The dynamics of chain closure in semiflexible polymers
Arti Dua and Binny J. Cherayila)
Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore-560012, India
The mean first passage time of cyclization ␶ of a semiflexible polymer with reactive ends is
calculated using the diffusion-reaction formalism of Wilemski and Fixman 关J. Chem. Phys. 60, 866
共1974兲兴. The approach is based on a Smoluchowski-type equation for the time evolution, in the
presence of a sink, of a many-body probability distribution function. In the present calculations,
which are an extension of work carried out by Pastor et al. 关J. Chem. Phys. 105, 3878 共1996兲兴 on
completely flexible Gaussian chains, the polymer is modeled as a continuous curve with a nonzero
energy of bending. Inextensibility is enforced on average through chain-end contributions that
suppress the excess fluctuations that lead to departures from the Kratky–Porod result for the
mean-square end-to-end distance. The sink term in the generalized diffusion equation that describes
the dynamics of the chain is modeled as a modified step function along the lines suggested by Pastor
et al. Detailed calculations of ␶ as a function of the chain length N, the reaction distance a, and the
stiffness parameter z are presented. Among other results, ␶ is found to be a power law in N, with a
z-dependent scaling exponent that ranges between about 2.2–2.4.
I. INTRODUCTION
A number of important genetic processes, including transcription, replication, and recombination, are believed in
some way to be under the structural control of the hair-pin
loops in DNA and RNA that are produced by base-pairing
within single-stranded polynucleotide segments.1– 4 Loop formation itself is controlled by such factors as solvent quality,
chain rigidity and degree of polymerization.5 These factors,
therefore, indirectly influence the course of gene expression
and regulation. Their effects on single chain dynamics have
recently been studied in the experiments of Libchaber and
co-workers6 on the thermodynamics and kinetics of chain
closure in synthetic single stranded DNA 共ssDNA兲. In these
experiments, single-stranded poly共deoxyadenosine兲 or
poly共deoxythymidine兲 of between 8 and 30 repeat units are
designed with terminal 5-base complementary sequences
having a fluorophore and a quencher at either end. In aqueous solution, the DNA strands cyclize whenever the ends are
close enough to allow base-pairing. The average time of
chain closure—a measure of the ease of loop formation—can
be determined from the extent of fluorescence quenching,
which varies as a function of the distance between the fluorophore and the quencher.
The results are sensitive to details of backbone rigidity,
and cannot be rationalized solely by treating the DNA molecule as a Gaussian coil. Such a model is expected to apply
only to chains whose overall length is many times larger than
the average persistence length, a condition unlikely to hold
for relatively short polynucleotide strands, which are intrinsically semiflexible. A theory of chain closure in these systems must therefore account explicitly for the effects of
chain stiffness to be generally valid. We are, therefore, moa兲
Electronic mail: cherayil@ipc.iisc.ernet.in
tivated to develop a model of chain dynamics that can be
used to determine cyclization times of semiflexible polymers. We do this following essentially the same methodolgy
as Pastor et al.,7 who adapted the many-body diffusion–
reaction formalism of Wilemski and Fixman8 –10 to treat flexible polymers; our approach is different only in that the polymer is now treated as an essentially inextensible continuous
curve of variable persistence length l. The small l limit of
this model describes the flexible Gaussian chain.
As discussed in Sec. II below, the dynamics of chains
with reactive, loop-forming end-groups can be described by
a Smoluchowski equation to which a sink term, representing
the loss of probability from chain closure, has been added.
Within this formalism, Wilemski and Fixman have shown9
that the closure time ␶ can be expressed as the integral of an
equilibrium time correlation function of the sink term, which
in turn can be related to the time correlation function of the
end-to-end distance. Section III discusses the evaluation of
this latter correlation function, following the introduction of
a continuum model for a semiflexible polymer. The calculation of ␶ is then shown to reduce to the numerical evaluation
of a single integral involving a sum over certain normal
modes. Section IV presents the principal results of this calculation. Derivations of some of the key expressions used in
the text are included in the Appendices.
II. TIME EVOLUTION OF CHAIN CONFORMATIONS
The rate of cyclization of a chain with reactive ends in a
neutral solvent 共assumed to be at the theta temperature so
that excluded volume interactions can be neglected兲 is governed by the time evolution of the chain conformations as a
whole. Since chain closure takes place on relatively long
time scales 共on the order of microseconds to milliseconds for
the DNA samples of Ref. 6, for instance兲, the dynamics of
the polymer—in the absence of reactions between its
ends—is effectively overdamped, and can be adequately described by the Smoluchowski equation. If ␺ ( 兵 r其 ,t) denotes
the probability density that the chain has the conformation
兵 r其 ⬅r1 ,r2 ,...rn at time t, where ri is the position of the ith
monomer in a chain of n monomers, then neglecting hydrodynamic interactions, this equation takes the form
n
冋
册
⬅D ␺ 共 兵 r其 ,t 兲 ,
共1兲
共2兲
where k is a rate constant, and S is the sink function, the
choice of which prescribes the minimum contact distance of
two ends of the chain when reaction takes place.
The chain dynamics described by Eq. 共2兲 is not necessarily equivalent to the experimental situation that obtains in
Ref. 6, where cyclization is generally reversible, and measurements are made under steady-state conditions. The treatment of steady-state chain closure will be discussed elsewhere.
To derive an expression for the closure time ␶, we introduce the probability ␹ (t) that the chain is unreacted at time t;
this probability is given by
␹共 t 兲⫽
冕
d 兵 r其 ␺ 共 兵 r其 ,t 兲 .
where
␮共 t 兲⫽
冕
d 兵 r其 S 共 兵 r其 兲 ␺ 共 兵 r其 ,t 兲 .
共5兲
␹ 共 t 兲 ⯝e ⫺t/ ␶ ,
where D 0 is the diffusion constant, defined as the inverse of
the friction coefficient per unit length of the polymer, and U
is the potential energy of the chain. The second line of this
equation provides the definition of the generalized diffusion
operator D.
This equation must be modified when the ends of the
chain are reactive. If the reaction 共in this case linkage of the
ends of the chain兲 is assumed to occur instantaneously and
irreversibly whenever the ends come within a distance a of
each other, the overall rate of reaction will be governed by
the rate of approach of the end groups. This process is still
diffusion controlled, so it can continue to be described by a
Smoluchowski-type equation, as in Eq. 共1兲 above. However,
Eq. 共1兲 is conservative, implying that the total time derivative of ␺ ( 兵 r其 ,t) is zero. This is no longer the case when
reactions at the ends of the chain permanently eliminate
closed loops from the distribution of chain conformations. To
account for this loss of probability over time, ␺ ( 兵 r其 ,t) can
either be made to satisfy appropriate boundary conditions,11
or an additional ‘‘sink’’ term can be introduced into the diffusion equation. The latter approach, which is somewhat
simpler to implement, is adopted here, and it leads to the
following equation:
⳵ ␺ 共 兵 r其 ,t 兲
⫽D ␺ 共 兵 r其 ,t 兲 ⫺kS 共 兵 r其 兲 ␺ 共 兵 r其 ,t 兲 ,
⳵t
共4兲
If ␹ (t) is now assumed to decay as a single exponential, such
that
⳵ ␺ 共 兵 r其 ,t 兲
⳵
⳵
⳵U
⫽D 0
•
⫹
␺ 共 兵 r其 ,t 兲
⳵t
⳵ ri ⳵ ri
i⫽1 ⳵ ri
兺
d␹共 t 兲
⫽⫺k ␮ 共 t 兲 ,
dt
共3兲
By differentiating Eq. 共3兲 with respect to time, and making
use of Eq. 共2兲 along with the result 兰 d 兵 r其 D ␺ ⫽0 共which
holds because of the vanishing of ␺ at the boundaries兲, one
finds that
共6兲
then the closure time is given by
␶⫽
冕
⬁
0
dt ␹ 共 t 兲 .
