Diffusion-controlled first contact of the ends of a polymer: Crossover between two scaling
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
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
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
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
?first= ?R? N2,
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-
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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.
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
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,
??Nb ? a?.
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-
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
CHEN, TSAO, AND SHENGPHYSICAL REVIEW E 72, 031804 ?2005?
??ln 2 − 1??
?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.
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?.
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PHYSICAL REVIEW E 72, 031804 ?2005?
a/?bN1/2? ? 1,
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
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,
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?.
2b? could be much larger
C. Scaling argument for the crossover between the Rouse and
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
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,
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
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
distance of R0, the two polymer ends must explore the entire
volume a3?. Hence, the time that it takes to do this is
3/a3. In order to find each other starting from a typical
3volume thoroughly ?i.e., searching those small regions of
? ? K? ?R0
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
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;
CHEN, TSAO, AND SHENG PHYSICAL REVIEW E 72, 031804 ?2005?
a ? aR
the chain ends could not find each other within the distance
a, and must continue the search beyond the time period ?R;
a ? aR
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-
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
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PHYSICAL REVIEW E 72, 031804 ?2005?
general trend, a further decreasing of a/b—bringing the sys-
tem to the “true” SSS region—would yield a similar plot to
Fig. 3?a? where ?tfirst?R/?firstbecomes a constant ??1?. This
asymptotic limit can be understood from the same explana-
tion in the above paragraph. For small a/b, regardless of the
initial distance between the ends to start with, the two ends
are very likely to miss each other in an end-approaching
event. Instead, the chain would relax from an initial end-to-
end separation of R to R0=?Nb, as the first step. This process
takes up ?Ron average. The first passage time, of closing for
two ends to start with a separation of distance R0, is ?SSSin
the SSS regime; this fact, together with ?SSS??Rin the SSS
regime, would completely undermine the initial relaxation
period from R to R0. Hence, ?tfirst??R?/?firstshould become an
R- and N-independent constant in the SSS regime. This in-
dependence can be explained by the reduced physical picture
of diffusion between two end monomers ?13? in an effective
spring potential, after the internal degrees of freedom being
integrated out. In the so-called “overdamped” limit, the ef-
fective free energy barrier, that two monomers must over-
come before approaching to each other in a distance smaller
than a, is much greater than kBT; the time duration for the
reaction to happen is largely dependent on the overall barrier
height and is not sensitive to the initial separation of the two
To further contrast looping formations operating in the
SSS and Rouse regimes, we have also considered the ?nor-
malized and conditional? distribution of tfirst/?firstfor a given
R. In the Rouse regime ?see Fig. 4?b??, the distribution func-
tions have a strong R dependence, as we discussed above,
reflected in these plots by the different slopes of the distri-
bution function in the semilogarithmic plots. The events that
start with a smaller R have significant statistics which are
different in the initial period tfirst/?first?1. Because the chain
ends are started with already close distance, the reaching of
R=a could happen in the “first” passage near the vicinity of
R=a within a relatively short time scale. A very interesting
limit is the large tfirstbehavior of the distribution functions
for large R’s, which tend to approach a common slope. Note
these starting distances are relatively far from R0, the relax-
ation from these positions to R0takes almost the same ?R.
Sokolov has recently analyzed the distribution function in
Fig. 4 by using a much more complicated mathematical pro-
In the near-SSS regime ?see Fig. 4?a??, all distribution
functions, no matter the values of R, collapse into a univer-
sal, simple exponential decay at large tfirst. This represents
the events that are characterized by the looping of the poly-
mer chain from R0to a distance a. Because a is small, most
trajectories must spend multiple number of passes in the vi-
cinity of R=a to thoroughly explore the regime. The number
of passes, estimated in Secs. III B and III C, could be as large
R-independent number in the SSS regime ?see Eq. ?6??.
?SSS/?R?b/?a?N?, is actuallya large,
In summary, we have studied the closing dynamics of a
polymer with two reactive ends, using both Monte Carlo
FIG. 3. ?Color online? Average first-passage closing time as a
function of the initial square end-to-end distance R2for ?a? a/b
=0.4 ?close to the SSS regime? and ?b? a/b=1.6 ?in the Rouse
regime?. Circles, squares, diamonds, and triangles correspond to
N=10, 20, 40, and 80.
FIG. 4. ?Color online? The normalized conditional distribution
functions of the first-passage time for fixed initial end-to-end sepa-
ration for N=80 in the ?a? SSS and ?b? Rouse regimes. In each plot,
from top to bottom near tfirst=0, curves correspond to R/?N
=?0.2,0.4?, ?0.4,0.6?, ?0.6,0.8?, ?0.8,1.0?, ?1.0,1.2?, ?1.2,1.4?,
?1.4,1.6?, and ?1.6,1.8?, respectively. The fluctuations of the curves
are good reflections of the error sizes of the data.
CHEN, TSAO, AND SHENG PHYSICAL REVIEW E 72, 031804 ?2005?
simulations and a scaling argument. The main focus has been
put on clarification of the crossover between the two scaling
regimes. We have performed carefully prepared Monte Carlo
simulations that sampled a fairly large parameter space and
accumulated significant statistics for this purpose. This has
allowed us to convincingly pin down the scaling regimes and
the crossover between them. We have shown how the first
passage time ?first of closing crosses over from the
a-dependent ?SSS?N3/2/a to the a-independent ?Rouse?N2
as N and a varies. Within a scaling argument, we have de-
rived the observed scaling relationship between ?SSSand N
as well as a, directly from a simple physical picture. The
same reasoning has also allowed us to determine the various
quantities that can be further used to characterize the poly-
The authors wish to acknowledge the financial support
from NSERC, the computational time allocation from Sharc-
net, and the critical reading of an earlier version of this paper
by Bae-Yeun Ha.
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DIFFUSION-CONTROLLED FIRST CONTACT OF THE…
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