The Dissertation Committee for Lei Huang Certifies that this is the approved version of the following dissertation:

Simulation studies of biopolymers under spatial and topological constraints

Committee:

_____________________________ Dmitrii E. Makarov, Supervisor _____________________________ Ron Elber _____________________________ Graeme Henkelman _____________________________ Rick Russell _____________________________ Thomas M. Truskett

Simulation studies of biopolymers under spatial and topological constraints

by

Lei Huang, B.E.; M.S.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

May 2008

iii

Simulation studies of biopolymers under spatial and topological constraints

Publication No._____________

Lei Huang, Ph.D

The University of Texas at Austin, 2008

Supervisor: Dmitrii E. Makarov

The translocation of a biopolymer through a narrow pore exists in universal cellular

processes, such as the translocations of nascent proteins through ribosome and the

degradation of protein by ATP-dependent proteases. However, the molecular details of these

translocation processes remain unclear. Using computer simulations we study the

translocations of a ubiquitin-like protein into a pore. It shows that the mechanism of

co-translocational unfolding of proteins through pores depends on the pore diameter, the

magnitude of pulling force and on whether the force is applied at the N- or the C-terminus.

Translocation dynamics depends on whether or not polymer reversal is likely to occur

during translocation. Although it is of interest to compare the timescale of polymer

translocation and reversal, there are currently no theories available to estimate the timescale

of polymer reversal inside a pore. With computer simulations and approximate theories, we

show how the polymer reversal depends on the pore size, r, and the chain length, N. We find

that one-dimensional transition state theory (TST) using the polymer extension along the pore

axis as a reaction coordinate adequately predicts the exponentially strong dependence of the

iv

reversal rate on r and N. Additionally, we find that the transmission factor (the ratio of the

exact rate and the TST rate) has a much weaker power law dependence on r and N. Finite-size

effects are observed even for chains with several hundred monomers.

If metastable states are separated by high energy-barriers, transitions between them will

be rare events. Instead of calculating the relative energy by studying those transitions, we can

calculate absolute free energy separately to compare their relative stability. We proposed a

method for calculating absolute free energy from Monte Carlo or molecular dynamics data.

Additionally, the diffusion of a knot in a tensioned polymer is studied using simulations

and it can be modeled as a one-dimensional free diffusion problem. The diffusion coefficient

is determined by the number of monomers involved in a knot and its tension dependence

shows a maximum due to two dominating factors: the friction from solvents and “local

friction” from interactions among monomers in a compact knot.

v

Table of Contents

Chapter 1 Introduction................................................................................................................1

Chapter 2 Computer simulations of the translocation and unfolding of a protein pulled mechanically through a pore ....................................................................................................12

2.1 Introduction.....................................................................................................................12 2.2 Model and methods.........................................................................................................15 2.3 Results.............................................................................................................................17

2.3.1 Comparison of translocation and stretching.............................................................17 2.3.2 The translocation time as a function of the pulling force.........................................22 2.3.3 The pore-size effect ..................................................................................................25 2.3.4 Comparison of the translocation of a ubiquitin-like protein and a homopolymer ...27

2.4 Discussion .......................................................................................................................28

Chapter 3 The rate constant of polymer reversal inside a pore ................................................30

3.1 Introduction.....................................................................................................................30 3.2. Model and Simulation Method ......................................................................................34 3.3. TST and Kramers’ theory estimates of the reversal rate. ...............................................36 3.4. Exact rate vs. TST and Kramers’ theory ........................................................................42 3.5. The dependence of the reversal rate on the pore radius and the chain length................46

Chapter 4 On the calculation of absolute free energies from molecular dynamics or Monte Carlo data .................................................................................................................................52

4.1. Introduction....................................................................................................................52 4.2. The algorithm.................................................................................................................54 4.3. Choosing the weight function w: general strategies and illustrative examples .............56 4.4. Examples of applying different methods .......................................................................59

4.4.1. Coupled harmonic oscillators ..................................................................................62 4.4.2. Uncoupled anharmonic oscillators ..........................................................................62 4.4.3. Ideal gas...................................................................................................................63 4.4.4. One-dimensional potential coupled to a harmonic oscillator bath ..........................64

4.5. Constructing better weight functions: Clustering algorithms ........................................65 4.6. Discussion ......................................................................................................................72

Chapter 5 Langevin dynamics simulations of the diffusion of molecular knots in tensioned polymer chains .........................................................................................................................76

5.1. Introduction....................................................................................................................76

vi

5.2. Methods..........................................................................................................................79 5.3. Simulations results .........................................................................................................84 5.4. Discussion. .....................................................................................................................90

Chapter 6 Summary..................................................................................................................93

Appendix A. Distribution of the knot escape time in the free diffusion model........................96

Appendix B. The drag force on the knot region in the sliding knot model ..............................98

References ..............................................................................................................................100

Vita .........................................................................................................................................106

1

Chapter 1 Introduction

In living organisms, DNA, RNA and proteins often exist in states where they are

subjected to mechanical forces caused by geometrical and/or topological constraints. The

work presented in this Thesis deals with the dynamics of biopolymers under such constraints

and focuses on two classes of problems. The first class is concerned with the spatial

confinement of proteins in the course of their translocation across pores. The second class is

concerned with the dynamics of knotted polymers.

Protein Translocation

The translocation of biopolymer through narrow pores is a ubiquitous biological

phenomenon. Examples include the infection of a cell by virus, the translocation of nascent

proteins through ribosome and protein degradation by ATP-dependent proteases [1,2]. For

example, misfolded proteins or partially damaged polypeptides are toxic [3,4] for a cell since

they cannot function properly. Consequently, they will be degraded by proteases after they

are detected by the cellular quality control system. During the degradation process,

polypeptides will be labeled with a tag, e.g., polyubiqutin, at one end, which enables them to

bind to the proteasome. The protein to be degraded will be unfolded first by being towed

though the channel of the proteasome whose narrowest width is 10-15 Å. The unfolded

polypeptide will be degraded efficiently by sequential hydrolysis [2]. There is experimental

evidence that ATP-dependent proteases actively unfold the proteins targeted for degradation

and can accelerate the unfolding rate by several orders of magnitude in comparison with

chemical denaturation [1,5,6,7], suggesting that the mechanism of co-translocational protein

unfolding is generally different from that of chemical or thermal denaturation. It has been

hypothesized that the cellular machinery accomplishes protein unfolding by pulling

2

mechanically at one end of the polypeptide chain labeled for degradation or translocation

[1,2,7,8,9].

In addition to its biological significance, biopolymer translocation can be utilized in a

number of bio- and nano-technological applications. Experiments with electrophoretically

driven translocation of biopolymers across certain biological and synthetic pores

[10,11,12,13,14] indicate that it is possible to develop biosensors based on the pattern of

electronic current during the translocation. One potential application of these techniques is

high throughput, low cost DNA sequencing. For example, Meller and coworkers have shown

that α-hemolysin (α-HL) can distinguish poly-nucleotide with different sequences [13] and

Astier et al. showed four different nucleotides can be identified with engineered α-HL [14].

The work described in this Thesis was largely motivated by the lack of information

about molecular details of the translocation process. While there had been considerable

theoretical interest in the general polymer translocation problem, most studies had been

limited to the case of unstructured homopolymers [15,16,17,18,19,20,21,22], with the

exception of Refs. [23,24]. In [23], the co-translocational folding dynamics of a new

synthesized protein was studied by computer simulations with a lattice model to mimic the

folding of a nascent protein coupled with translocation through the channel of a ribosome. As

a complementary method to experiment and theory, computer simulations can provide the

molecular details that are not available in either experiment or theory.

To accomplish this, however, a major challenge has to be addressed, which is concerned

with computational feasibility of such simulations. Although all-atom models can provide

atomistically detailed information, their use is often computationally prohibited for a typical

translocation system consisting of a protein and a pore (e.g. a-hemolysin). Specifically,

atomistic, explicit-solvent simulations of such a system would involve 105 or more atoms. As

an alternative, using coarse-grained models can significantly reduce the computational cost.

3

For example, in the model developed by Sorenson and Head-Gordon [25,26,27], following

the work by Thirumalai and co-workers [28,29], each residue, generally including 10~25

atoms, is represented by only one bead. In the same spirit, it is possible to construct a

coarse-grained model for the α-hemolysin pore [30] or even represent it by a simple

analytical potential. The effective potentials in such a model are constructed to account for

the solvent effects implicitly. In addition, the friction imposed by the solvent on the protein is

incorporated by using Langevin dynamics method.

Even with the significant computational savings provided by using minimalist models,

significant computational challenge still exists. Intuitively, the mechanism of translocation

can be studied directly by performing simulations of dynamics, i.e., by launching a series of

Langevin dynamics trajectories after assigning proper coordinate and velocity for each

particle and investigating the translocation pathway of each trajectory. However, the

translocation process commonly involves overcoming energetic and entropic barriers induced

by the pore. Even with minimalist model, the slow, biologically relevant time scales

associated with barrier crossing events are rarely accessible via direct simulations of protein

dynamics [31]. The situation is further complicated by the fact that single translocation

trajectory - even if it can be computed - does not provide information about the dominant

translocation pathway(s), and so a statistical analysis of a large number of long trajectories is

required.

This time scale problem is not specific to the translocation problem only. For example,

the mechanical unfolding of proteins by pulling their ends apart as in atomic force

microscope (AFM) pulling experiments [31,32,33,34,35] is often studied by using the steered

molecular dynamics (SMD) method [32, 34]. In SMD, one effectively attaches a spring to

one end of the protein and pulls at that spring with a constant velocity. The force response of

the protein is calculated as the spring constant times the spring extension,

4

)( vtddkf folds +−=

Here, ks is the stiffness of the spring and d is the domain extension. The domain

extension d is defined as the distance between the two residues between which the stretching

force is measured. dfold is the initial extension of the folded domain and v is the loading rate.

However, typical simulation time scales in SMD are about six orders of magnitude shorter

than the experimentally relevant time scales. To simulate the unfolding within

computationally accessible times, the protein domain is pulled up to six orders of magnitude

faster with two orders of magnitude stiffer spring constants. The resulting simulated

unfolding forces are much larger than those observed experimentally [36,37]. To compare

SMD results with experiments, the former have to be extrapolated to much lower

forces/pulling speeds, which generally is a very difficult task [31,32,33,34,35].

Some approaches to bridge the timescale gap between experiments and simulations have

been developed. Recent studies [31] showed that mechanical unfolding of proteins in many

cases can be quantitatively understood by using a simple one-dimensional picture, where it is

viewed as Langevin dynamics governed by the potential of mean force G(R), defined as

)(ln)( RpTkRG B−= . R is a “mechanical” reaction coordinate and usually chosen to be equal

to the distance between the residues between which the pulling force is applied and p(R) is

the probability distribution of R. Potential of mean force method [38] is widely used, e.g., in

the study of ion permeation in ion channels [39] and in the crystal nucleation [40]. The

unperturbed potential of mean force G(R) can be computed using the weighted

histogram/umbrella sampling method [38,41,42] from a series of equilibrium molecular

dynamics (MD) simulations performed with different constraints. The rate of protein

unfolding, ku(f), under any specific external force can be estimated from transition state

theory (TST) [43]. Moreover, the validity and limitations of this method in the context of

5

mechanical unfolding have been critically assessed previously [31,32,33,34]. Other

approaches for computing G(R) have been proposed, such as the methodology based on

Jarzynski’s identity between free energies and the irreversible work [44,45,46].

In this dissertation, the method based on a reduced-dimensionality view is also used to

study the co-translocational unfolding of a ubiquitin-like protein (See Chapter 2). Using this

approach has the additional benefit that it can be compared to the experimental results, which

are commonly interpreted in terms of one-dimensional free energy profiles [31,47]. Recently,

methods have been developed for optimal estimation of G(R) from the experimental data

[48].

However it is important to understand the intrinsic limitations of such one-dimensional

models. In this dissertation, reduced-dimensionality approaches are tested and compared to

numerically exact calculations for the particular problem of polymer reversal illustrated in

Fig. 1.1.

Fig. 1.1. A scheme shows the reversal of a polymer confined in a narrow pore.

During the translocation of a short polymer, the translocation dynamics depends on

whether the polymer can reverse its direction while it is inside the pore. The polymer reversal

is expected to significantly affect the interpretation of the results of DNA sequencing for

short DNA segments. It is therefore of interest to compare the relative timescale of polymer

translocation and polymer reversal in a pore. In this thesis we use Langevin dynamics

simulations to study how the timescale of polymer reversal depends on pore size and chain

k

6

length. The goal of this study has been two-fold: Firstly, as stated above, the time scale of the

reversal is of interest per se, in the context of the translocation dynamics. Secondly, it allowed

us to quantitatively test reduced-dimensionality methods for computing barrier-crossing rates

in biopolymers.

Barrier crossing rates in biomolecules are commonly interpreted in terms of simple,

one-dimensional free energy landscapes and TST [31,47,49,50] due to the simplicity and high

computational efficiency. However, such equilibrium free energy landscapes are mostly of

value only insofar as they provide information about dynamics and rates. Whether or not a

one-dimensional free energy barrier can provide any reasonable estimate of a transition rate is

generally unclear. Variational transition state theory [51] asserts that TST applied to a

one-dimensional free energy profile along any specific reaction coordinate in principle

provides an upper bound on the true rate constant. However, in practice the reaction

coordinate has to be good enough to make such an upper bound meaningful. Poor choice of

the reaction coordinate, e.g., in the case of mechanical protein unfolding, leads to unphysical

results [52]. It has also been argued that biopolymers have very complex energy landscapes,

involving multiple minima and saddle points and so no single path dominates the transition

rate. Consequently, it is very challenging to find a good reaction coordinate. Intuition can be a

poor guide for choosing reaction coordinates in complex systems. With a few exceptions

[53,54,55], the performance of reduced dimensionality models in describing

multidimensional dynamics and barrier crossing in biomolecules has not been addressed in

the literature. The model problem of polymer reversal inside a pore provides an opportunity

to investigate this gap between proper multidimensional rate theories and simple

one-dimensional free energy barrier models. In this dissertation forward flux sampling

[56,57,58], a method similar to transition path sampling [59], is used to calculate the “exact”

transition rates from simulations. The relative displacement of two ends of polymer along

7

pore axis is chosen as the reaction coordinate and the performance of approximate

one-dimensional theories, such as TST and Kramers’ theory [43], are evaluated by

comparison with “exact” transition rate.

The possibility of polymer reversal in the course of a translocation process leads to an

additional challenge, illustrated in Fig. 1.2. Consider the case, where a di-block copolymer is

confined in a nano bottle in an imaginary experiment. There are two meta-stable states, A and

B, corresponding to different polymer directions. In order to characterize the dynamic and/or

thermodynamic properties of the polymer one needs to know the relative probabilities of

these two states. Due to confinement, it is hard to observe the transitions between the two

states directly as the two energy basins are separated by a high barrier. A possible solution is

to resort to the sampling methods that can boost transitions that cross high energy-barriers,

such as the replica exchange method [38,60] and the accelerated molecular dynamics [61,62].

All of them are concerned with calculating the phase-space distribution of the system directly.

Thinking from a different perspective, one may wish to estimate the free energy difference

between the two states immediately without having to simulate the rare transitions between

the two states. In other words, we want to calculate the absolute free energy for each of the

two states separately by performing relatively short time simulations, one for state A and one

for state B. Unfortunately, it is normally expected that the absolute free energy/partition

function cannot be computed directly with a satisfactory accuracy [38].

Fig. 1.2. Two meta-stable states, A (left) and B (right), of a di-block copolymer confined in a “bottle”.

8

In thermodynamics, the Gibbs free energy is given by G=U-TS. It is straightforward to

calculate the average potential energy U. Therefore, the difficulty of calculating the absolute

free energy lies in the calculation of entropy. Several approximate methods address the

problem of computing the absolute entropy directly from a Monte Carlo (MC) or molecular

dynamics (MD) simulation. The entropy can be estimated by considering the covariance

matrix of atomic fluctuations near A or B [63,64,65,66] or by using a harmonic expansion of

H(p,q) near the minima corresponding to A and B (see [67,68] and Refs. therein; It is also

possible to take anharmonicity into account [67]) – both approaches essentially assume that

the probability distribution is Gaussian for each basin. A number of (in principle) exact

algorithms employing sampling in energy space have been proposed [69,70]. Veith, Kolinski

and Skolnick [71] proposed an exact method of evaluating equilibrium constants from Monte

Carlo or molecular dynamics data. Their method however requires accurate evaluation of

probability density in a relatively small region of configurational space and may therefore be

computationally expensive.

In this thesis, we present a procedure for the calculation of absolute partition functions

QA(B) from MD trajectories (or MC data) for A(B). The method is in principle exact, is

applicable to any long-lived molecular conformations and can be used in conjunction with

any available (e.g., commercial) MD or MC method.

Dynamics of molecular knots

In microscopic world, molecular knots tied in individual polymer strands have attracted

attention of many physicists, chemists and molecular biologists [72,73,74,75,76,77,78,79,80,

81,82,83,84,85,86]. In mathematics knots are defined as closed, self-avoiding curves in

3-dimensional space [72]. Open knots, which have two ends, are more common in nature than

9

closed knots. As indicated by de Gennes [80], such knots are spontaneously tied and untied

by thermal fluctuations in long polymer. Recently, the unknotting time and dynamics of a

trefoil knot tied on a polyethylene chain embedded in a melt of similar but unknotted chains

was studied by molecular dynamics simulations [87]. Furthermore, knots were found exist

both in DNA [74] and in proteins [88] (there are 273 knotted structures found in 32853

entries in Protein Data Bank). The importance of knots as topological defects that affect

polymers’ dynamics has been recognized in a number of contexts. They may, e.g., impede

DNA replication (see, e.g., Ref. [73] and references therein) or lead to long-time memory

effects in polymer melts [80,87]. From a polymer theory perspective, a number of fascinating

issues exist that deal with the scaling properties of random knots (see, e.g., Refs. [73,82,84]).

