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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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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