Effect of hydrodynamic interaction on polymeric

PHYSICAL REVIEW E 82, 041910 共2010兲

Effect of hydrodynamic interaction on polymeric tethers Suman G. Das, Dimitri Pescia, Mithun Biswas, and Anirban Sain* Physics Department, Indian Institute of Technology-Bombay, Powai 400076, India 共Received 12 December 2009; revised manuscript received 16 September 2010; published 14 October 2010兲 Weak bonds are ubiquitous in biological structures. They often act as adhesive contacts within an extended structure, for example, the internal bonds in a folded protein or a DNA/RNA loop. They also act as linkers between two structures, for example, a protein grafted in a cell membrane or a protein linking the cell membranes of two neighboring cells. Typically, the breakage of a bond depends on the strength of the binding potential and viscosity of the medium. But when extended structures couple to the bond, as in the above examples, the dynamics of the structure also has to be considered in order to understand the bond breakage phenomenon. Here we consider a generic model, a stretched polymer 共an extended structure兲 tethered to a soft bond and study how the dynamics of the polymer, in addition to thermal noise, influences bond breakage. We also explore how the hydrodynamic interaction due to the fluid medium, which couples the distant parts of the polymer, change the bond breakage rate. We find that hydrodynamic interaction enhances the breakage rate and also makes the motion of the unstable collective mode of the polymer more coherent. DOI: 10.1103/PhysRevE.82.041910

PACS number共s兲: 82.39.⫺k, 82.20.Uv, 02.50.Ey, 05.40.⫺a

I. INTRODUCTION

In the last decade, breakage of weak bonds received a lot of attention, mainly in the context of single molecule experiments 关1兴. Although weak noncovalent bonds decide stability of most macromolecular structures encountered in biological and other soft-matter systems, breakage of such bonds have been typically addressed using ad hoc effective one particle descriptions 关2兴. In all the examples 关3兴 mentioned in the abstract, the two molecules that form the bond are also attached to other degrees of freedom. For example, in RNA, hairpin loop forms when base pairing 共weak bonding兲 occurs between two distant bases along the RNA strand 关4兴. Rate of thermal breakage of this base pairing bond depends on the length of the hairpin loop and numerical solutions showed that longer the loop faster is the breakage 关4兴. In this paper we will focus on a many-body bond breakage problem, similar in spirit to the above. Our system consists of a stretched polymeric tether, fixed at one end and tethered to a weak bond on the other end 共see Fig. 1兲. To understand bond breakage in such situations one has to go beyond simple Kramers theory, which 关5兴 models the breakage of a bond as the escape of a particle from a potential trap, under the effect of thermal activation 共kBT兲 and friction 共␨兲 of the medium. Kramers theory yields the bond breakage rate 共r兲, or its inverse 共␶ = r−1兲 which is the average lifetime. r ␻ = ␭+ 冑Vo exp共−Eb / kBT兲, where Eb is the height of the potential ⬙ barrier, V⬙ and ␻2o are the double derivatives of the trapping potential at the barrier top and bottom, respectively. In the overdamped limit, which is relevant for most biological situations, the prefactor ␭+ = V⬙ / ␨. In general, to understand the breakage of such intramolecular bonds one has to understand how the collective modes of the macromolecule couple to the bond. In Ref. 关6兴 one of us had considered a model where the polymer of Fig. 1 has been replaced by a one-dimensional 共1D兲 harmonic