Due to the topology constraint, the existence of a knot will decrease the entropy of the

polymer chain and shrinking a knot will increase the enthalpy of the knot area. To balance

entropy and enthalpy, knots in long circular polymers are suggested to be localized by

theoretical and numerical studies [89,90], i.e., the amount of polymer engaged within the knot

is very small compared to the total length of the polymer. In an open knot, a very recent

theoretic study [91] predicted that a tight state of the knot in a long worm-like polymer [92]

can be a metastable state, i.e., starting from a knotted conformation, the knot will

spontaneously shrink or expand to a well-defined size and then diffuse along the polymer

until it slips off at either end. How the size of a knot depends on the external forces applied at

two ends was studied by Monte Carlo simulations for knotted polymer chains with finite

length [93]. Meanwhile, with the development of single molecule manipulating technique,

molecular knots have been created and observed at a single molecule level [94,95]. In

particular, knots tied in DNA chains with optical tweezers were retained localized by

applying fixed tensions at both ends of DNA and they were seen to undergo diffusive motion

[95] induced by thermal fluctuations. The diffusion coefficients of different types of knots

10

have been measured [95] and they were found to have no tension dependence for various

tentions (0.1-2pN). Those experiments have motivated several theoretical and simulation

studies of knot dynamics in polymers [79,83,96]. Vologodskii [79] has used Brownian

dynamics simulations to study knot diffusion in DNA with pulling force as 0.5pN and found

that the computed diffusion coefficients for different types of knots agrees with the

experimental values within a factor of two. More recently, Metzler et al. [83] have considered

various knot diffusion mechanisms and have predicted that the diffusion should become faster

near the chain ends. However, the reliability of those theories remains to be corroborated due

to the lack of experiments data. The aim of the present work is to undertake a more

systematic study of the effects of the knot type, the tension in the chain, and the polymer’s

flexibility on the knot diffusion with computer simulations. It is also of interest to compare

the knot in a tensioned polymer with theoretical results for ideal tight knots, which have been

studied theoretically [78,97].

The layout of this dissertation is as follows:

Chapter 2: “Computer simulations of the translocation and unfolding of a protein pulled

mechanically through a pore”. This Chapter is concerned with translocation of an initially

folded protein through a narrow pore. Langevin dynamics simulations is used to study the

translocation of a ubiquitin-like model protein through a pore. We show how the

co-translocational unfolding mechanism depends on the pore diameter, the magnitude of the

driving force and on whether the force is applied at the N- or the C-terminus of the chain.

Chapter 3: “The rate constant of polymer reversal inside a pore”. This Chapter is

concerned with the problem of the reversal of a model polymer inside a pore. Using the

forward flux sampling method, we calculate the “exact” reversal rate and study its

dependence on the polymer length and the pore dimensions. We further compare the exact

11

rate with 1-D transition state theory and with Kramers' theory.

Chapter 4: “On the calculation of absolute free energies from molecular dynamics or

Monte Carlo data”. A new method is proposed for calculating absolute free energies from MC

or MD data. The method is based on the identity that expresses the partition function Q as a

Boltzmann average: 1/Q = <w(p,x)exp[βH(p,x)]>, where w(p,x) is an arbitrary weight

function such that its integral over the phase space is equal to 1. Several ways to choose

weight functions are compared and their limitations are discussed.

Chapter 5: “Langevin dynamics simulations of the diffusion of molecular knots in

tensioned polymer chains”. Using langevin dynamics, we study how the diffusion coefficient

of a knot in a tensioned polymer depends on the applied tension, the flexibility of the polymer

chain and on the type of the knots. We further propose a simple one-dimensional model that

explains both experimental and simulation results.

Chapter 6 contains the summary of the main results of this dissertation.

The materials of Chapter 2-5 have been published [30,98,99,100,101].

12

_____________________________________ aLarge portions of this chapter have been previously published as reference 30 and 98.

Chapter 2 Computer simulations of the translocation and unfolding of a

protein pulled mechanically through a porea

2.1 Introduction

Protein degradation by adenosine triphosphate (ATP)-dependent proteases and protein

import into the mitochondrial matrix involve the threading of proteins though narrow

constrictions whose dimensions are too small to accommodate folded proteins [1,2]. The

ensuing unfolding process is sometimes orders of magnitude faster than chemical

denaturation of the same proteins [1,5,6,7], suggesting that unfolding in the cell may occur

via pathways different from those probed in chemical/thermal denaturation studies [8]. It has

been hypothesized that the cell machinery accomplishes unfolding by mechanically pulling at

the end of the polypeptide chain that is labeled for degradation or translocation [1,2,7,8,9].

Very little is known about the molecular details of such a process. Most current insights

into the unfolding-via-translocation mechanisms [8,9] are inferred from the single molecule

pulling experiments (reviewed in Refs. [102,103]), which probe the mechanical unfolding

process induced by stretching polypeptide chains.

One, however, expects significant differences between these two cases. It is known

[8,9,47,104,105] that the direction and the geometry of the applied force may dramatically

affect the mechanical unfolding mechanism. Since the mechanical pulling experiments

involve the application of a force at the ends of the chain while translocation involves a

distributed force arising from the interaction between the protein and the pore, the resulting

unfolding mechanisms should generally be different.

Here, we use simulations to study the mechanisms of protein unfolding induced by

13

translocation along a cylindrical pore and compare them with the mechanical unfolding as

observed in single-molecule pulling studies. Because fully atomistic simulations are

computationally prohibitive, we have resorted to the use of a minimalist off-lattice model of a

ubiquitin-like domain with N=68 residues [25,26,27,28,29,106]. Our study is different from

previous theoretical studies of polymer translocation [15,17,18,20,21,22,107,108] (in

particular, DNA translocation), which were mostly limited to unstructured homopolymer

models.

We have already exploited [52] the minimalist model of ubiquitin used here to study the

mechanical stretching of ubiquitin. We found this model to have good qualitative agreement

with the more detailed molecular-dynamics simulations of ubiquitin [47,49] as well as with

experimental single molecule pulling studies of ubiquitin, [104,109,110] although the model

tends to somewhat underestimate the unfolding forces and barriers.

Even after the computational savings provided by the use of minimalist models, slow,

biologically relevant time scales associated with barrier crossing events are rarely accessible

via direct simulations of protein dynamics [31]. In a simulation, one can speed up the

translocation by applying a very large force, in the spirit of the steered molecular dynamics

(SMD) method [32]. However, the extrapolation of the SMD results to the lower-force regime

is a very difficult task [31,32,33,34,35]. Hence, here we are following the approach that has

been used and extensively tested in our previous work on the mechanical unfolding of

proteins [31,47, 49,52].

Specifically, when a constant external force f is applied, the protein experiences a

potential of mean force equal to Gf(z)=G0(z)-fz. Here, z is the reaction coordinate coupled to

the force and G0(z) is the protein’s free energy as a function of this reaction coordinate (in the

absence of the force). The choice of the reaction coordinate depends on the process

considered, as shown in Fig. 2.1. In the case of translocation, z ≡ z1 (or zN) is the displacement

14

of the N- (or C-) terminus, whichever is pulled mechanically, along the direction of pulling.

In the case of mechanical stretching, 1| |Nz z z≡ − is the protein extension along the direction

of the stretching force f.

We then assume that the translocation process is slow (as compared with the time scale

of the protein’s internal dynamics) and adopt the simplified view that the translocation

dynamics can be viewed as one-dimensional diffusive motion along z in the potential Gf(z).

The validity and limitations of this view in the context of mechanical unfolding have been

critically assessed previously [31,32,33,34,52]. The key quantity in our theory is therefore the

free-energy profile G0(z). In the following, we describe our calculation of G0(z) for the

translocation process and compare it with that for the mechanical stretching case. The latter

has been studied in detail in our article [52], and here we reproduce some of our previous

results for comparison. Once we know G0(z), we examine the translocation mechanism at

different values of the driving force f applied at either the N- or the C-terminus of the chain.

In addition, we examine how the translocation mechanism depends on the pore diameter.

Fig. 2.1. The mechanical stretching reaction coordinate is the component of the end-to-end distance vector in the direction of the stretching force. The translocation coordinate is the displacement of the chain end along the axis of the cylindrical pore, relative to the pore entrance. The force f applied to this end acts along the axis. The plots of protein structures here and in other figures were created by using the PYMOL software (W. L. DeLano, PYMOL, DeLano Scientific, San Carlos, CA, 2002).

15

2.2 Model and methods

The model. We used the off-lattice model of Sorenson and Head-Gordon [25,26,27], which

in turn builds on the earlier work of Thirumalai’s group [28,29]. Models of this type have

previously been used to study the folding of confined proteins [111,112,113], protein

aggregation [114], and mechanical unfolding of proteins [52,115]. Within this model, each

residue in a protein is represented by a single bead that can be of three types: hydrophobic (B),

hydrophilic (L), or neutral (N). The interaction potential, as a function of the position ri,

i=1, …, N, of each residue, is given by:

V(r1, r2, …, rN) = Vbond + Vbend + Vdih + Vnon-bonded + Vpore

Here, the potential Vbond accounts for the connectivity of the chain and assumes that each

bond is a stiff harmonic spring,

Vbond = 2

2(| | ) / 2

N

b ii

k σ=

−∑ u

where ui = ri - ri-1 is the bond vector and σ = 3.8 Å is the bond length. The force constant of the

spring is kb = 100εh/σ2, where εh is a unit of energy that represents a typical energy scale of the

hydrophobic interaction. The bending potential imposes the constraints inherent to the peptide

bond geometry:

Vbend = 1

20

2( ) / 2

N

ii

kθ θ θ−

=

−∑

where θ0 = 1050 is the equilibrium bending angle, θ i is the angle between ui and ui+1, and kθ =

20εh/(rad)2 is the spring constant. The dependence of the energy on the dihedral angles iϕ

formed between the di = (ui × ui+1) and di+1 = (ui+1 × ui+2) vectors is incorporated in Vdih:

Vdih = 2

2(1 cos ) (1 cos ) (1 cos3 ) 1 cos

4

N

i i i i i i i ii

A B C D πϕ ϕ ϕ ϕ

−

=

+ + − + + + + + ∑

16

The parameters Ai, Bi , Ci and Di are determined by the dihedral sequence of the chain. There

are three dihedral conformations: helical (H) with A=0, B=C=D=1.2εh, extended (E) with

A=0.9 εh, B=D=0, C=1.2 εh, and turn (T) with A=B=D=0, C=1.2 εh. The energy Vnon-bonded

describes the interaction between sequence-distant residues that are not covalently bonded.

This term accounts for excluded volume interactions as well as the attractive forces between

hydrophobic residues and is taken to be a sum of pairwise potentials:

Vnon-bonded = 12 6

1 2| | 3

4 Hi j ij ij

S Sr rσ σ

ε− ≥

−

∑

where the value of the parameters S1 and S2 depends on the type of the residues i and j: S1 =

S2 = 1 for BB interactions; S1 = 1/3, S2 = -1 for LL and LB interactions, and S1 =1, S2 = 0 for

NN, NB, and NL interactions.

Following the work of Head-Gordon’s group [26], the bead sequence used for the

ubiquitin-like model is, LBLBLBLBLBNNLNBBBBBBBBNNLLBBLBBLLBNNLBBBB-

BNLBLBLLBNLBBLBBLNBBLBLBLBL and its dihedral sequence is, EEEEEEEEHTHEH

TEEEEEHHEHHHHHHHHHHEHTEEEEETHEEEETEETHHHHHHHEHHEEEEE. For

sufficiently low temperature, this 68-residue chain folds in the course of a Langevin

dynamics simulation assuming a ubiquitin-like conformation.

Interactions between the chain and the wall: The potential poreV : Within our model, the

interaction between the protein with a cylindrical pore whose radius is porer is given by:

102 2

2

11.01

1ipore lz

Ci i

pore

AVe x y

rα

−

= − + ++

∑

17

where A= 20 hε and l=20.0/σ. Since this potential is not exactly a hard-wall repulsive

interaction, the parameter rpore only roughly describes the pore dimension.

Simulating the protein dynamics: Following previous studies [17,28,29] it was assumed that

the dynamics of each atom in the chain are governed by the Langevin equation of the form

({ }, )( ) ( ) ( )j

i ii

Vm t t tξ

∂= − − +

∂r F

r r Rr

&& & ,

where ri is the position of the i-th atom, m is its effective mass, ζ = 0.05 (σ2/mεh)-1/2 is the

friction coefficient, and R(t) is a random δ−correlated force satisfying the fluctuation-

dissipation theorem. This equation was solved by the velocity Verlet algorithm as described in

Ref. [116].

Obtaining free energy profiles: The replica exchange method [60,117] was used to improve

sampling statistics and avoid the trapping of the system in metastable states over the

simulation timescale. The free energy as a function of the reaction coordinate z is given by

( ) ln ( )BG z k T p z= −

where ( )p z is the sampled probability distribution of z. To obtain the global shape of G(z) far

away from minimum of G(z), the weighted histogram/umbrella sampling method was used

[38,41,42]. All simulations reported here were performed at 0.26B hk T ε= .

2.3 Results

2.3.1 Comparison of translocation and stretching

Free-energy profiles Gf(z) are shown in Figs. 2.2–2.4 for the case of mechanical

18

stretching (Fig. 2.2) and for pulling the same protein through a pore by applying the force

either at its N-(Fig. 2.3) or C-terminus (Fig. 2.4).

When one end of the chain is moved along the pore, G0(z) increases monotonically until

the entire protein is inside the pore, after which G0(z) remains constant. At this point, the

protein has achieved its maximally extended state attainable for a given pore diameter. When

it first happens, the leftmost end of the chain is located at the tunnel entrance (Fig. 2.1).

Therefore the value of the translocation reaction coordinate z≡z1(N)≈45σ measured relative to

the tunnel entrance is the same as the protein extension z≡|zN-z1|. It is then not surprising that

the free-energy cost G0(z≈45σ) of achieving the same extension is similar for both

translocation and stretching. In the case of stretching, G0(z) rises abruptly if the domain is

extended past z~45σ because at this point further chain extension involves a high enthalpic

cost associated with a deformation of molecular bonds.

While the overall free-energy cost of extending the protein is similar in each of the three

cases presented in Figs. 2.2-2.4 the shapes of G0(z) and, consequently, the force-induced

unfolding mechanisms reflected in the shape of Gf(z) are different in each case. For the case

of translocation, the free energy profile Gf(z) is a “downhill ramp” for large values of z,

favoring translocation thermodynamically(see Figs. 2.3 and 2.4). The kinetics of translocation,

however, depends on the applied force. For modest forces, squeezing the protein into the pore

requires surmounting one or several free-energy barriers. Similarly, the native-like (small z)

and the extended (large z) conformations of protein mechanically stretched by a force are

separated by one or more barriers (Fig. 2.2). The local minima of Gf(z) correspond to

unfolding intermediates, which are different in all three cases.

19

Fig. 2.2. The potential of mean force Gf(z) is plotted as a function of the mechanical stretching

reaction coordinate at different values of the stretching force. The force is measured in dimensionless units of f0 = εh/σ. The darkness of each point reflects the probability of observing the corresponding contact in the equilibrium ensemble of conformations corresponding to the given extension z. Secondary structures (helices and strands), to which residues i and j belong, are shown along the i and j axes so that clusters of contacts on the map correspond to the proximity of secondary structure elements.

When the protein is pulled at its N-terminus (Fig. 2.3), the resulting translocation

proceeds via three distinct intermediates represented by structures 1, 2, and 3 in Fig. 2.3. For

a sufficiently high pulling force, the rate-limiting step corresponds to the barrier encountered

between structures 1 and 2 and involves the peeling of a strand from the rest of the structure

and its entrance into the pore. At lower force, however, the rate-limiting step corresponds to

the transition between the last intermediate, structure 3, and the extended state.

20

Fig. 2.3 The potential of mean force Gf(z) is plotted as a function of the translocation coordinate equal to the position of the N-end of the chain along the pore at different values of the stretching force. The pore radius is pore r=σ. The free energy is measured in units of εh and the force is measured in units of f0 = εh/σ (see the Methods Section) The minima of Gf(z) correspond to translocation intermediates 1-3, whose structure is shown along with the corresponding contact maps. The native-like structure 0 not shown here is similar to structure 0 shown in Fig. 2.2.

Similarly, when the C-end of the protein is pulled (Fig. 2.4), several intermediates are

observed too. As the force is increased, the last surviving barrier is that occurring early in the

translocation process (the one between structures 1 and 2 in Fig. 2.4), while at low forces the

rate-limiting step takes place late in the translocation process and involves the squeezing of

21

the tail of the chain into the pore. An inspection of the intermediate structures encountered in

the translocation process reveals that in each case the protein unravels from the end at which

it is pulled. This is easy to observe in the contact maps of the intermediate structures. A

contact map here is a plot containing the points {i, j} for each pair of residues i and j such

that |i-j|>3 and |ri-rj|<d, where d=1.97σ=7.5Å In the case where the N-terminus

(corresponding to the residue i=1) is pulled, the contacts {i, j} with low values of i or j are

destroyed first and the ones in the upper right corner of the map (corresponding to i and j

close to N=68) survive last, indicating that the residues that are close to the N-terminus are

the ones which first become separated from the rest of the domain. Similarly, when the

C-terminus (i.e., the 68th residue) is pulled, this end becomes separated from the rest of the

domain first and the local structure involving the residues close to the N-terminus (i.e., with

low values of i and j) is the last to disappear. These findings support the view that, in the

course of translocation, the protein unfolds sequentially from the end containing the targeting

sequence [1,5,6].

Certain similarities exist between the mechanical unfolding mechanism shown in Fig.

