transition between the coil and rod

also appears in polymers of xed p when the nature of the solvent changes [15].

Our goal of this chapter is to characterize the polymer completely, by solving for T (s)

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and its integral, R(s), in each of these four extreme cases. We will accomplish this in

the following steps:

1. Use the fact (Lemma 3.2) that in the lowest energy state, in which the Brownian

motion is negligible, the forced polymer lies in a plane, to show that the tangent

vector follows a 2-dimensional formula. This is equivalent to the case in which

both ζ and p → ∞.

2. Integrate the tangent vector to obtain the deterministic limiting K-curve, and

investigate how fast this curve approaches a straight line.

3. Use the theory of large deviations to show that as the drift term of equation (5.2)

remains constant but the diffusion term goes to zero, the polymer converges to

the K-curve in probability.

4. Then, prove the convergence of the model in each of the four extreme cases:

• Fixed p and ζ → 0: The limit is the original, unforced Kratky-Porod

model.

• Fixed p and ζ → ∞: The polymer is stretched in the direction of the

force.

• ζ, p → ∞ together: The limit is the K-curve.

• ζ, p →0 together: The polymer shrinks to a point.

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5.1 Characterization of the K-Curve

Let us begin, then, by considering the case in which the forced worm-like polymer is

not affected by any thermal forces (for example, if the temperature is absolute zero).

Then, the equation (5.1) becomes

dT (s) = ζ

p(I −T (s)⊗T (s)) f ds. (5.3)

Let us assume that the ratio between the force parameter and persistence length is

constant: ζ/ p = C . Also, let us denote the tangent vector in this case by x, rather

than T , since it is the solution to a deterministic ordinary differential equation.

Lemma 5.1. Let x(s) represent the unit tangent vector to the forced polymer in the

lowest energy state, so that it satises equation (5.3). Let x(0) be the initial direction

of the polymer, f the unit vector in the direction of the force eld, and z 0 = x(0) · f .Then x(s) has the following deterministic formula:

x(s) = g(s) f + γ (s) f ⊥ (5.4)

where the component functions g and γ are given by

g(s) = tanh Cs + tanh −1 z 0 (5.5)

and

γ (s) =

1 −g(s)2 (5.6)

and the unit vector f ⊥ is given by the Gram-Schmidt process:

f ⊥ = x(0) −z 0 f

1 −z 20. (5.7)

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Proof. Recall, as shown in Lemma 3.2, that in the lowest energy state, the polymer is

oriented so that it is completely contained in the plane spanned by the initial direction

vector x(0) and the force vector f . In other words, for all s∈[0, L],

x(s)∈S 2 ∩Span X (0), f . (5.8)

Thus, the tangent vector can be written in terms of scalar components, g(s) and γ (s),

each of which is the projection of x(s) onto one of the unit vectors f or f ⊥, which

is taken to be the Gram-Schmidt orthonormalization of T (0), according to equation(5.7). Thus, f ⊥ f ⊥ and f ⊥ ∈ SpanT (0), f . This is expressed in equation (5.4),

where g(s) = x(s) · f and γ (s) = x(s) · f ⊥. Additionally, the locus of possible points

for x(s) is a great circle along the sphere; this set can be parametrized by a single

scalar, so that the behavior for x(s) can be described by a differential equation in

one variable only. The fact that x(s) lies on the unit sphere imposes the following

restriction on g(s) and γ (s):

g(s)2 + γ (s)2 = 1 (5.9)

since x(s) ·x(s) = 1. We can solve the above equation for γ in terms of g to ob-

tain equation (5.6). Therefore, the behavior of x can be determined from solving a

differential equation for g(s).

Since x(s) obeys equation (5.3), and since g(s) = x(s) · f , we can take the inner

product of the differential version of equation (5.3) with the vector f to obtain a

one-dimensional ordinary differential equation:

dg(s) = C (I −x(s)⊗x(s)) f · f ds. (5.10)

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This simplies (by converting every instance of x to g) to

dg(s) = C (1 −g(s)2) ds. (5.11)

This ODE is not linear, but it is separable, so it can be solved explicitly. We nd

that

dg1 −g2 = C ds (5.12)

tanh −1 g = Cs + K (5.13)

g(s) = tanh( Cs + K ). (5.14)

Using the initial condition that g(0) = T (0) · f = z 0, we nd that

K = tanh −1 z 0 (5.15)

and so equation (5.5) is satised, completing the proof.

Notice that since for all y, tanh 2 y +sech 2 y = 1, we have an explicit formula for γ (s):

γ (s) = sech Cs + tanh −1 z 0 . (5.16)

This formula for the tangent vector x(s), in terms of its orthogonal components,

can be integrated with respect to arc length to obtain the formula for the position

vector of the force-driven polymer, in the deterministic stiff limit (the K-curve). This

curve will have varying shapes, depending on the constant C and the parameter z 0,which measures the cosine of the angle that the force vector f makes with the initial

direction vector x(0), but it can be described by a single formula.

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Lemma 5.2. In the low-energy limit, the force-driven Kratky-Porod model depicts a

bent chain polymer, whose position Rf (s) satises a system of equations similar to

those governing x(s).

Proof. We know from Lemma 5.1 that the tangent vector lies in a plane, and its

two orthogonal components are described by equations (5.5) and (5.16). Thus, we

can integrate each component with respect to s, and the result is the components of

R f (s). Again, since the position vector in this case is deterministic, let us denote it

by X (s) (so that dX = x ds). Write the components as

X (s) = h(s) f + η(s) f ⊥. (5.17)

Recall that the initial condition of X is that X (0) = 0. Then, h has the following

formula:

h(s) =

s

0g(σ) dσ (5.18)

= s

0tanh( Cσ + tanh −1 z 0) dσ (5.19)

= 1C

logcosh(Cs + tanh −1 z 0)

cosh(tanh −1 z 0). (5.20)

Similarly, η is the integral of γ with η(0) = 0:

η(s) = s

0γ (σ) dσ (5.21)

= s

0 sech(Cσ + tanh −1

z 0) dσ (5.22)

= 2C

tan −1 tanh Cs + tanh −1 z 0

2 − 2C

tan −1(tanh((tanh −1 z 0)/ 2)) .

(5.23)

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Therefore, equations (5.17), (5.20), and (5.23) give the denition of the position vector

X (s) for each s∈[0, L].

Notice that equation (5.20) gives a formula for the component of X (s) in the direction

of the force equal to 0 when s = 0. The following plots, made with Mathematica ,

show how h(s) varies as s ranges over the interval [0 , L], for various parameter values,

in the more interesting cases in which the polymer bends towards the force as its

magnitude increases.

On these plots, ‘C’ is really log2 C , so that the parameter could be adjusted on a

linear scale, and the axes are oriented so that the force always points up (on the

positive y-axis). Also, in order to non-dimensionalize the length, we let L = p.

Thus, the length of each polymer in the graphic is exactly its persistence length.

In later chapters, we will make the problem dimensionless by considering the ratio

between each length scale ( p, ¯R, and Rg) and the overall contour length L.

This gives a picture of how the polymer, in the deterministic limit, behaves and

changes with the direction of the force, expressed as x(0) · f , and with the ratio of

parameters C = ζ/ p. However, this only occurs if there is no Brownian motion in

the system, that is, if there are no thermal forces acting on the molecule, or if all

thermal forces are negligible. In terms of the parameters of the polymer, this case

occurs when both ζ and p grow towards innity. In other words, ζ

→ ∞means that

the system is acted on by an irresistible force, and p → ∞ means that the polymer

is innitely stiff; that is, it is an immovable object.

Since it is impossible to cool the molecule to absolute zero, or make it innitely stiff,

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Figure 5.1: Plots of the K-curve for various values of C and z 0 = −0.9

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there will always be some thermal force term, albeit a small one, in equation (5.1).

Thus, if is a small parameter, we can re-write the equation in this form, where T (s)

is the solution:

dT (s) = √ dQ(s) + C (I −T (s)⊗T (s)) f ds (5.24)

and let T 0(s) be the solution X (s) to the equation without Brownian motion, namely,

the process for which we solved in Lemma 5.1:

dT 0(s) = C (I −T 0(s)⊗T 0(s)) f ds. (5.25)

This approach is used in [35] and [36] to examine solutions to perturbed stochastic

differential equations. Notice that, in order to keep the perturbed equation (5.24)

consistent with equation (5.1), we must let = −1 p . It follows that as →0, p → ∞

and ζ → ∞ as well. This corresponds to the extreme case which we are currently

examining. Our intention is to show that as → 0, the solutions T converge in

probability to the deterministic solution T 0. This we will show using methods from

the theory of large deviations.

Lemma 5.3. For each > 0, let T (s) be the solution to equation (5.24) above, and

let T 0(s) be the solution to the unperturbed equation (5.25). Then, as →0,

T →T 0 (5.26)

in probability.

Proof. We prove the assertion by imitating some steps of Theorem 1.1 in Chapter 4

of [36] and lling in the details. Let T and T 0 be as above. Dene the term m (s),

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for each s∈(0, L], to be the following measure of the square of the distance between

the two processes:

m (s) = E sup0≤σ≤s

T (σ) −T 0(σ) 2 . (5.27)

We wish to nd a bound for m (L) in terms of , so that we can obtain convergence

in probability, and our hope is to use Doob’s L2 maximal inequality. However, this

inequality only applies to submartingales, and the quantity T (s) −T 0(s) is a semi-

martingale, since it is the solution to a stochastic differential equation driven by a

semimartingale (the spherical Brownian motion). Thus, the bound will require addi-

tional effort. Let M + A and M 0 + A0 be the Doob-Meyer decompositions for T

and T 0, respectively, and notice that since T 0 has no Brownian motion, M 0 = 0 a.s.

Then the process

T (s) −T 0(s) − A (s) −A0(s)

is a martingale, and

T (s) −T 0(s) + A (s) −A0(s)

is a submartingale. Let us now apply the Doob L2 maximal inequality:

m (s) ≤ E sup0≤σ≤s

T (σ) −T 0(σ) + A (σ) −A0(σ) 2 (5.28)

≤ 4E T (s) −T 0(s) + A (s) −A0(s) 2 (5.29)

≤ 8E [ T (s) −T 0(s) 2] + 8E [ A (s) −A0(s) 2]. (5.30)

The last step above follows from the parallelogram inequality: a + b 2 ≤ 2( a 2 +

b 2).

