Generalized Langevin Equation as a Model for Barrier Crossing

Jan 3, 2019 - Langevin equation with a friction force that depends not only on the ...... dynamics simulation code and to Dave Thirumalai for helpful...
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Generalized Langevin Equation as a Model for Barrier Crossing Dynamics in Biomolecular Folding Published as part of The Journal of Physical Chemistry virtual special issue “Deciphering Molecular Complexity in Dynamics and Kinetics from the Single Molecule to the Single Cell Level”. Rohit Satija† and Dmitrii E. Makarov*,†,‡ Department of Chemistry and ‡Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas 78712, United States

J. Phys. Chem. B Downloaded from pubs.acs.org by IOWA STATE UNIV on 01/24/19. For personal use only.

†

ABSTRACT: Conformational memory in single-molecule dynamics has attracted recent attention and, in particular, has been invoked as a possible explanation of some of the intriguing properties of transition paths observed in single-molecule force spectroscopy (SMFS) studies. Here we study one candidate for a nonMarkovian model that can account for conformational memory, the generalized Langevin equation with a friction force that depends not only on the instantaneous velocity but also on the velocities in the past. The memory in this model is determined by a time-dependent friction memory kernel. We propose a method for extracting this kernel directly from an experimental signal and illustrate its feasibility by applying it to a generalized Rouse model of a SMFS experiment, where the memory kernel is known exactly. Using the same model, we further study how memory affects various statistical properties of transition paths observed in SMFS experiments and evaluate the performance of recent approximate analytical theories of non-Markovian dynamics of barrier crossing. We argue that the same type of analysis can be applied to recent single-molecule observations of transition paths in protein and DNA folding.

1. INTRODUCTION Single-molecule measurements of biomolecular folding and dynamics are often interpreted using the model where the dynamics of the reaction coordinate (usually equated to the experimental observable x) is one-dimensional diffusion subjected to a potential of mean force U(x).1−4 The latter is related to the equilibrium distribution peq(x) of the reaction coordinate U (x) = −kBT ln peq (x)

transition paths, i.e., short segments of the molecular trajectories spent crossing the folding free energy barrier. Specifically, it was proposed9,10 that, while eq 2 may be a good description of the folding and unfolding rates, various properties of transition paths such as their temporal duration11−22 (i.e., transition path time) and their average shape23−28 may be more sensitive to memory effects. Indeed, molecular simulations,29−38 theoretical considerations,39 and a few experimental studies40−42 indicate significant memory and anomalous diffusion effects in protein dynamics. Quantifying memory effects in single-molecule dynamics is generally a difficult task.43,44 Many distinct mathematical models result in anomalous diffusion,45−47 and which one provides the most accurate description of biomolecular dynamics remains an open question. Here, we explore the generalized Langevin equation (GLE)

(1)

and the equation of motion along this coordinate is the Langevin equation mx ̈ = −U ′(x) − γx ̇ + ζ(t )

(2)

where γ is a friction coefficient and ζ(t) is a Gaussian random noise with zero mean, which satisfies the fluctuation− dissipation theorem ⟨ζ(t )ζ(t ′)⟩ = 2γkBTδ(t − t ′)

mx ̈ = −U ′(x) −

(3)

t

ξ(t − t ′)x(̇ t ′) dt ′ + ζ(t )

(4)

as a candidate for a model of the dynamics of reaction coordinates in folding. Here ξ(t) is a friction memory kernel and ζ(t) is a Gaussian random noise, which has zero mean and which satisfies the fluctuation−dissipation relationship:

The friction coefficient is related to the diffusion coefficient D through the Einstein relationship, D = kBT/γ. Equation 2 describes a Markovian stochastic process (i.e., dynamics without memory). Biomolecules are, however, polymers, and the dynamics of monomers within polymers is known to be distinctly non-Markov.5−8 Such non-Markov effects have attracted recent attention in connection with experimental measurements of the properties of molecular © XXXX American Chemical Society

∫0

Received: November 16, 2018 Revised: January 3, 2019 Published: January 3, 2019 A

DOI: 10.1021/acs.jpcb.8b11137 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B ⟨ζ(t )ζ(t ′)⟩ = kBTξ(t − t ′)

(5)

0 = ⟨− U ′[x(t )]x(0)⟩ −

The GLE (eq 4) reduces to the ordinary Langevin equation (eq 2) when the memory kernel decays faster than other relevant dynamical time scales in the system, such that the kernel can be approximated by a delta function ξ(t ) = 2γδ(t )