共7兲
Equation 共7兲 can be re-expressed as an integral over an equilibrium time correlation function involving the sink function,
as shown in Ref. 7, using results from Ref. 9. For completeness, a sketch of the derivation is provided in Appendix A. In
this section, we only state the result
␶⫽
冕 冉
⬁
dt
0
冊
C共 t 兲
⫺1 ,
C共 ⬁ 兲
共8兲
where
C共 t 兲⫽
冕 冕
dR
dR0S 共 R 兲 G 共 R,t 兩 R0,0兲 S 共 R 0 兲 ␺ eq共 R0兲 .
共9兲
Here G(R,t 兩 R0,0) is the conditional probability that a chain
with the end-to-distance R0 at time t⫽0 has the end-to-end
distance R at time t; ␺ eq(R0 ) is the equilibrium distribution
of the end-to-end distance; and S(R) is a sink function,
which is assumed to depend only the magnitude of the separation between the ends of the chain. An explicit expression
for this sink function will be provided later. As shown in
Appendix A, the derivation of this equation assumes the limit
k→⬁, which in physical terms means that chain ends are
assumed to react immediately and irreversibly when they
satisfy the distance constraints imposed by the sink function.
The closure time is therefore independent of k, and is actually a first passage time.
The key unknown in Eq. 共9兲 is the conditional probability G(R,t 兩 R0,0). For a completely flexible polymer, this
probability is Gaussian, because R is the sum of a large
number of independent, random bond vectors, and the central limit theorem applies.7 For a semiflexible polymer, on
the other hand, successive bond vectors are no longer uncorrelated, but R can still be written as the sum of a large
number of independent normal modes,12 so in this case too,
for moderate to relatively high degrees of stiffness, it is reasonable to describe G(R,t 兩 R0,0) by a Gaussian.13 共Without
this assumption the resulting mathematics would become
quite intractable.兲
A Gaussian distribution is completely specified by its
mean ␮ m and variance ␴ 2 ; for a polymer, these take the form
␮ m ⫽ 具 R共 t 兲 典 eq⫽ ␾ 共 t 兲 R0 ,
共10兲
␴ 2 ⫽ 具 R2 共 t 兲 典 eq⫺ 具 R共 t 兲 典 eq• 具 R共 t 兲 典 eq ,
共11兲
where the angle brackets 具 (¯) 典 eq denote an ensemble average over ␺ eq . The function ␾ (t) is defined as
␾共 t 兲⫽
具 R共 t 兲 •R共 0 兲 典 eq
.
具 R 2 典 eq
共12兲
冉
3
2 ␲ 具 R 2 典 eq
冋
冊
册
共13兲
For a semiflexible polymer 具 R 2 典 eq is given by14
冋
册
N
1
1⫺
共 1⫺e ⫺2pN 兲 ,
p
2pN
共14兲
where N⬅nl is the contour length of the chain, and p⬅1/l is
the inverse persistence length.
Substituting Eq. 共13兲 into the expression for the correlation function C(t) 关Eq. 共9兲兴, and carrying out the angular part
of the integrations over the vectors R and R0 , one can show
that
C 共 t 兲 ⫽16␲ 2
⫻
冕
冉
⬁
0
冋
⫻exp ⫺
⫻
冊
3
3
1
2
2 ␲ 具 R 典 eq 共 1⫺ ␾ 2 共 t 兲兲 3/2
dRR 2 S 共 R 兲
冕
⬁
0
dR 0 R 20 S 共 R 0 兲
共 R 2 ⫹R 20 兲
3
2 具 R 2 典 eq 共 1⫺ ␾ 2 共 t 兲兲
册
共15兲
The further evaluation of C(t) requires a specific choice
for the sink function. Here we restrict our attention to the
Heaviside sink, defined as
⫽0;
共16兲
where a is the closest the two ends of the chain may approach before irreversible ring formation takes place. Although this choice does not permit the analytic evaluation of
the integrals in Eq. 共15兲, it is possible to obtain what is
expected to be an accurate approximate expression for C(t)
by expanding the integrand to first order in the small parameter
x 0⫽
3a 2
,
2 具 R 2 典 eq
共17兲
and then integrating term by term. This produces
C共 t 兲⫽
16x 30
9 ␲ 共 1⫺ ␾ 兲
2 3/2
冋
1⫺
册
6x 0
⫹¯ .
5 共 1⫺ ␾ 2 兲
共18兲
To the same order of approximation, this expression may be
rewritten in the more convenient resummed form as
⫺3/2
共19兲
,
冊
3R 2
,
2a 2
共20兲
III. TIME CORRELATION FUNCTION OF THE END-TOEND DISTANCE
The semiflexible polymer is modelled as a continuous
curve with a nonzero energy of bending. Points on this curve,
at time t, are described by the Cartesian vector r(s,t), where
s is a real variable that defines where a given point 共monomer兲 is on the curve with respect to an origin located at its
mid-point; for a chain of contour length N, s lies between
⫺N/2 and N/2. In terms of the monomer coordinates r(s,t),
and in units where the thermal energy k B T⫽1, the configurational part of the Hamiltonian of this model of a semiflexible chain can be written as16 –19
冕
N/2
⫺N/2
ds
冏
冏 冕 冏
⳵ r共 s,t 兲 2
⫹␩
⳵s
N/2
⫺N/2
ds
冏
⳵ 2 r共 s,t 兲
⳵s2
2
⫹ ␯ 0 共 兩 u⫺N/2兩 2 ⫹ 兩 uN 兩 2 兲 ,
共21兲
3p
,
2
共22兲
where
␯⫽
␩⫽
3
,
8p
␯ 0⫽
3
4
and
R⭐a
R⬎a,
冊
where N is a normalization constant. The two expressions
are not identical, however; the coefficient of x 0 in Ref. 15 is
4/3, not 4/5, as it is above. In our calculations, C(t) is determined exclusively from the resummed form of the Heavide sink, Eq. 共19兲.
To calculate ␶ it now only remains to specify the form of
the correlation function ␾ (t). This is discussed in the following section.
H⫽ ␯
sinh关 3 ␾ RR 0 / 具 R 2 典 eq共 1⫺ ␾ 2 共 t 兲兲兴
.
3 ␾ RR 0 / 具 R 2 典 eq共 1⫺ ␾ 2 共 t 兲兲
S 共 R 兲 ⫽1;
4
1⫺ ␾ 2 ⫹ x 0
5
S 共 R 兲 ⫽N exp ⫺
3 共 R⫺ ␾ 共 t 兲 R0兲 2
⫻exp ⫺
.
