Page 1

Diffusion-controlled first contact of the ends of a polymer: Crossover between two scaling

regimes

Jeff Z. Y. Chen,1Heng-Kwong Tsao,2and Yu-Jane Sheng3

1Department of Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

2Department of Chemical and Materials Engineering, National Central University, Jhongli, Taiwan 320, Republic of China

3Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan, 106, Republic of China

?Received 10 May 2005; published 8 September 2005?

We report on Monte Carlo simulations of loop formation of an ideal flexible polymer consisting of N bonds

with two reactive ends. We determine the first-passage time associated with chain looping that yields a

conformation in which the end monomers are separated by a distance a—the reaction radius. In particular, our

numerical results demonstrate how this time scale crosses over from ?first?N3/2/a to the a-independent ?first

?N2as N is increased. The existence and characteristics, of the two scaling regimes and the crossover between

the two, are further illuminated by a scaling argument.

DOI: 10.1103/PhysRevE.72.031804PACS number?s?: 82.35.?x, 61.41.?e, 05.70.Ln

I. INTRODUCTION

Loop formation of a polymer chain is a dynamic process

by which two monomers along the chain approach each other

within a small distance. Subsequently, the interaction of

these two monomers occur as the result of the normally

short-ranged interaction ?with a force range a? between the

reactive monomers. The locations of these interacting mono-

mers, along the contour of the polymer chain, could be as

distant as the entire chain length in the case of two interac-

tive ends. Starting from an open configuration where the two

reactive monomers are separated by a physical distance that

could be much greater than a, the polymer undergoes con-

figurational fluctuations that bring together ?or separate? the

two ends—which is a process that is solely determined by

the entire chain.

Abiopolymer, for example, may require loop formation as

a primary step for acquiring a desired structure to perform its

biological functioning. Understanding the looping dynamics

of a relatively simple system that depends on a few essential

physical parameters can form a first step toward gaining

much insight into a wide range of biological and physical

processes such as protein folding ?1? and DNA replication

?2?. The recent advance in single-molecule manipulations on

this kind of systems has allowed one to probe chain closing

times ?3?. In reality, a biopolymer can carry many reactive

groups and the reaction between two groups are usually fur-

ther complicated by the participation of other molecules in

the system. There are also examples in synthetic polymers

where loop formation of a polymer is an important process

?4?.

On the theoretical side, much effort has been paid to

studying single-loop formation of a polymer with two reac-

tive ends ?5–14?. One of the main issues is whether or not the

diffusion-controlled reaction of two ends is more compli-

cated than the dynamic behavior represented by the autocor-

relation function of the end-to-end vector. Of particular in-

terest is the characteristic time that required for the two ends

to approach each other and react. In the case of instantaneous

reaction, the focal point has been on the mean first-passage

closing time, ?first, for the reactive monomers to close within

a distance of a from each other for the first time starting from

an open configuration ?averaged over all initial conforma-

tions that are typically assumed to follow an equilibrium

distribution?. The closing dynamics in this case could be

complicated by the internal dynamic modes, which lead to

the rapid motion of chain ends ?5?. The competition, between

local equilibration at the length scale a ?7,11? and the global

conformation fluctuations, gives rise to more than one scal-

ing regimes for ?first. To complicate things further, a typical

theoretical approach to this problem usually relies on making

several approximations ?5–7,11,12?. For example, the as-

sumption of “local equilibrium” by Szabo, Schulten, and

Schulten ?SSS? leads to

?first= ?SSS? N3/2/a

?1?

for an ideal flexible chain when a is small ?7?. On the other

hand, by keeping a/b finite, it has been argued that for a long

ideal flexible polymer, ?firsthas another scaling behavior

?5,6?

?first= ?R? N2,

?2?

where ?Ris the Rouse relaxation time representing the global

relaxation of the polymer, which scales as N2?15?; note that

?Rdoes not depend on a.

