Table Of ContentUnderstanding looping kinetics of a long polymer molecule in solution. Exact solution
for delocalized sink model
Moumita Ganguly∗ and Anirudhha Chakraborty
School of Basic Sciences, Indian Institute of Technology Mandi, Kamand, Himachal-Pradesh 175005, India.
The fundamental understanding of loop formation of long polymer chains in solution has been an
important thread of research for several theoretical and experimental studies. Loop formations
7 are important phenomenological parameters in many important biological processes. Here we give
1 a general method for finding an exact analytical solution for the occurrence of looping of a long
0 polymerchains in solution modeled by using a Smoluchowski-like equation with a delocalized sink.
2 Theaveragerateconstantforthedelocalizedsinkisexplicitlyexpressedintermsofthecorresponding
rateconstants for localized sinkswith different initial conditions. Simpleanalytical expressions are
n
a provided for average rate constant.
J
5
It is well known that loop formation is an important step in several biological processes such as control of gene
1
expression [1, 2], DNA replication [3], protein folding [4] and RNA folding [5]. With the progress of single molecule
] spectroscopic methods, the kinetics of loop formation has regenerated attention of both experimentalists [6–8] and
h theoreticians [9–15],. The advancement in single molecule spectroscopy have made it possible to look into the fluctu-
c
ationsnecessaryforloopformationatthe singlemoleculelevel[16,17]. Loopformationinpolymersareactuallyvery
e
complex and exact analytical solution for the dynamics is not possible. All the theories of loop formation dynamics
m
are in general approximate [10, 12]. In this paper, we give a general method for finding an exact analytical solution
-
t for the problem of looping of a long chain polymer in solution. We start with the most simplest one dimensional
a
description of the end-to-end distance of the polymer. The probability distribution P(x,t) of end-to-end distance of
t
s a long open chain polymer at time t is given by [18, 19],
.
t
a ∂P(x,t) 4Nb2 ∂2 2 ∂
m = + x−k −S(x) P(x,t), (1)
∂t τ ∂x2 τ ∂x s
- (cid:18) R R (cid:19)
d
wherexdenotesend-to-enddistance. τ istherelaxationtimetoconvertfromonetoanotherconfiguration,lengthof
n R
thepolymerisgivenbyN and‘b’denotesthebondlength. Thetermk denotestherateconstantofallotherchemical
o s
c reactions (involving at least one of the end group) apart from the end-to-end loop formation. The occurrence of the
[ loopingreactionisgivenbyaddingthesinkS(x)terminthe R.H.S.oftheaboveequation. Theloopformationwould
occur approximately in the vicinity of the point x=0. Therefore it is interesting to analyze a model, where looping
1
occurs at a particular end-to-end distance modelled by representing sink function S(x) by a Dirac Delta function
v
6 [19]. Although delocalized sink function provides a more realistic description of the process of looping. The purpose
9 of the current work is to provide an exact solution for rate constants for a general delocalized sink function. It will
9 be further seen that the rate constant for the generalised sink problem in the absence of all other chemical reactions
3 (involving at least one of the end group) apart from the end-to-end loop formation can be evaluated in terms of the
0
corresponding rate constants for localized sinks, with suitable sink positions and initial conditions. To solve Eq. (1),
.