*[email protected] 1539-3755/2010/82共4兲/041910共6兲

chain. This model could be solved analytically, and it was shown that the influence of the collective modes cannot be mimicked by an effective one particle description of the trapped particle with a renormalized effective friction or mass. Reference 关6兴 also showed that the effective drag on the escaping particle is not the sum of the direct drag on the escaping particle and the drag on the polymer, but lesser. Most importantly, the calculation led to change in the prefactor ␭+ appearing in the rate formula. Despite the common wisdom that the escape rate from a trap is dominated by the exponential dependence on the barrier height, the crucial importance of the prefactor was first pointed out by Kramers in his classic 1940 paper 关5兴. He computed the nonequilibrium, stationary, probability distribution function P共x , v兲 of a single escaping particle in the phase-space, which led to a major correction 共up to few orders magnitude兲 to the prefactor, as compared to the previous equilibrium theories. Analogously, on the many-body front, one of our work 关7兴 which dealt with the breakage of a bond in the interior of a 1D chain of particles interacting through Lennard-Jones potential, succeeded in substantially reducing the previously reported mismatch 关8兴 between simulation and theory. Here we generalize the bond breakage problem to a Rouse polymer in three-dimension 共3D兲 and also investigate the effect of hydrodynamic interaction 共HI兲 on the breakage rate. Hydrodynamic effect on bond breakage has been studied by Szymczak and Cieplak 关9兴 in the context of force extension experiments on proteins. The protein contained a subloop 共see Fig. 2兲 which resulted from formation of a weak internal bond between two distant segments along the contour of the

z fixed end

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FIG. 1. A stretched polymer fixed at the left end and tethered to a weak bond on the right. 041910-1

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II I F

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FIG. 2. A protein forming a loop, via an internal bond 共dotted line兲 between the ith and jth monomers, is pulled by constant force F and is fixed at the right end. Fluctuations of all the three segments, denoted by I,II, and III, contribute to breakage of the bond.

protein. When the protein was subjected to a constant stretching force, at some point the weak bond broke due to the joint effect of tension and fluctuation and the loop opened up. Through a Brownian dynamics 共BD兲 simulation they showed that the lifetime of the bond reduces 共or equivalently the bond breaks at a faster rate兲 when the effect of HI is included. The system which we consider in Fig. 1 is qualitatively similar but has weaker fluctuation effects. In Fig. 2 bond-breaking fluctuations arise from three subsystems, namely the two handles of the loop 共segments I and III兲 and the loop 共segment II兲 itself, where as in our system 共Fig. 1兲 it comes from only one chain segment. Also since hydrodynamic interaction falls off inversely with interbead distance, HI is much weaker in Fig. 1 than in Fig. 2, because the chain is extended in Fig. 1. Although both fluctuations and HI are weaker in our system, nevertheless it is analytically tractable. Further, Szymczak and Cieplak 关9兴 explained the HI induced faster breakage, qualitatively, in terms of concerted motion of the beads due to HI. This effect can be seen quantitatively in our model. Here we attempt to understand, theoretically, the effect of HI on the unstable collective mode of the system, which causes breakage. Our calculation indeed shows enhancement of breakage rate due to HI. It also shows how the displacement profile of the unstable collective mode of the tether gets modified and become faster in the presence of HI. As it turns out later, it is the growth rate of the unstable breaking configuration which determines the breakage rate. So we need to understand how hydrodynamic drag influences the growth of the unstable collective mode rather than the relaxation time scales of the stable modes of the system. It is known that, so far as equilibrium fluctuations are concerned, for a stretched polymer with its two ends fixed at two distant walls, the relaxation of the collective modes become faster due to HI 关10,11,13兴. But bond breakage or equivalently escape from a potential trap is essentially a nonequilibrium process. In fact right after the escape, the relaxation of the stretched polymer in a viscous fluid is also a nonequilibrium process 关14兴 and our intuitions about equilibrium fluctuations cannot be applied, in principle, to these cases. Instead we need to understand how HI affects the many-body nonequilibrium probability distribution P共兵x j其 , 兵v j其兲 of the escaping polymer. Toward this we first solve the many-body escape problem without hydrodynamic interaction, using Rouse model, and then incorporate HI through the Oseen tensor, as in the Zimm model 关15兴. Note that P共兵x j其 , 兵v j其兲 above contains both position and velocity, whereas the Rouse model does not have accelera-

tion terms 共overdamped兲. So we generalize Rouse model by retaining the mass terms in the equations of motion of the beads. This generalization allows us to employ the phasespace formalism, developed by Langer 关16,17兴 in the context of nucleation phenomena. As such, the effect of the massive tethers on the escape dynamics could be interesting and has been studied 关6兴 for a 1D harmonic chain, albeit without HI. The paper is organized as follows. Section II describes the model and Sec. III gives the analytical results in the absence of HI and in the overdamped limit. Section IV shows the calculation of the escape rate for a massive polymer in the presence of HI. We end with a discussion on the implications of our results, while details of the escape rate formalism for a massive polymer is given in the Appendix. II. MODEL

