Excluded Volume Effects on the Intrachain Reaction Kinetics - The

Apr 18, 2008 - ... chain relaxation times τ1 of excluded volume chains that are in excellent agreement with the available Brownian dynamics simulatio...
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J. Phys. Chem. B 2008, 112, 6250-6258

Excluded Volume Effects on the Intrachain Reaction Kinetics† Ji-Hyun Kim,‡ Woojin Lee,‡ Jaeyoung Sung,*,§ and Sangyoub Lee*,‡ Department of Chemistry, Seoul National UniVersity, Seoul 151-747, South Korea, and Department of Chemistry, Chung-Ang UniVersity, Seoul 156-756, South Korea ReceiVed: August 10, 2007; In Final Form: February 8, 2008

On the basis of the recently developed optimized Rouse-Zimm theory of chain polymers with excluded volume interactions, we calculate the long-time first-order rate constant k1 for end-to-end cyclization of linear chain polymers. We first find that the optimized Rouse-Zimm theory provides the longest chain relaxation times τ1 of excluded volume chains that are in excellent agreement with the available Brownian dynamics simulation results. In the free-draining limit, the cyclization rate is diffusion-controlled and k1 is inversely proportional to τ1, and the k1 values calculated using the Wilemski-Fixman rate theory are in good agreement with Brownian dynamics simulation results. However, when hydrodynamic interactions are included, noticeable deviations are found. The main sources of errors are fluctuating hydrodynamic interaction and correlation hole effects as well as the non-Markovian reaction dynamic effect. The physical natures of these factors are discussed, and estimates for the magnitudes of required corrections are given. When the corrections are included, the present theory allows the prediction of accurate k1 values for the cyclization of finite-length chains in good solvents as well as the correct scaling exponent in the long-chain limit.

1. Introduction Intrachain reactions occurring in a variety of polymer systems, including proteins and nucleic acids, have attracted much attention because the study of intrachain reactions provides valuable information on the conformational structure and dynamical behavior of polymer chains as well as on the reaction dynamics itself.1,2 In particular, fluorescence spectroscopic techniques have been widely used to measure the end-to-end cyclization rates of chain polymers, and the molecular weight dependence of the rate constant has been determined.3,4 A general theory for describing the diffusion-influenced kinetics of intrachain reactions was first advanced by Wilemski and Fixman.5 By utilizing the so-called closure approximation, they could derive analytic expressions for the reaction rate for several types of intrachain reactions. The accuracy of the Wilemski-Fixman (WF) theory has been assessed by several authors. Doi derived a variational principle for the asymptotic rate constant and showed that use of the simplest trial function reproduces the WF result.6 Battezzati and Perico7 and Weiss8 carried out systematic perturbation analyses to examine the validity of the WF theory. In particular, Weiss clearly showed that the WF theory can be reproduced by making a factorization approximation to the resulting multiple integrals in a specific way. More specific aspects of the intrachain reactions have been investigated also. Stampe and Sokolov investigated the effects of electrostatic interaction between the charged end groups on the cyclization rate.9 Dua and Cherayil considered the effect of backbone rigidity on the dynamics of chain closure.10 Hyeon and Thirumalai also considered the effect of chain stiffness on the kinetics of interior loop formation.11 They analytically †

Part of the “Attila Szabo Festschrift”. * Corresponding authors. E-mail: [email protected] (S.L.) and [email protected] (J.S.). ‡ Seoul National University. § Chung-Ang University.

calculated the interior distance distribution function based on a mean field approach and used it to obtain a Kramers’ type rate expression for interior looping that is valid when the looping time is much larger than the chain relaxation time. Bandyopadhyay and Ghosh utilized a non-Markovian reaction-diffusion equation to investigate the memory effect in the fluorescence resonance energy transfer.12 Sung et al. generalized the WF theory to treat the relaxation kinetics of reversible cyclization reactions.13 In the present work, we will consider the excluded volume (EV) effects on the intrachain reactions based on the recently developed optimized Rouse-Zimm (ORZ) theory of chain polymers.14 Excluded volume interactions between nonadjacent monomeric units on the chain backbone cannot be neglected except in polymer melts or in theta solutions. In ref 14, we showed that the ORZ theory enables the theoretical prediction of various frictional and dynamical properties of chain polymers such as intrinsic viscosity, translational diffusion coefficient, and intrinsic dynamic viscoelasticity within a unified framework, and the results are in good agreement with experiments. There have been essentially two approaches that deal with the EV effects on the intrachain reactions. The first one is the WF-type formulation that utilizes the Gaussian Green functions.5 Although Wilemski and Fixman used the boson representation method15 by which explicit consideration of excluded volume potential is possible, a systematic perturbative improvement beyond the Gaussian formulation is prohibitively complicated. Debnath and Cherayil16 combined the WF rate theory with the so-called generalized random walk model for the chain with nonlocal interactions to investigate their effects on the cyclization rate. In their model, the nonlocal effects are incorporated into a description of the chain that is effectively local and are characterized by a single scaling exponent depending on the solvent quality. Although their theory can describe the chain closure dynamics in poor, theta, and good solvents just by changing the scaling exponent, it neglects the effects of

10.1021/jp076426i CCC: $40.75 © 2008 American Chemical Society Published on Web 04/18/2008