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By the same inequality,

E [ A (s) −A0(s) 2] ≤2E [ T (s) −T 0(s) 2] + 2E [M (s) 2] (5.31)

and since the driving process is Gaussian, the variance of the martingale part is

simple:

E [ M (s) 2] = s (5.32)

since the speed of the Brownian motion is √ . Therefore, we have the following bound

for the squared difference m :

m (s) ≤24E [ T (s) −T 0(s) 2] + 16 s. (5.33)

To evaluate this expression, let us consider the processes T and T 0 in integral form:

T 0(s) = e3 + C s

0V (T 0(σ)) dσ (5.34)

T (s) = e3 + Q(√ s) −Q(0) + C

s

0V (T (σ)) dσ (5.35)

where, for simplicity, we have written

V (x) = ( I −x⊗x) f . (5.36)

Thus, we obtain this bound for m (s):

m (s) ≤24E Q(√ s−Q(0)) + C s

0[V (T (σ)) −V (T 0(σ))] dσ

2

+16 s. (5.37)

Using the parallelogram inequality again, we can simplify the above to nd

m (s) ≤48E Q(√ s) −Q(0) 2 +48 C 2E s

0[V (T (σ)) −V (T 0(σ))] dσ

2

+16 s.

(5.38)

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Let us now consider each term on the right-hand side of the above inequality. The

rst term is the variance of a spherical Brownian motion, which satises

Q(√ s) −Q(0) = s. (5.39)

For the second term, notice that the integrand is bounded: since V (T (s)) is a unit

vector, the integrand has norm at most 2. Recall that for any bounded function F

on a compact interval [ a, b],

b

aF (x) ds

2

≤(b−a) b

asup

a≤x≤bF (x) 2 dx (5.40)

with equality occurring precisely when F is constant on [a, b]. The boundedness of

the integrand also allows us to use Fubini’s theorem, so that we can interchange the

expected value and the integral. Therefore,

E s

0[V (T (σ)) −V (T 0(σ))] dσ

2

≤s s

0E sup

0≤σ≤sV (T (σ)) −V (T 0(σ)) 2 dσ.

(5.41)

Next, notice that the function V is Lipschitz continuous: for any x, y ∈S 2,

V (x) −V (y) = (I −x⊗x) f −(I −y⊗y) f (5.42)

= (x · f )x −(y · f )y (5.43)

≤ (x · f )x −(x · f )y + (x · f )y −(y · f )y (5.44)

≤ (x · f ) x −y + y x · f −y · f (5.45)

≤ (x · f ) x −y + y x −y · f (5.46)

≤ [(x · f ) + ( y · f )] x −y (5.47)

≤ 2 x −y . (5.48)

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Therefore,

E s

0[V (T (σ)) −V (T 0(σ))] dσ

2

≤4s s

0E sup

0≤σ≤sT (σ) −T 0(σ) 2 dσ.

(5.49)

But the integrand is now exactly m (s). Thus by combining the differences in the

drift and diffusion terms, we obtain

m (s) ≤64 s+ 192C 2s s

0m (σ) dσ (5.50)

and so the bound depends on the integral of m itself. This is a common expressionin the theory of differential equations. If we divide the above inequality through by

s (if s > 0), then we nd that

m (s)s ≤64 + 192C 2

s

0

m (σ)σ

σ dσ (5.51)

which is exactly the condition for Gr¨onwall’s lemma with constant coefficients. Ap-

plying the lemma yields

m (s)s ≤64 exp 192

s

0σ dσ (5.52)

which, when multiplied back through by s, yields the desired bound for m (s) that

depends on only:

m (s) ≤64se96C 2 s 2

. (5.53)

To use this bound to show convergence in probability, let δ > 0 be xed, and consider

m (L), so that the whole interval [0 , L] is covered. Then, by Chebyshev’s inequality,for any s∈[0, L],

P ( T (s) −T 0(s) > δ ) ≤ m (L)

δ 2 =

64Le96C 2 L 2

δ 2 (5.54)

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which yields the conclusion that T →T 0 in probability (moreover, in L2), as desired.

5.2 Results for the Extreme Cases

Thus, the theory of large deviations allows us to prove that the tangent vector diverges

from the tangent to the K-curve by only a small amount when is small, and so when

the effect of the thermal forces goes to zero, the stochastic process governing the

tangent vector converges to a deterministic limit. This provides us with the theoretical

machinery necessary to prove the convergence of the position vector of the polymer

in this extreme case. The other extreme cases can be proven by more elementary

means. Therefore, we can summarize these results in the following theorem, which is

the second main result of this dissertation.

Theorem 5.4. The forced Kratky-Porod model has the following behavior:

1. Fix p and let ζ → 0. Then the polymer Rf (s) converges to the Kratky-Porod

model R(s).

2. Fix p and let ζ → ∞. Then the polymer converges in probability to a rigid rod

in the direction of the force: Rf (s) →sf .

3. Let C = ζ / p be constant, and let ζ → ∞. Then, R f (s) converges in probability

to the curve

X (s) = s

0x(σ)dσ (5.55)

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The integrand is bounded above by 1, since it is a projection of a unit vector. This

yields a deterministic bound for the difference of solutions:

T (s; ζ ) −T (s; 0) ≤ ζ

ps, a.s. (5.62)

This shows uniform convergence of T (s; ζ ) to T (s; 0). To prove convergence of the

position vectors, recall that Rf (s) = s

0 T (σ; ζ ) dσ and R(s) = s

0 T (σ; 0) dσ. Thus,

the difference between the position vectors satises

R f (s) −R(s) ≤ ζ

p

s2

2 , (5.63)

and therefore, R f →R uniformly, a.s.

For case (2), let p still be xed, but now let ζ → ∞. Then, the spherical Brownian

motion term will become negligible, as it is dwarfed by the drift term. We wish to

show that, as ζ → ∞, T (s; ζ ) → f in probability. Then it will follow, from the

Bounded Convergence Theorem, that R f (s) →sf .

We show this by means of a time change (actually, an arc length change) in thevariable s. Let

T (s) = T sζ

(5.64)

Q(s) = Qsζ

(5.65)

so it then follows, by the chain rule, that

dT (s) = 1ζ p dQ(s) +

1 p V (T (s)) ds. (5.66)

As ζ → ∞, the Brownian motion term vanishes, and T (s) converges in probability

to the deterministic tangent vector, x(s) as in Lemma 5.1, with C = −1 p , by a

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perturbation argument similar to that of Lemma 5.3. Recall the asymptotic behavior

of x(s) as s → ∞, by examining the component functions, g(s) in the direction of f ,

and γ (s) in the perpendicular direction, as given by equations (5.5) and (5.6):

lims→∞

g(s) = 1 (5.67)

lims→∞

γ (s) = 0 (5.68)

regardless of the initial direction parameter z 0, since C > 0. Thus it follows that

lims→∞

T (s) = f , (5.69)

in probability. We complete the proof of this part by returning to the original tangent

process, T (s). Let s∈[0, L] be xed, and recall that

T (s) = T (sζ ). (5.70)

Thus as ζ

→ ∞, T (sζ )

→ f , and therefore, T (s)

→ f in probability. It follows, by

the Bounded Convergence Theorem, that

R f (s) →sf (5.71)

in probability as well.

For case (3), we have shown, in Lemma 5.3, that in the case in which ζ/ p = C

and both parameters go to innity, the tangent vector process T (s) → T 0(s) in

probability. This process T 0(s) is equal in distribution to the solution x(s) of equation

(5.56). Therefore, by the Bounded Convergence Theorem, we can interchange limit

and integral again, and as →0, that is, as ζ , p → ∞, Rf (s) converges in probability

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to the deterministic K-curve, described by X (s), the solution to (5.55). This limit

has the formula given by equations (5.17) and following, as proven in Lemma 5.2.

Finally, case (4) is similar to the result from Chapter 2 about the coiled polymer in

the limit that p →0. In this case, however, we also let ζ →0 at the same time, so

that ζ / p = C , a constant. Then, T (s) has the following form:

dT (s) = 1

pdQ(s) + C (I −T (s)⊗T (s)) f ds. (5.72)

By another time change argument, similar to the one in case (2), the second term is

negligible, and the tangent vector process converges in probability to an innitely fast

Brownian motion on the sphere, with a uniform stationary distribution, as p → 0.

As the speed of the Brownian motion increases, any two tangent vectors along the

polymer become uncorrelated, since for all s < t ∈[0, L],

limp →0

E [T (s) ·T (t)] = limp →0

e−|t−s|/ p = 0 . (5.73)

The mean-square end-to-end length ¯R

2

, which is always non-negative, is the doubleintegral of the tangent-tangent correlation function:

R2 = L

0 L

0E [T (s) ·T (t)] dsdt (5.74)

and since the integrand converges to zero in probability, the double integral does the

same, by the Bounded Convergence Theorem. Thus since the average length of the

molecule vanishes, for all s ∈ [0, L], R(s) → 0 a.s., and the polymer shrinks to a

point.

This completes our summary of the extreme cases of the forced Kratky-Porod model,

for which we can nd an approximate solution to the differential equation for the

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tangent vector process T (s) and the position vector R f (s). While the equation (5.1)

cannot be solved explicitly and analytically for all values of the parameters ζ and

p, we can construct the forced wormlike chain by numerical methods, then compare

the results that we obtain for physical characteristics of the polymer, such as mean-

square end-to-end distance and radius of gyration, to the results for the unforced

model, calculated in Chapter 2. This will lead us to formulate a conjecture about

the behavior of the polymer in the general case, not only for the highest and lowest

values of the parameters.

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CHAPTER 6

NUMERICAL RESULTS FOR THE FORCED POLYMER

We have now stated and proven two theorems regarding the forced Kratky-Porod

model: one on the convergence of the discrete polymer models, and one on its behavior

in extreme cases. These two rigorous results together give an adequate mathematical

description of the semi-exible polymer in this model. However, they do not tell us

much about the behavior of the polymer in the general situation, for any values of

the dimensionless force parameter ζ and persistence length p. Since the differential

equation for the tangent vector process cannot be solved explicitly in all cases, our

analysis still leaves several questions unanswered. For example:

• How do the root-mean-square end-to-end distance and radius of gyration of the

polymer vary with both parameters?

• Can these physical quantities be described by a simple formula, as in the case

of the original Kratky-Porod model (Chapter 2), and is it consistent with these

results?

• How do the results of this model compare to the force-extension results of the

chemical physics literature?

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This dissertation will now take on a more expository and also more speculative turn,

as we formulate answers to these and other questions. This chapter will focus on a

numerical Monte Carlo routine, run in MATLAB, designed to construct a polymer

according to the forced Kramers chain model for a large number N of segments, so

that it closely approximates the forced wormlike chain. We will also analyze the

output of the program, which is the value of the mean-square end-to-end length R2

and mean-square radius of gyration Rg for various parameter values. We close the

chapter by summarizing the numerical data in a phase diagram, which augmentsthe theoretical work of Chapter 5. In the following chapter, we will make a pair of

conjectures to provide a physical explanation for these data.