2.2 and the translocation pathways of Fig. 2.3 and 2.4. In particular, in each case the first step

of unfolding involves the separation of the two terminal parallel strands (the transition

between the native-like structure 0 and structure 1). However, in the case of translocation this

step is followed by the structure 1-to-structure 2 transition that involves a substantial barrier.

This barrier is not found in the mechanical unfolding case. The physical origin of the

structure 1-to-structure 2 transition depends on which chain end is being pulled. For example,

when the force is applied at the N-terminus, the local structure destroyed in this transition

involves one of the α helices; the same structure survives until late in the unfolding process in

both the mechanical unfolding case and in the case of the C-terminus-driven translocation.

For sufficiently high forces, these observations are consistent with the view that the local

22

stability of the part of the protein that enters first the translocation pore determines the overall

resistance of the protein to mechanical unfolding [1,7].

Fig. 2.4 The potential of mean force Gf(z) is plotted as a function of the translocation coordinate equal to the position of the C-end of the chain along the pore at three different values of the stretching force. Other details about the plot please refer to Fig. 2.3.

2.3.2 The translocation time as a function of the pulling force

It is possible to measure experimentally the time it takes a polymer to get across a

nanometer-sized pore. One technique (see, e.g., Refs. [10,11,12,118,119,120,121,122,123])

23

involves using an electric field to drive a charged polymer across a transmembrane protein

channel pore such as α-hemolysin while simultaneously measuring the ionic current across

the channel. Whenever a single polymer is inside the pore, it partially or completely blocks

the current; the duration of such blocking events directly reports on the time of polymer

translocation. It is therefore of interest to examine the dependence of the translocation time

on the driving force (that would be proportional to the electric field).

We note that the present model is by no means an accurate description of a typical

transmembrane protein pore. It does not adequately account for the pore geometry and the

distribution of the electric potential along the pore. It also assumes that the pore is long

enough that it can accommodate the entire protein inside. Furthermore, the details of the

translocation mechanism in such pores would depend on the location of the charged groups

along the polypeptide chain (whereas the present model corresponds to the case of charged

groups localized at a chain terminus).

Another complication is that translocation generally involves several intermediates

separated by multiple barriers. It therefore cannot be generally characterized by a single

first-order rate constant. Unless a single rate-limiting step can be identified, one may need to

go beyond the calculation of the equilibrium potential of mean force and study translocation

dynamics. A detailed study that addresses these issues will be reported elsewhere, while here

we limit ourselves to a crude estimate of the effect of the driving force on the translocation

time, which results from the assumption that the overall speed of translocation should be

correlated with the overall barrier ∆Gu(f)=maxGf(z)-minGf(z) encountered along the reaction

coordinate. The dependence of this translocation barrier on the applied force is shown in Fig.

2.5. Despite the fact that the free-energy cost of translocation is the same regardless of which

chain end is pulled (implying that the reversible work done by the pulling force would be the

same in both cases), at finite values of the force f translocation driven via pulling at the

24

N-terminus is slower than that in the case of pulling at the C-terminus.

Fig. 2.5.The unfolding barrier ∆Gu(f) is plotted as a function of the pulling force applied to the

N-terminus (solid line) and C-terminus (dashed line). The free energy is measured in units of εh and the force is measured in units of f0 = εh/σ.

In mechanical pulling studies the unfolding free-energy barrier is often assumed to be a

linear function of the force: [124,125,126]

zfGfGu ∆−∆=∆ )0()( 0

The coefficient ∆z has the simple meaning of the extension corresponding to the transition

state relative to the native state. This approximation does not hold in Fig. 2.5. Instead, the

slope of the function ∆Gu(f) undergoes an abrupt change corresponding to the transition from

the “late” transition state (low force) to the “early” transition state (high force) scenario,

which is accompanied by an abrupt change in ∆z.

25

2.3.3 The pore-size effect

The data reported above are for the case where the pore is narrow enough that any

tertiary structure is lost inside it. The dimensions of various translocation channels span a

considerable range, and the width of each individual channel also often varies along the

channel. Fig. 2.6 shows the free-energy profile G0(z) for three different values of the pore

radius, rpore= σ(case 1), 1.5 σ (case 2), and 2 σ (case 3), for the protein that is pulled at its

N-terminus. The difference between cases 1 and 2 is quantitative—the narrower the pore the

higher the free-energy barrier—but not qualitative: The number of intermediates and their

structures (Fig. 2.3 for case 1; data not shown for case 2) are very similar in these two cases.

Fig. 2.6. The potential of mean force G0(z) for different values of the pore diameter.

Case 3 is different. The pore is now too wide for the protein to attain a linear, extended

conformation without any tertiary structure. In Fig. 2.7, which shows Gf(z) for this case at

different values of the force, one finds only one intermediate (structure 1), which is very

similar to structure 1 in Fig. 2.3. In other words, the initial step of the translocation process,

in which a strand becomes separated from the rest of the domain that still remains outside the

pore, is the same for wide (case 3) and narrow (case 1) pores. However, further stages of

26

translocation in case 3 are different. As seen from Fig. 2.7, the largest barrier associated with

translocation involves the squeezing of a partially folded protein into the pore (structure

1-to-structure 2 transition in Fig. 2.7). Once the protein is inside the pore, it assumes the

partially folded structure (structure 2) and moves along the pore without further unfolding.

Notice that this final structure (structure 2) is different from any of the intermediates

observed for the narrow pore (cases 1 and 2).

Fig. 2.7. The potential of mean force Gf(z) is plotted as a function of the translocation coordinate

equal to the position of the N-end of the chain along the pore at different values of the stretching force for pore size r=2σ Also shown are representative structures encountered in the course of translocation, along with the corresponding contact maps.

A remarkable feature observed in Fig. 2.6 is that for z≤17σ the shape of G0(z) is

independent of the pore size. As a consequence, in the large force limit (corresponding to the

27

early transition state in Fig. 2.3; see the case of f =1.5εh/σ) the shape, the height, and the

location of the translocation barrier is the same regardless of the pore size (cf. Fig. 2.3 and 2.7

for f =1.5εh/σ). We conclude that the translocation time will be independent of the pore size

(in the range of the pore sizes studied) in this regime. The translocation barrier and,

consequently, the translocation time will become dependent on the pore size when the force is

lower (cf. Figs. 2.3 and 2.7 for f =0.5εh/σ).

2.3.4 Comparison of the translocation of a ubiquitin-like protein and a homopolymer

Given that most theoretical work on translocation has previously focused on

homopolymers, it is instructive to compare the translocation of ubiquitin with that of an

unstructured, random-coil-like homopolymer of the same length. This comparison is shown

in Fig. 2.8. Fig. 2.8a presents the translocation free energy profiles G0(z) for both cases. The

free energy cost of squeezing a homopolymer into the pore is much lower than that for

ubiquitin. In the homopolymer case, the barrier in the potential G0(z)-fz (not shown)

disappears at a low force f≈0.4εh/σ≈7.2pN, while in the case of ubiquitin, a substantial barrier

exists even at a much higher force of f≈1.5εh/σ.

The difference between the two cases is qualitative rather than just quantitative. Our

homopolymer essentially behaves as a random coil and the free energy cost required to

accommodate it inside the pore has largely an entropic origin since the entropy of the

polymer constrained by the pore is lower than that of the free random coil. The situation is

different in the case of a folded domain. As it enters the pore, the resulting change in the

entropy is a result of two opposite trends: Unfolding of the domain is associated with an

entropy increase; However confinement of the unfolded domain within the pore results in a

decrease in entropy. Depending on the pore radius, the entropy reduction due to confinement

28

may or may not be larger than the entropy of unfolding. Consequently, the protein inside the

pore may have entropy that is higher than its entropy outside the pore.

Fig. 2.8 a) Free energy G0(z) and b) entropy S(z) as a function of the translocation coordinate z for

ubiquitin and for a homopolymer model that consists of neutral beads.

This is indeed the case here. Fig. 2.8b shows the entropy S(z) of both ubiquitin and the

homopolymer as a function of the translocation coordinate. The entropy was calculated by

using the relationship S(z)=(<V(r1,…,rN)>z1=z-G0(z))/T, where <V(r1,…,rN)>z1=z is the

polymer’s energy with the z-displacement of its first bead constrained at z. We see that, unlike

the case of the homopolymer, the entropy of the domain inside the pore is higher than that of

the domain outside (for the value of the pore radius used). The main origin of the free energy

barrier in the case of ubiquitin comes from the energetic cost of denaturing the protein.

2.4 Discussion

We have studied here how a protein unfolds when being pulled across a long, narrow

cylindrical pore. The observed unfolding mechanism is different from that probed by single

molecule mechanical unfolding experiments. It depends on the applied force, on the pore

29

diameter, and on whether the C- or the N-terminus is pulled. The translocation kinetics is

expected to have complex pulling force dependence instead of a simple exponential function

in two-state system. Also the free energy cost in protein translocation has an enthalpy origin

contrary to that in the translocation of a homopolymer.

It is important to note several other limitations of our study that may prevent its direct

comparison with experiments. First, we assume there are only purely repulsive interactions

between pore and monomers, which is obvious not true. Second, our study assumes a pore

long enough that it can accommodate the entire protein inside. The long-pore model would be

inadequate for many pores including the α-hemolysin pore. Our assumption that translocation

is a slow, barrier crossing type of process may be incorrect, especially in a case where the

pore is wide and the resulting free energy barrier is low. If the translocation time scale is

comparable with that of the internal dynamics of the protein then the simple picture of

one-dimensional diffusion in the equilibrium potential of mean force is no longer applicable

and a full-blown simulation of translocation dynamics may be required.

30

_____________________________________ bLarge portions of this chapter have been previously published as reference 99.

Chapter 3 The rate constant of polymer reversal inside a poreb

3.1 Introduction

The physics of polymers confined in pores has received considerable theoretical

attention in recent years [15,18,20,21,98,108,111,127,128,129,130,131,132,133], largely in

connection with the single-molecule experiments, in which DNA and peptides were driven

electrically across synthetic or biological pores [11,12,134,135,136,137,138,139,140].

Translocation of biopolymers through protein pores is believed to have many biological

implications such as the function of proteasomes and protein synthesis [1,2,8,141].

Confinement of a polymer within a pore often breaks ergodicity on an experimental

and/or biological time scale. For example, if one end of the chain enters the pore first, it is

often unlikely that the polymer will reverse its direction so that the head-first progression is

commonly assumed in most models of polymer translocation. Moreover, one often invokes a

simple one-dimensional view of translocation, in which the polymer is assumed to move

along a one-dimensional free energy landscape that describes its interactions with the pore

[21,30,98,111,142]. If the pore is narrow enough that the polymer chain cannot reverse its

direction on a time scale of the translocation event then this free energy landscape is not the

true landscape but one that is obtained by ignoring reversed polymer configurations. If now

the pore becomes wider or the polymer becomes shorter, the likelihood of its reversal

increases. In the opposite regime of a wide pore, the polymer is free to tumble inside the pore

so that the true free energy landscape should be used.

These considerations show the importance of knowing the time scale of polymer reversal

relative to the typical time it dwells inside the pore. One example where these two time scales

31

can be comparable is the case of translocation of short peptides with a typical length of

20N ≤ through the α-hemolysin pore, whose diameter is comparable with the characteristic

size of the peptide [121,132]. To our knowledge, no theory has been proposed so far to

estimate the time-scale of polymer reversal and its dependence on the pore size and the

polymer length, although polymer reversal events have been observed in a recent simulation

study [143].

This chapter is concerned with computing the polymer reversal rate for the simple case of

an unstructured, flexible polymer chain inside a neutral pore (no attractive polymer-pore

interactions), as shown in Fig. 3.1. Our goal is two-fold. Firstly, we would like to understand

the dependence of the timescale of reversal on the polymer length and the pore size. Our

second goal is concerned with the general problem of computing transition rates in systems

with complex energy landscapes such as biomolecules. One of the most successful and

practically useful approaches to computing chemical reaction rates is multidimensional

transition state theory (TST), which goes more than 70 years back [144]. Application of TST

involves identifying the transition state, i.e., the lowest saddle point separating the reactants

and the products on the system’s potential energy surface, a task that becomes

computationally nontrivial for large systems [145,146,147,148]. It has however been argued

that that biopolymers have very complex energy landscapes, involving multiple minima and

saddle points and so no single path dominates the transition rate. In this case it has been

proposed to stochastically sample the ensemble of transition paths [59]. A variety of such

transition path sampling methods have been developed (see Ref. [59] for a review) but their

practical use is often limited because of their high computational cost.

On the other hand, barrier crossing rates in biopolymers are commonly interpreted in

terms of simple, one-dimensional free energy landscapes. For example, the unfolding of

proteins [125,126,149] and RNA [150,151,152] subjected to mechanical forces is routinely

32

described as a crossing of a barrier on a one-dimensional potential of mean force G(z), where

z is the reaction coordinate whose choice is based on physical intuition. For mechanical

pulling experiments, the choice of z as the extension of the molecule appears natural. Similar

models have also been proposed for translocation of biomolecules through pores [30,98,153].

Because one-dimensional free energy landscapes are relatively easy to compute from

molecular dynamics simulations using, e.g., umbrella sampling [38], such free energy

calculations have recently been used to predict the rates of mechanical unfolding

[31,47,49,50]. Conversely, one can fit experimental rates of unfolding or polymer

translocation to obtain the best approximation for G(z) [48,153,154,155]. Extraction of

equilibrium one-dimensional free energy landscapes from either non-equilibrium simulations

or experiments [32,33,34,44,45,46,151,156,157] has recently received considerable attention.

It is however important to remember that such equilibrium free energy landscapes are

mostly of value only insofar as they provide information about dynamics and rates. Whether

or not a one-dimensional free energy barrier can provide any reasonable estimate of a

transition rate is generally unclear. Variational transition state theory [51] asserts that TST

applied to a one-dimensional free energy profile along some selected reaction coordinate in

principle provides an upper bound on the true rate constant. However in practice the reaction

coordinate has to be good enough to make such an upper bound meaningful. Poor choice of

the reaction coordinate, e.g., in the case of mechanical protein unfolding, leads to unphysical

results [52]. Intuition can be a poor guide for choosing reaction coordinates in complex

systems. With a few exceptions [53,54,55], the performance of reduced dimensionality

models in describing multidimensional dynamics and barrier crossing in biomolecules has not

been addressed in the literature.

The present model problem of polymer reversal in a pore provides an opportunity to

investigate this gap between proper multi-dimensional rate theories and simple

33

one-dimensional free energy barrier models, which is the second goal of this chapter. On one

hand, polymer reversal has all the attributes of a complex transition requiring transition path

sampling, since much of the reversal free energy barrier is of an entropic origin and it is

unlikely that the dynamics would be dominated by a single saddle point. On the other hand,

the distance 1Nz z z= − between the ends of the polymer chain measured in the direction of

the pore (Fig. 3.1a) provides an intuitive choice for the reaction coordinate and the potential

of mean force ( )G z is readily computed. Using TST for this potential, a simple estimate of

the polymer reversal rate constant is obtained. The TST result overestimates the rate constant

since it ignores recrossings of the transition state [43]. Indeed, such recrossings are observed

in Fig. 3.1g, which shows the dynamics of the reaction coordinate, z(t)= zN(t)-z1(t). Within the

simple one-dimensional picture, one possibility of estimating the effect of recrossings is to

use Kramers’ theory [43]. To do so one assumes that the dynamics along z obeys a Langevin

equation describing a particle moving in the potential G(z) and subjected to a stochastic force

and a velocity-dependent friction force. In order for this approach to be useful in practice, one

has to be able to estimate the parameters of the Langevin equation from dynamics at a time

scale that is much shorter than that of reversal. Several methods have been proposed for

estimating the friction coefficient from molecular dynamics simulations [31,32,33,34].

In what follows we compute the exact reversal rate by using the forward flux approach

[56,57,58] and compare it with TST and with Kramers’ theory. We find that while TST is off

by ~2 orders of magnitude for the parameters chosen (the error generally increasing with the

increasing polymer length), it reproduces reasonably well the exponentially strong

dependence of the rate constant on the polymer length and the pore diameter. Although

Kramers’ theory somewhat improves upon the TST estimate, it still significantly

underestimates the effect of recrossings and its dependence on the polymer length.

34

Fig. 3.1. From a to f shown a typical trajectory of polymer reversal observed in simulations. Also shown in (a) is the definition of reversal reaction coordinate, equal to the difference zN-z1 of the positions of the 1st and last monomers measured along the pore axis. (g) An example of a trajectory zN(t)-z1(t) showing several chain reversal events.

3.2. Model and Simulation Method

The potential for a polymer chain in a pore. Our model of a polymer chain consists of N

beads. Adjacent beads are connected by stiff harmonic bonds described by the potential

21(| | ) / 2bond b i iV k σ−= − −r r , (3.1)

35

where ri=(xi, yi, zi) are the bead coordinates, σ is the equilibrium bond length,

2100 /bk ε σ= , and ε is a parameter that sets the energy scale. Excluded volume effects are

incorporated by using a pairwise repulsive potential defined as

( ) ( )

12 6

1( , ) 44nonbonded i j

i j i j

V σ σε

= − + − −

r rr r r r

(3.2)

for 1/62, 2j ij i σ≥ + − ≤r r and zero otherwise.

An infinite cylindrical pore with radius porer is aligned in the z direction. The interaction

between each monomer and the pore is described by a potential equal to

( )42 2 1/2( )

1000 i i porepore i

x y rV ε

σ

+ −=

r (3.3)

for ( )1/22 2i i porex y r+ ≥ and zero otherwise.