2 具 R 2 典 eq共 1⫺ ␾ 2 共 t 兲兲
具 R 2 典 eq⫽
9␲
冉
which is of exactly the same functional form as the expression derived by Doi15 using the following Gaussian sink
function:
3/2
1
共 1⫺ ␾ 2 共 t 兲兲 3/2
16x 30
冉
From these definitions one can show that
G 共 R,t 兩 R0,0兲 ⫽
C共 t 兲⬵
u共 s,t 兲 ⬅
⳵ r共 s,t 兲
.
⳵s
共23兲
The first term in Eq. 共21兲 describes the connectivity of the
chain, 共which is purely entropic in origin兲, while the second
describes its bending energy. By representing the chain in
this form, in terms of first and second derivatives of the
monomer position, the tangent vector u(s) 关Eq. 共23兲兴 should,
strictly speaking, be a unit vector, meaning 兩 u(s) 兩 ⫽1. This
constraint is difficult to enforce rigorously, so as a matter
convenience it is replaced by the mathematically less demanding constraint 具 兩 u(s) 兩 2 典 ⫽1, where the angular brackets
refer to an equilibrium average over the conformations of the
chain. This constaint is now handled by the last two terms in
Eq. 共21兲, which act effectively to suppress the excess fluctuations in the chain ends that lead to departures from the
Kratky–Porod result for the end-to-end distance.19
In the foregoing continuum representation of the chain,
the end-to-end distance R(t) at time t can be written as
R共 t 兲 ⫽r共 N/2,t 兲 ⫺r共 ⫺N/2,t 兲 .
共24兲
The equilibrium correlation function of this distance at two
different times, therefore, becomes
具 R共 t 兲 •R共 0 兲 典 ⫽ 具 r共 N/2,t 兲 •r共 N/2,0 兲 典 ⫹ 具 r共 ⫺N/2,t 兲
•r共 ⫺N/2,0 兲 典 ⫺ 具 r共 N/2,t 兲 •r共 ⫺N/2,0 兲 典
⫺ 具 r共 ⫺N/2,t 兲 •r共 N/2,0 兲 典 .
共25兲
To evaluate the averages that appear on the right-hand side of
the above equation, we need an expression for r(s,t), the
position of the sth monomer at time t. This function satisfies
a differential equation that can be derived from the application of Hamilton’s principle to the chain Lagrangian,
N/2
dsṗ2 ⫺H, where p(s) is the momentum of
L⫽( 12 m) 兰 ⫺N/2
the monomer at s. This differential equation, as shown for
instance in Ref. 17, is given by
冉
冊
⳵
⳵4
⳵2
⫹2D 0 ␩ 4 ⫺2D 0 ␯ 2 r共 s,t 兲 ⫽ ␪共 s,t 兲 ,
⳵t
⳵s
⳵s
共26兲
L⫽2 ␩
冏
⫽0,
Q n 共 s 兲 ⫽A cos共 ␤ n s 兲 ⫹B cosh共 ␣ n s 兲 ⫹C sin共 ␤ n s 兲
⫹D sinh共 ␣ n s 兲 ,
冏
冏
s⫽N/2
⳵ r共 s,t 兲
⳵ r共 s,t 兲
⫺␩
␯0
⳵s
⳵s2
s⫽⫺N/2
2
⫽0.
共29兲
2
具 ␪共 s,t 兲 ␪共 s ⬘ ,t ⬘ 兲 典 ⫽ ␦ 共 s⫺s ⬘ 兲 ␦ 共 t⫺t ⬘ 兲 I,
␨
共30兲
I being the unit tensor.
Equation 共26兲 is solved by
冕
N/2
⫺N/2
⫹
冕
ds 1 G 0 共 s,s 1 兩 t 兲 r共 s 1 兲
N/2
⫺N/2
ds 1
冕
t
0
dt 1 G 0 共 s,s 1 兩 t⫺t 1 兲 ␪共 s 1 ,t 1 兲 , 共31兲
where the Green’s function G 0 is the solution to
冉
冊
⳵
⳵
⳵
⫹2D 0 ␩ 4 ⫺2D 0 ␯ 2 G 0 共 s,s 1 兩 t 兲 ⫽ ␦ 共 s⫺s 1 兲 ␦ 共 t 兲 .
⳵t
⳵s
⳵s
共32兲
4
2
冋 冋冉 冊 册 册
冋 冋冉 冊 册 册
␯
⫾
2␩
i ␤ n ⫽⫺
共28兲
The variable ␪(s,t) is a random force with white noise statistics whose correlation satisfy
r共 s,t 兲 ⫽
␣ n⫽
共27兲
⫽0,
G 0 共 s,s 1 兩 t 兲 ⫽ ␪ 共 t 兲
兺
n⫽0
Q n共 s 兲 Q n共 s 1 兲 e
⫺D 0 ␭ n t
,
共33兲
where ␪ (t) is the step function, and Q n (s), n⫽0,1,2,... are a
complete orthonormal set of eigenfunctions that are the solutions to the following eigenvalue equation:
LQ n 共 s 兲 ⫽␭ n Q n 共 s 兲 ,
n⫽0,1,2,...,
共34兲
where ␭ n are the eigenvalues, and L is the Hermitian operator
␯
2␩
␯
⫿
2␩
2
⫹
␯
2␩
␭n
2␩
2
⫹
1/2 1/2
共37兲
,
␭n
2␩
1/2 1/2
.
共38兲
The parameters ␣ n and ␤ n are related to each other and to
the eigenavlues ␭ n by the equations
␯
␣ 2n ⫺ ␤ 2n ⫽ ,
␩
共39兲
␭ n ⫽2 ␩ ␣ 2n ␤ 2n .
共40兲
and
Because Q n (⫺s) is also a solution Eq. 共34兲, the linear combinations Q n (s)⫹Q n (⫺s) and Q n (s)⫺Q n (⫺s) are solutions as well, so the eigenfunctions may be chosen with definite parity. In general, then, they are given by Q n (s)
⫽A cos(␤ns)⫹B cosh(␣ns) 共even parity solutions兲, and by
Q n (s)⫽C sin(␤ns)⫹D sinh(␣ns) 共odd parity solutions兲. The
application of the boundary conditions in Eq. 共27兲, gives the
following expression for Q n (s):
Q n共 s 兲 ⫽
冉 冊冋
冉 冊
冉 冊冋
冉 冊
Cn
N
⫹
By direct substitution, the solution to Eq. 共32兲 is found to be
⬁
共36兲
where A, B, C, and D are constants to be determined from the
boundary conditions, and ␣ n and ␤ n are parameters that are
obtained from the characteristic equation of the matrix M.
They are defined as
s⫽⫾N/2
⳵ r共 s,t 兲
⳵ 2 r共 s,t 兲
␯0
⫹␩
⳵s
⳵s2
共35兲
The eigenfunctions Q n (s) satisfy the same boundary conditions as defined in Eqs. 共27兲–共29兲.
Equation 共34兲 is a simple fourth-order differential equation that may be solved by rewriting it as a system of four
first-order differential equations in a new set of dependent
variables. In matrix notation, this system is of the form
⳵ X(s)/ ⳵ s⫽M•X(s). From the solution to this equation, one
can show that12
and satisfies the following boundary conditions:
⳵ 3 r共 s,t 兲
⳵ r共 s,t 兲
⫺␩
␯
⳵s
⳵s3
⳵4
⳵2
.