Intrachain Reaction Kinetics

J. Phys. Chem. B, Vol. 112, No. 19, 2008 6251

hydrodynamic interactions and applies only to the limiting case of long chains. A second approach for dealing with the EV effect on the chain cyclization is that based on the renormalization group (RG) theory. Friedman and O’Shaughnessy calculated the long-time cyclization rates of chain polymers with and without hydrodynamic and EV interactions.17,18 For free-draining polymers, they found that the rate is diffusion-controlled and the product of the rate constant k1 and the longest chain relaxation time τ1 is a universal constant independent of the inherent reactivity and the size of the reacting ends. On the other hand, in good solvents where both hydrodynamic and EV interactions play important roles, they found that a “mass action” law holds asymptotically; that is, k1 is proportional to the equilibrium contact probability of chain ends. However, the RG approach also applies only to the limiting case of long chains. It cannot explain the dependence of the cyclization rate on the inherent reactivity and the size of the reacting ends as well as the crossover behavior from the diffusion-controlled to the mass action law regimes that are important for chains of finite contour length. Accuracies of the various theoretical approaches have been examined against computer simulation results.19-27 In particular, Rey and co-workers provided extensive simulation results for the dependence of k1 on the reacting group size, chain length, and hydrodynamic and EV interactions and evaluated the accuracies of the WF theory and the RG theory of Friedman and O’Shaughnessy.21,22 Experiments as well as computer simulations are usually carried out for the cyclization of chains that are not too long. For example, Lapidus et al. investigated the chain-length dependence of the end-to-end contact rates of relatively short polypeptide chains.28-30 Therefore, it is necessary to formulate a rate theory which describes the EV effects on the polymer chains of finite length. The major advantage of the present ORZ-WF theory is that it can describe various aspects of the cyclization kinetics of finite length chains, such as the dependence on the reacting group size, chain length, and hydrodynamic and EV interactions, within a unified framework. We compare our theoretical results calculated in the ORZWF framework with the simulation results of Rey and his coworkers.19-22 The calculated rate constants are found to be in good agreement with simulation results in the free-draining limit. However, when hydrodynamic interactions are included, noticeable deviations are found. The main sources of errors may be traced to fluctuating hydrodynamic interaction and correlation hole effects that are neglected in the WF type formulation utilizing the Gaussian Green function. Another source of error is the neglect of the non-Markovian reaction dynamic effect. We provide the estimates for the magnitudes of required corrections for these errors. 2. ORZ-WF Theory of Chain Cyclization Kinetics We adopt the optimized Rouse-Zimm model for chain polymers.14 A single polymer chain consists of N + 1 beads. Let ψ(rN+1,t) denote the probability density that the chain has not reacted by time t with the beads located at rN+1[ ≡ (r0, r1, ... , rN)]. ψ(rN+1,t) evolves in time according to the reaction kinetic equation given by

∂ ψ(rN+1,t) ) [D - K]ψ(rN+1,t) ∂t

(1)

where D denotes the many-body generalized Smoluchowski operator and K is the sink function that counts the change due to reaction between two end beads. The sink function is assumed

to have the form K(rN0) ) κRδ(RN0 - σ)/4πσ2, where RN0 (≡ |rN0| ) |rN - r0|) denotes the distance between the end beads and κR is a parameter representing the inherent reaction strength. From extensive Brownian dynamics (BD) simulation results, Rey and co-workers calculated the long-time first-order rates k1 for cyclization reaction of chain polymers of varying length with and without hydrodynamic and EV interactions in the limit κR f ∞.19-22 In the WF formalism,5 with ˆf(s) denoting the Laplace transform of f(t), k1is given by

k-1 ˆ 1(-k1) 1 )χ

(2)

Here, χˆ 1(s) is a sink-sink correlation function defined as31 -1 -1 χˆ 1(s) ) k-2 eq 〈K(s +L ) K〉 - s

(3)

The angular bracket denotes the average over the equilibrium distribution ψeq [that is, 〈‚‚‚〉 ≡ ∫ drN+1ψeq(rN+1)(‚‚‚)], and L is an operator defined by the relation, Dψeqφ ) -ψeq L φ. The term keq denotes the equilibrium rate constant: keq ) 〈K〉. In terms of the end-to-end distance Green function, G ˆ (r,s|r0) ≡ Peq(r0)-1〈δ(RN0 - r0)(s + L )-1δ(RN0 - r)〉, χ1(t) can be expressed as

χ1(t) ) [Peq(σ)]-1G(σ,t|σ) - 1

(4)

In the presence of EV interactions, G is not Gaussian and its explicit analytic expression is not available. To go further, we thus need some approximations. In the Rouse-Zimm theory,32,33 the evolution equation for ψ(rN+1,t) in the absence of reaction is approximated by

∂ψ

)

∂t D1

∑

γ)x,y,z

[( ) ( ) ( ) ∂

∂Xγ

T

‚H‚

∂ψ

+

∂Xγ

3

∂

T

b20 ∂Xγ

]

‚H‚A‚Xγψ (5)

Here, the superscript T denotes the transpose and D1 is the diffusion coefficient of a single bead. Xγ is the (N + 1) dimensional column vector defined by

() () ()

x0 x Xx ) 1 l xN

y0 y Xy ) 1 l yN

z0 z Xz ) 1 l zN

(6)

b0 is the Kuhn statistical length, that is, the root mean squared bond length in the unperturbed state. A is the (N + 1) × (N + 1) structure matrix that takes into account the topological connectivity of the chain and the equilibrium bond-bond correlations. H is the (N + 1) × (N + 1) dimensionless diffusion matrix whose off-diagonal terms are the pre-averaged Oseen tensors:

〈〉

(H)ij ) δij + (1 - δij)h*

b0 Rij

(7)

with Rij ) |rij| ) |ri - rj|. The reduced friction coefficient h* () ζ1/6πηsb0) measures the hydrodynamic interaction strength with ζ1 () kBT/D1) denoting the friction coefficient for a single bead and ηs being the solvent viscosity. In the original Zimm theory, the diffusion matrix H and the structure matrix A do not take into account the EV effects. In ref 14, we showed how one can incorporate the nonuniform

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Kim et al.

chain expansion effects due to EV interactions into the expressions for H and A by adopting the ORZ theory of Bixon and Zwanzig.34-36 We believe that the ORZ approach as detailed in ref 14 provides the best available theoretical framework for interpreting the experimental data on linear viscoelastic properties of perturbed polymers. Nevertheless, the chain distribution is Gaussian in the ORZ approximation. It is thus expected that the ORZ theory may not give accurate results for very long polymers. Nevertheless, Bixon and Zwanzig36 showed that the ORZ theory could provide reasonable estimates for the intrinsic viscosity of stiff polymers, of which the actual chain distributions are far from Gaussian. In the ORZ theory, the structure matrix is calculated as A ) aT‚U‚a, where a is the N × (N + 1) position-to-bond transformation matrix, (a)ij ) δij - δi-1,j (1 e i e N and 0 e j e N), and U is the N × N matrix whose inverse is constructed with equilibrium bond-bond correlation functions as (U-1)ij ) 〈li ‚ lj〉/b02 with li ) ri - ri-1 (1 ei eN). It was shown that 〈li ‚ lj〉 is related to the mean-square interbead distances as