ξ(t − t ′)

d⟨x(t ′)x(0)⟩ dt ′ dt ′ (8)

(6)

0 = Cfx(t ) −

∫0

t

̇ (t ′) dt ′ ξ(t − t ′)Cxx

(9)

or, in Laplace space, ξ (̂ s) =

Cfx̂ (s) ̂ (s) − Cxx(0) sCxx

(10)

where the hat denotes the transform (e.g., ∞ f ̂ (s) = ∫ f (t )e−st dt ). This gives the memory kernel in 0

Laplace space; numerically inverting the Laplace transform,58 the memory kernel in the time domain can be further recovered. Although eq 10 is exact (in the overdamped GLE case) for any potential U(x), it is instructive to consider the case of a 1 harmonic potential well, U (x) = 2 κx 2 . This potential is a good model when, for example, x(t) describes fluctuations of a molecule around its equilibrium structure or fluctuations of the intramolecular distance within an unfolded protein.59 In this case, eq 10 reduces to the known expression:57,60 ξ (̂ s) = κ

̂ (s ) Cxx ̂ (s ) Cxx(0) − sCxx

(11)

The limit s → 0 of this expression, if it exists, gives the Markov approximation to the GLE ̂ ̂ (0)/Cxx(0) = κCxx ̂ (0)/⟨x 2⟩ = κ ξ (0) = κCxx

∫0

∞

χ (t ) d t (12)

where the normalized autocorrelation function χ (t ) = ⟨x(t )x(0)⟩/⟨x 2⟩

(13)

was introduced. Indeed, the Markov approximation, i.e., replacement of eq 4 by eq 2, amounts to approximating the memory kernel by the delta function ξ(t ) ≈ 2δ(t )

∫0

∞

̂ δ(t ) = 2γδ(t ) ξ(t ) dt = 2ξ (0)

(14)

̂ so that γ = ξ(0) is the friction coefficient in the Markov limit. ̂ Note, however, that ξ(0) may be infinite, as it is in the case where the memory kernel is a power lawa Markov limit is never attained in this case. Consider now the limit s → ∞, which, in the time domain, corresponds to short-time dynamics. In particular, consider the short-time behavior of the mean square displacement

2. METHODS: ESTIMATING THE MEMORY KERNEL FROM DATA Our method of estimating the friction memory kernel from a trajectory x(t) is closely related to the approaches described by Daldrop et al.52 and by Debnath et al.57 We neglect the inertial effects and assume the overdamped limit of the GLE, eq 4, which amounts to setting its left-hand side to zero:

∫0

t

Introducing the position autocorrelation function Cxx(t) = ⟨x(t)x(0)⟩ and the force−position correlation function Cf x(t) = ⟨−U′[x(t)]x(0)⟩, eq 8 can be written as

Establishing whether the GLE is a viable (and, in particular, better than the ordinary Langevin equation) description of an experimental trajectory x(t) is a nontrivial task. The Markovianity test proposed recently48 allows one to assess the significance of memory effects but offers no way to estimate the parameters of the underlying non-Markov model. Because the probability of a particular realization x(t1), x(t2), x(t3), ... of a non-Markov process cannot be written as a product of pairwise propagators, a maximum likelihood type of approach49 for estimating the model parameters is not feasible in the case of a GLE. It is possible to estimate the memory kernel directly from simulations,50−53 but with the exception of recent work from the Netz group,52 such methods usually require modified simulations in which, e.g., the position x is constrained to a fixed value. They are, therefore, inapplicable to experimental signals. A further complication stems from the limited time resolution of the experimental data and/or the limited sampling rate of the experimental trajectories, which prohibits measuring quantities related to, for example, instantaneous velocities associated with reaction coordinates. In what follows, we describe a method for extracting the memory kernel directly from unbiased trajectories x(t), which only requires slowly varyingand thus experimentally accessiblequantities (such as the autocorrelation function of x). We illustrate this method using a toy model of a singlemolecule force spectroscopy experiment. This model, similar to the generalized Rouse model (GRM) proposed by the Thirumalai group,54 (i) captures the essential features of the free energy landscape traversed by a biomolecule in the process of its unfolding and refolding under mechanical stress, (ii) includes non-Markov effects caused by polymer dynamics, and (iii) can be described exactly by a GLE with a memory kernel that can be estimated analytically. By analyzing folding/ refolding trajectories obtained from this model, we further show how memory affects experimentally observable properties of transition paths (such as their shapes and transition path times) and test analytical approximations for the distributions of transition path times proposed recently.55,56 We envisage that a similar analysis can be applied to experimental trajectories obtained from single-molecule force spectroscopy data.