4 ⫺2 ␯
⳵s
⳵s2
Q n共 s 兲 ⫽
⫺
cos共 ␤ n s 兲
sin共 ␤ n N/2兲
册
共 even parity兲 ,
共41兲
册
共 odd parity兲 ,
共42兲
␣ n 2 cosh共 ␣ n s 兲
,
␤ n sinh共 ␣ n N/2兲
Dn
N
⫹
1/2
1/2
sin共 ␤ n s 兲
cos共 ␤ n N/2兲
␣ n 2 sinh共 ␣ n s 兲
,
␤ n cosh共 ␣ n N/2兲
where C n and D n are unknown, but can be chosen to ensure
normalization of the eigenfunctions. The further application
of the boundary conditions in Eqs. 共28兲 and 共29兲 leads to a
pair of eigenvalue equations that correspond to the even and
odd parity solutions; these equations are
冉 冊冋冉
␤ 3n sinh共 ␣ n N/2兲 cos共 ␤ n N/2兲
␣ 2n
Dn
N
⫽
sec2 共 ␤ n N/2兲 ⫺ 2 sech2 共 ␣ n N/2兲
N
2
␤n
⫹ ␣ 3n cosh共 ␣ n N/2兲 sin共 ␤ n N/2兲 ⫹2 p 共 ␣ 2n ⫹ ␤ 2n 兲
⫻sin共 ␤ N/2兲 sinh共 ␣ n N/2兲 ⫽0,
␤ 3n
共 even parity兲 ,
共43兲
cosh共 ␣ n N/2兲 sin共 ␤ n N/2兲
⫺ ␣ 3n sinh共 ␣ n N/2兲 cos共 ␤ n N/2兲 ⫺2 p 共 ␣ 2n ⫹ ␤ 2n 兲
⫻cos共 ␤ n N/2兲 cosh共 ␣ n N/2兲 ⫽0,
共 odd parity兲 .
共44兲
From Eqs. 共31兲 and 共33兲, an expression for the time correlation function of the monomer position can now be derived as
具 r共 s,t 兲 •r共 s,0兲 典
兺 冕⫺N/2ds 1 Q n共 s 兲 Q n共 s 1 兲 具 r共 s 1 ,0兲 •r共 s,0兲 典
n⫽0
⬁
⫽
N/2
兺 冕⫺N/2ds 1 冕0 dt 1 Q n共 s 兲 Q n共 s 1 兲
n⫽0
⬁
⫻e
⫺D 0 ␭ n t
⫹
N/2
t
⫻具 ␪共 s 1 ,t 1 兲 •r共 s,0兲 典 e ⫺D 0 ␭ n 共 t⫺t 1 兲 .
共45兲
The second term in this equation vanishes by virtue of the
Gaussian character of the random force ␪(s,t), which has a
zero mean, so the sole contribution to the monomer time
correlation function comes from the first term. The evaluation of this term requires an expression for the equilibrium
average 具 r(s 1 ,0)•r(s,0) 典 , which is shown in Appendix C to
be given by
具 r共 s 1 ,0兲 •r共 s,0兲 典
⫽
1
min兵 s,s 1 其
⫺ 2 关 1⫺ 共 e ⫺2ps ⫹e ⫺2ps 1 ⫺e ⫺2p 兩 s⫺s 1 兩 兲兴 .
p
4p
共46兲
Substituting Eqs. 共43兲–共46兲 into Eq. 共25兲 and carrying out
the integration s 1 , we obtain the following expression for the
time correlation function of the end-to-end distance:
⬁
具 R共 t 兲 •R共 0 兲 典 ⫽
冉
Dn
2T 1 e ⫺D 0 ␭ n t
N n⫽1
p
兺
odd
⫻
冋
冉 冊册
2T 2
␤ 2n
⫻ 1⫺
⫺
冊
共 1⫹e ⫺2pN 兲 T 3
␣ 4n
␤ 4n
␣ 2n
,
␣n
tanh共 ␣ n N/2兲 ,
␤n
T 2 ⫽tan共 ␤ n N/2兲 ⫺
␤n
tanh共 ␣ n N/2兲 ,
␣n
␣ 3n
T 3 ⫽tan共 ␤ n N/2兲 ⫺ 3 tanh共 ␣ n N/2兲 ,
␤n
with
共 ␣ 2n ⫹ ␤ 2n 兲 ␤ n
册
⫺1
.
共49兲
There are no contributions to 具 R(t)•R(0) 典 from n even.
The explicit expression for ␾ (t), as given by Eq. 共12兲,
can be found from Eqs. 共47兲–共49兲. This is then substituted
into Eq. 共19兲 to determine C(t), which along with C(⬁), is
used in Eq. 共8兲 to obtain the closure time ␶.
The above procedure can only be implemented numerically. It begins by assigning a definite value to the stiffness
parameter z⬅pN by assigning definite values to the contour
length N and the inverse persistence length p. Variations in z
are effected by varying p at constant N. Having fixed z, Eq.
共44兲 is solved for the parameter ␣ n , using Eq. 共39兲 to express ␤ n in terms of ␣ n , at different fixed values of the mode
number n. The solutions are obtained numerically using
Mathematica. The nth eigenvalue ␭ n is then calculated from
Eq. 共40兲. The time correlation function of the end-to-end
distance is next calculated from Eq. 共47兲 by first assigning a
definite value to the time t 共its units are determined by D 0 ,
which is arbitrarily set to 1, as in Ref. 7兲, and then summing
over modes using the previously determined values of ␣ n ,
␤ n , and ␭ n in conjunction with Eqs. 共48兲 and 共49兲. The sum
typically converges within the first 500 modes. The function
␾ (t), for the given value of t, is now obtained from Eqs. 共47兲
and 共12兲; ␾ (t) in turn is used to determine C(t) from Eq.
共19兲 after assigning a definite value to the reaction distance a
共which, like N, is expressed in units of l, the persistence
length.兲 The process is repeated for different fixed values of
t. C(⬁) is likewise calculated from Eq. 共19兲 by taking its t
→⬁ limit analytically. Knowing C(t) and C(⬁), the closure
time ␶ is now calculated from Eq. 共8兲 by numerical integration using routines available in Mathematica. The entire calculation can be repeated for different values of z, N, and a.
IV. RESULTS AND DISCUSSION
A. The flexible limit
␤ n 共 1⫺e ⫺2pN 兲
⫹
2p ␣ 2n
共47兲
where
T 1 ⫽tan共 ␤ n N/2兲 ⫹
⫹
共 3 ␣ 2n ⫺ ␤ 2n 兲 T 2
冊
共48兲
As a check of the general methodology developed in the
foregoing sections, we first study our model in the flexible
limit, where prior analytical and simulation results are available for comparison.7,11,15,20–24 This limit corresponds, in
general, to values of z much larger than unity. In practice, we
set z to N, thus fixing the persistence length l at 1, the fundamental bond length. That this assignment does correspond
to the flexible chain limit is verified by noting that the eigenvalues ␭ n calculated from Eqs. 共39兲, 共40兲, and 共44兲 are nearly
identical to the known eigenvalues of the continuum Rouse
model. With z set to N, we now estimate ␶ for different
values of N and a. The results are shown in Table I in the
column marked DC, along with results from the simulations
of Pastor et al.7 共the column marked Sim.兲, the results calculated by Pastor et al. using a certain approximation in the
Wilemski–Fixman model7–10 共the column marked WF2兲 and
the results of two other calculations using the Wilemski–
Fixman model to be discussed shortly 共the columns marked
TABLE I. Closure time ␶ for different N and a as estimated by different
theoretical approaches.