The size of the pore is fully characterized by the parameter porer . However when

reporting our results in the following Sections, we will be using an effective geometric radius

of the pore / 2porer r r σ= + ∆ + that contains two additional correction terms. The term r∆

accounts for the fact that each monomer can travel farther than porer away from the pore axis

because the pore potential is soft. It is estimated by requiring that the effective volume

accessed by a monomer is the same as for a hard-wall pore with the radius porer r+ ∆ :

( )/

0 02 2porepore B

r r r rV r k T

r rre dr rdrπ π

=∞ = +∆−

= ==∫ ∫ (3.4)

The second correction term accounts for the finite monomer “radius” equal to / 2σ

and is similar to the correction used by Luijten and Cacciuto [158]. This correction is

important for reproducing the correct scaling relationship for the pore size dependence of the

36

free energy of confinement [159] (see Section 3.5) when using values of porer that are only

moderately large. In the range of the parameters used, the overall correction is virtually

conatsnt, 0.66porer r σ≈ + .

Polymer dynamics. We assume that the chain obeys the Langevin equation of the form

( ) ( ) ( )i ii

Vm t t tξ∂

= − − +∂

r r Rr

&& & , (3.5)

where m is the monomer mass, ξ is the friction coefficient for each monomer [whose value

is chosen to be ξ =2.0(σ2/mε)-1/2], and R(t) is a random δ−correlated, Gaussian-distributed

force satisfying the fluctuation-dissipation theorem. We report all of our results using

dimensionless units of energy, distance, time, and force set by ε , σ , 2 1/2=(m / )τ σ ε , and

/ε σ , respectively. All of the simulations were performed at the same temperature equal to

1.0 / BT kε= .

3.3. TST and Kramers’ theory estimates of the reversal rate.

The distance

1Nz z z= − (3.6)

between the chain ends projected onto the direction of the pore axis (see Fig. 3.1a) provides

an intuitive choice for the reversal “reaction coordinate”. Fig. 3.2 shows the potential of mean

force (i.e. the free energy profile) ( )G z computed for pores of different size. We used the

standard umbrella sampling/weighted histogram method [38,41] to calculate G(z) from a

series of constrained Langevin dynamics simulations.

The free energy G(z) has two minima corresponding to the polymer aligned along the

37

pore in either direction. We can view one minimum (say, the left one) as “reactants” and the

other as “products” of the reversal transition. In transition state theory, the rate of barrier

crossing can be calculated according to the formula:

[ ][ ]

reactants

exp ( ) /2 exp ( ) /

TS BTST

B

G z k TukG z k T

−=

−∫. (3.7)

Here 0TSz = is location of the transition state, the integral in the denominator is carried out in

the vicinity of the “reactant” minimum, and the velocity factor is given by 2 Bk Tu zπµ

= =& ,

where / 2mµ = is the reduced mass associated with relative motion of the two end

monomers.

-16 -12 -8 -4 0

0

3

6

9

G(z

)/kBT

z

N=20 r=1.66σ r=1.86σ r=2.06σ

Fig. 3.2 The free energy G(z) plotted as a function of the reaction coordinate z=zN-z1 for a polymer confined inside pores of different radius. G(z) is a double well symmetric with respect to z=0. Here we show only the left side of the double well. The kink observed at small values of |z| is due to the discrete nature of the end beads, which are in close proximity at z≈0.

38

TST overestimates the rate constant k as it ignores recrossings of the transition state and

assumes that every trajectory that crosses from the reactant side to the product side is reactive.

However such recrossings are observed in Fig. 3.1g. The effect of recrossings can be

quantified by the transmission factor defined as

/ TSTk kκ = (3.8)

In the following Sections we will compute the exact rate constant and thus estimate κ . As

such a calculation is computationally expensive, it is useful to explore other alternatives for

approximating this factor. One possibility is to assume that one-dimensional trajectories z(t)

can be described by the generalized Langevin equation (GLE) of the form

0

( ) ( ) / ( ) ( ) ( )t

z t dG z dz K t z d f tµ τ τ τ= − − − +∫&& & , (3.9)

where ( )K t is the friction kernel and ( )f t is a stochastic force that is related to the kernel via

the fluctuation-dissipation theorem, (0) ( ) ( )Bf f t k TK t= . It is somewhat counterintuitive

that the effective mass µ does not depend on the length or other properties of the polymer

and is simply equal to m/2. This is because the kinetic energy must satisfy the equipartition

theorem, 2 2 21/ 2 / 2 / 2N Bz z z k Tµ µ= + =& & & , and 2 2

1 / 2 / 2 / 2N Bm z m z k T= =& & .

Two further approximations can be made. In the overdamped case, one can neglect the

acceleration term in Eq. 3.9. If the characteristic memory time of the kernel ( )K t is short

compared to the relevant dynamics timescale then memory can be neglected and one can

write

0 ( ) / ( ) ( )dG z dz z t f tµγ= − − +& , (3.10)

where

39

1

0

( )K t dtγ µ∞

−≈ ∫ (3.11)

Then the transmission factor is obtained from Kramers’ theory [43]:

2

214

b

b b

ωγ γκ

ω ω γ= + − ≈ , (3.12)

where

( ) /b TSG zω µ′′= − (3.13)

is the upside down barrier frequency. Several methods of estimating the effective friction

coefficient have been previously proposed [31,32,33,34]. The method used here is based on

equilibrium fluctuations of the reaction coordinate in the “reactant” state. Specifically,

consider the correlation function

2( ) ( ) (0)C t z t z z= − (3.14)

We assume that between 0 and t the polymer does not undergo the reversal transition. In other

words, we are considering the dynamics on a time scale much shorter than that of the

transition of interest. Using quadratic approximation for ( )G z in the vicinity of z and

assuming overdamped regime one finds from Eq. 3.9:

0

( ) ( ) ( ) ( ) 0t

G z C t K t C dτ τ τ′′− − − =∫ & , (3.15)

or, in the Laplace space [160]:

ˆ( ) ( )ˆ ( ) ˆ(0) ( )G z C s

K sC sC s

′′=

−, (3.16)

where the hat denotes Laplace transform. Using Eq. 3.11, we get the following estimate of the

friction coefficient:

40

0

( )ˆ( ) ( )(0)ˆ (0) /(0) (0)

C t dtG z G zCK

C Cγ µ

µ µ

∞

′′ ′′= = =

∫ (3.17)

To illustrate the utility of Eq. 3.17, consider the simplest dynamical model of a polymer,

the Rouse chain [161], which can be thought of as a collection of beads connected by

Hookean springs placed in a viscous liquid, with the excluded volume and hydrodynamic

effects ignored. The probability distribution of the end-to-end distance vector 1N= −r r r for a

Rouse chain is Gaussian,

3/2 2 2 2

2 2

1( ) exp2 2

x y zpN Nπ σ σ

+ + = −

r ,

so that the resulting one-dimensional potential of mean force corresponds to a harmonic

oscillator,

22

1( ) ln ( )2

BB

k TG z k T p z zNσ

= − = . (3.18)

The correlation function for end-to-end distance fluctuations is given by [161]

2 21,3,5,...

8( ) (0) expp p

tC t Cpπ τ=

= −

∑ . (3.19)

Here 2 2(0) / 3C z Nσ= = and the relaxation times are given by

2 2

2 23pB

Nk Tp

ξ στ

π= , (3.20)

where ξ is the monomer friction coefficient. Application of Eq. 3.17 gives, for the effective

friction coefficient:

1a Nµγ ξ= , (3.21)

where 1 1/ 36 0.0278a = ≈ . This friction coefficient is proportional to the monomer friction

41

coefficient and grows linearly with the number of monomers.

The dynamics of the end-to-end distance of the Rouse chain can thus be approximated by

that of a damped harmonic oscillator, with the potential given by Eq. 3.18 and the friction

coefficient from Eq. 3.21. Obviously this description is not exact since the correlation

function for a damped harmonic oscillator (in the overdamped limit) should exhibit a single

exponential decay,

( ) (0)exp G tC t Cµγ

′′ = −

, (3.22)

while Eq. 3.19 is multi-exponential. Nevertheless, the amplitude of the second term in Eq.

3.19 is 9 times smaller than that of the first one so that the main contribution to Eq. 3.19

comes from the slowest relaxation mode of the chain. If we simply neglect all the terms with

p>1 and equate the remaining exponential term to Eq. 3.22, we obtain a slightly different

estimate for γ :

2a Nµγ ξ= , (3.23)

where 2 4

8 0.02743

aπ

= ≈ , which is very close to Eq. 3.21.

These considerations suggest that a damped harmonic oscillator model may be a reasonable

description for the dynamics of the end-to-end distance of a Rouse chain. Of course, it is also

possible to incorporate memory effects, since the memory kernel K(t) can be computed from

Eqs. 3.16 and 3.19.

We note that the same type of analysis has been applied to the relaxation dynamic of a

polymer chain squeezed inside a pore by Arnold et al. [143], who showed that it can be well

approximated in terms of a damped harmonic oscillator. They have further demonstrated that

the linear dependence of the effective friction coefficient similar to that in Eqs. 3.21 and 3.23

holds even for a confined chain.

42

3.4. Exact rate vs. TST and Kramers’ theory

We now compare TST (Eq. 3.7) and Kramers’ theory (Eq. 3.12) rate estimates with the

exact (to within statistical errors) rate. We used two approaches to compute the exact rate. For

wide enough pores and short enough chains, we have been able to run a simulation long

enough to observe chain reversals directly (see Fig. 3.1g) and to compute the rate k by fitting

the probability distribution ( )p t dt of the waiting time t between two successive reversal

events to an exponential function, ( ) ktp t dt ke dt−= . For longer chains and narrower pores we

have used the forward flux method [56,57,58].

Fig. 3.3. (a) (Left) The reversal rate constant plotted as a function of the pore radius for a chain

consisting of N=20 monomers. The rates from transition state theory, Kramers’ theory, the forward-flux sampling method and brute-force simulations are compared. See text for the details of the calculations. (b) (Right) The reversal rate constant plotted as a function of the pore radius for a chain consisting of N=35 monomers. Comparison of transition state theory, Kramers’ theory, and the forward-flux sampling.

1.7 1.8 1.9 2.0 2.1 2.2

10-6

10-5

10-4

10-3

Brute-force simulations Forward-Flux-Sampling Kramers' theory TST

k

r2.1 2.2 2.3 2.4 2.5 2.6 2.7

10-7

10-6

10-5

10-4

10-3

Forward-Flux-Sampling Kramers' theory TST

k

r

43

Fig. 3.3 compares the two approximations with the exact rate constant for different pore

radius and different chain length. For the parameters used in Fig.3.3, TST overestimates the

exact rate roughly by two orders of magnitude and Kramers’ theory by one order of

magnitude.

It is somewhat surprising that Kramers’ transmission factor (Eq. 3.12) with the friction

coefficient of Eq. 3.17 estimates the effect of recrossings rather poorly. To understand the

origin of this discrepancy better, let us recall another formulation of the exact rate based on

correlation functions (see, e.g., Refs. [43,162,163,164]). In this formulation, the rate constant

is written in terms of the flux-position correlation function:

11 (0) ( (0) ) ( ( ) ) ( )

plateaur TS TS t t kk p z z z z t z k tδ θ −−

<= < − − >≡ =

& (3.24)

Here pr =1/2 is the probability for the system to be in the “reactant” basin of attraction and

( )xθ is the step function equal to 1 for x>0 and zero otherwise. At short time, 0t → , all

trajectories that originate at the transition state and move in the direction of the products

appear to be reactive and contribute into Eq. 3.24. This corresponds to the TST

approximation so that (0) TSTk k= . At longer times some of the trajectories recross and thus

are not selected by the step function ( ( ) )TSz t zθ − . Eventually, k(t) reaches a plateau value

( )plateau TSTk t k k= < , which does not change as long as the time is still short compared to 1/k.

This plateau value is the exact rate constant. It is convenient to define the time-dependent

transmission factor

( ) ( ) / TSTt k t kκ = (3.25)

such that (0) 1κ = and ( )tκ κ= for 1pateaut t k −< = .

In Fig. 3.4, we plot the transmission factor κ(t) computed for the polymer directly from

44

Eq. 3.24 and compare this with ( )Langv tκ computed from the one-dimensional Langevin

equation with the friction coefficient estimated according to Eq. 3.17. As expected, the

plateau value for ( )Langv tκ coincides with Kramers’ transmission factor given by Eq. 3.12. At

short times, ( )tκ nearly coincides with ( )Langv tκ . However at longer times, ( )tκ becomes

lower than ( )Langv tκ and achieves a different plateau valueκ . Being close to zero, this plateau

is swamped by statistical errors. This illustrates the common difficulty of using the

correlation function approach: Even though it gives the exact result for the rate in principle,

in practice, when the transmission factor is small, it may be very difficult to get it to converge.

This is why we have used the forward flux method rather than the correlation function

approach in all of our calculations of the rate constant k.

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

κ(t)

t

simulation in 1-D PMF full simulation

a)

0 20 40 60 80 1000

1

2

3

4

5

6

C(t)

t

simulation in 1-D PMF full simulation

b) Fig. 3.4. Comparison of the actual dynamics along the reaction coordinate z(t) with those approximated by a one-dimensional Langevin equation in the potential of mean force G(z). For 1D Langevin dynamics, the effective friction coefficient was estimated from Eq. 3.17. The data are for N=20 and r=2.16σ. (a) The time-dependent transmission factor κ(t) defined by Eqs. 3.24-25. (b). The position autocorrelation function used to evaluate the friction coefficient according to Eq. 3.17.

While the Langevin equation estimate for the transmission factor, ( )Langv tκ , disagrees

45

significantly with ( )tκ , the position autocorrelation function (Eq. 3.14) agrees reasonably

well with that computed using one-dimensional Langevin equation (Fig. 3.4b).

It is impossible to fit the entire time dependence of the transmission coefficient ( )tκ

using a Langevin equation with a constant friction coefficient γ . Specifically, in order to

obtain the correct plateau value κ from Eq. 3.12, one would have to choose the value of

γ to be an order of magnitude higher than we estimated from Eq. 3.17. However if such high

value is chosen, the short-time behavior of ( )Langv tκ will differ significantly from ( )tκ .

Moreover, the position autocorrelation function ( )C t will display incorrect behavior. These

observations indicate that, for the purposes of estimating the transmission factor,

one-dimensional Langevin equation with a constant friction coefficient is a rather poor

description of the dynamics along the reaction coordinate, even though it may capture other

properties (such as C(t)) reasonably well.

As is further discussed in the next Section, the scaling behavior of the transmission

factor κ with the polymer chain length is also different from that predicted from Kramers’

theory. Since the effective friction coefficient is directly proportional to N [143], the Kramers

transmission factor in the overdamped regime (see Eq. 3.12) should be inversely proportional

to N (note that we found the barrier frequency bω to be only weakly dependent on N). In

contrast, the actual transmission coefficient exhibits a much stronger dependence on N.

It is not surprising that the simple Langevin equation fails to accurately account for

barrier recrossings. Near the transition state where 0z ; the two end monomers are near

one another (see Fig. 3.1c-e) each experiencing a rough energy landscape due to the crowding

of the monomers of the doubled back chain inside the pore. This leads to transiently trapped

chain conformations near the top of the barrier that are not captured by the simple Langevin

46

equation, with the friction that was estimated from the relaxation dynamics of the chain that

is aligned along the pore.

It is possible that allowing for position dependence of the friction coefficient and

memory effects will improve a Kramers’-type estimate of the rate constant [55]. We did not

attempt to do this because accurately determining position dependence of the friction

coefficient is a rather difficult task that defeats the purpose of using the Langevin equation as

a quick and simple way of estimating the effect of recrossings.

3.5. The dependence of the reversal rate on the pore radius and the chain length

20 25 30 35 40 4510-8

10-7

10-6

10-5

10-4

10-3

10-2

Forward-Flux-Sampling Kramers' theory TST

k

N

0 10 20 30 40 50 60 700

3

6

9

12

∆G

TS/k

BT

N

r=2.16σ r=2.46σ r=2.66σ

The dependence of the reversal rate constant on the pore radius is already shown in

Fig. 3.3. Fig. 3.5 shows the chain length dependence, indicating that k(N) is nearly

exponential. To further understand these dependencies, we write the rate as a product of three

Fig. 3.5. The reversal rate constant plotted as a function of the polymer length for

2.16r σ= .

Fig. 3.6. The reversal free energy barrier ∆GTS≡G(0)-min G(z) plotted as a function of the chain length for different values of the pore radius. See Fig. 3.2 and Section 3 for the description of G(z).

47

terms:

( , )( , ) ( , ) ( , ) exp TSTST TST

B

G N rk N r k N r N rk T

κ κ ν ∆

= × = × × −

(3.26)

Here ( ) min ( )TS TSG G z G z∆ = − is the barrier height and TSTν is the TST prefactor, which is

found from Eq. 3.7:

[ ]1

exp ( ) / , ( ) ( ) min ( )2

TSz

TST Bu G z k T dz G z G z G zν

−

−∞

= −∆ ∆ = − ∫

(3.27)

Let us now examine each term separately.

The free energy barrier TSG∆ . As seen from Fig. 3.6, the barrier is a linear function of N.

Following the standard argument [159], in the limit where the pore diameter is much larger

than the monomer size TSG∆ should depend only on the ratio of two length scales of the

problem: The first one is the Flory dimension of the chain given by FR aNν= , where

3 / 5ν ≈ . The second length scale is the pore radius r. Thus one expects [159]

( , ) FTS B

RG N r k Tr

α ∆ ∝

(3.28)

Then the only way the barrier can be an extensive property of the chain, i.e., TSG N∆ ∝ , is if

1/ 5 / 3α ν= ≈ so that

5/3( , ) /TSG N r N r∆ ∝ (3.29)

We however found that our data for different polymer lengths and pore sizes could not be

fitted by a universal scaling formula of Eq. 3.29.While the barrier is a linear function of N

(see Fig. 3.6), its r-dependence is stronger than that predicted by Eq. 3.29. To understand this

finding, we note that in the transition state (z=0) the polymer is close to a ring conformation.