We consider a chain of N connected monomers, each having mass M 共set to unity here兲, friction ␨ and position Rn. Each monomer is connected to two nearest neighbors through harmonic interaction. One end of the polymer is fixed and the other end is trapped in a potential well with a barrier 共see Fig. 1兲. The equations of motion 共EOM兲 are ¨ +R ˙ = 兺 H . 关k共R ␨−1R n n nm m+1 − 2Rm + Rm−1兲 + fm兴 m

for n = 1,2, . . ,N − 1,with boundary conditions 共BC兲: R0 = 0,

at the fixed end and

¨ +R ˙ = 兺 H . 关k共R − R 兲 − ⵜ V + f 兴, ␨−1R N N Nm N−1 N RN N m

共1兲

for the trapped end. a

The trapping potential V has a generic form V共X兲 = 21 X2 a + 32 X3 with a maximum and a minimum. Here X = 兩RN − Rmin兩 is deviation of the end point RN from the potential minimum Rmin and 兵fm其 is the thermal noise that obeys a Gaussian distribution controlled by ␨ and kBT. The system has two static equilibrium points, one stable 共s兲 and the other unstable 共u兲, near the minimum and the maximum of the potential respectively, where the pull due to the stretched polymer is balanced by the potential trap. The potential energy difference Vu − Vs = ˜EB is the effective barrier height. In Eq. 共1兲, H is the 3N ⫻ 3N dimensional Oseen tensor approximating the HI 关15兴 among the beads of the polymer and is given by Hij共m,n兲 = I H共n,n兲 = , ␨

1 共␦ij + rir j/r2兲, 8␲␩r where r = Rm − Rn .

共2兲

␨ = 6␲␩a is the stokes friction on each bead, where ␩ and a are the fluid viscosity and the hydrodynamic 共Stokes兲 radius of the bead, respectively. Note that each H共m , n兲 is a 3 ⫻ 3 matrix.

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For Rouse model the force-free spring length is zero but in a thermal environment it has an isotropically oriented equilibrium bond length b0, satisfying 21 kb20 = 23 kBT. Consequently the average end-to-end distance 共eed兲 vector is zero, although the average scalar eed is not zero. So any external stretching force at the polymer end can break the isotropy and produce a uniformly stretched static configuration. III. ANALYTICAL RESULTS

In the spirit of Kramers 关5兴, we approximate the trapping potential as V共RN兲 = 21 Vs⬙共RN − Rs兲2 , Vs⬙ ⬎ 0 near the stable 共s兲 potential well at Rs, and as V共RN兲 = EB + 21 Vu⬙共RN − Ru兲2 , Vu⬙ ⬍ 0, near the unstable 共u兲 potential maximum at Ru. The EOM for the 3N dimensional vector R = 共R1 , R2 , . . , RN兲 can be written as ¨ +R ˙ = H关AR + f兴. ␨−1R

共3兲

The nonzero elements of the 3N ⫻ 3N matrix A are given by Ann = −2I and An,n+1 = An+1,n = I for n = 1 , 2 , . . N − 1, while ANN = 共−1 + V⬙ / k兲I. In Langer’s formalism 关16兴 one computes the probability current for the escaping particle using the nonequilibrium ˙ 其兲 which is obtained by solvprobability density P共兵Rn其 , 兵R n ing the relevant Fokker-Plank equation. The upshot is a ␭ Kramers-like formula for the escape rate ⌫ = 2␲+ Re−Eb/kBT.