The seeming discrepancy between ?SSSand ?Rhas also

inspired numerical studies in an effort to understand the dif-

ference between the two. However, earlier simulations were

limited to a relatively small parameter space and the results

are not conclusive. The computer simulations of Paster,

Zwanzig, and Szabo confirmed the N3/2dependence of ?SSS

for a single value of a, leaving the inverse a-dependence

unchecked ?11?; furthermore, if they had extended their

simulations for the exactly same a to a much larger N, they

would have seen the crossover to a different scaling behav-

ior. In contrast, the simulations of Podtelezhnikov and Volo-

godskii exhibited the N2dependence of ?R, but the results

also suggested an a-dependent coefficient that does not exist

in ?R?16?. In a theoretical treatment supplemented by simu-

PHYSICAL REVIEW E 72, 031804 ?2005?

1539-3755/2005/72?3?/031804?7?/$23.00©2005 The American Physical Society031804-1

Page 2

lations, Portman showed that ?SSSand the mean first-passage

time, determined according to the Wilemski and Fixman ap-

proximation, are the lower and upper bonds for ?first, respec-

tively ?17?. Because ?Ris one of the predictions based on the

Wilemski and Fixman approximation, this indicates the pos-

sible existence of the crossover between the two types of

scaling behavior in Eqs. ?1? and ?2? ?11?.

The main purpose of this paper is threefold, first we pro-

vide extensive computer simulations of polymer loop forma-

tion in a wide parameter range. The results can be used to

provide numerical evidences convincingly supporting the

two scaling regimes ?see Sec. III A? and demonstrating the

crossover between these ?see Sec. III B?. This is computa-

tionally demanding and the current computational power has

just allowed us to consider sufficiently large N and relatively

small a. For consistency with the physical models used in

most previous theoretical treatments, we model a polymer as

a free-jointed chain with no excluded volume interaction or

hydrodynamic effects. In particular, we study how distinct

scaling regimes of ?firstemerge. Our simulations show a

crossover from ?SSSto the a-independent ?Rat certain values

of N and a. Unlike earlier simulations, our simulations pro-

vide direct evidence of the crossover.

Second, we also give a scaling argument for the observed

crossover between the two scaling regimes ?see Sec. III C?.

The theory is conceptually based on a simple physical pic-

ture, without the involvement of approximations and com-

plex mathematical derivations used previously to arrive at

the two scaling relations, in Eqs. ?1? and ?2?. We derive the

relevant time scales and length scales that are responsible for

the competitions between the two scaling regimes, in consis-

tence with existing theories of polymer dynamics.

Finally, we examine the conditional distribution functions

of ?firstwith a fixed initial end separation R for two typical

values of a. The analysis further clarifies the physical picture

of crossover between the SSS anf Rouse regimes ?see Sec.

III D?.

II. MODEL AND SIMULATION DETAILS

The polymer model used in this study is a typical freely

jointed chain, consisting of N bonds of fixed length b ?15?. In

a simulation implementation, a randomly selected monomer

is rotated about the axis defined by the vector connecting the

two nearest-neighbor monomers; the lengths of the con-

nected bonds are unaffected by the rotation. The rotational

angle was selected as a random number between ?−?,+??,

where ?=?0??/20 ?18?. All time scales in this work are

measured in terms of a MC step ?MCS?; within each MCS all

monomers along the chain have the probability to move

once.

An initial configuration is generated by a random walk

with step length b ?=1?. The chain is then subject to MC

moves. The terminal monomers, labeled 0 and N, are consid-

ered to have made a contact when they fall within the reac-

tive range, R?a, where R is the end-to-end distance. In the

diffusion-controlled reaction, the two ends are assumed to

instantaneously react and trap each other ?i.e., the chain

closes? as soon as R?a; the simulation for a single closing

event then stops. A first-passage time, tfirst, the time for the

chain to close for the first time from a given initial confor-

mation, is then recorded. We repeat this simulation with a

new initial conformation generated by a fresh random seed

every time. For each set of ?a,N?, a total of M=400 closing

events have been observed, with the exception of the data

points for N?300 where M=50. The average first-passage

time ?firstis then an algebraic average of tfirstobserved in

these M events.