0 = −U ′(x) −

∫0

⟨Δx 2(t )⟩ = ⟨[x(t ) − x(0)]2 ⟩ = 2⟨x 2⟩[1 − χ (t )] 2k T = B [1 − χ (t )] κ

(15)

Using eq 11, we have

t

ξ(t − t ′)x(̇ t ′) dt ′ + ζ(t )

(7)

χ ̂ (s ) =

Multiplying the above equation by x(0) and averaging over the thermal noise, we obtain

ξ (̂ s) κ + sξ (̂ s)

≈

1 κ − + ... s ξ (̂ s)s 2

(16)

If B

DOI: 10.1021/acs.jpcb.8b11137 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B ξ (̂ ∞) ≡ γ∞

where the first term accounts for the elasticity of the polymer chain. The memory kernel is independent of the potential Uee; it thus can be determined by, for example, setting Uee = 0 such that the potential of the mean force is that of a harmonic oscillator with a spring constant κ = κ0/(N − 1). In the Laplace space, then, the memory kernel is given by eq 11, with the autocorrelation function of the end-to-end distance given by the known expression61,62

(17)

is finite, then, taking the inverse Laplace transform of eq 16, one recovers the short-time behavior: χ(t) ≈ 1 − κt/γ∞. Using eq 15, one finds, then, that the mean square displacement undergoes diffusive dynamics at short times ⟨Δx 2⟩ ≈ 2

kBT t ≡ 2D∞t γ∞

(18)

̂ The plateau value ξ(∞) is, therefore, the effective friction coefficient that would be inferred from the short-time dynamics of the system. In the non-Markov case, this value ̂ is different from the Markov limit ξ(0) of the friction coefficient. ̂ Equation 18 cannot be used when ξ(∞) = 0. Consider, for example, the power law friction kernel of the form55 ξ(t ) =

ηα t

Cxx(t ) =

where upn =

(19)

2kBT tα ηα Γ(1 + α)

and

ÄÅ ÉÑ Å pπ Ñ λp = 4 sin 2ÅÅÅÅ ÑÑÑÑ ÅÇ 2N ÑÖ

Using eq 10, we have extracted the memory kernel from Brownian dynamics simulations (i.e., from numerical integration of eq 22) for the GRM with different potentials Uee and compared it to the exact resultsee Figure 1. In this figure, as well as in the rest of the paper, kBT is used as the energy unit, 10 kBT /κ0 is the length unit, and γ0/κ0 is the time unit. The exact memory kernel is independent of the potential Uee, but the accuracy of its numerical estimate depends on this potential, on the length of the trajectory used, and on the data sampling rate. This dependence is important to understand if

̂ = ηα/s1−α, where 0 < α < 1. Taking its Laplace transform, ξ(s) eqs 15 and 16 give ⟨Δx 2⟩ ≈

ÄÅ É 2 1 yÑÑÑ ÅÅ pπ i cosÅÅÅ jjjn + zzzÑÑÑ ÅÅÇ N k N 2 {ÑÑÖ

(24)

(25)

−α

Γ(1 − α)

N−1 ij κ0λpt yz 1 1 zz(u − u )2 ∑ expjjjj− zz pN p1 βκ0 p = 1 λp γ 0 { k

(20)

The short-time dynamics is, therefore, subdiffusive in the case of a power-law memory kernel.

3. RESULTS: APPLICATION TO THE GENERALIZED ROUSE MODEL (GRM) Here we study the dynamics of the toy model of a onedimensional Rouse chain whose ends interact with one another via some potential and which may be subjected to an external force. The total potential energy of the chain, which is composed of beads with coordinates x1, x2, ..., xN, is given by N−1

UGRM(x1 , ..., xN ) =

∑ κ0(xn+ 1 − xn)2 /2 + Uee(xN − x1) n=1

(21)