6.1 Numerical Methods

In order to obtain the values of the mean-square end-to-end length R2 and mean-

square radius of gyration R2g for given values of ζ and p, I have written a procedurein MATLAB that constructs a discrete polymer according to the forced Kramers chain

model, in which the bond lengths and angles are the same, but in which the torsional

angles follow the Boltzmann-Gibbs distribution. The code appears in Appendix A,

but a description of the procedure follows below.

The program takes the following input parameters:

N , the number of segments in the polymer. In our rst run of the routine, weused N = 1000, but in subsequent runs, to obtain a more accurate result, we

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have used N = 4000. This value was used to obtain the data discussed in this

chapter.

L, the total arc length of the polymer. This is constant for every run, and in order

to simulate a strand of DNA or a ber of actin, we have used a molecular length

of 10−7 m. As it turns out, the results obtained from the code are independent

of this parameter.

‘lp’, the persistence length p. This parameter measures the decay of tangent-tangent

correlations along the polymer, and in the routine, we let it vary from 1/256 of

the contour length L to 256 times L, in multiplicative increments of 2. Thus,

we can consider a dimensionless parameter p/L to be the variable.

‘zeta’, the dimensionless force parameter. This is the magnitude of the external force

divided by the thermal energy τ = kB T and multiplied by L. As with p, we

let it vary by a power from 2 from one set of runs to the next, so the resulting

display is a log-log plot. As with p/L , we let ζ range between 1/256 and 256.

This corresponds to a force in the piconewton range when τ is the energy at

room temperature, which is consistent with the chemical physics experiments

[19], [37].

‘z0’, the inner product of the initial unit tangent vector with the unit the force vector,

ˆf . The initial segment of the polymer is xed to be ( L/N )e3, so this quantityis equal to the z -component of the unit force vector, and it determines the

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potential energy. Different angles of inclination between the force and the xy-

plane will yield different results for the RMS end-to-end length; in the program,

we used inclination angles of ι = 0, π/ 4, and −π/ 4.

‘runs’, the number of runs for each value of ζ and p for which the inner product of

the vector R(L) with itself is averaged together. In order to make the routine

run efficiently, we have used 1000 runs for each pair of parameters. Each run

uses its own random seed to generate the torsional angles, which are random

variables with the Boltzmann-Gibbs distribution.

In addition to the input parameters, the code uses various other variables and func-

tions to keep track of the behavior of the polymer:

‘phi’, a vector of length N −1, which contains the values of all the torsional angles φ1

through φN −1 of the polymer. These angles are use to calculate the quantities Z

(the current value of rotation matrix process) and ‘fr’ (the rotated force vector)

deterministically.

Q, a 3 ×N matrix containing the coordinates of all the segment vectors that are

individually calculated according to the Boltzmann-Gibbs distribution.

R, a 3 ×N matrix containing the positions of the beads on the polymer. This is a

running total of the segments, and when the radius of gyration is calculated, a

column representing R0 = 0 is appended to the beginning of the array.

’a’, the segment length L/N ; ‘kappa’, the curvature parameter 2L/ p; and ‘theta’,

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the bond angle κ/ √ N , which are all necessary in constructing the polymer

numerically, since it is discrete.

‘mu’, which is exactly as dened in Chapter 4: µ = L f sin( κ/ √ N )τ , or more simply,

ζ sin θ. This is the coefficient in the exponent of the Boltzmann-Gibbs distri-

bution.

‘psi’, which I call a phase shift in the code. This represents φ0k , the most likely

value of the torsional angle at that point on the polymer. This can be seen as

a phase shift, because if the inner product Qk+1 · f is represented as the cosine

of the angle between the vectors, then it can be expressed as cos( φk −φ0k) plus

a constant (that divides out when the density function is normalized), that is,

the cosine of the difference between the actual value of the torsional angle and

its most likely, or preferred, value.

‘rot’, the rotation matrix H k calculated deterministically from the torsional angle

φk , is a nested function in the routine.

‘g’, the probability density function for the Boltzmann-Gibbs distribution, is the

other nested function. It is dened to be

g(φ) = exp( µ cos(φ −ψ)) (6.1)

‘R2’, or R2, is one of the output values: the mean-square end-to-end length of the

constructed polymer.

‘Rcm’, the center of mass vector, or average location of all the beads, is used to

calculate the radius of gyration.

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‘Rg2’, the mean-square radius of gyration, is the other output value of the routine.

Now that we have described the variables and functions in the code, we must demon-

strate how this Monte Carlo routine works. The key to constructing the polymer

is the distribution of the random variables. On a computer, random numbers are

actually generated by a deterministic algorithm, and the routine ‘rand’ in MATLAB

takes an initial seed and uses it to compute a uniformly distributed pseudo-random

number in the interval [0 , 1]. However, we wish to calculate a random variable φ with

the Boltzmann-Gibbs distribution, with density function as given in equation (6.1).

This function is not built into MATLAB; only the uniform and normal distributions

are.

There is, however, a means of calculating a random variable with any given density

function, known as the rejection method , so named because a pair of random numbers

is generated, and it is rejected if it does not lie under the desired density curve. The

procedure runs as such:

1. Generate two uniformly distributed random variables: u1 ∈ [0, 2π] and u2 ∈[0, eµ]. The former is the uniform distribution on the set of possible values of

φ (as in the model without a force), and the latter is on the set of possible

ordinates of the density function g(φ). Scaling the variable u2 in this way

eliminates the need to normalize the density function.

2. Determine if the ordered pair ( u1, u2) is acceptable: that is, u2 < g(u1), meaning

that the point ( u1, u2) lies beneath the density curve, then take u1 to be the

value of the random variable φ. If not, reject the pair.

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3. Repeat steps (1) and (2) until an acceptable value of u1, which is used for φ, is

generated.

The number of random numbers that must be generated depends on the shape of

the density curve; a atter curve will yield a smaller area in which an ordered pair

(u1, u2) in the rectangle [0 , 2π]×[0, eµ] lies above the curve u2 = g(u1), while a sharper

curve leaves only a narrow band of acceptability. The shape of the curve is in turn

determined by the value of µ: if µ = 0, as in the model without a force, the density

curve is identical to the uniform distribution. However, as µ → ∞, the density

approaches a delta function, in which φk = φ0k for all k, and the torsional angles

become deterministic. Since µ is directly proportional to both ζ and θ, a exible

polymer combined with a strong force will lead to this deterministic scenario. When

running the routine, we have noticed that the run time is longer when ζ is large and

p is small, precisely because the rejection method has to be run multiple times.

The plots in gure 6.1 show the probability density function g(φ), shifted so thatφ0 = 0, and normalized so that the maximum value of each curve is 1. As with the

plots of the K-curve in the previous chapter, the sliding parameter in Mathematica ,

while called µ, is really log2 µ. In the rst plot, in which µ = 2−5, we see that the

density function does not differ much from that of the uniform distribution. For

µ = 0, we see a bell-curve like distribution, which tightens as µ increases to 29/ 2, in

which case we see that the torsional angle has no discernible probability of deviating

from the most likely state by π/ 4 radians. If we let µ increase to 29, the density curve

has the appearance of a delta function, and so the torsional angle becomes nearly

deterministic.

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Figure 6.1: Probability density curves for various values of µ

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Now that we have determined how the random variables are calculated, let us describe

how they are used to build the polymer. Recall that we keep a variable Z in the

program to represent the process of rotation matrices, Z N k . At the beginning of each

run, it is initialized to equal the identity matrix. Once a suitable value of φk has

been found, the function ‘rot’ computes the rotation matrix H k deterministically.

Then, the variable Z is updated according to the rule: Z N k = Z N

k−1H k . With the

new value of Z in place, the (k + 1) st column of the array Q, that is, the vector

Qk+1 , is taken from the third column of Z (recall that we follow the conventionthat Z = ( M 1|M 2|T )). This procedure repeats until all N columns of the array Q,

representing the N segment vectors, are determined. Then, the columns of the array

R, representing the positions of the beads, are computed as the cumulative sum of

the columns of Q. The last column of R, which is the vector RN N , points from the

starting point, or xed end, of the polymer, to its free end. Therefore, the square

of the norm of this vector is the output ‘R2’ of the run. After a number of runs,

called ‘runs’ in the program, the output values ‘R2’ from all of the runs are averaged

together. The resulting value is taken to be the computed mean-square end-to-end

length for that particular pair of parameters ζ and p. However, it is more useful to

take the square root of the result, which is the RMS length of the polymer, and then

divide that length by the total polymer length L (the total arc length), so that the

resulting value lies in the interval [0 , 1]. Thus, a scaled RMS value of 0 corresponds

to a polymer that coils back upon itself, and a value of 1 corresponds to a rigid

rod-shaped polymer.

This routine was run for several pairs of parameter values, and a surface plot, showing

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the dependence of the root-mean-square end-to-end length on the persistence length

and the force parameter, was generated, one for each chosen direction of the force.

In each case, the initial direction of the polymer was along the positive z -axis, but

the direction chosen for the force affected the shape of the polymer. A force with

a negative z -component caused the polymer to bend more than one with a positive

z -component. The more that a polymer bends, the less the length of its end-to-end

vector will be. To demonstrate this behavior, three sets of runs were taken, one with

a value of z 0 = + 1/ 2, one with z 0 = 0, and one with z 0 = − 1/ 2, so the lastcase will see a polymer that bends the most. In each scenario, the same number

of runs (1000) and number of segments in the polymer (4000) was used, and the

parameters ζ and p took on the same values. Both the force parameter ζ and the

ratio of persistence length to contour length p/L were integer powers of 2, ranging

form 2−8 to 28. These values were chosen to demonstrate the full range of outcomes

of the routine; any larger value of p shows no change in the polymer length over the

range of values of ζ , since the force is not strong enough to pull a polymer of that

stiffness away from its initial position along the positive z -axis. Any smaller value

of p is impractical to use, unless we also increase N , which would drive up the run

time of the routine. This is because each segment must deviate from the previous

one by an angle of κ/ √ N . The largest value of κ that I used in the simulation was

29/ 2, which gives an angle of

512/ 4000 = 0.358 radians, or about 20.5 degrees. This

means that it is impossible for the polymer to be stretched straight, but rather, in

the presence of a strong force, it forms a zig-zag pattern. The maximum length of

such a polymer is the product of the number of segments, the length of each segment,

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and pulls it in its direction, causing R/L to increase monotonically and non-linearly

towards 1. The most prominent feature on each plot is a trough in the surface,

running from the corner of high ζ and p into the valley of low ζ and p, where R is

near zero. This trough corresponds to a critical region of the graph in which neither

the intrinsic stiffness nor the external force dominates the behavior of the polymer,

but the two properties interact so that the polymer bends. This effect, described in

the previous chapter as the K-curve, is more salient when the initial direction of the

polymer and the direction of the unit force vector are farther apart (that is, z 0 < 0).In other words, if the force pulls the polymer downwards, below the xy-plane, then

the polymer will bend more, and the end-to-end vector will be shorter, causing a

deeper trough on the plot. These results are consistent with the four extreme cases

described in Theorem 5.4, and they also include the intermediate behavior.