␶
a
WF3b
WF4c
DCd
393
169
130
797
381
306
1306
673
558
162
143
123
365
331
293
644
591
530
233
168
136
500
378
317
863
674
578
N
a
Sim.
WF2
50.0
0.1
0.5
1.0
0.1
0.5
1.0
0.1
0.5
1.0
570⫾30
174⫾10
110⫾ 5
1070⫾70
410⫾20
250⫾10
1800⫾80
680⫾30
450⫾20
677
233
163
1313
498
367
2104
848
646
75.0
100.0
Pastor–Zwanzig–Szabo model 共PZS兲, Ref. 7, in the Wilemski–Fixman approximation, with ␾ (t) and ␭ n calculated from Eqs. 共50兲 and 共51兲, respectively.
b
PZS model with ␾ (t) calculated from Eq. 共52兲. The eigenvalues used in the
calculation are the discrete Rouse eigenvalues.
c
PZS model with ␾ (t) calculated from Eq. 共53兲. The eigenvalues used in the
calculation are the continuum Rouse eigenvalues.
d
Present calculations.
a
WF3 and WF4, respectively.兲 The values of N and a used in
our calculations were chosen to coincide with the values reported in Ref. 7.
It is clear from the table that there can be significant
differences between all five sets of predictions, although they
are all consistent in suggesting 共a兲 that ␶ decreases with increase in a at fixed N, and 共b兲 that ␶ increases with increase
in N at fixed a. These trends are physically reasonable: On
the one hand, increasing a at fixed N brings the reaction
radius closer to the equilibrium end-to-end separation, making it easier on average for the chain to cyclize; and on the
other, increasing N at fixed a takes the reaction radius farther
away from the end-to-end separation, making it harder on
average for the chain to cyclize.
The quantitative differences between these predictions
are less easily rationalized, however. One reason for the differences may be the function ␾ (t), the time correlation function of the end-to-end distance. In Ref. 7, ␾ (t) is written as
␾共 t 兲⫽
8
N 共 N⫹1 兲 odd
兺n
冉
冊
1 1
⫺ exp共 ⫺D 0 ␭ n t 兲 ,
␭n 4
共50兲
where the eigenvalues ␭ n are given by
␭ n⫽
3
共 2 sin ␪ n /2兲 2 .
b2
共51兲
Here b is the bond length, ␪ n is n ␲ /(N⫹1), and the summation is carried out over a finite number of modes. The
factor of (2 sin ␪n/2) 2 in Eq. 共51兲 coincides with the eigenvalues of the discrete Rouse chain, but the coefficient 3/b 2 is
extraneous, and has been put in by hand. 共On dimensional
grounds alone the presence of the bond length b in this equation would be suspect, but presumably it is really a dimensionless length of some kind. In any event, it is set to 1 in the
calculations, so it effectively drops out of the calculations.兲
Although the factor of 3 in Pastor et al.’s definition of the
discrete Rouse eigenvalues should be omitted when these
eigenvalues are used in the denominator of Eq. 共50兲, this
factor should be retained when the eigenvalues are used in
the argument of the exponential 共provided D 0 is set to 1兲.
The normalization of ␾ (t) in Eq. 共50兲 appears to have
been taken care of by the additive factor involving the coefficient of 1/4. 关That is to say, ␾共0兲, as obtained from Eq. 共50兲,
is unity.兴 But it would have been more natural to normalize
␾ (t) by writing it as
␾共 t 兲⫽
兺 odd n ␭ ⫺1
n exp共 ⫺3D 0 ␭ n t 兲
兺 odd n ␭ ⫺1
n
,
共52兲
with D 0 ⫽1 and ␭ n given by (2 sin ␪n/2) 2 . The closure times
␶ predicted by the use of this expression for ␾ (t) in the
Wilemski–Fixman model along with the resummation approximation of Ref. 7 lead to the values shown under column
WF3 in Table I. These values are all considerably lower than
those in WF2, particularly at a⫽0.1, indicating a fairly sensitive dependence of ␶ on ␾ (t).
In our calculations, ␾ (t) is obtained from the large z
limit of Eq. 共47兲 after normalization. It cannot be expressed
in closed form, but for a given time t and with z⫽N, it is
found to yield the same numerical value as the following
expression:
⬁
␾ 共 t 兲 ⫽8
1
兺 2 2 e ⫺3n ␲ D t/N .
odd n n ␲
2 2
0
2
共53兲
Since the calculations use a continuum model of the chain, it
may be more appropriate to compare our estimates of ␶ with
the estimates obtained using Eq. 共53兲 in the Wilemski–
Fixman model, rather than either Eqs. 共50兲 or 共52兲. Values of
␶ calculated with Eq. 共53兲 are shown in column WF4 in
Table I, and they are seen to be quite comparable to the
figures in the column marked DC. 共Because of the way ␾ (t)
enters into the expression for ␶ 关Eqs. 共8兲 and 共19兲兴, small
numerical differences in its value can be magnified. So although the eigenvalues that generate WF4 and DC are nearly
the same, the final estimates for ␶ need not be.兲
Further inspection of Table I indicates that the greatest
deviations between the various estimates of ␶ occur for the
smallest value of a. These differences tend to become
smaller as a is increased beyond 1. Table II shows this trend
quantitatively in four of the above models 共WF2, WF3, WF4,
and DC.兲
A simulation study by Srinivas et al.25 on fluorescence
energy transfer between chromophores attached terminally to
Rouse chains has recently been carried out using the
Wilemski–Fixman approach. The simulation methodology
was checked by applying it to the polymer cyclization problem and comparing the results with those obtained by Pastor
et al.7 for the Heaviside sink of infinite strength. The two
sets of results were found to be in agreement. The results in
Ref. 25 are obtained with a ␾ (t) based on the following
formula derived by Wilemski and Fixman:
⬁
␾共 t 兲⫽
8
␲ 2 odd
4
兺n n 2 exp共 ⫺␭ n t 兲 .
共54兲
The eigenvalues here are defined as ␭ n ⫽3D 0 n 2 ␲ 2 /N 2 b 2 ,
with D 0 and b set to 1.
TABLE II. Dependence of closure time ␶ on reaction radius a for l⫽1 at
N⫽100.
TABLE III. Scaling exponents in the variation of ␶ with N at different
z and a.
␶
a
WF2
a
WF3
b
WF4
0.1
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
2104
848
646
547
477
422
376
337
303
273
247
1306
673
558
484
432
387
348
314
284
257
233
644
591
530
476
428
385
347
313
283
257
233
c
z
d
a
Scaling exponent
DCa
0.1
0.5
1.0
0.1
0.5
1.0
0.1
0.5
1.0
0.1
0.5
1.0
2.43
2.37
2.39
2.30
2.28
2.30
2.22
2.20
2.22
2.18
2.16
2.20
DC
1.0
863
674
578
510
454
406
365
328
297
269
244
a
PZS model as defined in footnotea Table I.