〈li ‚ lj〉 )

{

2 〈Ri,i-1 〉 (i ) j) 2 2 2 2 〉 + 〈Ri-1,j 〉 - 〈Ri,j 〉 - 〈Ri-1,j-1 〉]/2 (i * j) [〈Ri,j-1

(8)

The EV effects are reflected in the expressions for 〈1/Rij〉 in eq 7 and 〈R2ij〉 in eq 8, whose explicit forms can be found in ref 14. With the ORZ approximation, the reduced Green function G(r,t|r0) has the form,

G(r,t|r0) )

{ [

( )

exp -

3 2 2π〈RN0 〉

1/2

1 1 × (1 - φ2)1/2 4πrr0φ

]}

] [

2 2 3 (r - φr0) 3 (r + φr0) exp 2 2 2〈RN0 〉 (1 - φ2) 2〈RN0 〉 (1 - φ2)

(9) Then the sink-sink correlation function is given

TABLE 1: Values of the Expansion Factor rR2 and EV Parameter z as a Function of Bead Numbers (N + 1) N+1

RR2

z

RS2(ORZ)a

RS2(MC)a

6 8 10 11 15 20 25 37 45 75 145

1.209 1.282 1.339 1.364 1.446 1.525 1.588 1.705 1.765 1.933 2.170

0.177 0.250 0.312 0.340 0.437 0.539 0.626 0.800 0.898 1.195 1.688

1.19 1.26

1.17 1.23

1.33 1.40 1.47 1.53 1.63

1.29 1.36 1.44 1.50 1.62

a Expansion factors of radius of gyration RS2(ORZ) calculated from the ORZ theory are also compared with the available MC simulation results RS2(MC) of ref 38.

the reacting group size parameter σ, chain length N, hydrodynamic interaction (HI), and EV effects. In order to evaluate the HI and EV effects, we need the HI parameter h* appearing in eq 7 and the EV parameter z that is required to calculate the equilibrium averages of the second-order interbead distances 〈R2ij〉 and the reciprocal interbead distances 〈R-1 ij 〉. Since we want to compare our theoretical results with the BD simulation results of Rey and co-workers,19-22 we use the parameter values corresponding to their simulations. They used the Osaki’s draining parameter h*Osaki, the value of which is 0.25.37 Since 1/2 * * 1/2 h* Osaki ) (3/π) h , we set the value of h to (π/3) 0.25 = 0.256. In most theories dealing with the EV effects on polymer properties, the EV variable z is regarded as an adjustable parameter that gives the best fit to the experimental expansion factors. Rey et al. calculated the mean squared end-to-end 2 distance 〈RN0 〉 for EV chains with varying numbers of beads.38 We thus evaluate the EV variable z for a chain with given length N from the Barrett’s expression for the expansion factor of 2 〉,39 〈RN0 2 2 R2R ) 〈RN0 〉/〈RN0 〉without EV )

by23

2 〉/(Nb20) ) [1 + 20z/3 + 12.6z2]0.2 (12) 〈RN0

χ1(t) ) exp[-2x0φ2/(1 - φ2)] sinh[2x0φ/(1 - φ2)] 2x0φ(1 - φ )

2 1/2

- 1 (10)

2 where x0 ) 3σ2/2〈RN0 〉. In eqs 9 and 10, φ is the normalized time correlation function of the end-to-end vector,

φ(t) )

b20

N

-t/τ ∑ c2kµ-1 k e

2 〈RN0 〉 k)1

k

(11)

where ck ) QNk - Q0k with Q denoting the matrix whose columns are eigenvectors of the matrix H‚A. That is, H‚A is diagonalized by the similarity transformation Q-1‚H‚A‚Q. The eigenvalue λk ) [Q-1‚H‚A‚Q]kk is related to the kth mode (dielectric) relaxation time as τk ) b20/3D1λk. The same Q can also diagonalize H and A separately, whose eigenvalues are denoted as νk ) [Q-1‚H‚Q-1T]kk and µk ) [QT‚A‚Q]kk, respectively. 3. Results and Discussion Determination of ORZ Parameters. By using the ORZWF theory, we calculate the long-time cyclization rate constant k1 for linear chain polymers and investigate its dependence on

Table 1 gives values of R2R and z for chains with varying numbers of beads. For the chains with N ) 10, 45, 75, and 2 145, there were no reported values of 〈RN0 〉, but Rey et al. 2 〉 versus N is showed that the double logarithmic plot of 〈RN0 2 2 linear and can be fitted to a formula, 〈RN0〉/b0 ) 0.9132N1.1742, which is in agreement with the known value for the Flory exponent (ν ) 0.588). With this fitting formula, we have 2 estimated 〈RN0 〉 for chains with N ) 10, 45, 75, and 145. For a short chain with N ) 10, this may not give a reliable estimate 2 2 for 〈RN0 〉. However, Rapaport reported that at least for 〈RN0 〉 40 the large N-limit behavior is reached for N > 10. In Table 1, the expansion factors of the gyration radius R2S calculated from the ORZ theory14 with the z values obtained as above are also given and compared with the available Monte Carlo results.38 The good agreements confirm the accuracy of the ORZ theory in predicting the structural properties of EV chains. Longest Chain Relaxation Time. In the free-draining limit, the end-to-end cyclization rate is diffusion-controlled and the long-time rate is inversely proportional to the longest chain relaxation time τ1.17,41 Figure 1 shows that the ORZ theory provides τ1 values in excellent agreement with the available BD simulation results22 not only in the free-draining limit

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J. Phys. Chem. B, Vol. 112, No. 19, 2008 6253

Figure 1. Double logarithmic plot of the longest relaxation time τ1 of excluded volume chains as a function of chain length: (a) the freedraining case and (b) non-free-draining case.

Figure 2. Double logarithmic plot of the long-time cyclization rate k1 of excluded volume chains as a function of chain length in the freedraining case.