In each of these plots, the force parameter ζ increases to the right, and the persistence

length p increases upwards. Thus, the extreme case of the K-curve corresponds to

the top right of each graph, in which we nd the trough, and the case of the shrinking

random coil corresponds to the bottom left of each plot. In the lower right of each

plot, ζ p, so the polymer is exible and pulled by a strong force, while in the upper

left, p ζ , so the polymer is stiff and the force is weak. Thus, while we observe that

R2 = 1 and R2g = 1/ 12 in these two parts of the graph, on both sides of the trough,

the physical situation is very different. In the former case, the polymer is a rigid

rod that aligns with the force vector, as the force dominates the model, while in the

latter, the polymer is a rigid rod that points in the initial direction, along the positive

z -axis. Thus, while these two regimes on the graph may appear similar, the molecule

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05

10 15 20 0

5

10

15

20

0

0.5

1

log2(lp /L)+9

log2 ζ+9

RMS length for z 0=sqrt(.5), N=4000, runs=1000

R b a r / L

Figure 6.2: Plot of RMS length for an initial angle of 45 degrees

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0 2 46 8 10 12 14 16 18 20

0

5

10

15

20

0

0.5

1

log2(lp /L)+

log2 ζ+9

RMS length for z 0=0, N=4000, runs=1000

R b a r / L

Figure 6.3: Plot of RMS length for an initial angle of 90 degrees

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05

10 15 20 0

5

10

15

20

0

0.1

0.2

0.3

0.4

log2(lp /L)+9

log2 ζ+9

Radius of gyration for z 0=sqrt(.5), N=4000, runs=1000

R g

/ L

Figure 6.5: Plot of radius of gyration for an initial angle of 45 degrees

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0 2 46 8 10 12 14 16 18 20

0

5

10

15

20

0

0.1

0.2

0.3

0.4

log2(lp /L)+

log2 ζ+9

Radius of gyration for z 0=0, N=4000, runs=1000

R g

/ L


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05

10 1520

0

5

10

15

20

0

0.1

0.2

0.3

0.4

log2(lp /L)+9

log2 ζ+9

Radius of gyration for z 0=−sqrt(.5), N=4000, runs=1000

R g

/ L


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05

10 1520

0

5

10

15

20

4

6

8

10

12

log2(lp /L)+9

log2 ζ+9

Ratio of Rbar 2 /Rg2

for z0=sqrt(.5), N=4000, runs=1000

R b a r 2

/ R g 2

Figure 6.8: Plot of the ratio R2/R 2g for an initial angle of 45 degrees

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0 24 6 8 10 12 14 16 18 20

0

5

10

15

20

4

6

8

10

12

log2 (lp /L)+

log2 ζ+9


for z0=0, N=4000, runs=1000

R b a r 2

/ R g 2


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05 10 15 20

0

5

10

15

20

4

6

8

10

12

log2(lp /L)+9

log2 ζ+9


for z0=−sqrt(.5), N=4000, runs=1000

R b a r 2

/ R g 2


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points in two different directions. As we move up and to the right along the graph, we

see that every pair of ζ and p falls either in one of these rigid rod regions, or within

the chasm, which corresponds to the bent polymer, and the height remains constant

along lines of slope 1, that is, ζ / p = C . This is consistent with the statement of the

previous chapter on convergence to the K-curve. The result for the force-dominated

regime is also consistent with Toan and Thirumalai, in the sense that the energy term

due to the force attains a global minimum when the polymer is stretched straight in

the direction of the force [19]. While they conclude that any polymer in this regimeacts as a freely-jointed chain, this is so that the discrete molecule may, in fact, be

stretched straight and not form a zig-zag due to the xed bond angle, as mentioned

before.

Moreover, let us consider the ratio R2/R 2g. Recall that for the random coil, or Rouse

model, this ratio equals 6, but for a rigid rod, the ratio is 12. In Figure 6.2ff, we

notice that the ratio is 6 in the lower left, in which both ζ and p are small, and 12

in either of the two rigid rod regimes in the upper left or lower right. Starting from

the lower left, as we let either parameter increase while keeping the other constant,

the ratio increases toward 12 along a sigmoidal curve. At the bottom of the trough,

the ratio is an intermediate value that decreases as the angle between f and T (0)

increases. This will be explained in more detail in the following chapter.

This describes the joint dependence of the size, or extension, of the polymer on both

the curvature parameter and the force. However, diagrams in the chemical physics

literature often contain two-dimensional plots demonstrating the dependence of the

extension on the force only (or sometimes vice versa). From these numerical data,

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scale (related to extension) as it varies across the full range of ζ , for each value of

p, while the graph on the right cuts off the smaller values of ζ (for which the data

contain larger numerical errors) and only includes every other value of p, with a

multiplicative step size of 4, to make the plot less cluttered.

From these curves, we see the following behavior:

If p > 1, then the function R(ζ ) has a single minimum, where the trough is located

on the surface plot, for either direction of the force that was considered. This

minimum occurs at the point at which the polymer is most bent, and the force

at which it occurs increases with the persistence length. Most notably, for the

higher values of ζ and p, the curve for 2 p trails the curve for p exactly by one

plot point.

If p < 1/ 4, then the function R(ζ ) is monotonically increasing for both directions

of the force vector. In these cases, the unforced polymer is coiled, so that when

the force pulls the polymer in its direction, it serves to straighten the molecule,

and the end-to-end length increases with the force; there is no value of the force

at which the polymer is bent, in which the internal stiffness and external force

balance each other.

In the intermediate cases, particularly for p = 1, the function has more than one

stationary point, as these persistence lengths make up the region of the plot

in which the trough meets the valley of random coil polymers. This is the

transitional region in which the polymer is somewhat stretched even without

the force, and stretched straight in the direction of a sufficiently large force.

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Thus, by examining the numerical data and observing the force-extension curves that

describe the results, we reach the empirical hypothesis that the root-mean-square

end-to-end length of the polymer has three types of behavior: straight ( R ≈L), bent

( R < L ), and coiled ( R L), and the type of behavior corresponds to one of four

regions on the graph of R/L versus p and ζ .

This can be seen on the phase diagram in Figure 6.2. This gure summarizes the

numerical results and is consistent with the main theorem of Chapter 5 for the extreme

cases, as well as the analytical result for the unforced polymer in Chapter 2.We now wish to compare our results with those of the chemical physics literature.

However, very few of the sources speak of root-mean-square polymer length or radius

of gyration in this context. Rather, several physicists use the quantity of extension ,

which is a measure of the root-mean-square size of the component of the end-to-end

vector R(L) in the direction of the force. This is often denoted by X and divided

by L so that is a dimensionless number in the range [0 , 1]; we will call this ξ [30].

Thus, the extension is equal to 1 in the case of high ζ and low p, and z 0 in the case of

low ζ and high p. Prasad, Hori, and Kondev describe a case, for a two-dimensional

polymer, in which the extension ξ is zero for a force of zero, then increases rapidly

and levels off at 1 as the force increases [30]. Their graph does not include the lower

left corner of our charts, because they use a linear scale for the force, rather than

our logarithmic scale. Marko and Siggia, on the other hand, do include a logarithmic

plot, and their curve of ξ versus force is sigmoidal in shape [18]. Both of these results

are consistent with our result for a more exible polymer.

Our next goal is to interpret these numerical results in a manner consistent with our

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0 ! " # $

lp

%

$

&

Rigid rod:

T(s) = e 3

Rigid rod:

T(s) = fRandom coil:

R(s) 0

Bent polymer

K-curve

Figure 6.13: Phase diagram for the forced Kratky-Porod model

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previous theorems. Thus, to refer to our questions at the beginning of this chapter, we

have answered the rst and third questions, but not the second. The following chapter

will strive to answer it and provide a theoretical support for the trends observed from

our computer simulation.

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CHAPTER 7

CORRELATION ANALYSIS

We now wish to join our numerical results regarding the mean-square end-to-end

length and radius of gyration from the previous chapter to the theorems 4.16 and 5.4

that we proved earlier. To do so, we seek a formula for R2 that holds for all ζ and

p, not only the extreme cases mentioned in Theorem 5.4. While we do not nd an

exact expression for R2, we do make two conjectures regarding the polymer length

that are consistent with our numerical results in several limiting cases.

To characterize the mean-square end-to-end length, we will begin by evaluating it in

the same way that we did in Chapter 2, by its denition:

R2 = E [R(L) ·R(L)] (7.1)

= E L

0T (s) ds ·

L

0T (t) dt (7.2)

= L

0 L

0E [T (s) ·T (t)] dsdt (7.3)

that is, the mean-square end-to-end length is the double integral of the tangent-

tangent correlation function. (Note that unlike in Chapter 2, here t represents a

value of the arc length parameter, not time.) In the unforced model, this correlation

was simply an exponential decay:

E [T (s) ·T (t)] = e−|s−t |/ p . (7.4)

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But now that the force has been added to the system, the correlation function cannot

depend on the intrinsic stiffness of the polymer only. For if the polymer were a rigid

rod pointing in the direction of the force, its autocorrelation function would be 1,

regardless of p. Thus, this formula must depend on the magnitude and direction of

the external force, and the persistence length itself may now be a function of the force.

Some formulations in the chemical physics papers, in which the polymer is dened as

a Hamiltonian system, yield an explicit formula for mean-square end-to-end distance

¯R

2

(for the unforced polymer only) [29] or extension ξ [18], [30], but this is moredifficult to derive from the stochastic differential equation.

If we had an explicit expression for the autocorrelation of the tangent vector as a

function of ζ and p, we could use it to nd the corresponding mean-square end-to-end

length, and in turn, the mean-square radius of gyration R2g (which is itself expressible

as a double integral of R2). Then, we can compare this to the numerical results from

the Monte Carlo simulation. However, we are only able to make conjectures about the

correlation function, which means that the theoretical results regarding the physical

length scales are merely speculative.

This provides us with the focus of this chapter. We will rst describe the correlation

function as the solution to a differential equation and to an eigenvalue problem,

because the principal eigenvalue of the operator that generates the equation for the

tangent vector is inversely related to the correlation length of the polymer. Then, we

will use the eigenfunction expansion to formulate our conjecture, and test it by tting

it to our numerical results. We show that this conjecture explains the behavior of the

mean-square polymer length well for the rigid rod or coiled molecule, but not for the

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bent polymer. By examining the tangent vector in the limiting cases of low and high

contour length L, we can make a more accurate claim about R for these cases.