1 we first use the following transformation
0
7 P(x,t)=F(x,t)exp(−k t) (2)
s
1
: and obtain the following simplified equation
v
i
X ∂F 4Nb2∂2F 2 ∂
= + (xF)−S(x)F (3)
r ∂t τ ∂x2 τ ∂x
a R R
The solution of Eq. (3) can be written as
∞ t
F(x,t|x ,0)=F (x,t|x ,0)− dx′ dt′F (x,t−t′|x′,0)×S(x′)F(x′,t′|x ,0), (4)
0 0 0 0 0
Z−∞ Z0
where the function F (x,t|x ,0)is the solutionin the absenceof any sink term, i.e., S(x)=0. Both F(x,t|x ,0)and
0 0 0
F (x,t|x ,0) corresponds to the same initial condition, which we consider here to be a Dirac delta function given by
0 0
∗[email protected]
2
the equation
F(x,t)=F (x,t)=δ(x−x ) att=0 (5)
0 0
In the following we use the method of Szabo, Lamm, and Weiss [11] where an arbitrary sink function S(x) can be
expressed as S(x)= ∞ dx′S(x′)δ(x−x′) and the integral can be discretized as shown below
−∞
R N
S(x)= k δ(x−x ), (6)
i i
i=1
X
where k [= ω S(x )] denotes the sink strengths, with ω depending on the scheme of discretization. So now Eq.(4)
i i i i
becomes
N t
F(x,t|x ,0)=F (x,t|x ,0)− k dt′F (x,t−t′|x ,0)×F(x ,t′|x ,0). (7)
0 0 0 i 0 i i 0
i=1 Z0
X
Taking appropriate Laplace transform of the above equation, we obtain
N
F˜(x,k +s|x ,0)=F˜ (x,k +s|x ,0)− k F˜ (x,k +s|x ,0)×F˜(x ,k +s|x ,0) (8)
s 0 0 s 0 j 0 s j j s 0
j=1
X
where
∞
F˜(x,k +s|x ,0)= dt exp[−(k +s)t]F(x,t|x ,0) (9)
s 0 s 0
Z0
∞
F˜(x,k +s|x ,0)= dt exp[−(k +s)t]F (x,t|x ,0)
0 s 0 s 0 0
Z0
ConsideringEq. (7)atthediscretepointsx=x ,x ,.....,x ,weobtainasetoflinearequations,whichcanbewritten
1 2 N
as
AˆPˆ =Qˆ, (10)
where the elements of the matrices Aˆ→{a }, Pˆ →{p } and Qˆ →{q } are given by
ij ij ij
a =k F˜ (x ,k +s|x ,0)+δ , (11)
ij j 0 i s j ij
P =F˜(x ,k +s|x ,0),
i i s 0
q =F˜ (x ,k +s|x ,0).
i 0 i s 0
One can solve the matrix equation i.e., Eq. (10) easily and obtain P˜(x ,k +s|x ,0) for all x . Here the quantity of
i s 0 i
interest is the survival probability F(t) of the open chain polymer, which is defined as
∞ N t
F(t)= P(x,t)dx=exp(−k t) 1− k F(x ,t′|x ,0)dt′ , (12)
s i i 0
Z−∞ " i=1 Z0 #
X
so that one can define the average rate constant k as [19]
I
∞
k = F(t)dt (13)
I
Z0
and also a long-time rate constant k as[19]
L
K =− lim(d/dt)InF(t). (14)
L
t→∞
Soonecaneasilyshowk−1 =F˜(0)andk isthenegativepoleofF˜(s)closesttotheorigin. Theaveragerateconstant
I L
k is thus given by
I
N
k−1 = lim(k +s)−1 1− k F˜(x ,k +s|x ,0) (15)
I s i i s 0
s→0 " #
i=1
X
3
where F˜(x ,k +s|x ,0) is to be obtained by solving Eq. (10), Which is straightforward if the Laplace transformed
i s 0
quantitiesF˜ (x ,k +s|x ,0)andF˜ (x ,k +s|x ,0)appearinginthematricesAˆandQˆ,respectively,canbeevaluated
0 i s j 0 i s 0
analytically. The easiest way to solve Eq. (10) is by Cramer’s method, and the solution for P →F˜(x ,k +s|x ,0)
j j s 0
is given by
P =detAˆ(j)/detAˆ, (16)
j
where det Aˆ represents the determinant of matrix Aˆ and Aˆ(j) is a matrix obtained by replacing the j-th column of
the matrix Aˆ by the column vector Qˆ. Substituting this solution into Eq. (14) one has the result
N
k−1 = lim detAˆ− k detAˆ(j) /{(k +s)detAˆ}. (17)
I s→0 j s
j=1
X
Usingsimple algebraicmanipulations,itis straightforwardto showthatthe numeratorofEq.(17)remainsunchanged
ifinalltheelementsofthematrices,thequantitiesF˜ (x ,k +s|x ,0)andF˜ (x ,k +s|x ,0)arereplaced,respectively,
0 i r j 0 i r 0
by △F˜ (x ,k +s|x ,0) and △F˜ (x ,k +s|x ,0), defined below.