Here Eb is the barrier height and R = 冑兩det Eu / det Es兩, where Eu and Es are the Hamiltonian matrices of the interactions, in the quadratic approximation, near the stable 共s兲 and unstable 共u兲 equilibrium points, respectively. It turns out that ␭+ comes from the unstable eigenmode exp共␭+t兲f共兵Rn其兲 of the corresponding unforced EOM. This connection is mentioned in Ref. 关17兴 and is elaborated in the Appendix. We first analytically solve for ␭+ in the absence of HI. The method will be demonstrated by solving for ␭+ in the Rouse limit, i.e., for the overdamped case 共␨ / M Ⰷ Vu⬙兲. To solve for ␭+ we use a continuum approach. In the continuum limit, the overdamped Rouse equations read

␨

⳵ R共n,t兲 ⳵2R共n,t兲 =k + f共n兲 for n 苸 关0,N兴, ⳵t ⳵ n2

冏

⳵ R共N兲 ⳵ R共n,t兲 =−k ⳵t ⳵n

冏

− ⵜV共X兲兩X=R共N兲 .

␨⳵tQn = k⳵2nQn + fn with the BC: Q0 = 0, and ␨⳵tQN = − k⳵NQN − V⬙QN + fN . 共5兲 This shows that the three spatial dimensions get decoupled and will have identical unstable modes. This is due to the quadratic dependence of the approximate potential on the distance Vu⬙共RN − Ru兲2. Another scenario could be that the trapping potential depends only on z and thus in the x-y direction the end RN is free. In either case the problem reduces to one dimension. Solutions for the escape rate are already available for 1D harmonic chain in Ref. 关6兴. The only difference here is that the springs in Rouse model have zero unstretched length, as opposed to the case of finite bond length considered in Ref. 关6兴. In order to compute ␭+ we need to solve the unforced version of Eq. 共4兲 共in the vicinity of the unstable stationary state兲. The unique unstable solution of Eq. 共4兲 turns out to be Qu共n , t兲 = Qu共0兲e␭+t sinh共␣un兲 where ␭+ = k␣2u / ␨, and ␣u satisfies the boundary condition tanh共␣uN兲 =

k␣u . 兩Vu⬙兩 − k␣2u

共6兲

We solve this equation numerically for ␭+. Note that ␭+ depends on Vu⬙ / k 关through Eq. 共6兲兴 as well as k. For the moderately damped case a similar equation as above results, except with one extra term in the denominator, originating from the acceleration term in the EOM. In Fig. 3 we plot ␭+ as a function of Vu⬙ 关dashed line in Fig. 3共b兲兴 and the bond breakage rate as a function of stretch 关dashed line in Fig. 3共d兲兴, in the moderately damped regime. The solid lines in Figs. 3共b兲 and 3共d兲 show the changes due to HI, which are discussed later while Figs. 3共a兲 and 3共c兲 remain same for both HI and no-HI cases. The stable solutions of Eq. 共5兲, are 2 of the form Q p共n , t兲 = Q p共0兲e−共k␣p/␨兲t sin共␣ pn兲 , p = 1 , 2 , . . , N − 1. The set 兵␣ p其 can be determined by solving the boundary k␣ equation tan共␣ pN兲 = 兩V⬙兩−kp ␣2 numerically. The solution of the u p homogeneous equations 共ignoring the noise兲 thus reads Qh共t兲 = Qu共0兲ek␣ut/␨ sinh共␣un兲 + 兺 Q p共0兲e−k␣pt/␨ sin共␣ pn兲, 2

2

p

with BC:R共0兲 = 0, and

␨

EOM, obtained by substituting Rn = Cn + Qn共t兲 into Eq. 共4兲 and using the quadratic approximation for the potential.