7.1 Analysis of the Correlation Function

The tangent-tangent correlation function can be considered as the solution to a dif-

ferential equation, just as with the correlation for the forceless polymer. When the

force is added, the differential equation gains a term, but it can still be derived in

the same manner as it is if f = 0. Let us then consider the forceless case and derive

a differential equation which the correlation between tangent vectors satises. To

distinguish this case from the model with the force, let us refer to the tangent vector

process by Q(s), since it is simply a spherical Brownian motion. Let s be xed, and

consider the change in E [Q(s) ·Q(t)] due to a change in t only. Then we have

dE [Q(s)

·Q(t)] = E [Q(s)

·dQ(t)] (7.5)

= E [Q(s) ·dZ (t)Q(0)] . (7.6)

But we know dZ (t), by our analysis of the process of Bishop frames and its driving

process:

dZ (t) = Z (t) κdB (t) + κ2

2 d[B, B ]t . (7.7)

This derivative is of Itˆo type, so it looks forward in time (that is, it is the limit of

Z (t + ∆ t) −Z (t) for ∆ t > 0), and thus its conditional expectation, with respect to

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the σ-eld F t , can be calculated easily:

E [dZ (t)|F t ] = Z (t)E κdB (t) + κ2

2 d[B, B ]t |F t (7.8)

= Z (t)E limN →∞

κ√ N

b[Nt/L ]+1 + limN →∞

κ2

2N b2

[Nt/L ]+1 |F t (7.9)

= Z (t) limN →∞

κ√ N L

E [b[Nt/L ]+1 |F t ] + κ2

2LE [b2

[Nt/L ]+1 |F t ] LN

(7.10)

= Z (t) limN →∞

κ√ N L

0 + κ2

2LD

LN

(7.11)

= κ2

2LZ (t)Dds. (7.12)

Recall that D stands for the diagonal matrix with entries ( −12 , −1

2 , −1), and that the

coefficient in front is simply 1/ p. Therefore, since

E [d(Q(s) ·Q(t))] = E [E [d(Q(s) ·Q(t))|F t ]] (7.13)

it follows that the derivative of the correlation satises

dE [Q(s) ·Q(t)] = 1

pE [Q(s) ·Z (t)DQ (0)] ds. (7.14)

But Q(0) = e3, so DQ(0) = −Q(0), and it follows, by the Markov property, that

dE [Q(s) ·Q(t)] = −1 p

E [Q(s) ·Q(t)] ds (7.15)

and the solution to this differential equation is precisely an exponential decay with

persistence length p.

Using a similar calculation, we can derive a differential equation for the quantity

E [T (s) ·T (t)], which takes the external force into account. The major difference is

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the fact that E [b[Nt/L ]+1 |F t ] is no longer zero, but rather depends on both the current

position of the Bishop frame and the force vector:

E [b[Nt/L ]+1 |F t ] = ζ κ p

[Z (t)−1 f , T (0)] (7.16)

and if we substitute this expression into equation (7.11), we obtain the following

differential equation:

dE [T (s) ·T (t)] = −1 p

E [T (s) ·T (t)] ds + ζ

pE [T (s) ·(I −T (t)⊗T (t)) f ]ds. (7.17)

Thus, the differential equation depends on a second expected value, and the corre-

lation function cannot be described as a constant exponential decay. In addition,

in the forced model, the correlation function is no longer independent of the start-

ing position. For example, consider the K-curve in the case in which C = ζ/ p is

large, and the angle between the initial direction vector T (0) and the force vector f

is wide (that is, z 0 is close to

−1). This polymer curves sharply at the origin before

straightening in the direction of f . Then, keeping |s −t| constant, if we let s and t be

close to zero, then the polymer is bent sharply, and the tangent-tangent correlation

is markedly less than 1. If, however, s and t are larger, so that R(s) and R(t) are

positioned on the straight part of the polymer, then the autocorrelation function is 1.

Therefore, the tangent-tangent correlation function is not a function of only the arc

length distance between the two points. Additionally, if we vary the initial direction,

so that it is closer to the force, the autocorrelation would be nearer to 1 in the former

case for small s and t. Thus, it holds that the tangent-tangent correlation depends

on the initial position T (0) as well. In order to describe the correlation function more

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accurately, we allow it to depend on a third parameter, marking the initial position

of the tangent vector. Let us denote

u(x,s,t ) = E x [T (s) ·T (t)] (7.18)

as the expected value of the inner product for a tangent vector process, solving the

stochastic differential equation as proven in Chapter 4, with initial condition T (0) = x

for a xed unit vector x. This correlation function, which is an expected value, can

be written as the double integral of the transition probability function:

u(x,s,t ) = S 2 ×S 2y ·z p(s,x,dy ) p(t −s,y,dz ) (7.19)

in which p(s,x,dy ) is the probability that the tangent vector moves from direction x

to the set dy on the sphere in an arc length difference of s. As with the unforced case

discussed above, we wish to nd a differential equation that the function u satises.

To do this, we describe the equation in terms of a differential operator L, for which we

will nd the principal eigenvalue that leads to a proposed formula for the correlationfunction u. Recall that the stochastic differential equation for T (s) is

dT (s) = 1

pdQ(s) +

ζ p

V (T (s)) ds (7.20)

where Q(s) is the standard Brownian motion on the unit sphere, and V (x) is the

vector eld mapping x to the projection of f onto the plane normal to x:

V (x) = ( I

−x

⊗

x) f . (7.21)

This drift-diffusion process is generated by the operator L, given by

L[T ] = 1

p∆ S 2 T +

ζ p

V (T ) (7.22)

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where ∆ S 2 is the spherical Laplacian. We claim that the correlation function u(s,x,t )

satises this differential equation:

∂ ∂s

+ ∂ ∂t

u(x,s,t ) = Lu(x,s,t ). (7.23)

with the following boundary conditions:

u(x,s,s + ) ≡ 1 (7.24)

u(x, 0+ , t ) = x

· S

2z p(t,x,dz ) (7.25)

= x ·E x[T t ]. (7.26)

We show that this is true by evaluating both sides of equation (7.23). Let us take

the partial derivatives of u with respect to s and t, using the formula (7.19):

∂ su(x,s,t ) = S 2 ×S 2y ·z ( ˙ p(s,x,dy ) p(t −s,y,dz ) − p(s,x,dy ) ˙ p(t −s,y,dz ))

(7.27)

∂ t u(x,s,t ) = S 2 ×S 2y ·z (s,x,dy ) ˙ p(t −s,y,dz ) (7.28)

Lu(x,s,t ) = S 2 ×S 2y ·z ˙ p(s,x,dy ) p(t −s,y,dz ) (7.29)

and equation (7.23) holds. Since L is a second-order elliptic operator, this equation

is a type of heat equation for the correlation function.

Our next goal is to express the tangent-tangent correlation function in terms of an

exponential decay. This forms the basis of our conjecture, namely, that there existconstants c0 and c1 so that for all x∈S 2, s, t∈[0, L],

c0 ≤ u(x,s,t )e−|t−s|/ c (f ) ≤c1 (7.30)

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where c(f ) is the correlation length, which is a function of the added force. This is,

in fact, the denition of a correlation length, and we claim that such a number exists.

We denote it by c to avoid confusion with p, an intrinsic property of the polymer.

Notice that c(0) = p. In order to nd c, we will investigate the eigenvalues and

eigenfunctions and relate the correlation length to the principal eigenvalue of the

operator L.

Recall that from the theory of linear operators, the eigenvalues of an operator L are

all real as long as L is self-adjoint. We claim that in this case, the operator L isself-adjoint with respect to a weight function m(x), which measures the distribution

of tangent vectors on the unit sphere. Once we show that for all C2 functions g and

h on the sphere,

S 2 L[g(x)]h(x)m(x) dx = S 2g(x)L[h(x)]m(x) dx, (7.31)

we can use a result of the operator theory (Mercer’s Theorem) that gives an eigen-

function expansion for the correlation function. (Since the weight is invariant in the

azimuthal angle φ, we will consider only the part of the operator that depends on the

polar angle θ.) This will aid us in making our conjecture for the correlation length.

We will build up to this result by rst considering the case for the forceless Kratky-

Porod model and then generalizing. If we let f = 0, then the operator L simplies

to the operator L0:

L0[T ] = 1 p

∆ T (7.32)

that is, simply a multiple of the Laplacian. Since the tangent vector is on the unit

sphere, the eigenfunctions of L0 are exactly the spherical harmonics Y ml (θ, φ) in terms

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of the angular coordinates on the sphere, when m = 0. The spherical harmonics take

the form

Y ml (θ, φ) = P ml (θ)eimφ (7.33)

where the functions P ml (θ) are the associated Legendre functions, often expressed

as polynomials of cosθ and sin θ. This comes as a result of separation of variables

when solving Laplace’s equation. When the force is added to the system, the spherical

harmonics are no longer the eigenfunctions of L, but we can use them as test functions

when approximating the principal eigenvalue. In order to investigate the effect that

the force f has on the eigenvalues, let us express the tangent vector T (s) in terms of

spherical coordinates; since we are free to choose the coordinate system, let us dene

the north pole of the sphere (where θ = 0) to align with the unit vector f , in order to

simplify our calculations. Then the projection term V , which is the additional term

that results from the external force in equation (7.20), is equal to

(I −T ⊗T ) f = f −( f ·T )T = f −T cosθ. (7.34)

To simplify this further, let us express the vectors f and T in rectangular coordinates,

so that we can subtract them component-wise:

f = (0 , 0, 1) (7.35)

T = (sin θ cos φ, sin θ sin φ, cos θ). (7.36)

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This eigenvalue problem is greatly facilitated if we can express it in Sturm-Liouville

form, for then we know that all the eigenvalues are unique and real. Thus, we wish

to prove the following claim.

Lemma 7.1. Given the operator Lθ as dened above, there exists a weight function

m(θ) for which the equation Lθ[g(θ)] = λ1g(θ) can be written

ddθ

(m(θ)g (θ)) = pλ1m(θ)g(θ). (7.45)

Proof. We wish to express the equation (7.44) in divergence form to match up with

(7.45), but (7.44) has one term in divergence form already. Let us rst expand the

operator:

Lθ[g(θ)] = 1

pg (θ) +

cos θsin θ −ζ sin θ g (θ) . (7.46)

The expression inside the parentheses in equation (7.46) can be used to calculate an

integrating factor, which will lead to the weight function m(θ). This factor will be

the exponential of the integral of the expression in parentheses:

exp cot θ −ζ sin θ dθ = exp[log sin θ + ζ cosθ] (7.47)

= eζ cos θ sin θ. (7.48)

Let m(θ) denote the integrating factor; then equation (7.46) can be written

m(θ)

Lθ[g(θ)] =

1

p

[m(θ)g (θ) + m (θ)g (θ)] . (7.49)

But the expression in the brackets is exactly the derivative of m(θ)g (θ). Thus, if

we multiply the above equation by p and set Lθ[g(θ)] = λ1g(θ), we obtain exactly

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equation (7.45). Therefore, m(θ) = eζ cos θ sin θ is the desired weight function, and we

can write

Lθ[g] = 1

p

(mg )m

(7.50)

which is a short-form expression for the operator Lθ in divergence form.