III. RESULTS AND DISCUSSION

A. Mean first-passage time for chain closing

Figure 1?a? shows the simulation data of ?firstas a function

of N for various values of a. To examine the N dependence

of ?first, we have replotted the same data set in Fig. 1?b?,

where the scaled first-passage time ?first/N2is displayed as a

function of?Nb/a ?i.e., the ratio of the root-mean-square

end-to-end distance, R0, to the interaction radius? for various

choices of a/b. As shown in the figure, data points corre-

sponding to different values of a/b converge to a constant

for large N, implying that ?first/N2is not dependent on N, nor

on a/b, in the limit of?Nb?a. Hence,

?first? N2

??Nb ? a?.

?3?

This numerically verifies that the scaling behavior of ?firstis

the same as that of the Rouse time in this limit.

While this seems to be straightforward, it has not been

fully confirmed numerically in the past for some reasons. For

example, earlier Brownian dynamics simulations of ?first

were limited to N?55 due to computational limitations and

failed to demonstrate the a-independent asymptotic result in

Eq. ?3? ?16? ?see also the apparent a-dependence of some

intermediate values of N in Fig. 1?b??. The a-independence

of Eq. ?3? implies that the first-passage time of closing, of a

sufficiently long chain, is independent of their force range a.

This can be contrasted with the action time between low-

molecular particles in the diffusion limited case where the

action time varies inversely with their reactive range. Unlike

low molecular systems, long-chain molecules show unique

dynamics due to the presence of conformation fluctuations

represented by, for example, i.e., the Rouse modes ?15?. The

Rouse time is a characteristic time that describes the fluctua-

tions of the chain ends; on average, it takes ?Rfor the chain

ends to have a chance to reach the vicinity of each other in

passing. The question then becomes would the two ends have

a chance to reach a distance of a ?which could be small?

during ?R? As will be demonstrated in Sec. III C, if the an-

swer is yes, ?Ris the characteristic time for the entire pro-

cess.

On the other hand, another distinctively different scaling

regime can be exhibited by our data for moderate N with

very small a/b. In order to perform a careful analysis, we

recall that the assumption of local equilibrium led SSS to

obtain ?7?

CHEN, TSAO, AND SHENGPHYSICAL REVIEW E 72, 031804 ?2005?

031804-2

Page 3

?first?N3/2

a?1 +??6

??ln 2 − 1??

a

bN1/2+ ¯?.

?4?

In

?Nb/a-dependent straight line, with a negative slope −1/2

on the double-logarithmic plot. The diamonds and squares

?a/b=0.4 and 0.5? in the vicinity of?Nb/a?10 indeed dis-

Fig.1?b?,the leadingterm wouldprojecta

play a similar behavior, but cannot be used to make a con-

clusive illustration. A better perspective can be gained if we

plot these data points in a different way; to illustrate a scal-

ing relation that would follow Eq. ?4?, we consider

?firsta/?bN3/2? as a function of a/?b?N? in a semilogarithmic

plot ?see Fig. 1?c??. Equation ?4? would then predict a

straight line with a slope given by?6/??ln 2−1? for small

values of a/?b?N?. Because the SSS regime is only valid for

very small a/b ?see next section?, the same data sets from

Figs. 1?a? and 1?b?, now plotted in Fig. 1?c?, are not adequate

for clear demonstration of the SSS scaling. We have con-

ducted additional MC simulations of systems with smaller

values of a/b ?=0.4,0.3,0.2,0.1?; the resulting ?firsthave

been plotted in Fig. 1?c? as filled symbols. These filled sym-

bols can be seen to approach a constant as a/?bN1/2? de-

creases, which confirms the existence of the SSS limit.

Moreover, the solid line in Fig. 1?c? gives an exact slope

?6/??ln 2−1?, representing the coefficient of the first-

correction term in Eq. ?3?. Hence, both leading terms in Eq.

?4? can be verified by our numerical data.

Note that in order to properly simulate the Brownian mo-

tion of the polymer in solution by using a Monte Carlo ap-

proach, the simulated displacement of a monomer must be

much smaller than the length scale that we are observing.