共7兲 共4兲

n=N

The stationary solution for the above equations, in the un2 forced limit 共 ⳵ R共n,t兲 = 0兲 is the uniformly stretched state: Rn ⳵n2 = Cn, where C is determined from the boundary condition kC = −ⵜV共CN兲. This is the statement of force balance between the elastic restoring force, coming from the entropic elasticity of the stretched Rouse polymer, and the trapping force from the potential. The trapping potential offers two stationary solutions, one stable 共C = Cs兲 near the minimum and the other unstable 共C = Cu兲, near the maximum. Fluctuations around these states Rn = Cn + Qn共t兲 obey 共in the continuum approximation and overdamped limit兲 the following

where 兵Q p共0兲 , p = 1 , 2 , . . . , N − 1其 and Qu共0兲 are the initial amplitudes of the stable and unstable modes, respectively. k␣2 The growth rate of the unstable breakage mode ␶+−1 = ␨ u in Eq. 共7兲 can be shown to be slower than the slowest 共stable兲 k␣21 Rouse mode of the polymer, namely, ␶+−1 ⬍ ␶−1 1 = ␨ . In Fig. 4 solutions of the transcendental Eq. 共6兲, and its counterpart for the stable modes, are shown graphically. The right and the left hand sides 共rhs and lhs兲 of the equation have been plotted separately as a function of x ⬅ ␣ and their intersection gives the solution for x. The rhs, denoted by f共x兲, has a singularity at xⴱ = 冑Vu⬙ / k. There is only one unstable solution x+ 关when lhs is tanh共xN兲兴 and several stable solutions x1 , x2 , . . . 关when lhs is tan共xN兲兴. Note that x+ ⬍ x1 ⬍ x2 in Fig. 4 indicates

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FIG. 3. 共Color online兲 共a兲 Curvature at the effective maximum, V⬙u versus D, the location of the trap with respect to the fixed end at the left 共see Fig. 1兲. The detailed form of the potential trap is a1 a2 V共x兲 = 2 共x − D兲2 + 3 共x − D兲3. In the plot D is normalized by Nb0, the equilibrium contour length of the polymer. Higher D amounts to higher stretch for the polymer. 共b兲 describes ␭+ versus Vu⬙, spanning over the moderately damped regime ␨ / M ⬃ Vu⬙. Solid and dashed lines represent presence and absence of HI, respectively, with ␩ = 1, where Stokes friction ␨ = 6␲␩a. ␭+ goes down with higher friction 共data not shown兲 as in the single particle Kramers case 共␭+ = V⬙u / ␨兲, in the overdamped regime. Note that existence of the metastable equilibrium and hence the escape requires V⬙u ⱖ k / N. Thus ␭+ is nonzero only for V⬙u ⱖ k / N. The parameters used for these plots are N = 30, M = 1 , k = 5, bead radius a = 0.1 and a1 = 15 and a2 = 10. 共c兲 shows how the stretch increases the exponential factor of the escape rate, which is proportional to ␭+ exp共−EB / kBT兲, by decreasing the effective barrier height. For the chosen range of stretch 共D兲 here, barrier heights range between 2 – 5kBT. 共d兲 shows the net escape rate versus D, as a competition between ␭+ and exp共−EB / kBT兲, which decreases and increases, respectively, with D. The exponential factor dominantes the competition and somewhat offsets the difference in ␭+ between HI and no-HI cases at small stretch 共D兲. Finally, hydrodynamic interaction enhances the escape rate, but by a small amount. Note that 共a兲 and 共c兲 are drawn using analytic expressions. −1 ␶+−1 ⬍ ␶−1 1 ⬍ ␶2 ⬍ ¯. As we find later, in the overdamped regime, this hierarchy of time-scales qualitatively does not change even with the addition of HI.

f(x) x1 x+

0

tanh(xN)

π/Ν

x* π/2Ν

f(x)

x

x2

tan(xN)

FIG. 4. 共Color online兲 Graphical solutions of transcendental Eq. 共6兲 共with ␣ replaced by x兲 and its counterpart for the stable modes. f共x兲, which has two branches, denotes the rhs of the equations, while tanh共xN兲 and tan共xN兲 are the lhs of the respective equations. The empty and the filled circles indicate the intersection points x+ and x1 , x2 , . . ., which are the unstable and stable solutions of the EOM.