Therefore, the eigenvalue problem can be classied as a Sturm-Liouville problem. Ac-

cording to the theory of partial differential equations, this means that the eigenvalues

increase to

∞ and the eigenfunctions are complete [38]. Since the weight function

m(θ) = eζ cos θ sin θ vanishes at the endpoints (namely, the north and south poles of

the sphere), the situation is known as a singular Sturm-Liouville problem, and so

rather than standard Dirichlet or Neumann boundary conditions on the endpoints,

the condition is that the solution g(θ) be nite at both θ = 0 and θ = π [38].

In addition, this weight function m(θ) is an L-invariant measure that is precisely the

weight with respect to which the operator Lθ is self-adjoint. To show this claim, let

g(θ) and h(θ) be two C2 functions on the interval [0 , π]. Then (suppressing dependence

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on θ in the functions to save space),

π

0gLθ[h]m dθ =

1 p

π

0g

1sin θ

(h sin θ) −ζ sin θh eζ cos θ sin θ dθ (7.51)

= 1

p π

0geζ cos θ(h sin θ) dθ −

ζ p

π

0geζ cos θ sin2 θh dθ (7.52)

= −1 p

π

0(geζ cos θ) h sin θ dθ +

ζ p

(geζ cos θ sin2 θ) h dθ (7.53)

= −1 p

π

0(g eζ cos θ sin θ −ζgeζ cos θ sin2 θ)h dθ

+ ζ

p π

0(g eζ cos θ sin2 θ

−ζgeζ cos θ sin3 θ + 2geζ cos θ sin θ cos θ)h dθ

(7.54)

= 1

p π

0(g eζ cos θ sin θ −ζgeζ cos θ sin2 θ) h dθ

+ ζ

p π

0(g eζ cos θ sin2 θ −ζgeζ cos θ sin3 θ + 2geζ cos θ sin θ cos θ)h dθ

(7.55)

= 1

p

π

0(g eζ cos θ sin θ −ζg eζ cos θ sin2 θ + g eζ cos θ cos θ

−ζg eζ cos θ sin2 θ + ζ 2geζ cos θ sin3 θ −2ζgeζ cos θ sin θ cos θ)h dθ

+ ζ

p π

0(g eζ cos θ sin2 θ −ζgeζ cos θ sin3 θ + 2geζ cos θ sin θ cos θ)h dθ

(7.56)

= 1

p π

0(g eζ cos θ sin θ −ζg eζ cos θ sin2 θ + g eζ cos θ cos θ)h dθ (7.57)

= 1

p π

0(g −ζg sin θ + g cot θ)mhdθ (7.58)

= π

0 Lθ[g]hmdθ. (7.59)

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The integration by parts is facilitated by the fact that the weight function m(θ) van-

ishes at the poles of the sphere, which are the endpoints of the interval of integration.

Therefore, we can apply Mercer’s theorem, which states:

Let Lbe a self-adjoint linear differential operator, with eigenvalues ( λn )n∈N

and eigenfunctions ( φn (x))n∈N . Then any C2 function p(t,x,y ) can be ex-

pressed as a series of eigenfunctions:

p(t,x,y ) = m(y) +

∞

n =1 e−λ n t

φn (x)φn (y). (7.60)

We can use this theorem to show that the correlation length c is equal to the inverse

of the principal eigenvalue λ1. First, from spectral theory, the transition probability

is dened if the nal argument is a point, not just a set, and this function p(t ,x,y ) is

absolutely continuous with respect to the measure m(dy) on the sphere. This measure

satises

p(t,x,dy ) = p(t,x,y )m(dy). (7.61)

We can now express the correlation function u(x,s,t ) in terms of the three-argument

transition functions and the measure ν :

u(x,s,t ) = S 2 ×S 2y ·z p(s,x,y ) p(t −s,y,z )m(dy)m(dz ). (7.62)

We wish now to show that c = λ−11 .

Lemma 7.2. The correlation length of the forced polymer c is exactly the multiplica-

tive inverse of the principal eigenvalue λ1 of the self-adjoint operator Lθ , which is

also the least positive eigenvalue of L.

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Lemma 7.3. For the singular Sturm-Liouville eigenvalue problem (7.45), the prin-

cipal eigenvalue λ1 is equal to

λ1 = ming∈C([0,π ])

QR [g] p

. (7.64)

This is a standard result from the theory of differential operators, and so the proof

has been omitted.

Notice, however, that λ1 is not the lowest possible value of the Rayleigh quotient

among all test functions. For example, if g were constant on [0 , π], then QR [g] =0. But, 0 cannot be an eigenvalue for an eigenfunction that satises the niteness

condition at the endpoints. To see this, consider the differential equation

(m(θ)g (θ)) = 0 (7.65)

whose solution g would have an eigenvalue of 0. Integrating both sides gives, for some

constant C ,

g (θ) = C m(θ)

(7.66)

or equivalently,

g(θ) = C π

0e−ζ cos θ csc θ dθ (7.67)

but this integral blows up at the endpoints. Thus, the niteness condition is violated,

and zero cannot be a true eigenvalue.

While there is no systematic way in which to nd the principal eigenvalue, we can

nd the value of the Rayleigh quotient for various test functions. We do not know

the exact eigenfunctions for the operator Lθ , but in the limiting case that ζ = 0, the

eigenfunctions are the Legendre polynomials. Thus, a good idea would be to use the

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l m P ml (θ)

0 0 1

1 0 cosθ1 1 sinθ

2 0 3 cos2 θ −1

2 1 sinθ cos θ

2 2 sin2 θ

3 0 5 cos3 θ −3cosθ

3 1 5 cos2 θ sin θ

−sin θ

3 2 cosθ sin2 θ

3 3 sin3 θ

Table 7.1: Associated Legendre functions P ml (θ) for l ≤3

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Legendre polynomials and associated Legendre functions P ml (θ) as our test functions.

Table 7.1 shows the rst few (un-normalized) associated Legendre functions, up to

l = 3.

The rst function, 1, gives the forbidden eigenvalue of 0, so it can be ignored. As an

example of calculating the Rayleigh quotient, let us consider the rst non-degenerate

case, P 01 (θ) = cos θ. It follows that

QR [P 01 ] = π

0 eζ cos θ sin3 θ dθ

π

0 eζ cos θ sin θ cos2 θ dθ (7.68)

= 2ζ cosh ζ −2sinh ζ

−2ζ coshζ + ( ζ 2 + 2) sinh ζ . (7.69)

It happens that this Rayleigh quotient leads to an asymptotically correct value for

the principal eigenvalue. Let us consider two limiting cases:

1. ζ → 0: This corresponds to the original model, in which f = 0. We should

expect to see the same eigenvalues as those of the Legendre polynomials in the

m = 0 cases, namely, l(l +1), so in our example of P 01 (θ) = cos θ, the eigenvalue

is 2. We can evaluate the limit of the Rayleigh quotient by expanding cosh ζ

and sinh ζ in their Maclaurin series

cosh ζ =∞

k=0

ζ 2k

(2k)! (7.70)

sinh ζ =∞

k=0

ζ 2k+1

(2k + 1)! (7.71)

and noticing that several of the lower-order terms cancel out. For P 01 , the terms

of third order in the numerator and denominator are the lowest-order terms

that do not cancel, and they provide a ratio of 2, equal to the quotient in the

unforced case.

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2. ζ → ∞: This corresponds to the deterministic case, in which the force is taken

to be innite. For ζ 1, the functions cosh ζ and sinh ζ can be assumed to be

equal. Thus, each hyperbolic function can be factored out, and the resulting

Rayleigh quotient is a rational function of ζ . In the case of P 01 above, we observe

that the Rayleigh quotient is zero, so the eigenvalue approaches zero in this case.

Thus, the desired behavior in both limits ( ζ 1 and ζ 1) is found when P 01 (θ) =

cos θ is used as a test function. Therefore, combining the value of the Rayleigh

quotient with its relationship to the principal eigenvalue and its reciprocal p, we nd

that our approximation for the correlation length of the polymer, as a function of f ,

is

c(f ) = p −2ζ cosh ζ + ( ζ 2 + 2) sinh ζ 2ζ coshζ −2sinh ζ

. (7.72)

7.2 Two Conjectures for the Correlation Function

Now that we have an approximation to the correlation length, let us apply both

this value and the boundary conditions of the differential equation (7.23) for the

correlation functions to obtain an approximate formula for u(x,s,t ), when s < t , so

that the increments of arc length are s and t −s. Our conjecture will use the leading

terms from the series in Mercer’s theorem and interpret the boundary conditions with

smooth exponential functions. Recall that the boundary conditions for u(x,s,t ) are

as follows:

1. If t −s = 0, then u(x,s,t ) = 1, independently of the starting position x;

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We are now ready to interpolate the boundary conditions and make the conjecture

for the approximate formula for the tangent-tangent correlation function:

Conjecture 7.4. The correlation function u(x,s,t ) = E x [T (s)·T (t)] can be expressed

by the following formula:

u(x,s,t ) = c(ζ ) c(ζ ) + ( z 0 −c(ζ ))e−s/ c + (1 −c(ζ )z 0)e−|t−s|/ c (7.80)

with z 0 = x · f , lc as in equation (7.72), and c(ζ ) as in equation (7.77).

If we integrate this formula over the region [0 , L] ×[0, L], we nd the approximate

formula for the mean-square end-to-end length. However, our formula (7.80) only

holds for s < t , but if we switch the roles of the two variables, we see that the same

formula holds on the subregion on which s > t . Thus, the integral over the square is

equal to twice the integral over the triangle 0 ≤ t ≤L, 0 ≤s ≤ t. The mean-square

length is thus given by:

R2 = 2 L

0 t

0c(ζ )(c(ζ ) + ( z 0 −c(ζ ))e−s/ c ) + (1 −c(ζ )z 0)e−|t−s|/ c dsdt (7.81)

= 2 L

0c(ζ )2tc(ζ )(z 0 −c(ζ )) c 1 −e−t/ c + (1 −c(ζ )z 0) c 1 −e−t/ c dt

(7.82)

= 2 L

0c(ζ )2t + (1 −c(ζ )2) c 1 −e−t/ c dt (7.83)

= c(ζ )2L2 + 2(1 −c(ζ )2) c L − c 1 −e−L/ c . (7.84)

Surprisingly, the dependence on z 0 vanishes after the integration in s, so this formula

does not tell us that R2 changes with the angle between the initial direction x and the

force f at all. This is not consistent with some areas of our plots that we generated

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in the previous chapter, as the trough, corresponding to the bent polymer regime,

deepens as z 0 decreases. However, if we hold z 0 constant, we observe the proper

asymptotic behavior in ζ :

• As ζ → 0, c(ζ ) → 0 and c → p. Thus the mean-square end-to-end distance

becomes

R2 = 2 p L − p 1 −e−L/ p (7.85)

which corresponds exactly with the formula for the original Kratky-Porod model,

given in Chapter 2.