PZS model as defined in footnoteb Table I.
c
PZS model as defined in footnotec Table I.
d
Present calculations.
b
B. Semiflexible chains
The effects of stiffness on chain closure times are studied by progressively decreasing the value of z from its flexible limit of z⫽100. The limit of complete rigidity is reached
when z→0, but we do not study this limit since rigid rods do
not cyclize. Nor do we consider the case zⰆ1 共the limit of
very restricted flexibility兲 since it is not clear that our assumption of a Gaussian distribution for the end-to-end distance remains valid under these conditions. Moreover, our
equations tend to become numerically unstable in this region
of z values. Accordingly, we restrict the further calculation of
␶ to the regime 1⭐z⭐100. In this regime, we investigate the
effect of semiflexibility on ␶ by plotting ␶ versus N 共on a
log–log scale兲 for different fixed values of z and a. Figure 1
shows one such plot for z⫽1 and a⫽0.1, 0.5, and 1.0. The
graphs are seen to be linear, so one can calculate a scaling
FIG. 1. Variation of closure time ␶ with chain length N at fixed z⫽1 共corresponding to a chain of limited flexibility兲 for different values of the reaction distance a 共0.1, 0.5, and 1.0兲. Open circles are the calculated values of
␶ using Eq. 共8兲 along with Eqs. 共12兲–共19兲. Full lines are lines of best fit
through the calculated points.
10.0
50.0
100.0
a
Present calculations.
exponent for the variation of ␶ with N. For the chosen values
of z and a, this exponent is roughly 2.4. At different z 共but
the same set of a values兲, ␶ continues to vary as a power law
in N, but with a different scaling exponent 共the graphs are not
shown, but they are qualitatively the same as Fig. 1兲. The
results are summarized in Table III, and they indicate, in
general, that the rate of change of ␶ with N increases as
stiffness increases.
In the opposite limit, i.e., as the chain becomes more
flexible, the exponent is seen to approach 2. Appendix B
calculates ␶ for the so-called harmonic spring model of the
chain, which is the same as the Rouse model, except that
only a single mode is retained in the summation of Eq. 共54兲.
Within the Wilemski–Fixman model using the resummed
form of the Heaviside sink, ␶ for this model can be calculated
exactly. The calculation shows that ␶ ⬃N 2 , independent of a.
Stiff chains of a given persistence length tend to become
more flexible as the length of the chain increases. One therefore expects the scaling exponents that describe ␶ for such
chains to cross over to values characteristic of flexible chains
as N increases at fixed l. To see if this trend emerges from
our calculations, we calculate ␶ as a function of N in the
range 50⭐N⭐1000 for fixed l⫽50 and a⫽1.0. At the smallest value of N, therefore, the chain can be regarded as stiff,
and thereafter, it becomes progressively more flexible. Figure
2 shows that the variation of ␶ with N is indeed characterized
by at least two distinct scaling regimes, one lying between
about 50⭐N⭐150, where we estimate the exponent to be
about 2.33 by linear least squares fitting, and the other lying
between about 150⭐N⭐1000, where we estimate the exponent to be about 1.80, again by linear least-squares fitting.
Some element of subjectivity obviously enters into the selection of the data points included in the fitting routine. If the
scaling exponent in the flexible limit is determined independently by studying the variation of ␶ with N at fixed l, rather
than at fixed z, one finds that for l⫽1 and a⫽1.0, this exponent is about 2.05. The discrepancy between this result
and the value of 2.20 obtained from a calculation at fixed z is
due to the difference between the two constraints. Varying N
at fixed z can generate l values less than 1 共when N is 50 and
z is 100, for instance兲, which we believe are not strictly compatible with the model as defined. Nevertheless, both ap-
FIG. 2. Variation of closure time ␶ with chain length N for a reaction radius
a⫽1 at fixed persistence length l⫽50 共corresponding to a chain of limited
flexibility when N⫽50兲. Filled circles are data points calculated using Eq.
共8兲 along with Eqs. 共12兲–共19兲. Straight lines are lines of best fit through the
putative semiflexible and flexible scaling regimes.
proaches yield qualitatively the same predictions.
As a further characterization of the effects of stiffness on
␶, we plot 共in Fig. 3兲 the variation of ␶ with 1/z at N⫽50 and
three different values of a 共0.1, 0.5, and 1.0兲. As expected, ␶
increases as l increases at all values of a for the given N. The
same qualitative behavior is observed for N⫽75 and N
⫽100, but these results are not shown. In the region of relatively high flexibility, the initial increase of ␶ is quite steep,
but thereafter it is much more gradual, and eventually ␶ approaches an asymptotic value as z→1. This asymptotic value
is found to be an increasing function of N. Figure 3 also
shows that ␶ is sensitive to the change in the contact distance
a. The decrease in a, for a given value of N, leads to a
significant increase in ␶.
FIG. 4. Variation of ␾ (t) 关as calculated from Eq. 共47兲兴 with t 共in units where
D 0 and b are unity兲 at three different values of z 共1, 2, and 100兲. Full circles
are data points obtained from the continuum Rouse expression for ␾ (t) 关Eq.
共53兲兴.
One final characterization of stiffness effects is provided
in Fig. 4, which shows the variation of ␾ (t) with t 共measured
in units where D 0 and b are unity兲 at three different values of
z 共1, 2, and 100兲. The smallest of these values corresponds to
chains that are comparatively stiff, while the largest corresponds to chains of essentially complete flexibility. Also
shown on the figure are values of ␾ (t) 共the full circles兲 at
different values of t as calculated from Eq. 共53兲, the continuum Rouse expression for the dynamical end vector correlation function. These data points are seen to fall exactly
over the curve with z⫽100. Curves for z in the range 10
⭐z⭐100 共not shown兲 also coincide almost exactly with this
last curve, so chain flexibility seems to set in at about z
⫽10.
Although part of the motivation for this study was to
rationalize the data of Libchaber and co-workers6 on chain
closure in DNA of different base sequences, the assumption
of irreversibility in the present calculations seems to preclude a direct comparison of their results with ours. The
generalization of our approach to the case of reversible chain
closure is being studied.
APPENDIX A: DERIVATION OF THE CLOSURE TIME
Formally, the solution to Eq. 共2兲 can be written as
␺ 共 兵 r其 ,t 兲 ⫽ ␺ eq共 兵 r其 兲 ⫺k
⫻
冕
t
0
冕
d 兵 r⬘ 其
dt ⬘ G 共 兵 r其 , 兵 r⬘ 其 兩 t,t ⬘ 兲 S 共 兵 r⬘ 其 兲 ␺ 共 兵 r⬘ 其 ,t ⬘ 兲 ,
共A1兲
where the Green’s function G satisfies
FIG. 3. Variation of ␶ with 1/z at N⫽50.0 for different values of a. Open
circles represent points calculated as in Fig. 1 from Eq. 共8兲, while the full
lines are guides to the eye.
DG 共 兵 r其 , 兵 r⬘ 其 兩 t,t ⬘ 兲 ⫽ ␦ 共 兵 r其 , 兵 r⬘ 其 兩 t,t ⬘ 兲 ␦ 共 t⫺t ⬘ 兲 .