Now we consider the effect of HI on the escape process. This requires us to solve Eq. 共1兲 共without the random force兲 to find its unstable mode. In the spirit of Zimm model we use the preaveraging approximation, i.e., substitute the 1 / r term in Eq. 共2兲 by 具1 / r典 which linearizes Eq. 共3兲. But unlike in the equilibrium case 关15兴, here 具1 / r典 has to be computed by averaging with respect to the nonequilibrium steady state distribution Pneq of the escaping Rouse polymer. Following the Kramers ansatz 共see Appendix兲, Pneq共⌿兲 = Peq共⌿兲␾共⌿兲 ˙ ; j = 1 , N兴 is the vector representing the where ⌿ ⬅ 关Q j , Q j state of the system in phase space, Peq共⌿兲 is the equilibrium distribution and ␾共⌿兲 is the nonequilibrium correction factor. As detailed in the Appendix, the problem boils down to computation of ␾共⌿兲. But the function ␾共⌿兲 is not available a priori; rather it depends on the unstable eigenmode 兵Ui其 and the corresponding positive eigenvalue ␭+ of the EOM 共see Appendix兲. This problem can be tackled numerically using an iterative scheme leading to a self-consistent solution for ␾共⌿兲 and ␭+. But this involves calculation of 具1 / r典neq which turns out to be a numerically daunting 2N-dimensional integral over the phase space 共even in 1D兲 and therefore cannot be done for reasonably large N. To make headway we approximate 具1 / r典neq by its equilibrium average. Following Ref. 关18兴, 具1 / r典eqlb = 兩f兩1 erf关 2兩冑f兩G 兴, where f = A共m − n兲 and G共m , n兲 2 k BT = 2k 共兩m − n兩 − 共m−n兲 N 兲. This expression was derived for a closely related problem: fluctuation of a partially stretched Rouse chain fixed at its both ends. Our present problem is slightly different, namely, one end of the chain is fixed and the other end is held by a leaky potential trap. The main feature of 具1 / r典eqlb is that it scales as the inverse square root of the interbead separation 共兩Rm − Rn兩−1/2兲 for small 兩m − n兩 while for distant beads the scaling is 兩Rm − Rn兩−1. Further, we restrict ourselves to 1D. Since the polymer here is stretched mainly in the z direction, the transverse component of 兩Rm − Rn兩 in the x-y plane will be much smaller than the longitudinal component along z. Thus, reduction to 1D may 关19兴 not take away much of the qualitative content of the hydrodynamic effect on bond rupture. Figure 3 shows our results for ␭+, as well as the escape rate, and shows that hydrodynamic interaction in general enhances the escape rate. As expected, higher stretch 共via higher D兲 results in higher escape rate 共mainly due to the lowering of barrier兲. Note that factors other than ␭+ in the rate formula depend on the statics and not dynamics. The effect of HI on the dynamics, i.e., on ␭+, should diminish at higher stretch. This is because at higher stretch the interbead separation increases and matrix elements of the H tensor falls off as 兩Rm − Rn兩−1. Consistently, in Fig. 3, at high D 共i.e., low Vu⬙兲, ␭+ for the HI and no-HI cases converge. Figure 5 shows how the displacement and momentum profiles of the unstable mode change due to HI and make the motion more coherent. V. DISCUSSION

Thus we have computed the enhanced breakage rate of a polymeric tether in the presence of HI, which qualitatively