• As ζ → ∞, c(ζ ) → 1, and so if c were xed, we would notice that R2 = L2.

However, c → ∞ as well, according to the asymptotic formula c ≈ ζ p/ 2,

derived from (7.72). This yields an asymptotic result for R2 as well, obtained

from expanding the exponential in (7.84):

R2

≈L2

− L2

4ζ 2 + O(ζ −3) (7.86)

and so the mean-square polymer length is L2 in the limit, regardless of p. This

suggests that this approximate formula does not account for the stiff polymer

in a strong force eld, or K-curve.

By examining the formula (7.84) , we notice that the root-mean-square polymer length

increases monotonically in both parameters. Thus, the trough that was so prominent

on the graphs of Chapter 6 would not appear on a surface plot based on equation

(7.84). Thus, we conclude that this approximation for the correlation function and

the mean-square end-to-end distance is not accurate in general, because it does not

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demonstrate the regime of the bent polymer, which we have observed both theoret-

ically (by plotting the K-curve) and numerically (by simulating the distributions of

torsional angles).

However, we can obtain a more satisfying result if we decide to keep constant the

parameters that we have been varying ( ζ and p) and vary the overall contour length

instead. That is, we consider the limits of R2 as L → ∞. According to Conjecture

7.4 and the subsequent equation (7.84), we see that

R2

L2 ≈1 −(1 −c(ζ )2) L3 c

+ o(L) (7.87)

for L 1, andR2

L2 ≈c(ζ )2 + 2(1 −c(ζ )2) c

L + o(L−1) (7.88)

for L 1.

Yet, we can make a second conjecture about the correlation function that yields a

limiting result for the mean-square end-to-end length which does take into account theorientation of the force, which disappears in the integration which leads to formula

(7.84). This is based on the fact that if L 1, then the tangent vector can be

expanded in an asymptotic power series in the arc length parameter, which can be

estimated; and if L 1, then the ergodic theorem yields that any two tangent vectors

will be independent and identically distributed. These give the following result:

Conjecture 7.5. The mean-square end-to-end distance of the polymer in the forced Kratky-Porod model has the following limiting behavior in the correlation length L:

1. As L →0, R2 has the form L2 −L3/ (3 c(f )) + L4 V (x) 2/ 4 + o(L);

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We can then integrate this formula twice to obtain R2:

R2 = L

0 L

0(1 + s∧t + st V (x) 2) dsdt (7.95)

= 2 L

0 t

0(1 + s + st V (x) 2) dsdt (7.96)

= 2 L

0t +

12

t2 + 12

t3 V (x) 2 dt (7.97)

= 212

L2 + 16

L3 + 18

L4 V (x) 2 (7.98)

which matches the formula in Conjecture 7.5. Recall that the magnitude of theprojection satises V (x) 2 = 1 −(x · f )2, so this formula does account for the initial

angle between the polymer and the force vector. In addition, this formula is consistent

with the forceless case, for if we remove the force, then c(f ) becomes p, and V (x)

goes to zero. Thus, the small- L formula in Conjecture 7.5 becomes formula (7.87).

For the case in which L → ∞, we observe that if both s and t − s p, then

the two tangent vectors T (s) and T (t) become nearly independent. By an ergodic

theorem, they both follow the distribution of T ∞, a tangent vector which is distributed

according to the invariant measure m on the unit sphere. Thus, it follows that

E [T (s) ·T (t)] = E [T ∞]2 = c(ζ )2. (7.99)

To nd the next-order term, we turn to the eigenfunction expansion of the transition

probability from Mercer’s theorem. This yields an exponential term whose coefficient

is the principal eigenvalue λ1. The correlation function thus has this asymptotic form:

u(x,s,t ) = c(ζ )2 + e−λ 1 |t−s| (7.100)

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and, when integrated, yields this formula for the mean-square end-to-end length:

R2 = L

0 L

0(c(ζ )2 + e−λ 1 |t−s|)dsdt (7.101)

= 2 L

0 t

0(c(ζ )2 + e−λ 1 (t−s))dsdt (7.102)

= 2 L

0tc(ζ )2 +

1λ1

(1 −e−λ 1 t ) dt (7.103)

= 212

c(ζ )2L2 + Lλ1

+ 1λ2

1(e−λ 1 L −1) . (7.104)

When we divide by L2 and replace λ1 with −1c , we obtain a formula equivalent to thatin Conjecture 7.5. This summarizes the pair of conjectures that strive to reconcile

the numerical results from the previous chapter with the theory of earlier chapters,

and it completes the picture of the forced Kratky-Porod model. In the next and nal

chapter, we turn our attention to various ways in which we can modify, extend, or

generalize our model to certain different physical situations.

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CHAPTER 8

FUTURE RESEARCH

Thus far, we have developed a complete picture of the extension of the Kratky-Porod

polymer model to the case in which a constant external force acts on the molecule,

and shown that the stochastic process that governs the tangent vector along the

polymer is a spherical Brownian motion with a drift term, which is proportional to the

projection of the force vector onto the normal development of the tangent vector. In

addition, we have examined the root-mean-square end-to-end length of the molecule,

both numerically and analytically, and shown how length scales for the size and shape

of the polymer vary with the curvature parameter κ and the magnitude and direction

of the force vector f . However, while this work constitutes a signicant extension

of the original Kratky-Porod model for a worm-like polymer chain in solution but

without a force, it is still not a sweeping generalization of the model. For the model

still has several restrictions: it is the limit of a discrete model which has a uniform

bond length and bond angle for all segments, and the force is also assumed to be

constant throughout space. Thus, out of the three degrees of freedom in choosing the

position of the next segment (stretching, bending, and twisting), only the twistingdegree is unconstrained.

Thus, in addition to completing the theory about the RMS polymer length and radius

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of gyration, future research in the area of models for semi-exible polymer chains can

be generalized by introducing new degrees of freedom to the model, or by making

a different set of assumptions. The Kratky-Porod model is designed to describe

polymer molecules in dilute solution, but if this description proves, through later

experimentation, to be inadequate, there are several directions in which an alternative

model can be formulated.

8.1 Alternative Models

The rst model which can be used to improve our description of the polymer’s behav-

ior does not allow any additional degrees of freedom for the polymer and its bonds,

but simply modies the force and potential energy. Rather than keep the force vector

constant for all locations in space, we can let the force eld be a function of position,

as is the case with electromagnetic elds. Thus it makes sense to speak of f (Rk) as

the force acting at the kth bead of the discrete model. Since this force now varieswith position, the potential energy is now more complicated; we can no longer write

it as simply −αf ·RN , but rather, the force vector at each bead must be taken into

account:

U = −αN

k=1

f (Rk−1) ·Qk = −αN

k=1

f (Rk−1) ·(Rk −Rk−1) (8.1)

which resembles the formula for the energy induced by a force on a continuous chain:

U = −α L

0f (R(s)) ·dR(s). (8.2)

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When calculating the conditional expectations of functions of the torsional angles in

this model, we must recognize that the force is variable. For example, the innitesimal

rotation matrix, which when scaled is the increment of the driving process, satises

E [bk(φk)|F k−1] = Lκ2τ √ N

[Z −1k−1f (Rk), e3] (8.3)

and the magnitude of the force, which is no longer constant, cannot be factored out

of the sum when calculating the driving process:

E [B N (s)] = κ2τ

[Ns/L ]

k=1

(Z N k−1)−1f (Rk−1), e3 ∆ s. (8.4)

When the limit is taken as N → ∞, we see that the force is now a stochastic process

that is adapted to the driving process, as follows:

E [B(s)] = κ2τ

s

0[Z (s)−1f (R(s)), e3]ds. (8.5)

Thus, we see how altering the force or the potential energy affects the calculations.

Since the force is no longer a constant, but is itself a stochastic process, the compu-

tations of the positions of the beads on the polymer, or of physical quantities such

as the mean-square polymer length, will be more complicated than the case that we

studied.

In this case, as well as the scenario with the constant force, the only degree of freedom

that was allowed at each bond was that of twisting; namely, the length of each segment

was xed, and so was the the angle between any pair of successive monomers, but

each segment was free to twist, or rotate, around the previous one. This is why the

model is called the freely rotating chain, to distinguish it from other systems which

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allow different degrees of freedom. The freely jointed chain, for example, makes each

segment a vector with a three-dimensional Gaussian distribution.

One such model that can be considered drops the condition that any pair of consecu-

tive segments must meet at the same angle, π −θ, thus allowing the bending degree

of freedom. Thus, while keeping the length of each segment xed at a, we can let

each segment be a random vector on the sphere of radius a. Fixing the rst segment

Q1 = ae3 as before, we can dene the potential energy of each segment to be

uk = αa 2(1 −Qk ·Qk+1 ) (8.6)

which is, asymptotically, proportional to the square of the angular distance between

the two segments. This is known as the nearest-neighbor interaction. Another for-

mulation would be to dene the potential energy based on the square of the angular

distance from each segment to the force vector:

uk = −αa f Qk+1 · f (8.7)

which is similar to our rst denition of potential energy, except the segment is dis-

tributed on a sphere rather than a circle. Here, the bond angle is free, not constrained.

Also, rather than change the potential energy, we could allow some bond exibility

by introducing an elastic restoring force, which allows the bond angle to deviate from

π −θ but causes a bond to move towards this angle when perturbed. This could be

examined with the balance-of-force approach of Chapter 3.

Another way to modify the model and alter the potential energy is to consider in-

teractions within the polymer that occur among non-consecutive segments. Several

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sources, such as [37], [39], and [40], describe a polymer as a self-avoiding random walk

in two or three dimensions, and the latter two sources consider this by means of hy-

drogen bonds. Thus, an excluded-volume interaction, also called a steric hindrance,

is added to the potential energy to assure that the molecule never coils back upon

itself or passes through itself, as in [41]. Flory describes this situation as early as

1974 [20] and observes that the mean-square end-to-end distance increases when this

consideration is added to the model. Also, we could add long-range Debye-H¨uckel

interactions, which arise from the distribution of electric charge within the polymer[18]. These introduce an energy term which is Gaussian in the large-force limit.