(Figure 1a) but also in the non-free-draining limit (Figure 1b). We take b0 and t1 (≡ b20/D1) as the units of length and time, respectively. The simulation results for τ1 were obtained by exponential fitting of the long time decay curve of the time correlation function φ(t) of the end-to-end vector.21,22 While fluctuating HI (FHI) is included in the simulations, the ORZ theory takes account of HI in the preaveraged form. However, it seems that FHI does not have a significant influence on τ1 values at least for the relatively short chains under consideration.42 The power-law exponent ντ1 defined by the relation τ1 ∼ (N + 1)ντ1 is calculated to be 2.18 and 1.74, in the absence and in the presence of HI, respectively. These values are in good agreement with the dynamical scaling predictions for ντ1, which are given by 2ν + 1 and 3ν in the respective cases with the Flory exponent ν ) 0.588.43 Cyclization Rate in the Free-Draining Case. With h* ) 0 and the EV parameter z determined as above-described, we can calculate the long-time cyclization rate constants k1 of EV chains in the free-draining limit from eqs 2, 10, and 11. The results are drawn as the solid line in Figure 2. In both cases, with σ ) 0.5 and 1.0, the ORZ-WF results are rather in good agreement with the simulation data represented by filled circles for chains with N + 1 ) 10, 25, 45, and 75. However, this agreements are due to some cancellation of errors. In the ORZ theory, nonuniform chain expansion due to EV interactions is taken into account, but the end-to-end distance distribution function Peq(RN0) is essentially Gaussian. However, Peq(RN0) for actual EV chains has a dip at short distance, which is called the correlation hole. There were many theoretical and simulation studies about the proper form of Peq(RN0).44 A widely accepted form for Peq(RN0) is the scaling function proposed by des Cloizeaux,45 κ δ PRG eq (RN0) ∼ ξ exp[-Cξ ]

with

2 〉)1/2RN0 (13) ξ ) (3/2〈RN0

The values of κ, δ, and C reported in ref 38 are κ ) 5/18, δ ) 2.5, and C ) 0.7366. A physical consequence is that there is an effective repulsion when the end groups get closer. This

retards the reaction rate. On the other hand, the WF theory does not take into account the non-Markovian reaction dynamic effect that increases the reaction rate when included. Hence, the errors in the ORZ-WF theory due to the neglect of the correlation hole and non-Markovian reaction dynamic effects tend to cancel each other. The rate retardation effect of the correlation hole is expected to be larger when the reaction radius σ is smaller, because the reacting groups must get closer against the effective repulsive force to undergo reaction. On the contrary, the non-Markovian reaction dynamic effect gets larger for larger σ and longer chains. Therefore, the ORZ-WF theory overestimates the cyclization rate when σ ) 0.5 (Figure 2a), while it underestimates the rate when σ ) 1.0 and N is large (Figure 2b). Later, we will reconsider these effects in more details. On the basis of the renormalization group (RG) theory, Friedman and O’Shaughnessy predicted that for free-draining polymers the product of the rate constant k1 and the longest chain relaxation time τ1 is a universal constant independent of the reaction radius σ.17,18 The first-order RG prediction for EV chains is k1τ1 ) 8/π3. The RG theory does not provide τ1 values, but the ORZ theory does. In Figure 2, the ORZ-RG predictions for k1 values are also presented as dashed lines. The results are in excellent agreement with BD simulations when σ ) 0.5 but deviate much when σ ) 1.0. The RG theory applies to the longchain limit where the magnitude of reaction radius is immaterial but cannot describe properly the cyclization kinetics of finite length chains. According to the RG theory, the power-law exponent νk1 in the relation k1 ∼ (N + 1)-νk1 is predicted to be 2ν + 1 = 2.18 for free-draining EV chains, which is in accord with the dynamical scaling relation for τ1; that is, τ1 ∼ N 2.18. The νk1 values obtained from BD simulations are 2.10 for σ ) 0.5 and 2.19 for σ ) 1.0,22 while the ORZ-WF values for νk1 are 2.22 and 2.38 for the respective cases. For the case with larger σ, the effects of finite chain length are more appreciable. Yeung and Friedman41 investigated the cyclization kinetics of a Rouse chain by BD simulation methods. From the plot of k1τ1 versus σ for Rouse chains of varying lengths, they showed the existence of the fixed point. At the fixed point with σ/b0 ) 0.75 (0.05, the value of k1τ1 is given by 0.57 (0.02, independent of the chain length. Off the fixed point, k1τ1 increases monotonically with σ for a given N. For a given σ smaller (larger) than the fixed point value, k1τ1 increases (decreases) monotonically with N. The first-order RG prediction17 for the fixed point value of k1τ1 was 0.516. Earlier, on the basis of the WF theory, Doi6 predicted the fixed point value of k1τ1 to be 0.46. Since BD simulations of EV chains are extremely timeconsuming, there has been no such study on the cyclization of EV chains. Hence, it must be worthwhile to present the corresponding ORZ-WF prediction. Figure 3 displays the variation of k1τ1 with σ for EV chains of varying lengths, which shows that k1τ1 ) 0.32 at the fixed point with σ/b0 ) 0.51. Cyclization Rate in the Presence of HI. In fact, this is the more realistic case since both EV and HI effects are important in good solvents. There are two opposite aspects in the effects of hydrodynamic interactions on the cyclization rate. HI retards the relative approach of the reacting ends at short distances. On the other hand, the HI makes the conformational search of the intervening beads less compact, which speeds up the diffusive approach of end beads at larger distances. One can thus expect that HI retards the cyclization rate for short chains but has the opposite effect for longer chains.13 According to

6254 J. Phys. Chem. B, Vol. 112, No. 19, 2008

Kim et al. Correlation Hole Effects. For EV chains, the end-to-end distance distribution function Peq(RN0) is not Gaussian as assumed in the ORZ-WF theory but is given by eq 13. We thus need to take account of the effective repulsive interaction between the two end beads at short distances. Unfortunately, the required correction cannot be estimated within the ORZWF theory because no explicit expression for χ1(t) is available. An alternative approach is to use the MFPT theory proposed by Szabo, Schulten, and Schulten (SSS),46 since we know k1 = 1/τMFPT in good solvents. With PRG eq (RN0) in eq 13, the SSS theory gives the following expression for τMFPT:

Figure 3. Double logarithmic plot of k1τ1 versus σ, calculated with ORZ-WF theory for excluded volume chains of varying length in the free-draining case.