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FIG. 5. 共Color online兲 Semilog plot of the unstable eigenmode 共Uq , U p兲 versus the bead index 共j兲, when the bond is close to rupture. This is obtained by solving the eigen value problem described in Sec. IV. The two cases, when HI is present and when it is absent, are shown by filled and open symbols, respectively. The displacement part 共Uq兲 and the momentum part 共U p兲 of the 2N dimensional eigenmode 共Uq , U p兲 are shown separately. The no-HI case 共open symbols兲 show exponential behavior as expected for large bead indices 共j兲, since sinh共␣u j兲 is the exact solution. With HI the profiles are qualitatively different and cannot be fitted by one exponential. In the no-HI case the exponential decay of the displacement and momentum of the beads away from the escaping end physically means that the distant beads do not participate in the escape process. On the other hand HI makes the collective mode more coherent, meaning the displacement and momentum of the beads are brought closer. For low N 共data not shown兲, this difference in the profiles are not easily distinguishable because few immediate neighbors of the escaping bead behave very similarly whether HI is present or not. For high N 共=30 in the figure兲 the eigenvector shows numerical instability 共random signs and nonmonotonicity兲 for small j-s, since the entries of the vector spans over large range of magnitudes. The parameters used here are N = 30, k = 5 , D / Nb0 = 1.48, a1 = 30, a2 = 40, ␩ = 1 , a = 0.1共␨ = 1.88兲 i.e., the moderately damped regime.

agrees with the BD simulation results reported in 关9兴. But the enhancement found by our theoretical calculation is much weaker than the enhancement seen in 关9兴. We believe this is due to three reasons: 共1兲 as explained in the introduction, in Fig. 2 the breaking bond is subjected to relatively stronger fluctuations than in Fig. 1. 共2兲 For the loop geometry, each bead has quite a few neighbors within a short distance, whereas for a stretched chain like ours, interbead distances are relatively large reducing the effect of HI, and 共3兲 our calculation suffers from preaveraging approximation. Nevertheless our results should be treated as the first theoretical attempt toward this many-body breakage phenomenon in the presence of HI. Bond breakage or escape rate is often interpreted in terms of the effective dynamic friction acting on the escaping particle. This is because in the single particle Kramers formula, in the overdamped limit, the rate decreases as friction increases. Even at the many-body level, i.e., for the polymer, this general trend is recovered 关6兴, although the value of the effective friction differs from the sum of the bare frictions acting on the individual monomers. In this context dynamic friction on macromolecules has been measured experimentally. But in this case 关12兴, the focus has been on equilibrium fluctuations which are in principle different from the dy-

namic friction faced by a macromolecule during nonequilibrium fluctuations. Also this experiment with stretched DNA had additional complications. There the friction had been indirectly measured using the phenomenological relation ␨ = k␶1, where k is the nonlinear elasticity and ␶1 is the relaxation time of the slowest quasinormal mode of DNA. The effect of elasticity has been found to be dominant over HI in such cases. On the other hand here, by choosing the Rouse model, we have avoided the complication of nonlinear elasticity and focused on the effect of HI only, albeit on a nonequilibrium phenomenon like bond breakage. The enhanced breakage rate 共or increase in ␭+兲 due to HI, can be interpreted as a reduction in the effective friction on the escaping bead, but not as an increase of the pulling force, because pulling force controls the effective barrier height which remains unchanged. Faster growth rate 共␭+兲 of the unstable collective mode also means that the bead moves out of the trap relatively fast due to HI. The difference between the HI and no-HI cases in the momentum and displacement profiles tells us that the immediate neighbors of the escaping bead behave similarly in both cases, but in the presence of long range HI the distant beads also participate in the escape process. Although this does not show up as a significant change of escape rate in our calculation, it results in marked coherence in the motion of the chain. ACKNOWLEDGMENT

D.P. and A.S. would like to thank Dr. Paolo Rios of EPFL Lausanne, Switzerland for useful comments. APPENDIX

Here we describe the general many-body Kramers formalism that allows us to compute the breakage rate in the moderately damped as well as overdamped regimes. The motion is described by the 2N dimensional state vector ⌿ ˙ ; j = 1 , N兴. Following 关16,17兴 the Fokker-Plank ⬅ 关Q j , Q j equation for the nonequilibrium probability density Pneq共⌿ , t兲, near the saddle point, at steady state 共⳵t Pneq共⌿ , t兲 = 0兲 can be written as