48

Therefore the reversal barrier can be crudely estimated as the difference between the

confinement free energy of the ring polymer and a linear polymer in a pore of the same

radius,

TS ring linearG G G∆ −; (3.30)

The case of the linear polymer is well studied in the literature [159] giving the scaling law of

the form of Eq. 3.29, 5/3linearG Nr−∝ . Using the argument used to obtain Eq. 3.29, the

confinement free energy of a ring polymer should satisfy the same scaling law,

5/3ringG Nr−∝ .

In Fig. 3.7 we examine the validity of this scaling law for ring polymers and for linear

chains. The free energy of confinement G is related to the mean radial force f the pore exerts

on a monomer [158],

1 /f N dG dr−− = . (3.31)

This force can be directly measured in the course of a simulation. In view of the linear scaling

of G with N, f is independent of the number of monomers N. If the free energy scales as 5/3r−

then the force should scale as 8/3r− .

Indeed, for linear polymers we find that ( )ln f− plotted vs. ln r is close to a straight

line and the slope ln( ) / lnd f d r− is close to the theoretical value -8/3. However for ring

polymers, the slope generally has an absolute value higher than 8/3. Furthermore, the slope

varies, becoming closer to the theoretical value in the limit of wide pore/long polymer. We

expect the theoretical scaling law to be recovered for wide pores and, respectively, long

chains. This regime is computationally demanding and not accessible by the present

simulations. The free energy of confinement is higher than that predicted by the scaling law

because of the additional contribution from the repulsive interaction between two strands of

49

the ring polymer. Therefore the requirement for the pore radius to be much larger than the

monomer size is more stringent for ring polymers. Finite size effects were also found to be

important for the relaxation dynamics of long polymer chains in Ref. [143], where the

asymptotic scaling behavior was found to be inaccessible by simulations.

2 3 4 5

0.1

1

0.1

1

-f

r

linear polymer N=20 N=45 N=80 N=200

ring polymer N=20 N=45 N=80 N=200 N=500

-f

Fig. 3.7. The mean radial compression force per monomer (Eq. 3.31) for a ring polymer and linear

polymer confined inside a pore of radius r. For the linear polymer, the straight line shown is the best fit for N= 200 and is described by the equation –f=3.4r--2.68, which is consistent with the theoretical scaling law. For the ring polymer, the straight line shown fits the data for N=500 and is described by–f=3.4r-2.93.

The prefactor TSν . As shown in Fig. 3.8, the TST prefactor can be fitted by the formula

0.37 1.110.731TSTv N r− −= (3.32)

In the harmonic approximation, this prefactor is equal to the attempt frequency [43],

50

0 / 2TSTν ω π≈ , (3.33)

where 0 ( ) /mG zω µ′′= is the harmonic oscillation frequency at the minimum mz of ( )G z .

For the pore size/chain length studied here, the error of using the harmonic approximation

does not exceed 20%. The quantity ( )mG z′′ has been extensively studied by Arnold et al. in

Ref. [143], where it was referred to as the effective spring constant effk . Simple scaling

arguments based on the “compression blob” picture [159,165] predict 1 1/3effk N r− −∝ . In

contrast, molecular dynamics simulations have led the authors of Ref. [143] to conclude

0.75 1.3effk N r− −∝ even for a chain as long as 2000 monomers. This would correspond to

0.375 0.650 N rω − −∝ . The discrepancy with the blob picture was attributed to finite size effects,

which persist even for very long chains. While the chain length dependence observed in Eq.

3.32 is similar to the finding of Ref. [143], the dependence on the pore radius r is different.

Several reasons may contribute to this discrepancy. Firstly, our calculations have been

performed for chains that are shorter than those in Ref. [143]. Secondly, since we are in the

regime where finite size effects are significant, the apparent scaling exponents are affected by

the particular interaction potential and by the precise definition of the effective pore radius

(see Section 2). Indeed, dropping the corrections to the effective pore radius and setting

porer r= as in Ref. [143], we find 0.75TST rν −∝ , which is closer to the result of Ref. [143].

Finally, the scaling reported here is for TSTν , which is somewhat different from

0 / 2ω π because of anharmonic effects.

The transmission factor. The transmission factor does not show any significant dependence

on the pore size. Its dependence on the chain length can be fitted by the power law

51

1.86.259 Nκ −= × , see Fig. 3.9. This dependence on the polymer length is much stronger than

one expects from Kramers’ theory (see Eq. 3.12) assuming that the friction coefficient is

directly proportional to N. Such a linear dependence for the friction coefficient is predicted

for the Rouse model (see Eqs. 3.21-23). The direct proportionality between γ and N has

also been confirmed by numerical simulations of Ref. [143] for a polymer inside a cylindrical

pore. Since we found the upside down barrier frequency bω to be nearly independent of N,

Kramers’ theory then predicts 1Nκ −∝ , which disagrees with the scaling behavior found

here.

1.5 2 2.5 3

0.06

0.08

0.1

0.12

0.14

0.16

νTST

r

N=20 N=28 N=35 N=45 N=60

20 25 30 35 40 45

0.005

0.010

0.015

0.020

0.025

0.030

κ

N

Fig. 3.8. The TST prefactor as a function of the pore radius for different chain length. The lines are described by the formula, υTST =0.731N-0.37r-1.11.

Fig. 3.9. The transmission factor plotted as a function of the polymer length for r=2.16σ. The solid line is described by κ=6.259N-1.8.

52

_____________________________________ cLarge portions of this chapter have been previously published as reference 100.

Chapter 4 On the calculation of absolute free energies from molecular

dynamics or Monte Carlo datac

4.1. Introduction

In many problems in chemistry, one wants to know the relative populations of two

different basins of attraction, say A and B, corresponding to the same Hamiltonian H. A and B

can for example correspond to two different isomers or to the folded and the unfolded states

of a protein. The basins are separated by a large free energy barrier so that the transitions

between A and B are rare and often cannot be captured by relatively short scale molecular

dynamics simulations. The equilibrium constant, i.e., the ratio of the equilibrium probabilities

pA and pB of finding the system in A and B is given by

exp exp exp B AB B B A B AAB

A A B B B

E Ep Q F F S SKp Q k T k k T

− − −= = = − = −

, (4.1)

where Qi, Fi, Si, and Ei are the partition function, the free energy, the entropy, and the energy

corresponding to the basin i = A or B.

How can one compute the above ratio? Since the energy of each configuration is

calculated in the course of a simulation, the difficulty lies in the evaluation of the entropies,

SA and SB. Leaving out prefactors, the partition function is given by:

( )

( )( )

exp( ( , ) / )A B

A B BQ d d H k T∈Γ

= −∫p,x

p x p x , (4.2)

where ( )A BΓ is the phase space region corresponding to A(B). Except for low-dimensional

cases, the direct quadrature evaluation of this integral is not feasible and the challenge is to

estimate this quantity from Monte Carlo (MC) or molecular dynamics (MD) data.

53

A number of approaches exist that attempt to achieve this. In Monte Carlo simulations,

one can try to design a Monte Carlo move that jumps across the barrier and accomplishes the

transition between A and B thus resulting in sampling of both A and B in a single simulation.

However this often requires detailed insight into the nature of the transitions between the

basins. Alternatively one can use a sampling scheme, in which transitions between A and B

are boosted to take place more frequently while still satisfying the detailed balance. Examples

of such approaches include replica exchange [38,60] and accelerated molecular dynamics

(see, e.g., Refs. [61,62]). While such improvements in sampling may potentially provide a

direct estimate of KAB, they would be an overkill in situations where all one needs to know is

the equilibrium properties of the system, which are determined by the thermodynamic

properties of each isomer (A and B). One should be able to estimate Eq. 4.1 without knowing

anything about transitions between A and B.

Several approximate methods address the problem of computing the absolute entropy

directly from a MC or MD simulation. The entropy can be estimated by considering the

covariance matrix of atomic fluctuations near A or B (Refs. [63,64,65,66]) or by using a

harmonic expansion of H(p,q) near the minima corresponding to A and B (see Refs. [67,68]

and references therein; It is also possible to take anharmonicity into account [67]) – both

approaches essentially assume that the probability distribution is Gaussian for each basin. A

number of (in principle) exact algorithms employing sampling in energy space have been

proposed [69,70]. Veith, Kolinski and Skolnick [71] proposed an exact method of evaluating

equilibrium constants from MC or MD data (further discussed below). Their method however

requires accurate evaluation of probability density in a relatively small region of

configurational space and may therefore be computationally expensive.

Here, we seek a procedure for the calculation of absolute partition functions QA(B) from

MD trajectories (or MC data) for A(B). Suppose we have used a commercial MD code to

54

record a molecular dynamics trajectory describing the dynamics of the species A. Then we

have restarted our simulation with a different initial condition corresponding to the species B

and recorded a trajectory within the basin B. How can one use this information to obtain the

best estimate of QA and QB? The method we seek has to satisfy the following two

requirements:

(i). It must use MD trajectories or canonical sampling data as its input. In other words, we are

not considering here different non-canonical sampling methods that could in principle

provide an estimate of QA and/or KAB.

(ii). Our method must be exact, at least in principle.

The algorithm of Veith, Kolinski and Skolnick [71] is the only one we know that

satisfies both of the above requirements. It will be shown to be a limiting case of a more

general and generally speaking more efficient method that will be presented below.

The rest of the chapter is organized as follows. In Section 2 we derive the central identity

upon which our method is based. Section 3 concerns with practical issues that arise when one

uses this identity. Several numerical tests of our method are presented in Section 4. Section 5

proposes a method to estimate the phase-space density of a system that has multiple basins of

attraction and shows how such estimates can lead to improved free energy calculations.

4.2. The algorithm

Suppose we have performed Monte Carlo (or molecular dynamics) sampling of the

phase-space corresponding to a given basin of attraction. By the basin of attraction, we do not

necessarily mean a potential well; it can by itself have a complex energy landscape. It is

simply the region of phase space that is accessible by a given MD trajectory or a set of data

from Monte Carlo. Other basins may be sampled by changing the initial conditions in the

55

simulation, e.g., when different basins correspond to different isomers of a molecule. As our

simulation does not “see” other basins by definition, in the following we will ignore them.

The following procedure is thus equally applicable to the calculation of the partition function

corresponding to a given basin and to the calculation of the total partition function given

complete sampling. The sampled phase-space points are distributed according to the

phase-space density

1( ) exp[ ( ) / ]BQ H k Tρ −= −p, x p, x (4.3)

normalized such that

basin

( ) 1d d ρ =∫ p x p,x (4.4)

From Eq. 4.3, the partition function corresponding to this basin is given by:

1 ( ) exp[ ( ) / ]BQ H k Tρ− = p,x p,x (4.5)

Since ( )ρ p, x can be found from simulation and H(p,x) is known, Q can be estimated from

Eq. 4.3 for any phase-space point (p,x). This observation is the basis of the method of Ref.

[71]. The practical problem with using Eq. 4.5 is that ( )ρ p, x must be accurately known.

Typically, Monte Carlo estimates of averages (i.e. the quantities that can be represented in the

form of integrals over ( )ρ p, x ) are much more accurate that those of the density ( )ρ p, x

itself. Since our expressions for the partition function or the free energy are not of the form of

phase-space averages, it is normally expected that they cannot be computed directly with a

satisfactory accuracy [38].

It is however possible to turn a single-point estimate of Eq. 4.5 into a phase-space

average. Since Eq. 4.5 is valid for any p and x, we can average this identity over any region

of phase space. Specifically, for any normalized weight function w(p,x) such that

( ) 1w d d =∫ p,x p x

56

we can average Eq. 4.5 over p and x using w(p,x) as a weight function. This gives 1 ( ) ( ) exp[ ( ) / ]BQ d d w H k Tρ− = ∫ p x p, x p,x p,x (4.6)

The right-hand side of Eq. 4.6 can be written as /1 BH k TQ we

ρ

− = (4.7)

where the brackets indicate averaging over the data points that are distributed according to

the density ( )ρ p, x :

... ( )(...)d dρ

ρ= ∫ p x p,x

If the Hamiltonian is of the form ( ) ( ) ( )H K U= +p, x p x where the kinetic energy K(p) is a

quadratic function of momenta then the momenta can be integrated out of Eq. 4.7 to obtain

( ) ( )p xQ Q Q= , where Q(p)= ( ) / BK k Td e−∫ pp is straightforward to evaluate,

1( ) / /( )1/ B BU k T U k TxQ d e weρ

−− ≡ = ∫ xx (4.8)

and w(x) and ( ) / ( )( ) /BU k T xe Qρ −= xx are functions of x only.

Eqs. 4.7-8 are the central result of this chapter.

4.3. Choosing the weight function w: general strategies and illustrative examples

In the rest of this chapter, we will assume that the momenta can be integrated out of

Eq. 4.7. We will then be concerned with the partition function Q(x) as defined by Eq. 4.8. We

will further suppress the superscript in Q(x) and denote this partition function simply Q.

57

Fig. 4.1. The probability distribution ( ) /U xe Qβ− (thick line) and three weight functions used to

calculate Q from Eq. 4.8. 2

1/ 22

12( ) xw x eπ

− =

, 62 ( ) 2.69478exp (5( 1))w x x = − − ,

6(2 )3 ( ) 2.1558 xw x e−= for x<0 and 3( ) 0w x = otherwise.

In principle, Eq. 4.8 is exact for any weight function. As a simple example, we have

calculated the partition function ( )U xQ dxe β∞

−

−∞

= ∫ for the double-well potential

2 2( ) ( ) / 3( 1) 0.9BU x U x k T x xβ ≡ = − − . The normalized probability distribution

corresponding to this potential is shown in Fig. 4.1 together with three different weight

functions (w1(x), w2(x), w3(x)) used to calculate Q from Eq. 4.8. We intentionally used weight

functions that are not optimal (see below): w1 is strongly delocalized, w2 is localized around

one of the wells and is essentially zero everywhere else, and w3 is appreciable only where

( )xρ is small. For each of the three weight functions, the exact value of the partition

function, Q=1.785, was recovered to within 0.5% using 50000 data points obtained via the

Metropolis algorithm [38].

58

As the dimensionality of a problem increases, use of an arbitrary weight function often

results in large statistical errors. What would then be a good choice of w(x)? Consider two

limiting cases. First, suppose that w(x) is a highly delocalized function so that the sampled

region of the configuration space Γ lies inside the region where w(x) is nonzero. Then

/ BU k Twe will be large when U is large and consequently the density 1( ) exp[ ( )]Q Uρ β−= −x x is

low. Since the tails of the distribution ( )ρ x are most poorly sampled, this choice of the

weight function will lead to large statistical errors.

In the opposite limit, w(x) is the delta-function, w(x)=δ(x-x0). Substituting this into Eq.

4.8 results in the single-point estimate 0( )10( ) UQ eβρ− = xx (cf. Eq. 4.5). Choosing w(x) to be

highly localized in some region is wasteful as any information outside this region is discarded

when performing the average. The procedure proposed by Veith, Kolinski, and Skolnick in

Ref. [71] can be viewed as a limiting case of our Eq. 4.8 where the weight function is chosen

to be highly localized in a small region of configuration space. More precisely one arrives at

their procedure by choosing the function w to be constant inside a small hypercube in

configuration space and by additionally replacing the system’s potential energy U(x) by a

constant energy corresponding to the center of the cube.

From the above discussion it is clear that the weight function should ideally be

sufficiently delocalized to retain maximum information contained in the sampled data points

yet it should be restrictive enough to remove contributions from the regions of the phase

space that are poorly sampled. In other words, w(x) should resemble ( )ρ x itself: It should

be large where ( )ρ x is large and zero where ( )ρ x is small. In fact, the exact ( )ρ x

provides the “perfect” weight function: If one sets / /BU k Tw e Qρ −= = in Eq. 4.8 then this

equation becomes 1 1Q Q− −= so the exact value of Q is recovered even if the data set

59

consists of only a single data point.

Of course, if we knew the exact ( )ρ x then we would know Q from Eq. 4.3 and our

task would be accomplished. Since we can always choose w(x) after having carried out the

MC or MD sampling, it is up to us to choose the weight to be localized in the regions of

space that are densely populated by the sampled configurations [71]. Furthermore, an

estimate for the probability distribution ( )ρ x provided by some other, approximate method

could serve as a weight function w(x). For example, an optimal Gaussian approximation for

ρ can be obtained by considering the covariance matrix of atomic fluctuations [63,64,65,66].

Specifically, it is given by

/ 2 1/ 2 11( ) (2 ) (det ) exp ( ) ( )2

gauss Nρ π − − − = − − − x σ x x σ x x (4.9)

Here N is the dimensionality of the configuration space and the covariance matrix is given by:

( )( )mn m m n n= − −σ x x x x (4.10)

where <…> denotes ensemble averaging. The entropy estimation methods of Refs. [63,64,

65,66] assume that this Gaussian approximation approximate the true ( )ρ x well enough and

consequently relate the entropy of the system to the eigenvalues of the covariance matrix.

Instead, we will use Eq. 4.9 to construct a weight function that can be subsequently used to

evaluate the (in principle) exact value of Q.

4.4. Examples of applying different methods

In what follows, we apply our method to several exactly solvable problems and

compare it to several approximations as well as to the exact result.