τPHI-RG (eq) ) SSS 2 〈RN0 〉

∫x∞ dx x-(3+κ)/2 eB(x) Γ(A,B(x))2 0

A

3δC Deff

Γ(A,B(x0))

(14)

2 where A ) (3 + κ)/δ, B(x) ) Cxδ/2, x0 ) 3σ2/2〈RN0 〉, and Γ(a,z) is the incomplete gamma function. Deff is the effective relative diffusion coefficient that is assumed to be given by

Deff ) 2D1[1 - h*〈b0/RN0〉] Figure 4. Double logarithmic plot of the long-time cyclization rate k1 of excluded volume chains as a function of chain length in the nonfree-draining case.

the BD simulation data of Rey and co-workers, this crossover effect of HI seems to occur at N ∼ 45 although statistical errors blur the picture. In the ORZ-WF theory, only preaveraged HI (PHI) effects are taken into account, with h* ) 0.256 in eq 7. It thus takes account of the long-range effect of HI that speeds up the reaction rate but significantly underestimates the rate retardation effect of the fluctuating HI (FHI) at short distances. Another major source of error in the ORZ-WF theory is that it does not take account of rate retardation due to the presence of the correlation hole, as mentioned already. Indeed, Figure 4 shows that the ORZ-WF theory overestimates the k1 values for all cases considered, and the errors are larger when σ is smaller, as expected. Before proceeding to describe how to estimate the magnitudes of required corrections for these errors, an important feature of the cyclization kinetics in good solvents needs to be mentioned. As predicted by the RG theory,17 the cyclization reaction in good solvents follows the exponential kinetics in the long-chain limit. In the framework of WF rate theory, this behavior can be understood as follows. Equation 2 can be rewritten as (k1τ1)-1 ) χˆ *1 (-k1τ1) by rescaling the time by τ1. As mentioned above, τ1 ∼ (N + 1)1.76 in the presence of HI. On the other hand, from the BD results depicted in Figure 4, k1 ∼ (N + 1)-νk1 with νk1 ) 2.00 and 2.23 for σ ) 0.5 and 1.0, respectively,22 while the corresponding ORZ-WF values for νk1 are 1.97 and 2.07. Hence, it is expected that k1τ1 f 0 in the long-chain limit, so that (k1τ1)-1 = χˆ *1(0) ) (τMFPT/τ1) with τMFPT denoting the mean first passage time (MFPT).46,47 The relation k1 ) 1/τMFPT means that the long-time rate constant is equal to average rate constant, which is possible only if the survival probability decays exponentially over the whole time range. Indeed, Figure 4 shows that within the ORZ-WF theory 1/τMFPT represented by the short-dashed curve almost coincides with the k1 represented by the solid curve. In passing, it should be noted that a larger value of k1τ1 indicates more dispersive nonexponential kinetics.41

(15)

where 〈b0/RN0〉 is given by14

〈 〉

b0 ) RN0

x πN6 [1 - 10(π - 4)z(1 + 2.1z)]

-0.1

(16)

Now the meaning of the notation τPHI-RG (eq) must be clear: it SSS is the MFPT (starting from equilibrium chain distribution) that is calculated by the SSS theory with the PHI approximation of eq 15 and the effective RG potential corresponding to PRG eq (RN0). A main drawback in the SSS theory is the lack of a systematic method for choosing Deff. The relative diffusion of the end beads is modulated by the motions of intervening beads, so that the relative diffusion coefficient depends on time:48,49 2 〉d 〈RN0 ln φ(t) D(t) ) 3 dt

(17)

The most widely used choice is Deff ) D(0), and with the ORZ approximation for the end-to-end vector time correlation function φ in eq 11, it is given by eq 15. Since D(t) decreases with the lower bound on the time, this choice for Deff makes τPHI-RG SSS exact MFPT. On the other hand, the MFPT, τORZ ˆ 1(0), WF (eq) ) χ calculated from the ORZ-WF theory is known to be the upper bound according to variational theories.6,49,50 At this point, it is of interest to investigate the initial distance dependence of the MFPT. Bicout and Szabo derived the following expression for the MFPT from an initial end-to-end distance r0 to the reaction radius σ:47

ˆ (σ,0|σ) - G ˆ (σ,0|r0)] τBS(r0) ) [Peq(σ)]-1[G

(18)

In ref 31, we showed that eq 18 can be obtained from a more rigorous expression, which takes account of the non-Markovian reaction dynamic effect, with WF-type approximation. Indeed, one has the relation, ∫ dr04πr02Peq(r0)τBS(r0) ) χˆ 1(0) ) τWF(eq). With the use of the approximation, Peq(RN0) = PORZ eq (RN0), eq 18 can be rewritten as ORZ τORZ BS (r0) ) τWF (eq) - ∆(r0)

(19)

Intrachain Reaction Kinetics

J. Phys. Chem. B, Vol. 112, No. 19, 2008 6255 Fluctuating Hydrodynamic Interaction Effects. As in the original Rouse-Zimm theory, the ORZ theory includes HI effects only in the preaveraged form. It is rather difficult to take account of the whole effects of FHI acting between every pair of beads. However, one may expect that FHI between the two reacting end beads is most important. The relative diffusion coefficient Deff between the end beads in eq 15 includes the preaveraged form of the Oseen or Rotne-Prager HI tensors. Hence, to consider the FHI effect, we may use the following expression for the distance-dependent longitudinal diffusion coefficient derived from Rotne-Prager HI tensor:

Figure 5. Initial distance dependence of the mean first passage times calculated by the ORZ-BS theory (eq 19) and the SSS theory (eq 21) for the case with N + 1 ) 75 and σ ) 0.5.

where ∆(r0) is given by

∆(r0) )

[

∫0∞ dt

exp[ - (1 + r˜20)x0φ2/(1 - φ2)] sinh[2x0r˜0φ/(1 - φ2)] 2x0r˜0φ(1 - φ2)1/2

-1

]

(20) with r˜0 ≡ r0/σ and the approximate φ in eq 11. On the other hand, the SSS theory with the approximation, Peq(RN0) = PORZ eq (RN0), gives PHI-ORZ (r0) ) τSSS

2 〈RN0 〉

y ∫ x 6Deff

0

0

dx x-3/2 exΓ(3/2,x)

(21)