共A2兲
If Eq. 共A1兲 is multiplied by S( 兵 r其 ) and the result integrated
over all monomer positions, one finds that
␮ 共 t 兲 ⫽ ␮ eq⫺k
冕 兵 其冕
d r
d 兵 r⬘ 其
冕
t
0
tions 兰 dRG(R,R⬘ 兩 t,t ⬘ )⫽ 兰 dR⬘ G(R,R⬘ 兩 t,t ⬘ )⫽1 allows
the time correlation function C(t⫺t ⬘ ) to be written as
dt ⬘
⫻S 共 兵 r其 兲 G 共 兵 r其 , 兵 r⬘ 其 兩 t,t ⬘ 兲 S 共 兵 r⬘ 其 兲 ␺ 共 兵 r⬘ 其 ,t ⬘ 兲 ,
共A3兲
where ␮ (t) has been defined in Eq. 共5兲; ␮ eq is defined in the
same way except that ␺ ( 兵 r其 ,t) is replaced by ␺ eq( 兵 r其 ).
Equation 共A1兲 is a nonlinear integral equation for the
distribution function ␺, and as such cannot be solved in
closed form. Wilemski and Fixman,11 therefore, introduced a
closure approximation to simplify the equation; this approximation takes the form
␺ 共 兵 r其 ,t 兲 ⬇ ␺ eq共 兵 r其 兲 ␻ 共 t 兲 ,
C 共 t⫺t ⬘ 兲 ⫽
冕 冕
dR
1
k
␻
˜ 共 s 兲⫽ ⫺
C̃ 共 s 兲 ␻
˜ 共 s 兲,
s ␮ eq
or equivalently
共A4兲
␮共 t 兲
.
␮ eq
共A5兲
It is easy to show using Eqs. 共A4兲 and 共A5兲 into Eq. 共A3兲
that
␻ 共 t 兲 ⫽1⫺
k
␮ eq
冕
t
0
C 共 t⫺t ⬘ 兲 ␻ 共 t ⬘ 兲 dt ⬘ ,
冕 兵 其冕
d r
冕
dRS 共 兩 R兩 兲 ␦ 共 R⫺rn 兲 .
冕 冕
dR
˜␹ 共 s 兲 ⫽s ⫺1 关 1⫺k ␮ eq␻
˜ 共 s 兲兴 .
H 共 t 兲 ⬅C 共 t 兲 ⫺C 共 ⬁ 兲 ,
冕 冕
共A15兲
共A16兲
t→⬁
共A7兲
Hence, from Eq. 共A13兲 and the Laplace transform of Eq.
共A15兲
冋
␻
˜ 共 s 兲 ⫽ s⫹
k
sH̃ 共 s 兲 ⫹k ␮ eq
␮ eq
册
⫺1
,
冋
˜␹ 共 s 兲 ⫽ 1⫹
册
k
H̃ 共 s 兲 ␻
˜ 共 s 兲.
␮ eq
共A18兲
Furthermore, from Eq. 共7兲, ␶ can be written as
共A8兲
␶ ⫽ ˜␹ 共 0 兲 ,
共A19兲
while from Eqs. 共A18兲, 共A17兲, and 共A15兲, respectively, we
have
冋
˜␹ 共 0 兲 ⫽ 1⫹
册
k
H̃ 共 0 兲 ␻
˜ 共 0 兲,
␮ eq
共A20兲
⫺1
␻
˜ 共 0 兲 ⫽k ⫺1 ␮ eq
,
d 兵 r⬘ 其 ␦ 共 rn ⫺R兲
共A17兲
which in turn when substituted into Eq. 共A14兲 produces
dR⬘ S 共 R兲 Ḡ 共 R,R⬘ 兩 t,t ⬘ 兲 S 共 R⬘ 兲 , 共A9兲
d 兵 r其
共A14兲
Now introduce the function
where
Ḡ 共 R,R⬘ 兩 t,t ⬘ 兲 ⬅
共A13兲
.
2
C 共 ⬁ 兲 ⬅ lim C 共 t 兲 ⫽ ␮ eq
.
When this expression is used in Eq. 共A7兲, C(t⫺t ⬘ ) becomes
C 共 t⫺t ⬘ 兲 ⫽
册
⫺1
where C(⬁) is the long time limit of C(t). From the definition 共A7兲 and the limit G( 兵 r其 , 兵 r⬘ 其 兩 t,t ⬘ )→ ␺ eq( 兵 r其 ) when t
→⬁, C(⬁) is seen to be given by
defines a sink–sink correlation function. In principle, this
function may depend on the details of the chain conformation as a whole, but in the present calculations it is a function
only of the magnitude of the separation between the ends of
the chain. Without loss of generality, one of the chain ends
may be located at the origin of coordinates 0. In this case,
S( 兵 r其 )⫽S( 兩 rn 兩 ), where rn is the position of the nth monomer
with respect to 0. The sink function may now be rewritten
identically as
S 共 兩 rn 兩 兲 ⫽
kC̃ 共 s 兲
␮ eq
共A12兲
Similarly, the Laplace transform of Eq. 共4兲 and the definition
Eq. 共A5兲 leads to
d 兵 r⬘ 其 S 共 兵 r其 兲 G 共 兵 r其 , 兵 r⬘ 其 兩 t,t ⬘ 兲
⫻S 共 兵 r⬘ 其 兲 ␺ eq共 兵 r⬘ 其 兲 ,
冋
␻
˜ 共 s 兲 ⫽s ⫺1 1⫹
共A6兲
where C(t⫺t ⬘ ), which is given by
C 共 t⫺t ⬘ 兲 ⫽
共A11兲
To proceed further, one now takes the Laplace transform of
Eq. 共A6兲, which produces
where
␻共 t 兲⬅
dR⬘ S 共 R兲 G 共 R,R⬘ 兩 t,t ⬘ 兲 S 共 R⬘ 兲 ␺ eq共 R⬘ 兲 .
共A21兲
and
⫻G 共 兵 r其 , 兵 r⬘ 其 兩 t,t ⬘ 兲 ␦ 共 rn⬘ ⫺R⬘ 兲 ␺ eq共 兵 r⬘ 其 兲 .
共A10兲
The function Ḡ(R,R⬘ 兩 t,t ⬘ ) satisfies the relations
兰 dRḠ(R,R⬘ 兩 t,t ⬘ )⫽ ␺ eq(R⬘ )
and
兰 dR⬘ Ḡ(R,R⬘ 兩 t,t ⬘ )
⫽ ␺ eq(R). The introduction of a Green’s function
G(R,R⬘ 兩 t,t ⬘ )
defined
by
Ḡ(R,R⬘ 兩 t,t ⬘ )
⫽G(R,R⬘ 兩 t,t ⬘ ) ␺ eq(R⬘ ), satisfying the normalization condi-
H̃ 共 0 兲 ⫽
冕
⬁
0
2
dt 关 C 共 t 兲 ⫺ ␮ eq
兴.
共A22兲
After substituting Eqs. 共A21兲 and 共A22兲 into Eq. 共A20兲, the
expression for ˜␹ (0) becomes
⫺1
˜␹ 共 0 兲 ⫽k ⫺1 ␮ eq
⫹
冕 冋␮ 册
⬁
dt
0
C共 t 兲
2
eq
⫺1 .
共A23兲
Since chain cyclization is assumed to take place essentially
instantaneously, the rate constant k is effectively infinite, so
that in this limit, using Eq. 共A23兲 in the definition of the
closure time, one finally obtains
␶ ⫽ ˜␹ 共 0 兲 ⫽
冕 冋
册
C共 t 兲
⫺1 .