60

In particular, we will consider two versions of the optimal Gaussian approximation

based on Eqs. 4.9-10. The effective harmonic potential that yields the probability distribution

of Eq. 4.9 is:

11( ) ( ) ( ) ( )2effU U

β−= + − −x x x x σ x x . (4.11)

The partition function corresponding to this potential is

( ) ( ) ( )/ 2 1/ 21 exp( ( )) 2 det expN

OG effQ d U Uβ π σ β = − = − ∫ x x x (4.12)

The equipartition theorem for a quadratic potential of the form (4.11) gives:

( ) ( )2NU Uβ

= −x x , (4.13)

which allows one to rewrite Eq. 4.12 as:

( ) ( )/ 2 1/ 22 2 det exp / 2N

OGQ U Nπ σ β= − + (4.14)

While Eqs. 4.12 and 4.14 are identical for a quadratic potential, Eq. 4.13 is not satisfied for an

anharmonic potential and Eqs. 4.12 and 4.14 lead to two different estimates for the true

partition function Q. Our experience is that QOG2 often results in a highly unphysical estimate

of Q when the “average” system configuration x is a high energy configuration belonging

to a region that is not frequently sampled by the system (e.g., if U(x) is a symmetric double

well then 0x = corresponds to the top of the barrier between the two minima). In such

situations 2OGQ provides a much better estimate of the partition function than 1OGQ . For the

examples below, we provide the values of both 1OGQ and 2OGQ .

Before reporting our results we note that for systems of high dimensionality N

computation of the partition function Q itself is an extremely daunting task. For example, for

a separable system with N =300 degrees of freedom, a systematic error of 1% in the partition

61

function for each degree of freedom will lead to a factor of 3001.01 20≈ error in the total

partition function. At the same time, since the entropy or average energy is additive rather

than multiplicative, the corresponding relative error may appear small. An order of magnitude

error in an equilibrium constant calculation is rarely acceptable and so the goal we are trying

to pursue here is rather demanding.

Table 4.1. Comparison of different methods used to estimate the partition function. See text for details. U(x) Qexact QOG1 QOG2 Q from Eq. 4.8

Eq. 4.15 9.391×1011 9.017×1011 9.366×1011 9.344×1011

Eq. 4.16 587.3 5.710 859.5 588.5

Eq. 4.17 0.5345 2.562×10-7 4.918×102 1.087×101

Eq. 4.18 6.592×10-17 5.592×10-17 7.533×10-17 6.558×10-17

Table 4.1 summarizes our calculations for several model potential described below. In

each case we provide the exact value Qexact for the partition function, the optimal Gaussian

estimates 1OGQ and 2OGQ , and the value Q computed by using the present method (Eq. 4.8)

with the weight function that is equal to the optimal Gaussian distribution,

/ 2 1/ 2 11( ) ( ) (2 ) (det ) exp ( ) ( )2

gauss Nw ρ π − − − = = − − − x x σ x x σ x x

obtained from the covariance matrix σ.

The model potentials and the details of each simulation are described below.

62

4.4.1. Coupled harmonic oscillators

12 2

1 1 11

1 1( ,..., ) ( )2 2

N

N i ii

U x x x x xβ−

+=

= + −∑ . (4.15)

This potential describes a linear harmonic chain with one end fixed ( 0 0x = ). The data

presented in Table 4.1 are for N=30. The Monte Carlo (MC) simulation consisted of

3,000,000 MC steps using the Metropolis algorithm [38], with configurations saved every

five MC steps. Obviously, for a harmonic potential both 1OGQ and 2OGQ are exact so any

deviations from the exact result Qexact are due to statistical errors. Indeed, we see that 1OGQ ,

2OGQ , and Q are all close to the exact value.

4.4.2. Uncoupled anharmonic oscillators

To test our method for a strongly anharmonic system, we have considered the separable

potential

303

1 301

( ,..., ) | |i ii

U x x a xβ=

= ∑ (4.16)

with 0.25ia i= . The data in Table 4.1 for this potential were obtained from 200,000 MC

steps. Since the true probability distribution is no longer Gaussian, both 1OGQ and 2OGQ are

different from the exact value. Our estimate for Q obtained by using the optimal Gaussian

weight function is very close to the exact value.

63

4.4.3. Ideal gas

45

1 451

( ,..., )0.5

ni

i

xU x xβ=

=

∑ (4.17)

This potential represents an ideal gas consisting of 15 atoms inside a 1 1 1× × cube. The walls

of the container are soft: The potential rises sharply but is not infinite when 0.5ix > . Use of a

hard-wall container is problematic when applying Eq. 4.8 if the weight function ( )w x spills

outside the container since U(x) is infinite there. At the same time, by choosing n large

enough in Eq. 4.17 we can make the walls arbitrarily hard yet we can avoid the above

difficulty. Here, we are using n=40.

For a hard-wall cubic container, the partition function is simply equal to 451 1= . For the

potential given by Eq. 4.17, 45 45[ (41/ 40)] 0.9862 0.534452exactQ = Γ ≈ ≈ .

The optimal Gaussian approximation replaces the (nearly) uniform distribution inside the

container by a Gaussian distribution with the same variance. For a one dimensional particle in

a box ( 0.5 0.5x− ≤ ≤ ) we have 0x = and 2 1/12x = so that the optimal Gaussian

approximation gives

2 2

45 45/(2 ) 7

1 4.76 106

x xOGQ dxe π∞

− −

−∞

= = ≈ ×

∫

and

/ 22 1 2812.51N

OG OGQ Q e= ;

The two values are drastically different from one another and from the exact answer. Similar

trends are seen in Table 1 for the potential of Eq. 4.17. The MC simulation included 400,000

steps with configurations saved every step. From Table 4.1 1OGQ and 2OGQ deviated from the

64

exact answer by 7 and 3 orders of magnitude and from one another by ~10 orders of

magnitude. At same time, the estimate of Q that used Eq. 4.8 with the optimal Gaussian

weight function was off by a factor of ~20, a considerable improvement over 1OGQ and

2OGQ .

Use of a better weight function (i.e., one that is closer to the exact ( )ρ x ) improves the

result of Eq. 4.8 drastically. For example, when the weight function was chosen to be the step

function defined by ∏=

=N

iiNi xwxxxw

11 )(),...,,...,( , where 1)( =ixw for 0.5 0.5x− ≤ ≤ and

0 otherwise, Q calculated from Eq. 4.8 was found to be within 0.1% from Qexact using the

same MC sampling data.

4.4.4. One-dimensional potential coupled to a harmonic oscillator bath

Here we consider a rugged potential coupled to a bath of harmonic oscillators: 230

21 30 1 1 12

2

1( ,..., ) ( ) ( )2

ii i

i i

cU x x U x x xβ ωω=

= + −∑

41 1 1 1 1( ) 0.5 0.6sin(30 ) 0.3cos(20 )U x x x x= + + (4.18)

where )1(4.0 −= ici and )1(8.0 −= iiω . The results shown in Table 1 were calculated by

using the data from a MC simulation including 250,000 moves.

The exact partition function can be easily estimated since the integration over the

harmonic oscillator modes can be performed analytically:

Qexact = ( 1) / 2 11 1 1

2

(2 ) exp( ( ))N

Ni

idx U xπ ω β

∞− −

= −∞

−∏ ∫

65

As seen from Table 4.1, both versions of the optimal Gaussian approximation do relatively well

estimating the partition function; Using these approximations to construct a Gaussian weight

function in Eq. 4.8 then results in an estimate for Q that is very close to the exact value.

4.5. Constructing better weight functions: Clustering algorithms

While for the model potentials described above our method fared quite well, we expect

that for problems of higher complexity and/or higher dimensionality choosing the weight

function to be Gaussian may potentially result in significant statistical errors.

As discussed in Section 3, the best choice of the weight function is one that is close to

the actual probability distribution ρ(x) of the data points. Therefore to improve our weight

function we need to find a better approximation for the probability distribution ρ(x) given the

sampled data. A number of techniques exist that strive to achieve this in the context of image

compression and pattern recognition. While we expect that some of the image compression

techniques can be adapted for our purpose, one should keep in mind that unlike two- or

three-dimensional images, the “image” formed by our sampled data is generally one in a

space of a much higher dimensionality. Discussed below is the use of one image compression

method, a clustering algorithm.

Generally, we expect a Gaussian weight function to become inadequate when the

potential energy landscape U(x) involves multiple basins of attraction sampled in a single

simulation. This situation is illustrated in Fig. 4.2 for the simple two-dimensional potential of

the form:

2 2 2( , ) ( 3.0) 0.8( 0.625 )U x y x y xβ = − + − (4.19)

66

Fig. 4.2. (a) Contour plot for the potential of Eq. 4.19. (b) Data points generated by Monte Carlo and

the contour plot of the exact equilibrium probability distribution ρ(x,y)=exp[-βU(x,y)]/Q.The clustering algorithm partitions the data points into two sets shown as triangles and crosses. (c). Contour plot of the optimal Gaussian weight function (Eqs. 4.9-10) for this potential. (d) Contour plot of the weight function obtained by using clustering (Eqs. 4.21-23).

This potential has two basins of attraction separated by a barrier, as shown in Fig. 4.2a. The

configurations (x,y) sampled by the Monte Carlo method will cluster around the two minima

of U(x,y) with very few configurations sampled in the vicinity of the barrier ( , ) (0,0)x y ; ,

see Fig. 4.2b. The optimal Gaussian weight function given by Eq. 4.9 however has its

maximum at ( , ) (0,0)x y = , see Fig. 4.2c. When evaluating Eq. 4.8 using this weight

function, the small number of high energy configurations in the vicinity of the barrier will be

given significant statistical weight and their contribution exp( )Uβ will be large since their

67

energy is high. Since sampling statistics are poor in this region ( ( , ) exp( ( , ))x y U x yρ β∝ − is

small), it is not surprising that the use of such a weight function results in large statistical

errors. In problems of low dimensionality one can still get away with a poor choice of the

weight function (cf. Fig. 4.1) but this is not so when the number of degrees of freedom

becomes large.

The above situation can be remedied by choosing a weight function that is a sum of two

Gaussians of the form of Eq. 4.9, each centered around the respective basin of attraction.

In order for this idea to work in a general case the basins of attraction need to be

identified; Those are the regions of the configuration space where the sampled configurations

x tend to cluster. A number of “clustering” algorithms that attempt to solve this problem have

emerged in recent years [166,167,168,169,170]. A clustering algorithm partitions the data into

clusters based on the distances between data points. Specifically, for a set {x(k)} of sampled

configurations (k =1, 2, … M, where M is the total number of data points) an MM distance

matrix dkm is constructed. Clusters are then formed from points that are sufficiently close to

one another.

An important property of clustering algorithms is that the computational expense

involved does not depend on the dimensionality N of the problem (except for the fact that the

CPU time required for evaluation of a distance between two points would typically be

proportional to N) but only on the number of the data points.

A natural definition of the distance between two data points, x(k) and x(m) would be

( ) ( ) ( ) ( ) 2

1( )

Nk m k m

km i ii

d x x=

= − = −∑x x (4.20)

However if the range spanned by two components of the same vector x is significantly

different then it may be necessary to rescale each component. For example, if in the (x,y)

68

vector 0 1x≤ ≤ and 50 10y≤ ≤ then the points (0, 10) and (1,10) would be considered to

be in close proximity despite the fact that their respective x components correspond to the

maximum and the minimum possible values of x. This kind of situation is remedied by

defining a rescaled vector

1 2

1 1 2 2

, ,...,max( ) min( ) max( ) min( ) max( ) min( )

N

N N

xx xx x x x x x

= − − −

x% .

The distance is then defined as ( ) ( )k mkmd = −x x% %

Once the simulation data {x(k)} are divided into n clusters containing M1, M2, … Mn data

points (M1 + M2 + … +Mn = M) , we construct the weight function as:

1( ) ( )

nj

jj

Mw w

M=

= ∑x x (4.21)

where

/ 2 1/ 2 11( ) (2 ) (det ) exp ( ) ( )2

Nj j jw π − − − = − − −

x σ x x σ x x (4.22)

and

( )( )cluster

( )j mn m m n n j= − −σ x x x x (4.23)

In other words, we construct the optimal Gaussian approximation for the data contained in

each cluster separately and define the overall weight function as its weighted average over all

clusters, the weights being proportional to the number of data points in each cluster.

Applying Eq. 4.8 we then find for the partition function:

/ 1

1 11/ B

n nj jU k T

j jj j

M MQ w e Q

M Mρ

−

= =

= =∑ ∑ (4.24)

where Qj is the partition function estimate obtained by using the weight function wj.

Each of the Qi’s by itself provides an estimate of Q. The difference between these

69

estimates may serve as a measure of the statistical error in using Eq. 4.24. In the test

problems that we have studied (see below) we saw that the exact value of Q lay between

1max jj n

Q≤ ≤

and 1min jj n

Q≤ ≤

suggesting that the cluster method can be used to obtain an upper and a

lower bound for Q. Although we do not expect this generally to be true, we can prove that

min maxj jQ Q Q≤ ≤ under the following – somewhat restrictive – assumptions:

(i) All low-energy basins of attraction are visited in the simulation

(ii) Each cluster corresponds to a different basin of attraction and there is very little

overlap between the Gaussian weight functions wj corresponding to different

basins of attractions

(iii) Local sampling is good so that within each basin of attraction the sampled

configurations are distributed according to a probability that is proportional to

exp( )Uβ−

In other words, this is a situation where locally each basin of attraction is sampled

adequately; however because of infrequent transitions among different basins the number of

data points within each basin deviates from the value expected on the basis of the Boltzmann

distribution.

Under these conditions we can write the probability distribution of the data points within

each cluster as:

1 ( )( ) UjQ e βρ ξ − −= xx

where the number of data points within this cluster is overestimated if 1jξ > or

underestimated if 1jξ < . This gives us /j jQ Q ξ= . Since the populations of clusters cannot

be all be overestimated (or all underestimated), jξ should be less than 1 for some clusters and

greater than 1 for others, which means that Q lies between the minimum and the maximum

70

values of Qj.

We now apply these ideas to two model potentials. The first one is the two-dimensional

potential of Eq. 4.19. Its contour plot is shown in Fig. 4.2a. We are using the k-mean

clustering algorithm as described in Ref. [171]. The sampled data points are partitioned into

two clusters as shown in Fig. 4.2b. The weight function of Eq. 4.21 based on this partitioning

(Fig. 4.2d) behaves similarly to the probability distribution ( , )x yρ , in contrast to the

optimal Gaussian weight function shown in Fig. 4.2c. The estimate for the partition function

obtained by using this method is compared to that calculated from Eqs. 4.8-9 without using

the clustering method as well as with 1OGQ and 2OGQ . The data were obtained from

6,000,000 MC steps, with configurations saved every 100 steps. As expected, the cluster

method provides significant improvement over the simple Gaussian weight function.

Our second test problem is the potential 2 2 2 26 20( 3) 4( 1) 4( 0.5) 2( 2) 2( , ) 1.25( ) 5 5 7.5 5 0.8( 0.625 )

3.5x x x xxU x y e e e e y xβ − − − − − + − += + + + + + − (4.25)

whose contour plot is shown in Fig. 4.3a. The simulation involved 12,000,000 MC steps with

configurations saved every 200 steps. Fig. 4.3b shows the data points obtained in this

simulation. Although the clusters corresponding to different potential minima can be easily

identified by visual inspection of the data points in Fig. 4.3b, the cluster algorithm that we

use fails to correctly identify those. Nevertheless, the weight function generated from the

clusters produced by this method (Fig. 4.3d) is an improvement over the simple Gaussian

(Fig. 4.3c) and results in a better estimate of the partition function, as seen from Table 4.2.

71

Fig. 4.3. (a) Contour plot for the potential of Eq. 4.25. (b) Data points generated by Monte Carlo and the contour plot of the exact equilibrium probability distribution ρ(x,y)=exp[-βU(x,y)]/Q. The clustering algorithm partitions the data points into five clusters shown in different colors. (c) Contour plot of the optimal Gaussian weight function (Eqs. 4.9-10) for this potential. (d) Contour plot of the weight function obtained by using clustering (Eqs. 4.21-23). Table 4.2. Comparison of clustering (Eqs. 4.21-23) method to other methods of estimating the partition function for two model two-dimensional potentials described in the text.

Potential Qexact QOG1 QOG2 Q from Eqs. 4.8-9 Q from Eqs. 4.21-23

Eq. 4.19 2.077 1.050×10-3 8.228 2.619 2.074

Eq. 4.25 3.091 1.176×10-1 7.347 3.236 3.089

72

4.6. Discussion

While the proposed method for calculating absolute free energies from Monte Carlo

or Molecular Dynamics data is in principle exact, in practice it has important limitations,

which are discussed below.

Our method assumes that the sampled data adequately represents the actual phase

space density ( )ρ p, x . In practice, this may be hard to accomplish. To illustrate the

consequences of insufficient sampling, consider the following example. Suppose that the

configuration space of an N-dimensional system is a hypercube whose volume is NV L= ,

where L is a linear dimension. Further suppose that the potential energy U is zero

everywhere inside this cube except inside a smaller hypercube whose volume is Nv l= (l<L)

and where the energy is given by , BU k Tε ε= − ? . The true partition function is then given

by

/exp( ( ) / ) ( 1)Bk TNBQ d U k T V v eε= − = + −∫ x x (4.26)

The two terms in Eq. 4.26 may be comparable in magnitude: While the contribution from the

smaller hypercube is proportional to its volume v and v < V, the exponential factor can make

the second term arbitrarily large. If our sampling data consists of M points then there is a

possibility than none of those is inside the volume v. More precisely, since the probability of

a randomly selected point to be within the volume v is v/V then, in order for at least one data

point to be within the volume v, the number of the data points must satisfy the inequality:

/ 1Mv V ≥ (4.27)

If the volume v corresponding to the low energy configurations remains unsampled then the

corresponding estimate for the partition function will be

73

estQ V≈ (4.28)

The resulting error /( 1)Bk TestQ Q v eε− −; can be arbitrarily large because of the factor

/ Bk Teε .

To avoid such an error, the number of data points must satisfy Eq. 4.27, or, equivalently:

( / )NM L l≥ , (4.29)

which means that the required number of data points must grow exponentially with the

dimensionality of the problem. Even if the linear dimension l per degree of freedom is not

much smaller than L, the likelihood of sampling the low-energy configurations within the

volume lN becomes negligible when the dimensionality is large. This means that for N that is

high enough an accurate estimate of Q may become impossible.