2 〉 and Deff is given by eq 15. In Figure 5, where y0 ) 3r02/2〈RN0 we compare τMFPT(r0) calculated from eqs 19 and 21 for the case with N + 1 ) 75 and σ ) 0.5. As expected, the SSS theory underestimates the MFPT for large r0 but gives the same results for small r0 as the WF theory. Since the correlation hole has a short-range effect, it is expected that the SSS theory gives the required correction. However, the WF theory gives better estimates for τMFPT(eq) than the SSS theory in general, as expected from the argument given above and also shown explicitly by Pastor et al. from BD simulations of Rouse chain cyclization.23 One may thus expect that the following relation would give an estimate for the rate constant with the correction due to the presence of the correlation hole (see Appendix A for a more detailed argument):

k1 = [τMFPT(eq)]-1 = [τORZ WF (eq) + (eq) - τPHI-ORZ (eq)]-1 (22) τPHI-RG SSS SSS Here, τPHI-ORZ (eq) is the MFPT (starting from equilibrium SSS chain distribution) calculated by the SSS theory with the PHI approximation of eq 15 and the effective potential corresponding to PORZ eq (RN0): 2 〉 〈RN0 τPHI-ORZ (eq) ) SSS 6Deff

∫x∞ dx x-3/2 ex[Γ(3/2,x)]2 0

Γ(3/2,x0)

(23)

k1values obtained from eq 22 are plotted as dot-dashed curves in Figure 4. The corrected results are clearly in better agreement with BD simulations but still deviate a little. This deviation appears primarily because of FHI, and we will discuss below how to estimate the required correction associated with FHI.

[

Deff(r) ) 2D1 1 -

( ) ( )]

b0 3h* b0 + h*3 2 r r

3

(24)

In writing eq 24, it has been noted that the hydrodynamic radius of a bead is given by h*b0. With the relative diffusion coefficient in eq 24 and PRG eq (RN0) in eq 13, the SSS theory gives the following expression for MFPT:

(eq) ) τFHI-RG SSS 2 〉 〈RN0

∞ dx ∫ A x 6δC D Γ(A,B(x )) 0

1

0

x-(3+κ)/2 eB(x)Γ(A,B(x))2 (25) 3 1 - Hx-1/2 + H3x-3/2 2

2 〉)1/2. Following the same argument where H ) h*(3b02/2〈RN0 that leads to eq 22 (Appendix A), we then expect that [τFHI-RG (eq) - τPHI-ORZ (eq)] would give the correction for the SSS SSS combined effects of correlation hole and FHI, and the better estimate for the rate constant is given by

k1 = [τMFPT(eq)]-1 = [τORZ WF (eq) + (eq) - τPHI-ORZ (eq)]-1 (26) τFHI-RG SSS SSS k1values obtained from eq 26 are plotted as long-dashed curves in Figure 4. The results are in much better agreement with BD simulations. It is of interest to examine the characteristics of the cyclization 2 〉) f 0. It kinetics in the long-chain limit with x0 () 3σ2/2〈RN0 is shown in the Appendix B that of the three terms contributing to the MFPT in eq 26 τFHI-RG (eq) dominates the other terms in SSS PHI-ORZ (eq)] the long-chain limit. That is, we have [τORZ WF (eq) - τSSS ORZ FHI-RG 3ν ν(3 (eq) ∼ N + κ). Hence, ∼ τWF (eq) ∼ N ln N while τSSS

(eq)]-1 ∼ N-ν(3+κ) k1∼[τFHI-RG SSS

(27)

This is in agreement with the RG prediction of Friedman and O’Shaughnessy.17 Physically, this means that the long-chain cyclization kinetics in good solvents is activation-controlled rather than diffusion-controlled. When the reaction radius is buried deep inside the correlation hole, the chain needs to climb up the potential wall to undergo reaction, which controls the overall reaction rate. Non-Markovian Reaction Dynamic Effects. There are two aspects of the non-Markovianity in the present problem. The first is the non-Markovian relative diffusion between end beads, which may be characterized by the generalized Smoluchowski equation of the type given in eq A1. The ORZ-WF theory describes this aspect of the problem in an appropriate manner through the end-to-end vector time correlation function φ(t) calculated according to eq 11.

6256 J. Phys. Chem. B, Vol. 112, No. 19, 2008

Kim et al.

The second is a more subtle one. In a recent work, it was shown that a fundamental assumption required for the validity of the WF theory is that the visit to the reaction surface at r ) σ in the absence of reaction is a stationary Markovian stochastic process, so that the joint probability density of multiple visits to σ at times (τ0, τ1, ..., τn) can be factorized as31

[∏

]

n

P(σ,τn;‚‚‚;σ,τ1;σ,τ0) )

G(σ,τm - τm-1|σ) P(σ,τ0) (28)

m)1

where G(σ,τm - τm-1|σ) denotes the conditional probability density that r(τm) ) σ given that r(τm-1) ) σ. This requirement of the reaction dynamic Markovianity fails badly for larger σ and longer chains. In ref 31, we formulated a rate theory that deals with the reaction dynamic non-Markovian effects. According to the theory, when the inherent reactivity is very large (κR f ∞), the survival probability and the MFPT are given by

Sˆ (s) ) [s + Ω ˆ ∞(s)-1]-1

τMFPT ) Ω ˆ ∞(0)

(29)

Hence, eq 2 for the long-time rate constant must be replaced by

ˆ ∞(-k1) k-1 1 )Ω

(30)

Although an exact formal expression for Ω ˆ ∞(s) was given in ref 31, its evaluation is a very formidable problem. Usually, the evaluation of τMFPT is an easier problem. If τMFPT and an expression for χ1(t) are known, k1can be evaluated as follows. Since the Markovian (WF) approximation for Ω∞(t) is χ1(t), we write

Ω∞(t) ≡ χ1(t)f(t)