0 i s j 0 i s 0
∞
△F˜ (x,t|x′,0)=F (x,t|x ,0)− dtexp[−(k +s)t](F (x,t|x′,0)−F (x,t=∞)), (18)
0 0 0 s 0 0
Z0
Inthe followingwewill usethe notationthatdenotesFst =F (x,t=∞). Denoting the modifiedAˆmatrix asmatrix
0 0
Bˆ with elements b →k △F˜ (x ,k +s|x ,0)+δ , the numerator of Eq. (17) becomes (detBˆ− N =1k detBˆ(j),
ij j 0 i s j ij j j
where in the matrix Bˆ(j) the j-th column of matrix Bˆ has been replaced by the column vector Q′ → {q′} with
P i
q′ = △F˜ (x ,k +s|x ,0).
i 0 i s 0
FormEq. (18),onehas △F˜ (x ,k +s|x ,0)=P˜ (x ,k +s|x ,0)− (k +s)−1Fst(x ), andhence the denominator
0 i r j 0 i s j s 0 j
of Eq. (17) can be rewritten as
N
(k +s)detAˆ= (k +s)detBˆ+ k detBˆ(j)′ (19)
s s j
j=1
X
where the matrix Bˆ(j)′ is obtained from matrix Bˆ by replacing the elements b of its jth column by the stationary
ij
values Fst(x ) for all the rows, i.e., i=1,...., N. Thus, on taking the limit s → 0, the final expression for the rate
0 i
constant k is given by [20]
I
N N
k−1 = detBˆ− k detBˆ(j) / k detBˆ+ k detBˆ(j)′ , (20)
I j s j
j=1 j=1
X X
So we have the rate constant for a delocalized sink, for the special case of a localized sink at a point x , i.e., S(x)
1
= k δ(x−x ), it can be expressed in a simple form as below
1 1
k ={k Fst(x )+k [1+k △F˜ (x ,k |x ,0)]}/{1+k [△F˜ (x ,k |x ,0)−△F˜ (x ,k |x ,0)]} (21)
I 1 0 1 s 1 0 1 s 1 1 0 1 s 1 0 1 s 0
For purely looping problem (no other chemical reactions involving end groups), i.e., k =0, the rate constant K for
s I
the general delocalized sink is given by Eq.(20) can be re-expressed in terms of the rate constant of Eq.(21). The
quantities that will appear in the final expression are denoted here as k (x ,x′) which is the rate constant (in case of
0 i
k =0)forsingleDiracδ-functionsink S(x)=k δ(x−x )ifthe end-to-enddistanceinitiallyis x=x′ andis givenby
s i i
k−1(x ,x′)=[k Fst(x )]−1+kd−1(x ,x′), (22)
0 i i 0 i i
where k (x ,x′) represents the overall rate constant in the limit k →∞), defined as
d i i
∞
k−1(x ,x′)=[1/Fst(x )]× dt[F (x ,t|x ,0)−F (x ,t|x′,0)]. (23)
d i 0 i 0 i i 0 i
Z0
4
After doing little bit of algebra one finally obtains the general rate constant given by
N N
k−1 =[−detCˆ+ detCˆ(j)]/[ detCˆ(j′)], (24)
I
j=1 j=1
X X
where the elements of the matrix Cˆ →{c } are given by
ij
k−1(x ,x ) for i6=j
c ={ 0 i j (25)
ij 0 for i=j.