In a different scenario, when the low-energy configurations of the system are known in

advance, one may start sampling from a low-energy configuration that lies within the volume

v. Metropolis sampling way never overcome the energy gap ε and escape the smaller

hypercube so that the estimated partition function will be exp( / )est BQ v k Tε= . Again, the

unsampled region of the configuration space results in an error estQ Q V v− = − that can be

arbitrarily large.

Of course, more efficient sampling methods (e.g. replica exchange [60] or accelerated

molecular dynamics [62]) may remedy the sampling problems; It is however important to

realize that out method - while it allows more accurate estimates of free energies given the

same data - is still only as good as the sampling is.

Based on similar considerations, evaluation of free energies of disordered systems of

high dimensionality (e.g., liquids) may present a difficulty. In Section 5 we have described

how one can in principle identify the basins of attraction in a system with a rugged potential.

74

Our method obviously applies only when the number of low-energy basins is much less then

the number of data points. However in disordered systems the number of basins may grow

exponentially with N, which means that all the basins cannot be sampled potentially resulting

in uncontrollable errors in the evaluation of Q.

In addition to the above limitations, statistical errors may result from using a poor

weight function w(x). Generally, the weight function should be chosen such that it excludes

high energy regions of the configuration space where sampling statistics are poor. The main

concern about using Eq. 4.8 is then that for a system of high dimensionality with a rugged

potential that contains many basins of attraction it may be very difficult to construct a weight

function that would avoid the energy barriers between the basins. Our proposal is to use

image compression techniques to construct an optimal estimate for the normalized probability

distribution ( )ρ x from the available data and use it as a weight function. We have used one

such technique, a clustering algorithm, to demonstrate the utility of this approach for two

illustrative examples. The computational expense involved in using such an algorithm would

scale linearly with the number of degrees of freedom and so we expect that the use of

clustering algorithms would remain feasible for systems of high dimensionality.

At the same time, a number of issues may potentially limit the applicability of clustering

methods. The clusters found by the clustering method do not necessarily correspond to the

individual basins of attraction, as seen in Fig. 4.3b. In some instances we observed that

adjusting the definition of the distance between data points or using different clustering

algorithms may result in a better partitioning of the data into clusters. However at this point,

choosing the most appropriate clustering method remains an art.

Furthermore, we expect the ideal clustering algorithm to automatically find the optimal

number of clusters. However finding the optimal number of clusters still remains an open

issue in the field of pattern recognition [172]. While visual inspection of Fig. 4.3b

75

immediately tells us how many clusters we see, the existing pattern recognition algorithms

are no match to the human brain.

Other image compression methods (e.g., wavelets) may prove useful in generating better

weight functions for our method. We hope to explore those in our future work.

76

_____________________________________ dLarge portions of this chapter have been previously published as reference 101.

Chapter 5 Langevin dynamics simulations of the diffusion of molecular knots

in tensioned polymer chainsd

5.1. Introduction.

Molecular knots tied in individual polymer strands have attracted attention of many

physicists, chemists and molecular biologists [72,73,74,75,76,77,78,79,80,81,82,83,84,85,86].

The importance of knots as topological defects that affect polymers’ dynamics has been

recognized in a number of contexts. They may, e.g., impede DNA replication (see, e.g., Ref.

[73] and references therein) or lead to long-time memory effects in polymer melts [80,87].

From a polymer theory perspective, a number of fascinating issues exist that deal with the

scaling properties of random knots (see, e.g., Refs. [73,82,84]). Recently, molecular knots

have been created and observed at a single molecule level [94,95]. In particular, knots tied in

DNA chains with optical tweezers were seen to undergo diffusive motion and the diffusion

coefficients have been measured for different types of knots [95]. Those experiments have

motivated several theoretical and simulation studies of knot dynamics in polymers [79,83,96].

Vologodskii [79] has used Brownian dynamics simulations to study knot diffusion in DNA

and found the computed diffusion coefficients for different types of knots to agree with the

experimental values to within a factor of two. Metzler et al. [83] have presented general

theoretical considerations of different mechanisms that may affect knot mobility. The aim of

the present work is to undertake a more systematic study of the effects of the knot type, the

tension in the chain, and the polymer’s flexibility on the knot diffusion.

Consider a knot tied in a polymer chain, whose persistence length lp is longer than the

77

distance between two neighboring monomers σ and whose contour length is much longer

than lp. Suppose the two chain ends are pulled apart with a force f. As the value of this force

is increased, three physical regimes are encountered:

1. The “blob” regime [83], /B pf k T l= . In this regime the force is too low to straighten

the chain so that locally, within a blob of size [159] /Bk T f∼ , the chain behaves as a

random coil that is unaffected by the force. If a knot is tied in such a chain, it will be

likely to collide with other segments of the chain and its size will fluctuate

significantly. The f=0 case has been addressed in Refs. [87,173] showing that knot

loosening and large size fluctuations can be important in the unknotting mechanism.

2. The “elastic regime”. In this regime, the force becomes high enough, /B pf k T l> , to

align the segments of the chain in the general direction of the force. Thermal

fluctuations are unlikely to cause collisions of different chain segments except for the

monomers constrained within the knot. In this regime, the knot size is determined by

the bending elasticity of the chain vs. the force. Imagine a knot tied in a guitar string.

The harder one pulls at the ends of the string, the smaller the knot. The higher the

bending stiffness (and, consequently, the persistence length), the larger the knot.

3. The tight knot regime. Finally, when the force becomes very high, the knot size will

no longer significantly change as its size will be dominated by the repulsive

interactions between the contacting monomers in the knot. This is similar to an “ideal

knot” in a flexible rope, where its size is determined by the thickness of the rope (see,

e.g., Refs. [78,97]).

The blob regime is not considered in the rest of this chapter. A double stranded DNA

with a persistence length of ~50 nm will be in the blob regime only at

78

forces / ~ 0.08 pNB pf k T l< that are lower than the forces used in the experiments described

in Ref. [95].

Between the elastic and the tight knot regimes, we commonly observe a turnover

behavior, where the knot diffusion coefficient first increases and then decreases as the applied

tension f is increased. This behavior can be understood if one assumes that the knot diffusion

is accomplished via concerted motion of a local knot region [83] so that the total friction drag

force that acts on the knot is proportional to the number N of monomers within the knot times

the friction coefficient 0ξ per single monomer. [A more precise definition of the effective

number N of monomers in the knot will be given in the subsequent Sections]. The knot

diffusion coefficient is then given by the Einstein formula:

0

Bk TDNξ

; (5.1)

In the elastic regime, increasing the tension reduces the knot length N thereby

accelerating the diffusion. As the tight knot limit is approached, the increased repulsive

interactions among the monomers within the knot region result in a “bumpier” energy

landscape for the knot translation, which can be interpreted as an increased “internal friction”

[174,175]. This leads to slower diffusion. Interestingly, we find here that in the tight knot

regime the diffusion coefficient depends only on the knot length N rather than separately

on the chain persistence length and the tension.

In the following Sections we will present our data and describe simple theoretical

arguments to rationalize our findings. We will further show that our results can shed light on

some of the experimental observations made in Ref. [95], such as the dependence of the

diffusion coefficient on the knot type and the apparent lack of its tension dependence.

The rest of this chapter is organized as follows. Section 2 describes how the simulations

79

were performed. Our results are presented in Section 3. Section 4 concludes with a

comparison of our results with experiments.

5.2. Methods

The model. Our model of a polymer chain consists of L=90 beads and is described by the

potential

1 2( , ,.., )L bond bend nonbondedV V V V= + +r r r , (5.2)

as a function of the positions ri of each bead. The first term

2 611

2

| |( (| | ) / 2 ( ) )L

i ibond b i i h

iV k k

bσ

σ −−

=

− −= − − +

∆∑ r rr r (5.3)

is an anharmonic potential that describes bond stretching. Here σ is the equilibrium bond

length, b∆ =0.25σ, 2100 /bk ε σ= , 2hk ε= , and ε is a parameter that sets the energy

scale. We use the bending potential

10 2

2( ) / 2

L

bend ii

V k kθ θ θ−

=

= −∑ (5.4)

to vary the polymer’s persistence length by adjusting the value of the dimensionless

bending stiffness k. Here 24.8 /k radθ ε= and iθ is the angle between the bond vectors

1i i−−r r and 1( )i i+− −r r , whose equilibrium value is 0θ π= . Finally, excluded volume

effects are incorporated by using a purely repulsive potential between the beads that are not

bonded:

( )12 6

| | 2

14 (( ) ( ) ) ( )4nonbonded i j

i j i ji j

V Sσ σε

− ≥

= − + −−−

∑ r rr rr r

, (5.5)

80

where S(x) is a step function defined as,

1/ 6

1/ 6( )1, 20, 2

S xxx

σσ

=

≤>

(5.6)

The dynamics of the chain were generated by solving the Langevin equation of the form

0( )( ) ( ) ( )stretch

i ii

V Vm t t tξ∂ +

= − − +∂

r r Rr

&& & , (5.7)

where m is the monomer mass, 0ξ is the friction coefficient for each monomer (whose value

is set to 0ξ =2.0(σ2/mε)-1/2), R(t) is a random δ−correlated, Gaussian-distributed force

satisfying the fluctuation-dissipation theorem, and 1( )stretch LV f z z= − − is a stretching

potential that describes a force f that pulls the 1st and the L-th monomers apart; The z-axis

coincides with the direction of the force. In the following, we report all of our results using

dimensionless units of energy, distance, time, and force set by ε , σ , 2 1/2=(m / )τ σ ε ,

and 0 /f ε σ= , respectively. All of the simulations were performed at the same temperature

equal to 1.0 / BT kε= .

The presence of a knot in the chain was monitored by using the program of Harris and

Harvey [176] that uses the method of Vologodskii et al. [177] to calculate the Alexander

polynomial.

The diffusion coordinate. To describe the movement of the knot as a one-dimensional

diffusion process, we first need to specify the coordinate along which it diffuses. Two

obvious choices exist: (i) Monitor the projection z of the knot position on the direction of the

applied force or (ii) Monitor the knot diffusion along the polymer chain by using a discrete

monomer index n as the diffusion coordinate. In the limit of a very high force the two

81

coordinates are equivalent as the shape of the chain away from the knot region is nearly a

straight line. In the blob regime the two diffusion coordinates would be drastically different.

Generally, the diffusion projected onto the axis z will appear to be slower than the diffusion

along the chain itself. In the range of forces used here, the difference is about 15% for the

lowest force used.

The choice (i) may be closer to the experimental measurements. However another, more

subtle point should be considered: To use definition (i) one has to specify the reference frame

with respect to which z is determined. This can be the laboratory frame, the chain’s center of

mass, or the position of one of the chain ends. The difference between these should disappear

in the limit of a very long chain, where the translations of the entire chain can be neglected.

However for practical reasons our chain cannot be too long in a simulation and for chains of

finite length all three definitions give different results. The “internal” diffusion coordinate (ii)

is however uniquely defined and can be used to determine the time it takes the knot to escape

off the chain ends. For this reason, we use the second choice for the diffusion coordinate here

while keeping in mind that some of the results may be affected by the particular way of

measuring the diffusion coefficient.

Finally, since knots have finite size, we need to specify how the knot position is

described in terms of a single point in space. To do so, we define the boundaries of the knot

region, nl and nr, as illustrated in Fig. 5.1. The knot coordinate along the chain is then defined

as n=( nl + nr)/2.

Determination of the knot diffusion coefficient. We have used two methods of computing

the knot diffusion coefficient. The first method uses the relationship

[ ]2( ) (0) /(2 )D n t n t= ∆ − ∆ , (5.8)

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where the square of the knot displacement ( ) (0)n t n∆ − is averaged over a series of

Fig. 5.1. Snapshots of the knots of different types studied here. Definition of the knot boundaries nl and nr is also illustrated. short-time simulations, with the knot initially located at n(0). Metzler et al. [83] have

considered various knot diffusion mechanisms and have predicted that the diffusion should

become faster near the chain ends. When the knot is close to a chain end, it can become

untied though a cooperative motion of the dangling chain segment; This untying mechanism

should become increasingly more likely as the distance from the chain ends becomes smaller

than the length of the knot itself [83]. In this regime, however, one cannot view the knot as a

point object and Eq. 5.8 cannot be used to determine the value of D. For this reason, the

diffusion coefficients reported here have always been calculated in the regime where such

boundary effects could be neglected (i.e., when the knot is sufficiently far from the chain

ends). In this regime, we find that the knot diffusion coefficient calculated from Eq. 5.8 is

insensitive to the knot’s initial location n(0). Furthermore, we found no significant

dependence of the diffusion coefficient on the overall chain length L.

83

An alternative way of determining D is to consider the probability distribution pesc(t) for

the time t it takes for the knot to escape off the ends of the chain, provided that at t=0 the knot

was placed in the middle of the chain, n(0) = L/2. To avoid the boundary effects mentioned

above, instead of considering the entire chain one can specify a chain segment ((L-l)/2,(L+l)/2)

such that the chain’s extremities are excluded from the consideration. We place the knot in the

middle of the chain, n(0) = L/2, and follow its dynamics until it reaches one of the segment

boundaries, n(t) = (L-l)/2 or (L+l)/2, for the first time. If the motion of the knot can be viewed

as free diffusion then the probability distribution pesc(t) of the time t it takes to reach a

boundary can be obtained by solving the free diffusion equation with absorbing boundary

conditions (see Appendix A) and the value of D can be obtained from a fit of the simulated

pesc(t). If the diffusion coefficient along the chain were not constant or there were a

deterministic biasing force driving the knot in the direction of, or away from the chain center,

then we would expect to see the simulated pesc(t) to deviate from the solution of the diffusion

equation with a constant D. As shown in Appendix A, we could not find any noticeable

deviations from the free diffusion model in the range of forces studied and the value of D

determined this way was the same as that estimated from Eq. 5.8. Furthermore, we found that

the boundary effects due to the chain extremities have no noticeable effect on pesc(t). In other

words, the probability for the knot to escape off the chain ends is still well described by the

solution of a one-dimensional diffusion equation with a constant D whose value is close to

that estimated from Eq. 5.8.

A tension in a knotted chain compacts the knot. We expect that the boundary effects

predicted in Ref. [83] would become pronounced at low or zero tension. However since in

this regime the knot size would become comparable with the relatively short chain length

used here, the diffusion coefficient in such low-force limit cannot be meaningfully extracted

from the simulations reported here.

84

5.3. Simulations results

Knot trajectories. In Fig. 5.2 we show the time dependence of the knot position n(t) for two

typical trajectories, one taken at low-force value and the other at a high-force value

corresponding to the tight knot regime. In the latter case, stalling events are observed, in

which the knot becomes trapped in a certain configuration and then escapes it through a

thermal fluctuation. Also shown in this figure is the time dependence of the instantaneous

knot length for the same trajectories. Two definitions of the knot length are used, one being

simply the contour length of the chain between the knot boundaries, nr - nl, and the other is

based on the sliding knot model and is defined below. It is seen that the knot size can

fluctuate significantly and that the knot tends to be tighter during the stalling events.

Dependence of the diffusion coefficient on the knot length. Several knot diffusion

mechanisms have been proposed by others [83,95], involving either cooperative motions of

large portions of the entire chain or local motions of a knot region. If the knot translation

Fig. 5.2. Typical knot trajectories n(t) at low and high forces (k=2 in each case). Circles indicate the stalling events that are observed in the high-force case. Lower panel shows the fluctuations in the instantaneous knot length N(defined by Eq. 5.9) for the same trajectories.

85

involves concerted motion of a chain segment that contains N monomers then the effective

friction coefficient for this segment should be ξ∼Ν ξ0, where ξ0 is the friction coefficient per

one monomer (defined in Eq. 5.7). We then expect the effective knot diffusion coefficient to

be approximately given by Eq. 5.1. If the local mechanism dominates then N should be of

order of the knot length, i.e. the number of monomers engaged in the knot. To test the validity

of Eq. 5.1 we then need a way of measuring the effective knot length N.

The definition of the knot length that we use here is based on the sliding knot model

described in Appendix B. In this model, the knot slides along the chain without changing its

shape while the chain ends are not moving. Even in this simple model, different chain

segments move with different velocities so N cannot be simply taken to be the number of

monomers that move. As shown in Appendix B, the total viscous drag force that acts on the

chain when the knot moves with a velocity v is equal to 0 Nvξ− , where N is given by:

1 knotted

unknotted

zN Lz

∆= − ∆

(5.9)

Here knottedz∆ and unknottedz∆ is the extension of the chain with and without knot,

respectively. In other words, the effective length of the chain segment involved in the knot

motion is the difference between the lengths of the unknotted and knotted chains.

Coincidentally, this measure of the knot length was used in Ref. [95] to estimate the knot

length from experimental DNA knot images. While Eq. 5.9 does not give the correct length of

the knotted chain segment [78,95], it turns out to be the proper knot length measure to be

used in Eq. 5.1, at least within the sliding knot model. Since, unlike the sliding knot model,

the chain fluctuates in our case, the knot length measure that we adopt in practice uses the

average chain extensions measured along the direction of the force for knottedz∆ and unknottedz∆ .

The instantaneous knot length reported in Fig. 5.2 is obtained by using the instantaneous

86

value of knottedz∆ instead of its mean.

To vary the knot length N we now change the bending stiffness k (see Eq. 5.4) while

keeping the applied tension constant. The resulting dependence of the knot diffusion

coefficient on k is shown in Fig. 5.3. As k is increased the chain becomes stiffer and the knot

length N becomes larger. According to Eq. 5.1, this should result in a decreasing value of D.

