Understanding 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. [1] K.Rippe, Trendsin Biochemical Science, 26, 733 (2001). [2] K.Rippe, P. H.V. Hippeland J. Langowski, Trendsin Biochemical Science, 20, 500 (1995). [3] S.Jun and J. Bechhoefer, Physicsin Canada, 59, 85 (2003). [4] M. Buscaqlia, J. Kubelka,W. A.Eaton, and J. Hofrichtev,J. Mol. Biol., 347, 657 (2005). [5] D.Thirumalai, N.Lee, S.A. Woodson, and D.Klimov, Annu.Rev.Phys.Chem., 52, 751 (2001). [6] M. A.Winnik, In Cyclic Polymers, Chapter 9 (Elsivier, New York,1986). [7] Z. Haung,H. Ji, J. Mays, and M. Dadmun,Langmuir, 26, 202 (2010). [8] R.R. Hudgins,F. Huang, G. Gramlich, and W. M. Nau, J. Am. Chem. Soc., 124, 556 (2002). [9] G. Wilemski and M. Fixman, J. Chem. Phys.,60, 866 (1074). [10] M. Doi, Chem. Phys.,9, 455 (1975). [11] A.Szabo, K. Schulten,and Z. Schulten,J. Chem. Phys.,72, 4350 (1980). [12] R.W. Pastor, R.Zwanzig, and A.Szabo, J. Chem. Phys., 105, 3878 (1996). [13] J. J. Portman, J. Chem. Phys., 118, 2381 (2003). [14] I.M. Sokolov, Phys.Rev.Lett., 90, 080601 (2003). [15] N.M. Toan, G. Morrison, C. Hyeon,and D. Thirumalai, J. Phys. Chem. B, 112, 6094 (2008). [16] T. E. Cloutier and J. Widom, Proc. Natl. Acad. Sci. U.S.A., 102, 5397 (2005). [17] D.Thirumalai and C. Hyeon,Biochemistry, 44, 4957 (2005). [18] K.Schulten,Z, Schultenand A.Szabo, Physica A 100, 599 (1980). [19] M. Ganguly, A.Chakraborty, Physica A (underrevision) (2017). [20] A.Samanta, S.K. Ghosh, Phys. Rev.E, 47, 4568 (1993). [21] K.L. Sebastian, Phys.Rev.A 46, R1732 (1992). [22] A.Szabo, G. Lamm, and G.H. Weiss, J. Stat. Phys., 34, 225 (1984).