The reduction of the displacement would also prompt much

longer computational time. To ensure that the simulated dis-

placement of a monomer is smaller than a/b, we have used

?=?1?0.01 as the maximal value for the random angle se-

lection in an MC move for a/b=0.3, 0.2, and 0.1. This se-

lection can be compared to ?=?0??/20 used above for

a/b?0.4. A major consequence of using a different value of

? is that the units of time are now different: the simulation

time it takes, in terms of MCS, to observe the same system

for a defined period of time, is now approximately ??0/?1?2

longer when ?=?1is used. Hence, the first-passage times

from these supplemented simulations have been multiplied

by a factor of ??1/?0?2in Fig. 1?c?, in order to take into

account the adjustment of units.

B. Crossover between the Rouse and SSS regimes

Having confirmed that both Rouse and SSS regimes in-

deed exist and can be associated with the respective scaling

behavior, we now turn to classifying the corresponding re-

gimes in the parameter space. Because two different time

scales are competing with each other for a given system,

?R?N2and ?SSS?N3/2b/a, the first-passage time for closing

is dictated by the longer of the two ?11?.

In the regime of the parameter space where ?R??SSS, or

N2?cN3/2b/a, the first passage time follows the SSS behav-

ior, where we have introduced a dimensionless numerical

constant c in the last equation for more precise specification.

This inequality hence yields the crossover condition,

aN1/2/b ? c.

?5?

This condition, along with the requirement that we are deal-

ing with a long-chain polymer,

FIG. 1. Mean first-passage time ?firstfor closing. Stars, crosses,

plus signs, circles, squares, diamonds represent MC simulation data

for a/b=2.0, 1.6, 1.0, 0.8, 0.5, and 0.4, respectively. In plot ?a?, the

original measurements are displayed; in ?b?, scaled ?first/N2is

shown as a function of the ratio R0/a where R0is the root-mean-

square end-to-end distance, i.e., R0=b?N; ?c? is a plot of ?firsta/N3/2

vs a/R0. In ?c?, our data have been supplemented with an additional

MC data set for smaller values of a/b ?=0.4, 0.3, 0.2, and 0.1?

?diamonds, up triangles, left triangles, and down triangles, respec-

tively?. The typical errors of the data are smaller than the sizes of

symbols in plot ?a?.

DIFFUSION-CONTROLLED FIRST CONTACT OF THE…

PHYSICAL REVIEW E 72, 031804 ?2005?

031804-3

Page 4

a/?bN1/2? ? 1,

?6?

specifies the SSS regime. The latter essentially comes from

the fact that the correction terms in Eq. ?4? should be small in

comparison with the leading scaling term. These two condi-

tions are quite restrictive; on the one hand, by combining

these two inequalities, we obtain

?a/b?2? c,

?7?

which is an indication that only small a/b systems would be

expected to display the SSS scaling, consistent with the ar-

gument made by SSS themselves ?7?. On the other hand, the

combination of these inequalities also leads to

a/b ??N ? cb/a,

?8?

where the two bounds have been set for N: the bound of

a/b??N that can be easily satisfied in most polymer sys-

tems and the other bound of?N?cb/a that restricts the SSS

scaling to the intermediate values of N—not all small a/b

systems would display the SSS scaling. In fact, for every

long polymer where?N?cb/a, we recover the Rouse re-

gime even for small a/b.

The filled symbols in Fig. 1?c? represent data points cor-

responding to aN1/2/b?1.5 where c=1.5 is an empirically

selected constant, used for the purpose of illustrating the SSS

regime. As N is increased ?hence lowering a/?b?N??, the

systems begin to move away from the SSS regime and the

data points start to deviate from the scaling behavior.

How important is the SSS regime in real systems, in view

of the restrictive conditions for validity in Eq. ?8?? For most

flexible chains, the force range of interaction a is of the same

order of magnitude as the polymer bond length b. As the

result,?N can easily exceed b/a. The Rouse time is probably

the only characteristic time observable in real flexible poly-

mers, according to the criterion for the validity of the SSS

regime in Eq. ?8?. However, in wormlike polymer systems,

the persistent length ?equivalent to

than a typical interaction range a. Double-stranded DNA

molecules are good examples of such systems. The SSS re-

gime then becomes significant ?13?. We have recently di-

rectly observed the SSS scaling for the first passage time of

closing in wormlike chains by numerical simulations ?14?.