Indeed, we observe a monotonically decreasing D(k) when the applied tension f is sufficiently

low. For high f, the observed dependence D(k) is non-monotonic, showing a maximum at

certain value of the chain stiffness.

a) b)

Fig. 5.3. (a). The dependence of the diffusion coefficient of the knot of type 31 on the bending spring constant k for different values of the tension f. The units are explained in the Methods Section. (b). Same data as in (a) plotted as the effective friction coefficient ξ=kBT/D vs. the knot length N. The dashed line is given by the equation ξ = ξ0Ν, where ξ0 is the friction coefficient per one monomer (cf. Eq. 5.7). Inset: Same plot with the ξ − scale blown up.

In Fig. 5.3b we plot the effective friction coefficient ξ=kBT/D as a function of the knot

length N for the same data. According to Eq. 5.1, we expect ξ to be proportional to N. Indeed,

the dependence ξ(Ν) is close to a straight line ξ(Ν)= ξ0Ν (shown as a dashed line in the inset

of Fig. 5.3b) for knots that are not too tight (i.e., for sufficiently large N). For tight knots

(small N) the behavior of ξ(Ν) is entirely different showing the opposite trend for more

87

compact knots to diffuse more slowly. This behavior of tight knots will be discussed below.

Dependence of the diffusion coefficient on the tension in the chain. The tension dependence

of the diffusion coefficient is shown in Fig. 5.4a for different values of the bending spring

constant k. When the chain is sufficiently stiff (i.e., its persistence length is long), D is a

non-monotonic function of the tension. The initial rise of D(f) at low forces is consistent with

Eq. 5.1 since an increased tension tightens the knot thus reducing its length N. This is further

illustrated in Fig. 5.4b, which shows the effective friction coefficient ξ as a function of the

knot length N for the same data: For stiff chains and large N (i.e., low force f) we observe that

ξ is an increasing function of N, behaving very similarly to the dependence ( )Nξ seen in Fig.

5.3b.

Both in Fig. 5.3b and in Fig. 5.4b we find that for certain values of the bending stiffness

and the force, the effective friction coefficient is somewhat lower than ξ= ξ0Ν (the points

below the dashed line). An effective friction coefficient that is higher than ξ0Ν can be

attributed to the contributions from the internal friction caused by monomer interactions

within the knot, as those are neglected in Eq. 5.1. However finding the effective friction

coefficient to be lower than ξ0Ν is somewhat surprising. Consideration of chain fluctuations

ignored in the sliding knot model may explain this observation. In particular, fluctuations of

the knot size effectively speed up the diffusion. Indeed, if the instantaneous knot length N

fluctuates significantly (cf. Fig. 5.2) then the observed value of D will be the mean diffusion

coefficient 0( / ) 1/BD k T Nξ= . If, for instance, the distribution of N is Gaussian (an

approximation that is consistent with simulations), then D will be higher than an estimate

obtained from Eq. 1 by using the mean knot length. The fairly small diffusion speedup found

here is roughly consistent with an estimate of D that takes the knot size distribution into

account.

88

Fig. 5.4. The dependence of the diffusion coefficient of the knot of type 31 on the applied tension f for different values of the bending spring constant k. (b). Same data as in (a) plotted as the effective friction coefficient kBT/D vs. the knot length N. The dashed line is given by the equation ξ = ξ0Ν. Inset: Same plot with the ξ − scale blown up.

Diffusion of tight knots. The knot diffusion coefficient depends on the properties of the chain

(such as the bending stiffness k) and the tension f. However in the tight knot limit (i.e., small

N) D depends only on the knot size N rather than separately on the tension or the chain

flexibility. That is, if we plot D (or ξ) vs. ( , )N k f for various f and k, all these dependences

will collapse onto a single curve. In particular, the curves ( )Nξ plotted in Figs. 5.3b and

5.4b are practically identical for 11N ≤ . Moreover, in this limit, unlike the large N case,

more compact knots move more slowly. How can we rationalize these findings?

When the knot is tight, “internal friction” of the chain, rather than viscous friction due to

the solvent, dominates its dynamics. The microscopic origin of such friction is the

“bumpiness” of the energy landscape of the knot caused by the intra-chain interactions

[174,175]. The knot moves via activated barrier crossing from one local minimum to another.

Indeed, stalling events where the knot is trapped in a local minimum configuration are readily

89

observed in Fig. 5.2 for the high force case. The barriers encountered in this process depend

on the magnitude of the tension in the chain. The higher the force f, the rougher the energy

landscape and consequently the slower the diffusion.

Consider now the interactions within a tight knot. The forces associated with the

bending potential Vbend in this limit become small as compared to the contribution from the

repulsive potential Vnonbonded, which prevents the knot from becoming even tighter. A compact

knot is a physical model of an “ideal” knot whose size can no longer be reduced [78,97]

except that our compact knots are somewhat compressible since the repulsive interactions are

continuous rather than hard-wall-type. The energy landscape in this limit is essentially

determined by the repulsive interactions of the monomers within the knot and it seems

plausible that it would be determined only by the knot size.

Dependence of the diffusion coefficient on the knot type. We have computed the diffusion

coefficient for several knot types (shown in Fig. 5.1) and for different values of k and f. The

results are shown in Fig. 5.5, where the effective friction coefficient /Bk T Dξ = is plotted

as a function of the knot length N. The diffusion of the knots of type 31, 51, 52, and 71 is well

described by the relationship Nξ ∝ : The bulkier the knot, the slower it moves. Moreover,

the ratio ( ) /N Nξ for low forces is very close to the friction coefficient 0ξ for a single

monomer, again pointing to the local diffusion mechanism described by Eq. 5.1, which

assumes a cooperative motion of N monomers in the knot region. The knot of type 41 seems

to be an outlier except at high forces, possibly because of the knot fluctuations or a higher

effective internal friction for this knot.

90

5.4. Discussion.

Since our polymer model does not directly describe a DNA, to compare our results with

the experimental findings of Ref. [95] we use reduced units of length and force. The

characteristic length scale is set by the polymer’s persistence length lp and the characteristic

force is set by /c B pf k T l= . Assuming lp = 50 nm, the forces used by Bao, Lee and Quake

are in the range (1 25) cf f−∼ . For such forces, they found the knot length to be 6 pN l; (for

the knot of type 31). To make a crude comparison with our results, consider the case k=1 in

Fig. 5.4. At this value of the bending stiffness, the persistence length of our polymer is ~ 5

monomers, which gives fc ~ 0.2 in the dimensionless units used in Fig. 5.4a. We see that the

experimental range of forces roughly corresponds to f < 5 in Fig. 5.4a. The highest force in

this range is close to the maximum of D(f).

To further validate this comparison we note that the knot length in this range of forces is

3 pN l∼ for the lowest force used (cf. Fig. 5.4b), which is comparable with the experimental

knot length (measured in units of lp).

These considerations suggest that the lack of tension dependence of the diffusion

coefficient reported by Bao, Lee, and Quake [95] may be due to the fact that the experimental

forces were close to the turnover regime, where – as we see from Fig. 4a – the force

dependence is weak.

The dependence of the effective friction coefficient ξ on the knot type observed in our

simulations is very close to that reported in the experimental study (see Fig. 5.3 in Ref. [95]).

Both the experimental curve ξ(N) and the dependences shown Fig. 5.5 are close to linear.

Moreover, the deviations of ξ(N) from a straight line follow the same pattern. Our results are

also consistent with the earlier simulation study by Vologodksii [79], which includes

91

electrostatic effects in DNA.

Fig. 5.5. The effect of the knot type on its diffusion: The effective friction coefficient ξ=kBT/D plotted as a function of the knot length for different types of knots and for different values of the tension f and of the bending stiffness k. The straight lines shown are least square fits with the knot type 41 excluded and are given by ξ=aN, where a= 2.13, 2.13, 3.26, and 5.33 for (k,f) = (1,2), (2,2), (2,6), and (2, 10), respectively.

As seen from Fig. 5.5, the linear dependence Nξ ∝ holds both at low forces (i.e., the

elastic regime) and at high forces (tight knot regime), although the slopes are different.

Therefore the linearity of this dependence alone cannot be used to distinguish between these

two regimes and to establish whether or not DNA knots are close to ideal.

While the simple model considered here provides useful insights into the general

problem of knot diffusion in tensioned polymers, a number of potentially important issues

pertinent to DNA and proteins have been left out, particularly the effect of twisting,

electrostatic effects, or of specific intra-chain interactions on the knot dynamics. These effects

92

may be particularly important in tight knots, where the strong constraints applied to the knot

monomers may lead to high sensitivity of the knot dynamics to the details of the molecule’s

energy landscape. We plan to address these issues in our future studies.

93

Chapter 6 Summary

1. The observed unfolding mechanism of a protein during the translocation process is

different from that probed by single molecule mechanical unfolding experiments. In the

course of translocation, the protein unfolds sequentially from the terminus, where pulling

force is applied. The mechanism of co-translocational unfolding depends on the applied force,

the pore diameter, and on whether the C- or the N-terminus is pulled. Compared with the

translocation of homopolymer, the free energy cost in the translocation of a protein is

dominated by enthalpy instead of by entropy. The kinetics of translocation also depends on

the applied force. For modest forces, squeezing the protein into the pore requires surmounting

one or several free-energy barriers. It therefore cannot be generally characterized by a single

first-order rate constant. Unless a single rate-limiting step can be identified, one may need to

go beyond the calculation of the equilibrium potential of mean force and study translocation

dynamics. Compared the free energy profiles of translocation coordinate for pores with

different size, the shape of G0(z) for z≤17σ is found independent of the pore size. As a

consequence, in the large force limit, the shape, the height and the location of translocation

barrier are the same regardless of the pore size. We expect that the translocation time will be

independent of the pore size (in the range of the pore sizes studied) in this regime. However,

the translocation barrier and, consequently, the translocation time will become dependent on

the pore size when the force is low.

2. The timescale of polymer reversal inside a narrow pore was studied using Langevin

dynamics simulations. The dependence of the rate constant of polymer reversal on the

polymer length and the pore diameter were explored and compared with the predictions of

simple one-dimensional theories that view polymer dynamics as Langevin dynamics

94

governed by a 1-D potential of mean force along the chosen reaction coordinate. The specific

reaction coordinate considered here is the distance of two ends of the chain measured along

the pore axis. Although we have found that these theories are far from exact, they capture the

dependence of rate of reversal, ( , )k N r , on the pore size, r, and the chain length, N,

reasonably well. More precisely, the rate constant can be factorized into the one-dimensional

transition state theory rate, ( , )TSTk N r , and a transmission factor, κ. The important point here

is that much of the N and r dependence of the rate constant is contained in ( , )TSTk N r . The

remaining dependence of the transmission factor is a power law, much weaker than the

exponentially strong dependence. Our study therefore provides an anecdotal support for the

utility of simple one-dimensional phenomenological models in studies of complex

biomolecular transitions such as those implicated in protein translocation or mechanical

stretching of proteins. We have also explored the simple phenomenological approach utilizing

Kramers' theory to estimate the prefactor and found it to be unreliable. It may be possible that

accounting for the position dependence of the effective friction coefficient and for memory

effects will improve such an estimate. Finally, we note that the scaling properties of the

reversal rate exhibit finite size effects even for fairly long chains/wide pores. Even for chains

as long as 500 monomers we observed significant deviations from the asymptotic scaling law.

Given that the number of monomers is of order of 100 for a typical single-domain protein, we

expect finite size effects to be important in polypeptide translocation experiments.

3. We have proposed a new method for calculating absolute free energies from Monte Carlo

or molecular-dynamics data. With introducing a clustering algorithm to attempt partitioning

the data into clusters corresponding to different basins of attraction visited in simulations, the

weight function is then constructed as a superposition of a series of Gaussians calculated for

95

each cluster separately. We show that this strategy is possible to improve upon the method of

estimating absolute entropies from covariance matrices.

4. Using Langevin dynamics simulations to study the diffusion of a knot along a tensioned

polymer chain, we have found that the diffusion processes can be modeled as a free diffusion

with constant diffusion coefficient in one dimension. It is shown that dependence of the knot

diffusion coefficient on the tension can be non-monotonic. This behavior can be explained by

the model, in which the motion of the knot involves cooperative displacement of a local knot

region. At low tension, the overall viscous drag force that acts on the knot region is

proportional to the number N of monomers that engaged in the knot, which decreases as the

tension is increased, leading to faster diffusion. At high tension the knot becomes tight and its

dynamics are dominated by the chain’s internal friction, which increases with increasing

tension, thereby slowing down the knot diffusion. This model is further supported by the

observation that the knot diffusion coefficient measured across a set of different knot types is

inversely proportional to N. We propose that the lack of tension dependence of the knot

diffusion coefficients measured in recent experiments is probably due to the fact that the

experimental values of the tension are close to the turnover between the high- and low-force

regimes. In a tight knot, the size of the knot is mainly determined by nonbonded repulsive

potential. A compact knot is a physical model of an “ideal” knot whose size can no longer be

reduced, except that our compact knots are somewhat compressible since the repulsive

interactions are continuous rather than hard-wall-type. We have observed the linear

dependence ξ∝ N holds both at low forces and high forces for various knot types. Therefore

the linearity of this dependence alone cannot be used to distinguish between these two

regimes and to establish whether or not DNA knots are close to ideal.

96

Appendix A. Distribution of the knot escape time in the free diffusion model

Suppose the knot’s dynamics can be described as one-dimensional motion along the knot

coordinate x. The knot starts in the middle of the chain at x=0 and is monitored until it

reaches one of the chain boundaries, ( ) / 2x t l= ± . We are interested in the probability

distribution pesc(t) of the time t it takes the knot to escape the chain segment

( / 2, / 2)l l− between the boundaries. To find this, we first calculate the probability density

p(x,t) for finding the knot at x at time t provided that it disappears irreversibly upon reaching

the boundaries. This is the solution of the one-dimensional diffusion equation

2

2

( , ) ( , )p x t D p x tt x

∂ ∂=

∂ ∂ (A1)

with the initial condition

( ,0) ( )p x xδ= (A2)

and absorbing boundary conditions at / 2x l= ± . The solution can be conveniently expressed

as a series:

21 ( )( , ) ( 1) exp44

n

n

x nlp x tDtDtπ

∞

=−∞

−= − −

∑ (A3)

The probability distribution of the knot escape time can be expressed in terms of the knot

survival probability

/ 2

/ 2

( ) ( , )l

l

S t p x t dx−

= ∫ (A4)

/ 2 / 2

( , ) ( , )( ) /escx l x l

p x t p x tp t dS dt Dx x= =−

∂ ∂= − = − − ∂ ∂

(A5)

Fig. A1 gives an example of the distribution of the knot escape time determined from a

97

simulation. The solid line is a fit that uses Eqs. A3-A5, with D being used as a fitting

parameter. The free diffusion model fits our data very well.

Fig. A1. The probability distribution of the knot escape time fitted by using the free diffusion model (solid line). The values of the bending stiffness and the force in the simulation are k=2, f=4. The knot was placed in the middle of the chain and monitored until its distance from the middle attained the value n(t)= ± l/2, where l = 40. The value of the diffusion coefficient obtained from this fit is D=0.0296.

98

Appendix B. The drag force on the knot region in the sliding knot model

Consider a continuous string with a knot tied in it. Here, we will assume that the knot

slides along the string without changing its shape, as illustrated in Fig. B1. The chain

segments that are far away from the knot region are not moving; In particular, the chain ends

are at rest. Assuming that the knot moves with a velocity v, we would like to calculate the

total viscous drag force that acts on the chain. To do so, it is convenient to switch to a moving

reference frame, in which the knot itself is at rest while each given point of the string is

moving with a constant velocity along the same curve ( )( ), ( ), ( )x s y s z s , which defines the

constant shape of the knot. Here 0s s vt= − is the position of the point measured along the

string. The shape of the knot curve is such that ( )( ), ( ), ( ) (0,0, )x s y s z s z= far away from the

knot region. In other words, the string is a straight line aligned along the z axis everywhere

except in the vicinity of the knot.

Fig. B1. The sliding knot model. The time dependence of the position of a selected point on the string is shown.

The absolute value of the velocity of any given point of the string in the moving frame is

equal to v while the velocity vector is given by:

99

( ), , , , , ,x y zds dx dy dz dx dy dzu u u vdt ds ds ds ds ds ds

= = −

% % % . (B1)

The velocity of the same point in the laboratory frame is

( ) ( ), , , , (0,0, ) , , 1x y z x y zdx dy dzu u u u u u v vds ds ds

= = + = − −

u % % % (B2)

The total viscous drag force on the chain is then given by

2

1

( )drag s ds sγ= − ∫f u , (B3)

where sγ is the friction coefficient per unit length of the string and 1 and 2 denote the chain

ends. Combining Eqs. B2 and B3 we find:

2

1

0,0, ( ) (0,0, )drag s sv ds dz v zγ γ

= − − = − ∆

∫f , (B4)

where unknotted knottedz z z∆ = ∆ − ∆ is the difference between the end-to-end distance of the

knotted and unknotted chains. The drag force is along the z-axis and its value is

proportional to the difference between the extension of the knotted and the unknotted chains.

For a discrete chain that consists of L monomers we can write 0 /s unknottedL zγ ξ= ∆ (where

0ξ is the friction coefficient per monomer) so that

0dragf Nvξ= − , (B5)

where the effective number of monomers involved in the knot motion is given by

1 knotted

unknotted

zN Lz

∆= − ∆

(B6)

100

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Vita

Lei huang was born in Qichun, Hubei, China, on July 3, 1978, the son of Xuegui Huang

and Shunchun Wang. After completing his work at Huanggang high school, Hubei, China, he

entered University of Science and Technology of China, Hefei, China in 1996. He got Bachelor

of engineering in department of polymer science and technology in 2001. He received Master

degree in chemistry from University of Science and Technology of China in 2004. In August

2004, he entered the department of Chemistry and Biochemistry, University of Texas at Austin

as a Ph.D student. He was awarded Ph.D degree in May 2008.

Permanent address: 58# New Street Road, Qizhou, Qichun, Hubei, 436315, China

This dissertation was typed by Lei Huang.