⳵

冋

⳵

兺ij ⳵ ␺i M ij 兺k − E jk共␺k − ␺uk 兲 + ␤−1 ⳵ ␺ j

册

Pneq共⌿兲 = 0. 共A1兲

Here M ij is the dynamical matrix and Eij is the potential energy matrix 共in harmonic approximation for the potential near the barrier兲. Both are 2N ⫻ 2N dimensional. Symboli−A共z兲 0 共z兲 cally, M = 共 01 −1 are ⌫ 兲, and E = 共 0 −m−1 兲, where m , ⌫ and A 共z兲 all N ⫻ N dimensional matrices. A is the z-sector of the potential energy matrix,

A共z兲 = k

冢

2

−1

0

¯

0

−1

2

−1

0

]

0

−1

2

−1

]

]

0

−1




−1

0

¯

0

−1 1−

V⬙ k

冣

,

m is the diagonal mass matrix with diagonal elements: 共M , . . . , M , M ⬘兲 and ⌫ is the diagonal dissipation matrix,

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DAS et al.

with elements: 共␨ , . . . ␨ , ␨⬘兲. We have kept the provision that the trapped bead may have a different effective mass and friction compared to the other beads in the polymer. In Sec. IV where hydrodynamic interaction is considered, the only change in the above analysis is to replace A by HA where H is the Oseen tensor. Now following the ansatz, as in Kramers’ escape problem, Pneq共⌿兲 = Peq共⌿兲␾共⌿兲 where Peq共⌿兲 = Z−1 exp关−␤E共⌿兲兴, and ␤ = 1 / KBT. The boundary conditions 共BC兲 on ␾ are: ␾共⌿兲 → 1 when ⌿ → ⌿s, the metastable well, and on the other side of the saddle point ⌿u, ␾共⌿兲 → 0. These enforce that away from the trap the escaping particle is eventually captured, whereas near the well the equilibrium distribution is attained. Now we briefly outline 共following 关16,17兴兲 how one solves for ␾共⌿兲, and using this ␾共⌿兲 how the stationary probability current out of the trap is calculated in the multidimensional phase space. An ansatz for ␾共⌿兲, satisfying the BC, is

␾共⌿兲 =

冑 冕 ␤ 2␲

⬁

u

exp关− ␤z2/2兴dz =

冑

共A3兲

ji

˜ = 共 0 −1 兲, and hence −M ˜ E = 共 共z兲0 −m−1 −1 兲. when M 1 −⌫ A −⌫m Ui are the components of the left eigenvector of the ma˜ E, corresponding to the unique positive eigenvalue trix −M ␭+, that describes the growth rate of a small deviation from the saddle point ⌿u. To prove this we will now show that ␭+ is the eigenvalue corresponding to the unstable solution of the linear equation of motion near the saddle point. For this, we first convert Eq. 共A3兲 to a right eigenvalue problem 共this is done just for convenience sake兲 ˜ E兲TU = ␭ U. − 共M +

共A4兲

Then we split the eigenvector U = 共Uq , Up兲, such that each of Uq and Up are N dimensional. Now we can construct U共t兲 = e␭+tU such that the above equation can be rewritten as ˜ E兲TU共t兲 = ⳵ U共t兲. − 共M t

共A5兲 共z兲

In terms of Uq and Up we get two equations: A Up = ⳵tUq and −m−1Uq − ⌫m−1Up = ⳵tUp. Applying an additional time derivative on the second equation and then eliminating Uq, we arrive at

冉冑 冊

␤ Erfc 2␲

˜ E =␭ U , − 兺 Ui M ij jk + k

␤ u 2

共A2兲

关m⳵2t + ⌫⳵t兴Up = − A共z兲Up ,

共A6兲

Here u = 兺iUi共␺i − ␺si 兲 is linear in deviations from the saddle point. In order to find 兵Ui其, we insert ␾共⌿兲 into Eq. 共A1兲 and find

which is same as Eq. 共4兲 with the random force set to zero and 兵Q j其 replaced by U p. Note that the boundary conditions of Eq. 共4兲 is contained in the first and the 共i = N兲th row of the matrix A共z兲.

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