(31)

with f(t) denoting the gauging function for the non-Makovianity. A useful approximation for f(t) was found to be31

f(t) = [1 + (D1t/b20)2]-R

(32)

where the exponent R is determined from the relation, τMFPT ) ∫∞0 dt χ1(t) f(t). Still, it is not possible to estimate the magnitude of the required correction for non-Markovianity, because we do not have an explicit expression for χ1(t) for EV chains with the FHI and correlation hole effects included. Hence, we will be content with ourselves by just giving an estimate of the nonMarkovian correction in the cyclization of the simple Rouse chain without HI and EV. Pastor et al.23 provided extensive BD simulation results for the Rouse chain cyclization. Their BD simulation values for τMFPT can be taken as the exact ones with the non-Markovian reaction dynamic effect included. For Rouse chains, exact expression for χ1(t) is also known, so that we can determine the parameter R of the non-Markovian dynamical gauging function f(t) in eq 32. We thus have an approximate expression for Ω∞(t), and the long-time rate constant k1 can be calculated from eq 30. By comparing this k1 value with the WF estimate obtained from eq 2, we can get the correction due to non-Markovianity. For chains with N ) 100, the WF estimates for k1τ1 are 0.411 and 0.515 for σ ) 0.5 and 1.0, respectively. The improved estimates with the non-Markovian correction included are 0.471 and 0.653 in the respective case. Hence, the required corrections due to non-Markovian reaction dynamics are 15% for σ ) 0.5 and 27% for σ ) 1.0. As expected, the non-Markovianity is enhanced for larger σ. Another interesting

point to note is that the improved estimates for the k1τ1 values appear to be near exact. According to the BD simulation study of Yeung and Friedman,41 the fixed-point σ value for Rouse chains is 0.75. If we take an average of the k1τ1 values obtained for σ ) 0.5 and 1.0, we get 0.56 that is in excellent agreement with the exact fixed-point k1τ1 value of 0.57. This confirms the usefulness of the non-Markovian dynamical gauging function. However, we expect that the non-Markovian reaction dynamic effect should be much smaller for chain cyclizations in good solvent. The WF rate theory is valid if the end-to-end distance distribution of the chain remains in local equilibrium over the spherical reaction surface at r ) σ throughout the whole time.23,51 As described above, the cyclization kinetics of EV chains in the presence of HI is less influenced by diffusion, which tells us that the chain distribution deviates little from equilibrium during the course of reaction. Therefore, the closure approximation made in the WF theory is expected to be not so bad for EV chain cyclization in good solvents. 4. Conclusion We have shown that the ORZ theory presented in ref 14 can be successfully applied to investigate the cyclization kinetics of EV chains. Once the EV parameter z is determined from the static structural data of the chains, the cyclization rates are calculated using the WF theory without any adjustable parameters. First, the ORZ theory predicts the longest relaxation time τ1 with excellent accuracy. In the free-draining case, the ORZWF theory gives the long-time rate constant k1 that is in good agreement with BD simulations. The ORZ-WF theory also predicts that the fixed-point values of k1τ1 and σ are 0.32 and 0.51b0, while the first-order RG prediction of the fixed-point k1τ1 value is 0.258. To check the actual values, extensive BD simulations would be required. In the non-free-draining case, the ORZ-WF estimates of k1 deviate significantly from BD simulation results. The main sources of errors are FHI and correlation hole effects. We have combined the WF and SSS theories to get the required corrections. Equation 26, which is the central result obtained for the cyclization in good solvents, has been found to give k1 values in very good agreement with BD simulations. In the long-chain limit, eq 26 predicts the same scaling behavior of k1 as the RG theory. The applicability of most previous theories dealing with the EV effects on chain cyclization is practically confined to the long-chain limit. For the chains of finite length, for example, the “mass-action-law” regime predicted by the RG theory of Friedman and O’Shaughnessy17 may not be observed. Since most experiments and computer simulations are carried out for relatively short polymer chains,1-4,52 the present ORZ-WF theory with the corrections due to FHI and correlation hole effects, as given in eq 26, is expected to be very useful in analyzing the kinetic results. However, in order to analyze the experimental results on the short polypeptide chains,28-30 the effects of chain-stiffness must also be taken into account. Work is in progress to include both chain-stiffness and EV effects within the unified theoretical framework. Acknowledgment. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2005-070-C00065). Appendix A: Derivation of eq 22 In this Appendix, we describe the physical basis of eq 22. In PHI-ORZ Figure 5, one can first see that τORZ (r0), which BS (r0) g τSSS is consistent with the known variational bounds on the MF-

Intrachain Reaction Kinetics

J. Phys. Chem. B, Vol. 112, No. 19, 2008 6257

PTs.49,50 τORZ BS (r0) fully takes into account the retardation effect (r0) uses the of D(t) on the first passage process, while τPHI-ORZ SSS short-time approximation for D(t) as mentioned in the paragraph following eq 17. Hence, τPHI-ORZ (r0) severely underestimates SSS the MFPT for large r0 but becomes coincident with τORZ BS (r0) as r0 gets closer to σ. Since the SSS theory gives the correct estimate of the initialdistance-dependent MFPT for small r0 and the correlation hole is rather short-ranged, its effect can be approximately taken into PHI-RG account as τORZ (r0) - τPHI-ORZ (r0)]. BS (r0) + [τSSS SSS To obtain eq 22, we then have to average [τORZ BS (r0) (r0) + τPHI-RG (r0)] over the initial equilibrium distriτPHI-ORZ SSS SSS ORZ PHI-ORZ (r0)] almost bution PRG eq (r0). For small r0, [τBS (r0) - τSSS vanishes, so that it does not produce any significant error to average the difference over the approximate distribution PORZ eq (r0) rather than over the exact PRG eq (r0). For intermediate to large r0, the difference becomes large but PRG eq (r0) is now similar to 2〉). Therefore, each term of [τORZ(r ) (r ) (with the same 〈R PORZ 0 0 eq BS (r0) + τPHI-RG (r0)] can be averaged over its own - τPHI-ORZ SSS SSS relevant distribution without significant errors, and eq 22 results. A further support for this approximate procedure comes from the excellent agreement of the results of eq 22 with the k1 values obtained from eqs 2 and 4 together with the Green function calculated numerically by solving the generalized Smoluchowski equation:

∂G(r,t|r0) 1 ∂ ∂ RG -1 ) D(t) 2 r2PRG Peq (r) G(r,t|r0) eq (r) ∂t ∂r ∂r r

PHI-ORZ In this appendix, we show how τORZ (eq), and WF (eq), τSSS in eq 26 depend on N, respectively, in the longchain limit. Let us first consider τORZ WF (eq), whose scaling behavior can be derived following a similar analysis of Doi in ref 6. We need a large N expression for 1 - φ(t). When N . 1, µk and τk appearing in eq 11 for φ(t) are well approximated as µk ≈ µ1k1+2ν and τk ≈ τ1k-zdν, respectively, where the dynamical exponent zd () ντ1/ν) is equal to 2 + ν-1 for the free-draining case and 3 for the non-free-draining case. Compared to the eigenvalues of the diffusion and structure matrices, the Rouse eigenmodes are perturbed to a less extent by HI and EV. We may thus approximate ck as ck2 ≈ 8/N for odd k and 2 〉 may be approximated as ck2 ≈ 0 for even k. Similarly, 〈RN0 2 2 〈RN0〉 ) (8b0/Nµ1) ∑oddk k-1-2ν. By putting these results into eq 11 and then replacing the discrete summation over odd k with an integral, which is justified when t , τ1, we obtain

(eq) τFHI-RG SSS

(1 - 2

[

() ]

π/4ν t 1 )ζ(1 + 2ν) Γ(2/zd) sin(2π/zd) τ1

2/zd

(t , τ1) (B1)

τ1 ln γ-2 ∼ N z d ν ln N 1.80

(B2)

with γ2 ) 2x0/3. Because of the weak logarithmic correction in eq B2, k1τ1f 0 in the largeN limit, which tells us that the longchain cyclization kinetics in the good solvent is not strictly diffusion-controlled even within the Gaussian formulation of the ORZ-WF theory. Let us then consider τPHI-ORZ (eq) given by eq 23. In eq 23, SSS the dominant contribution to the integral comes from small x range, so that ex[Γ(3/2,x)]2 in the integrand can be approximated as Γ(3/2)2. In the small x0 limit, we thus obtain

(eq) = τPHI-ORZ SSS

2 〈RN0 〉 1/2 -1/2 π x0 ∼ N3ν 6Deff

(B3)

For τFHI-RG (eq) in eq 25, the denominator of the integrand SSS looks problematic, but in the small x0 limit with large N and fixed σ, H is also a small parameter. In such a limit, eB(x)Γ(A,B(x))2 in the integrand can be approximated by Γ(A)2, and we obtain

(eq) = τFHI-RG SSS

Appendix B: Scaling Behaviors of the MFPTs in eq 26

-1-2ν

τORZ WF (eq) )

(A1)

This form of generalized Smoluchowski equation is rigorous only when the effective potential is harmonic or constant. Otherwise, it is expected to be phenomenologically valid when the potential deviates slightly from the harmonic potential or its gradient varies slowly. In addition, D(t) in eq A1 must be calculated from an expression generalizing eq 17 in consistence with the effective RG potential, which itself is a formidable task. Thus, in the calculations for verifying eq 22, we have just used eq 17 to calculate D(t) with φ(t) in the ORZ approximation as given in eq 11.

1 - φ(t) ≈

where ζ(z) is the Riemann zeta function and the squarebracketed factor is the result of the integration (1/2)∫ ∞0 dk k-1-2ν(1 - exp[-kzdνt/τ1]). Substituting eq B1 into the expression for χ1(t) in eq 10, we obtain the large N expression for ˆ 1(0) as τORZ WF (eq) ) χ

2 〈RN0 〉

2Γ(A) -(1+κ)/2 x0 ∼ Nν(3+κ) 3δC Deff(σ) 1 + κ A

(B4)

where Deff(σ) is given by eq 24. From the scaling behaviors (eq) indicated in eqs B2, B3, and B4, it is seen that τFHI-RG SSS dominates the other terms in eq 26 in the large N limit. References and Notes (1) Mita, I.; Horie, K. JMS-ReV. Macromol. Chem. Phys. 1987, C27, 91. (2) Eaton, W. A.; Mun˜oz, V.; Hagen, S. J.; Jas, G. S.; Lapidus, L. J.; Henry, E. R.; Hofrichter, J. Annu. ReV. Biophys. Biomol. Struct. 2000, 29, 327. (3) Winnik, M. A. Acc. Chem. Res. 1985, 18, 73. (4) Winnik, M. A. In Photophysical and Photochemical Tools in Polymer Science; Winnik, M. A., Ed.; Reidel: Dordrecht, The Netherlands, 1986. (5) Wilemski, G.; Fixman, M. J. Chem. Phys. 1974, 60, 866; 1974, 60, 878. (6) Doi, M. Chem. Phys. 1975, 9, 455; 1975, 11, 107. (7) Battezzati, M.; Perico, A. J. Chem. Phys. 1981, 74, 4527; 1981, 75, 886. (8) Weiss, G. H. J. Chem. Phys. 1984, 80, 2880. (9) Stampe, J.; Sokolov, I. M. J. Chem. Phys. 2001, 114, 5043. (10) Dua, A.; Cherayil, B. J. J. Chem. Phys. 2002, 116, 399. (11) Hyeon, C.; Thirumalai, D. J. Chem. Phys. 2006, 124, 104905. (12) Bandyopadhyay, T.; Ghosh, S. K. J. Chem. Phys. 2002, 116, 4366. (13) Sung, J.; Lee, J.; Lee, S. J. Chem. Phys. 2003, 118, 414. (14) Kim, J.-H.; Lee, S. J. Chem. Phys. 2004, 121, 12640. (15) Fixman, M. J. Chem. Phys. 1966, 45, 785; 1966, 45, 793. (16) Debnath, P.; Cherayil, B. J. J. Chem. Phys. 2004, 120, 2482. (17) Friedman, B.; O’Shaughnessy, B. Phys. ReV. A 1989, 40, 5950. (18) Friedman, B.; O’Shaughnessy, B. Macromolecules 1993, 26, 4888. (19) Ferna´ndez, J. L. G.; Rey, A.; Freire, J. J.; de Pie´rola, I. F. Macromolecules 1990, 23, 2057. (20) Rey, A.; Freire, J. J. Macromolecules 1991, 24, 4673. (21) Ortiz-Repiso, M.; Freire, J. J.; Rey, A. Macromolecules 1998, 31, 8356. (22) Ortiz-Repiso, M.; Rey, A. Macromolecules 1998, 31, 8363. (23) Pastor, R. W.; Zwanzig, R.; Szabo, A. J. Chem. Phys. 1996, 105, 3878. (24) Srinivas, G.; Yethiraj, A.; Bagchi, B. J. Chem. Phys. 2001, 114, 9170.

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