The matrix Cˆ(j) is obtained from the matrix Cˆ by replacing only its j-th column by the column vector D = {d },
i
where
d =k−1(x ,x ). (26)
i 0 i 0
Similarly, the matrix Cˆ(j) is same as matrix Cˆ, but for the elements of the j-th column all of which are replaced by
unity. So for k = 0, the rate constant k for looping of a long polymer modelled using a sink of arbitrary shape,
s I
described by a set of N localized sinks, can be obtained if the corresponding localized sink rate constant k (x ,x′)
0 i
defined in Eq. (22) can be evaluated for arbitrary values of initial end-to-end distance (x′), sink position (x ), and
i
strength (k ). So in the absence of any sink term
i
−(γ/2)(x−x′e−γt)2
F (x,t|x′,0)=[(γ/2π)(1−e−2γDt)]1/2× exp , (27)
0 (1−e−2γDt)
(cid:20) (cid:21)
where D = 4Nb2 and γ = 1 and therefore
τR 2Nb2
Fst(x )=(γ/2π)1/2 exp[−(γ/2)x 2] (28)
0 i i
So k can be expressed as
d
∞
k−1(x ,x′)=(γD)−1 dy[1−exp(−2y)]−1/2[exp[γx2e−y/(1+e−y)] (29)
d i i
Z0
−exp(γ{x ,x′e−y−(e−2y/2)[x2+(x′)2]}/(1−e−2y))].
i i
Although in general, it might be difficult to evaluate the above integration analytically, simplified expression can be
derived in the case where the sink position is at the origin (x =0), the expression for k (x ,x′) simplifies to [20]
i d i
|x′|
k−1(0,x′)=(π/2γ)1/2D−1× dxexp(γx2/2)×{1−erf[(γ/2)1/2x]}, (30)
d
Z0
The case of initial end-to-end distance of the poplymer is zero ( i.e.,x’=0) we get [20]
|xi|
k−1(x ,0)=(π/2γ)1/2D−1× dxexp(γx2/2)×{1+erf[(γ/2)1/2x]}. (31)
d i
Z0
One can derive a generalized expression [21]
|xi|
k−1(x ,x′)=(π/2γ)1/2D−1× dxexp(γx2/2)×{1+erf[(γ/2)1/2x]}. (32)
d i
Zx′
So k−1(x ,x′) can be expressed as a linear combination of k−1(0,x′) and k−1(x ,0) and this helps a lot in the
d i d d i
calculation using a sink of arbitrary shape. So far we have considered only the initial condition F(x,0)= δ(x−x ).
0
Now we will consider the case where initial condition has the following distribution, i.e., F(x,0) = Fst(x). In this
0
case, the rate constant k for the generalsink is again given by Eqs. (24) and (25),but the expressionfor d given by
I i
Eq.(26) is to be replaced by the following expression
d =[k Fst(x )]−1+[kst(x )]−1, (33)
i i 0 i d i
5
where
kst(x )=(γD)[(γ/2π)x2]1/2 exp[−(γ/2)x 2]/φ(x ), (34)
d i i i i
with
φ(x )=erf{[(γ/2)x2]1/2}+[(γ/2)x2]1/2exp[−(γ/2)x2] (35)
i i i i
∞ ∞
1
× −2+In2+(γ/2)x2 (−1)k[(γ/2)x2]k+m/[(k!m!(m+ )(k+m+1)]
i i 2
" #
k=0m=0
X X
The cardinal result of this work is Eq. (24), where the overall rate constant for looping of a long chain polymer for
a generalizedsink is expressedin terms of localized sink rate constants with different values for the sink position and
initial positions. The simple analyticalexpressionfor averagerate constantfor a generalisedsink function is derived.
Oneoftheauthor(M.G.)wouldliketothankIITMandiforHTRAfellowshipandtheotherauthorthanksIITMandi
for providing CPDA grant.
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