1

2b? could be much larger

C. Scaling argument for the crossover between the Rouse and

SSS regimes

So far, we have provided the numerical evidences for the

existence of and the crossover between the Rouse and SSS

regimes. The previous theoretical approaches used to arrive

at these results, however, are highly mathematical ?5–7,11?,

and the approximations used are not always transparent ?12?.

In this section, we develop a simple and consistent physical

picture through a scaling argument ?19?, that supports both

?Rand ?SSS.

Consider the end-to-end vector of the polymer chain. The

global dynamic behavior of this vector is governed by the

first normal mode of the entire polymer ?15?, which projects

the slowest relaxation time ?the Rouse time?; the tip of the

vector, however, undergoes rapid Brownian motion within a

small region much smaller than R0, the root-mean-square

end-to-end vector. In particular, we write ? for the amount of

time that it takes to explore a regional space of dimension a

?assumed here ?aRto be defined below? by the tip of end-

to-end vector. Now, because the dynamic motion is diffusive,

? ? a2/D0,

?9?

where D0is the effective diffusion constant associated with

the end monomer. Within such a small region and during ?,

the Brownian motion of the end-to-end vector tip contains

the fastest normal modes of the polymer chain; another way

of viewing this is that, during this period of time, the end

monomer “feels” the presence of a small number of con-

nected neighboring monomers only, not the entire chain.

Thus, D0is proportional to kBT/? ?the Einstein relation?

where ? is the friction constant experienced by the end

monomer.

Now, during the Rouse time, a much larger time scale

than ?, the end-to-end vector fluctuates globally, covering

the entire space of dimension R0. In total, the volume R0

contains K small regions, each having volume a3, where K

=R0

distance of R0, the two polymer ends must explore the entire

R0

volume a3?. Hence, the time that it takes to do this is

3

3/a3. In order to find each other starting from a typical

3volume thoroughly ?i.e., searching those small regions of

? ? K? ?R0

3

a3

a2

D0

?b

aN3/2?b2

D0?,

?10?

which gives a simple explanation of the SSS time in Eq. ?1?,

within the definition of the units of time represented by the

quantity in the square brackets of the above equation. This

expression is identical to Eq. ?29? in Ref. ?11?.

This scaling argument is based on the assumption that Eq.

?10? is greater than the Rouse time, ?R?N2—the polymer

ends need to find each other in the vicinity ?governed by the

Rouse time?, before they can thoroughly explore the region

of a grain size a3. Therefore, ?Ris the lower bound for ?first,

the first-passage time of reaction between the two ends. Re-

turning to the discussion in the second paragraph of Sec.

III B, we see that the SSS regime is restricted to the param-

eter space given by Eq. ?8?.

Another interesting question is, during the Rouse time,

how much detail in space would the two polymer ends ex-

plore. The answer to this question is actually already con-

tained in Eq. ?10?. Equating the left-hand side of the equation

with the Rouse time ?R, we can take a on the right-hand side

as the dimension of the space, aR, explored by the polymer

ends during the Rouse time. This yields

aR? b/?N,

?11?

which is a dimension much smaller than the Kuhn length b.

In light of the above discussion, the crossover condition ?Eq.

?5?? can be related to the physical meaning of aRas well;

when

CHEN, TSAO, AND SHENG PHYSICAL REVIEW E 72, 031804 ?2005?

031804-4

Page 5

a ? aR

?SSS?,

?12?

the chain ends could not find each other within the distance

a, and must continue the search beyond the time period ?R;

when

a ? aR

?Rouse??13?

the chain ends have already found each other during the

Rouse time ?R. Equation ?12? is consistent with the right-

hand side of Eq. ?8?.

As a final note, aRis also a characteristic length scale at

which we need to consider all Rouse modes, fast and slow,

for the description of the motion of end-to-end vector tip ?5?.

This can be contrasted with two limiting cases. First, when

we consider the dynamical properties at a length scale a that

is much smaller than aRduring time ? ?in the SSS regime?,

only the fastest Rouse modes contribute to the dynamics.