One other degree of freedom that could be allotted to the model is that of stretching;

that is, the bond length a could be made variable. That is, the length of each

segment could be a Gaussian random variable ak , and the segment Qk could then

be distributed on a sphere of radius ak , either uniformly or according to a potential

energy. In this model, the bonds can be likened to linear springs, as the potential

energy of the system will affect both the magnitude and direction of the segment

vectors.

Rather than change the constraints on the model, we can also consider additional

limiting cases. For example, in order to balance the effects of advection and diffusion

(whose coefficients are −1/ 2 p and ζ / p, respectively) in equation (1.16), we can let the

parameters ζ and p go to innity together, but at different rates, so that ζ /

p = C .

Thus the two terms (diffusion and drift) are proportional, and neither term dominates

the other in the limit. While no trends in the calculations of R and Rg were observed

in this case (which corresponds to a line of slope 1/2 in the log-log plot), a further

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investigation can be made of this situation, in which both sources of motion of the

tangent vector are balanced.

Finally, we can consider the dynamics of the forced polymer. This was the original

motivation for introducing the force, so that we could develop a mathematical ex-

planation for the physical results of Hallatschek, Frey, and Kroy, as the polymer is

stretched or released by a force [27]. This can be done in one of two ways: either, we

can follow the procedure described in Chapter 3 and solve for the positions of the tor-

sional angles based on the balance of forces and torques (which is a very complicatedcomputation best left to a computer for simulation), or we can simplify the model by

limiting the number of possible states. One such way to do this is to consider libra-

tions about the three states ( cis , gauche +, and gauche -) of the rotational isomeric

state (RIS) model; in short, the freely rotating chain is most likely to be oriented in

one of three states with the lowest energy, but may deviate from any of the states,

and this makes it more likely for the polymer to ip from one state to another than

in the more rigid RIS model.

These are several ways, though not the only ones, in which the freely rotating chain

polymer model, adapted to include an external force, can be modied to allow for a

greater amount of internal freedom for the segments, for a variable force, or for the

polymer to move in time.

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8.2 Conclusion

The freely rotating chain, combined with its continuous analogue, the Kratky-Porod

model, is a model for polymer molecules in dilute solution that has been in use

for decades, but it was only recently that results about the model, which had been

known empirically, were proven using the tools and methods of stochastic calculus

and probability theory. These results, such as the fact that the tangent vector along

the polymer traces out a Brownian motion along the sphere, are now mathematical

truths, and they can be extended to the case in which a uniform force eld, in addition

to the thermal noise, acts on the polymer. In this case, the stochastic processes that

govern the position of the polymer are modied to include the force, and the spherical

Brownian motion gains a drift term that reects the fact that the polymer is being

pulled to line up in the direction of the force. The results regarding the physical length

scales, such as root-mean-square length and radius of gyration, which were derived

for the original Kratky-Porod model, have also been altered to allow for the force.Depending on the strength of the force and the intrinsic stiffness of the polymer, the

molecule can be found in the regime of a random coil, a stiff, bent curve, or a rigid rod

extending in the direction either of the force or of the initial segment. As we deepen

our understanding of these theoretical models, we can apply them to the natural

world and gain a greater knowledge of the behavior of long polymer molecules, from

protiens to DNA, with which we come into contact in everyday life.

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Appendix A

MATLAB CODE

The following routine, ‘ffrcmodel.m’, constructs one run of the forced Kramers chain

by means of Monte Carlo methods.

function [R2,Rg2]=ffrcmodel(N,L,lp,zeta,z0,seed)

%Forced "freely" rotating chain

%N = number of segments in polymer

%L = total (arc) length of polymer

%lp = persistence length

%zeta = force parameter

%z0 = initial direction (cos(theta))

%seed: for random number generator

rand(’state’,seed);

%R2 = square of the end-end distance (for taking averages)

%Rcm = center of mass

%Rg2 = radius of gyration (squared)

%Constants

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a=L/N; %Common bond length

kappa=sqrt(2*L/lp); %Curvature parameter

theta=kappa/sqrt(N); %Common supplement to the bond angle

f=[0 sqrt(1-z0^2) z0]’; %Unit vector in direction of force

%Variables

phi=zeros(N-1,1); %Vector of the torsion angles

Q=zeros(3,N); %Array of segmentsR=zeros(3,N); %Array of positions of the beads

%Build the polymer step by step

Q(:,1)=[0;0;a];

Z=eye(3);

mu=zeta*sin(theta);

%Coefficient in the exponent of the Boltzmann-Gibbs distribution

for k=2:N

%Calculate phi by the rejection method

fr=Z’*f; %Rotated unit force vector

psi=atan2(fr(1),-fr(2)); %Phase shift

while phi(k-1)==0

u1=2*pi*rand;

u2=exp(mu)*rand;

if u2<g(u1,mu,psi)

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(1-cos(theta))*sin(phi)*cos(phi) 1-(1-cos(theta))*cos(phi)^2

-sin(theta)*cos(phi);

-sin(theta)*sin(phi) sin(theta)*cos(phi) cos(theta)];

end %rot

function dG=g(phi,mu,psi) %Density function for the torsional angles

dG=exp(mu*cos(phi-psi));

end %g

The following routine, ‘testffrc.m’, gathers data from several runs of ‘ffrcmodel.m’ for

various values of the input parameters.

function [A,B]=testffrc

%Test the routine "ffrcmodel" to examine the mean-square end-to-end

%length of the polymers under various conditions

%A = matrix of mean-square length for each lp, zeta

%B = matrix of radius of gyration for each lp, zeta

format long

%Constants

N=4000; %Number of bonds

L=1e-7; %Length of the molecule

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runs=1000; %Number of runs for each lp & zeta

dmax=50; %Maximum dimension

z0=0; %Initial direction (cos(theta))

%Variables

lp=L*2^-8; %Persistence length

lpmax=L*2^8;

zmin=2^-8; %Force parameterzeta=zmin;

zmax=2^8;

seed=1000; %Seed for random number generator

R2=zeros(runs,1);

Rg2=zeros(runs,1);

A=zeros(dmax);

B=zeros(dmax);

x=1; %Counter for lp

y=1; %Counter for zeta

%Run the tests

while lp <= lpmax

while zeta <= zmax

for k=1:runs

[a,b]=ffrcmodel(N,L,lp,zeta,z0,seed);

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BIBLIOGRAPHY

[1] O. Kratky and G. Porod: Diffuse Small-angle Scattering of X-rays in ColloidSystems, J. Colloid Sci. 4 (1949), pp. 35-70.

[2] H. A. Kramers: The Behavior of Macromolecules in Inhomogeneous Flow, J.Chem. Phys 14 (1946), pp. 415-424.

[3] J. G. Kirkwood and J. Riseman: The Intrinsic Viscosities and Diffusion Con-stants of Flexible Macromolecules in Solution, J. Chem. Phys 16 (1948), pp.565-573.

[4] J. G. Kirkwood and J. Riseman: The Rotatory Diffusion Constants of FlexibleMolecules, J. Chem. Phys 17 (1949), pp. 442-446.

[5] P. E. Rouse: A Theory of the Linear Viscoelastic Properties of Dilute Solutionsof Coiling Polymers J. Chem. Phys 21 (1953), pp. 1272-1280.

[6] M. Doi and S. F. Edwards: The Theory of Polymer Dynamics , Clarendon, New

York (1986)[7] M. V. Volkenstein: Congurational Statistics of Polymeric Chains , Interscience,

New York (1963)

[8] P. J. Flory: Statistical Mechanics of Chain Molecules , Interscience, New York(1969)

[9] H. Yamakawa: Modern Theory of Polymer Solutions , Harper & Row, New York(1971)

[10] P. Kilanowski, P. March, and M. Samara: Convergence of the Freely Rotating

Chain to the Kratky-Porod Model of Semi-Flexible Polymers, Pending (2010)[11] A. C. Maggs: Actin, a Model Semi-exible Polymer, arXiv:cond-mat/9704054v1

[12] D. Andelman and J. Joanny: Polyelectrolyte Adsorption, arXiv:cond-mat/0011072v1

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[13] I. Borukhov et al.: Elastically-driven Linker Aggregation Between Two Semi-

exible Polyelectrolytes, arXiv:cond-mat/0101149v1[14] R. R. Netz and D. Andelman: Neutral and Charged Polymers at Interfaces,

arXiv:cond-mat/0203364v2

[15] Z. Dogic et al.: Elongation and Fluctuations of Semi-exible Polymers in aNematic Solvent, arXiv:cond-mat/0401189v1

[16] Z. Dogic et al.: Isotropic-Nematic Phase Transition in Suspensions of Filamen-tous Virus and the Neutral Polymer Dextran, arXiv:cond-mat/0402345v1

[17] F. Huang et al.: Phase Behavior and Rheology of Sticky Rod-like Particles,

arXiv:0811.0669v1[18] J. F. Marko and E. D. Siggia: Stretching DNA, Macromolecules 28 (1995), pp.

8759-8770.

[19] N. M. Toan and D. Thirumalai: Theorey of Biopolymer Stretching at HighForces, arXiv:0909.1831v1

[20] P. J. Flory: Spatial Conguration of Macromolecular Chains, Nobel Lecture(1974)

[21] A. D. Drozdov: Stiffness of Polymer Chains, arXiv:cond-mat/0407716v1

[22] D. Chaudhuri: Equilibrium and Transport Properties of Constrained Systems,arXiv:cond-mat/0610399v1

[23] T. G. Kurtz and P. E. Protter: Weak Convergence of Stochastic Integrals andDifferential Equations, Lecture Notes in Mathematics 1627 (1995), pp. 1-41.

[24] S. N. Ethier and T. G Kurtz: Markov Processes: Characterization and Conver-gence , Wiley, New York (1986)

[25] S. M. Chitanvis: Theory of Semi-exible Polymers, arXiv:cond-mat/0103039v1

[26] R. L. Bishop: There is More than One Way to Frame a Curve, American Math-ematical Monthly , 82 (1975) pp. 246-251.

[27] O. Hallatschek, E. Frey, and K. Kroy: Tension Dynamics in Semiexible Poly-mers, Part I: Equations of Motion, arXiv:cond-mat/0609647v1

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[28] O. Hallatschek, E. Frey, and K. Kroy: Tension Dynamics in Semiexible Poly-

mers, Part II: Scaling Solutions and Applications, arXiv:cond-mat/0609638v1[29] A. Lamura, T. W. Burkhardt, and G. Gompper: Semi-Flexible Polymer in a

Uniform Force Field in Two Dimensions, arXiv:cond-mat/0110141v1

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