C共 0 兲
⬁
dt
0
共A24兲
The closure time for the harmonic spring 共HS兲 model
within the PZS reformulation of the Wilemski–Fixman approximation is derived as follows: recall that in this approximation the time correlation function C(t) is given by
C共 t 兲⫽
冉
9␲
1⫺ ␾ 2 ⫹
4x 0
5
冊
Therefore,
兺
0
冉
1
冑1⫺y
冊
共B9兲
⫺1 .
Therefore,
␶ HS⫽
1
C
冋
1
冑1⫺2 ␣
冉 冉
⫺1⫺2 ln cos
共B10兲
1 ⫺1
sin 共 冑2 ␣ 兲
2
冊冊 册
.
共B11兲
APPENDIX C: DERIVATION OF THE CORRELATION
FUNCTION OF THE MONOMER POSITION
k
共B2兲
,
where X⬅1⫹4x 0 /5. The harmonic spring model is defined
by retaining only the first mode in Eq. 共53兲 关which defines
the function ␾ (t)兴, i.e.,
8
␾ HS共 t 兲 ⫽ 2 exp共 ⫺3 ␲ 2 D 0 t/N 2 兲 ⬅Be ⫺Ct .
␲
共B3兲
From the above equation, and from the definition of the closure time as the time integral from 0 to ⬁ of Eq. 共B2兲, it
follows that:
⬁
1
1
共 2k⫹1 兲 !! k
S共 ␣ 兲⫽
␶ HS⬅
␣ ,
2C
2C k⫽1
kk!
兺
The quantity of interest is the average 具 r(s 1 ,0)•r(s,0) 典
关see Eq. 共46兲兴. It can be obtained from the expression for the
mean-square distance between any two monomers on the
chain, which can be written in terms of the correlation of
tangent vectors as
具 关 r共 s 兲 ⫺r共 s ⬘ 兲兴 2 典 ⫽
s
s⬘
ds 1
s
s⬘
ds 2 具 u共 s 1 兲 •u共 s 2 兲 典 .
共C1兲
具 u共 s 1 兲 •u共 s 2 兲 典
⫽
B2
.
␣⫽
2A
冕 冕
The tangent vector correlation function in turn is given by
1
Z
冕 冕 冕 冕
du f
du2
du1
dui u2 •u1 K 共 u f ,u2 ;N/2,s 2 兲
⫻K 共 u2 ,u1 ;s 2 ,s 1 兲 K 共 u1 ,ui ;s 1 ,⫺N/2兲 ,
共B4兲
共C2兲
where the propagator K, in general, is given by
Using the identities
共 2k⫹1 兲 !!⫽
dy
共B1兲
.
冉 冊
⬁
1
冑␲
y
x
The change of variables y⫽sin2 ␪ and the use of the trigonometric identities cos ␪⫽1⫺2 sin2(␪/2) and sin ␪⫽2 sin共␪/
2兲cos共␪/2兲 allows this integral to be evaluated at once. The
result is
⫺3/2
C共 t 兲
共 2k⫹1 兲 !! ␾ 2
⫺1⫽
C共 ⬁ 兲
k!
2X
k⫽1
冕
S 2 共 ␣ 兲 ⫽⫺4 冑␲ ln共 cos共 21 sin⫺1 共 冑2 ␣ 兲兲 .
APPENDIX B: THE HARMONIC SPRING MODEL
16x 30
S 2共 x 兲 ⫽
2 k⫹1
冑␲
⌫ 共 k⫹3/2兲 ⫽
2 k⫹1
冑␲
共 k⫹1/2兲 ⌫ 共 k⫹1/2兲
K 共 u f ,ui ;N/2,⫺N/2兲 ⫽
冕
u共 N/2兲 ⫽u f
u共 ⫺N/2兲 ⫽ui
D 关 u共 s 兲兴 exp关 ⫺S 关 u兴兴 ,
共C3兲
共B5兲
with
and
⌫ 共 k⫹1/2兲 ⫽ 冑␲ 2
⫺2k
共 2k 兲 !
k!
the sum S( ␣ ) can
⫹(1/冑␲ )S 2 ( ␣ ), where
S 1 共 ␣ 兲 ⫽ 冑␲
S 2 共 ␣ 兲 ⫽ 冑␲
⬁
兺
k⫽1
⬁
兺
k⫽1
be
共B6兲
written
as
(2/冑␲ )S 1 ( ␣ )
共 2k 兲 !
共 ␣ /2兲 k ,
共 k! 兲 2
共 2k 兲 !
共 ␣ /2兲 k .
k 共 k! 兲 2
共B7兲
The sum S 1 is known; in closed form, it is given by
S 1 ⫽ 冑␲
冋
1
冑1⫺2 ␣
册
⫺1 .
共B8兲
Since S 1 can be expressed in terms of a derivative with respect to ␣ of S 2 , it follows that S 2 can be calculated as:
S 关 u兴 ⫽ ␯
冕
N/2
⫺N/2
ds 兩 u共 s 兲 兩 2 ⫹ ␩
冕
N/2
⫺N/2
⫹ ␯ 0 共 兩 uN/2兩 2 ⫹ 兩 u⫺N/2兩 2 兲 .
ds
冏 冏
⳵ u共 s 兲
⳵s
2
共C4兲
Z is the configurational partition function of the semiflexible
chain, and is identical to the numerator of Eq. 共C2兲 except
for the absence of the factor u1 •u2 .
The propagator K(u f ,ui ;N/2,⫺N/2) can be evaluated
by minimizing the action S 关 u兴 , i.e., by finding the trajectory
ū(s) such that ␦ S 关 u兴 / ␦ u⫽0. The required trajectory satisfies
the equation
⳵ 2 ū共 s 兲 ␯
⫺ ū共 s 兲 ⫽0,
⳵s2
␩
共C5兲
which is readily solved. The solution, when substituted in
Eq. 共C4兲, gives the following expression for the minimized
action:
3
S 关 ū兴 ⫽
关共 u21 ⫹u22 兲 cosh共 2p 共 s 2 ⫺s 1 兲兲
4 sinh共 2 p 共 s 2 ⫺s 1 兲兲
⫺2u1 •u2 兴
共C6兲
Since S is quadratic in u, the substitution u(s)→u(s)
⫹ū(s) in Eq. 共C3兲 leads to
K 共 u f ,ui ;N/2,⫺N/2兲 ⫽exp关 ⫺S 关 ū兴兴 K 共 0,0;N/2,⫺N/2兲 .
共C7兲
The above expression for the propagator can be used in Eq.
共C2兲 to calculate 具 u(s 1 )•u(s 2 ) 典 by carrying out the integration over ui , u f , u1 , and u2 . The result is
具 u共 s 1 兲 •u共 s 2 兲 典 ⫽e ⫺2p 兩 s 1 ⫺s 2 兩 .
共C8兲
Integration over s 1 and s 2 , as given in Eq. 共C1兲, produces
the following expression
具 关 r共 s 兲 ⫺r共 s ⬘ 兲兴 2 典 ⫽
兩 s⫺s ⬘ 兩
1
⫺ 2 关 1⫺e ⫺2p 兩 s⫺s ⬘ 兩 兴 . 共C9兲
p
2p
The left-hand side of the above expression can be expanded
and rearranged to obtain Eq. 共46兲.
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