Second, when we consider the “global” dynamic properties

of the polymer at a length scale R0that is much greater than

aR, only the slowest Rouse mode contribute to the dynamics

?15?. Hence, aRis a characteristic length scale that reflects

the division between the fast and slow Rouse modes in poly-

mer dynamics.

D. Distribution function of tfirst

The physical picture presented above can be further aug-

mented by the distribution function of the first passage time

of closing. To provide sufficient statistical data for a

distribution-function analysis, we have concentrated on two

cases ?a? a/b=0.4 and ?b? a/b=1.6 for N varying from N

=10 to N=80. As can be viewed from the two scaling plots

of Fig. 1, systems ?a? and ?b? belong to the near-SSS and

Rouse regimes, respectively, for the mentioned N values. In

total, 106independent closing events, represented by tfirst,

have been recorded for the analysis performed in this section.

The histogram of tfirstis then plotted in Fig. 2.

Wilemski and Fixman ?6? suggested that the distribution

function for tfirstshould become simple exponential, f?tfirst?

?exp?−tfirst/?first?, in the large tfirstlimit. Indeed, all open

symbols for the near-SSS regime ?a/b=0.4? in Fig. 2 form a

straight line in a semilogarithmic plot, consistent with the

suggested exponential form. The distribution functions for

the four values of N, however, are different in the Rouse

regime, a/b=1.6, where the short-time behavior, in a rela-

tively significant region in tfirst/?first, displays the nonexpo-

nential form; the distribution function contains a fast initial

decay, indicating more short-tfirstevents.

Where do these short-tfirstevents come from? To further

dissect the physical picture, we have examined the mean

first-passage time for closing ?tfirst??R? as a function of an

initial end-to-end distance R. Computationally, to obtain the

distribution function discussed above, 106MC dynamic tra-

jectories have been produced, with the initial configurations

randomly adopted from independent equilibrium states. As

such, the initial end-to-end distance follows an equilibrium

density distribution function

Gaussian—the unnoticeable deviation comes from the dis-

crete nature of the adopted simulation model. We selected all

that areveryclose to

trajectories with a matching initial end-to-end distance to a

prespecified R range for this part of the analysis. ?tfirst??R? as

a function of R for both ?a? a/b=0.4 and ?b? a/b=1.6 is

displayed in Fig. 3 in a reduced form. In the Rouse regime,

we are dealing with systems where the capturing radius, a, is

relatively large in comparison with R0. Events that start with

an R shorter than R0contain two possible types of trajecto-

ries, those with the ends to separate farther from each other

and those with the ends to approach closer to each other.

Because of the relatively large a, the probability, that the two

ends capture each other on a simple path consisting of

mainly the approaching motion, is much greater than that in

smaller a systems. These short-tfirstevents are particularly

more frequent in systems with smaller N ?or, larger a/R0?.

Hence, on average, for a fixed R?R0to start with, smaller N

systems produce a smaller ?tfirst??R?, which can be seen in

Fig. 3?b?. These short-tfirstevents significantly contribute to

the initial, fast-decaying portion of the distribution functions

of a/b=1.6 systems in Fig. 2.

Now, in small a/b systems ?in the SSS regime?, these

short-tfirstevents are much rarer. This is reflected by another

dominating feature in Fig. 3: the reduction of the overall

variation of ?tfirst??R?/?firstas a/b decreases from 1.6 to 0.4

?note the two different vertical scales?. The spreading of the

data points in the vicinity of R2/N?1 in Fig. 3?b? for a/b

=1.6 can be contrasted with the narrow range of variations in

Fig. 3?a? for a/b=0.4. As can be projected based on the

FIG. 2. The distribution function of the first-passage time tfirstin

a near-SSS ?open symbols? and Rouse ?shaded symbols? regimes

for various N. To clearly display the curves, each set of data in this

plot has been shifted by a factor of 10 from the lower neighboring

set. The errors of the data are similar to the sizes of the plotted

symbols.

DIFFUSION-CONTROLLED FIRST CONTACT OF THE…

PHYSICAL REVIEW E 72, 031804 ?2005?

031804-5