Figure 1. Memory kernels extracted from the end-to-end distance of a 1D Rouse chain using eq 10 with different interaction potentials Uee acting on the chain ends. Here, the intrachain stiffness is κ0 = 100 and the number of monomers is N = 50. All trajectories were sampled using a time step of 0.5γ0/κ0 with a total simulation time of 106γ0/κ0. The high-frequency (large s) errors are due to finite sampling rate (i.e., these errors are reduced if the time resolution of data collection improves). The low-frequency (small s) errors are a result of statistical errors in estimating the correlation functions Cxx(t) and Cfx(t) (i.e., these errors are reduced as the simulated trajectories become longer). For a free chain, i.e., Uee(x) = 0, the correlation functions decay rapidly; as a result, convergence is achieved even with a trajectory of modest length. This decay time is longer when the dynamics involves crossing a barrier, as in the cases of Uee(x) = 3(x2 − 1)2 and Uee(x) = 48(x−12 − x−6)2 − 9.2x, requiring much longer trajectories to achieve the same accuracy. For the same simulation time, the error in estimating the low-frequency limit of the memory kernel increases progressively with the barrier height, which is, respectively, ∼2kBT and ∼6kBT for the above two potentials. The inset shows the friction kernel in the time domain obtained via numerical inversion of the Laplace transform. The kernel has a delta-function component, which is not shown. The gray dashed−dotted line shows the memory kernel ̂ = γ∞ + estimated by fitting eq 10 with the function of the form, ξ(s) a/(s + λ), which corresponds to exponentially decaying memory, ξ(t) = 2γ∞δ(t) + ae−λt.

where κ0 is a spring constant that defines the Kuhn length of the chain and Uee is the additional potential acting on the chain’s end-beads. In the next section, we will introduce a specific form of Uee suitable to mimic the unfolding and refolding of a biopolymer under a stretching force, as in singlemolecule force spectroscopy studies. Here we will consider two different potentials, along with the Uee = 0 case, with the goal to investigate how the method of estimating the memory kernel depends on the system’s potential of mean force. The beads of the chain obey the overdamped Langevin equation of the form 0 = −γ0xṅ − ∂UGRM /∂xn + ζn

(22)

where γ0 is a monomer friction coefficient and ζn(t) is a deltacorrelated Gaussian random noise with zero mean satisfying the relationship ⟨ζn(t)ζm(t′)⟩ = 2kBTγ0δ(t − t′)δnm. As was shown previously,6 the dynamics of the end-to-end distance x = xN − x1 in this chain satisfies a GLE, eq 7, exactly, with the potential of mean force given by U (x ) =

1 κ0 x 2 + Uee(x) 2N−1

(23) C

DOI: 10.1021/acs.jpcb.8b11137 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

4. RESULTS: PROPERTIES OF TRANSITION PATHS IN SIMULATED MECHANICAL UNFOLDING OF THE GENERALIZED ROUSE MODEL The model introduced in the previous section allows us to examine some of the generic features introduced by memory in single-molecule force spectroscopy experiments.13,28,63,64 To do so, we set the potential acting on the end monomers of the chain (see eq 21) to be ÄÅ ÉÑ ÅÅÅij σ yz12 ij σ yz6ÑÑÑ Uee(x) = 4εÅÅÅjj zz − jj zz ÑÑÑ − fx ÅÅk x { k x { ÑÑÑÖ (26) ÅÇ This potential includes a Lennard-Jones interaction between the end monomers and a stretching force f that is pulling these monomers apart. The corresponding potential of mean force (PMF) for the dynamics of the end-to-end distance x is ÅÄÅ ÑÉ ÅÅij σ yz12 ij σ yz6ÑÑÑ 1 κ0 2 Å U (x ) = x + 4εÅÅjj zz − jj zz ÑÑÑ − fx ÅÅk x { 2N−1 k x { ÑÑÑÖ (27) ÅÇ For the value of the force f used in this work, the PMF exhibits two minima: the minimum corresponding to the shorter distance x mimics the “folded state”, where the end monomers stick to each other because of the Lennard-Jones attraction, while the minimum corresponding to the higher extension of the chain corresponds to the “unfolded” state where the end monomers do not interact with each other (Figure 2). The

one wishes to apply this method to experimental data. In particular, consider the low-frequency behavior (s → 0), which corresponds to the long-time (t → ∞) tail of the memory kernel in the time domain. The Laplace-transformed memory ̂ kernel reaches a plateau ξ(0), which, according to eq 10, should be proportional to the integral of the force−position ∞ ̂ = −∫ dtCfx(t )/Cxx(0). The nucorrelation function, ξ (0) 0

merical noise in estimating the correlation function Cf x(t) may shift the estimated integral and thus alter the estimated plateau value, as is observed in Figure 1 for the cases of nonzero potential Uee: in those two cases, the dynamics involves barrier crossing, and as a result, the time scale of the decay of Cf x(t) becomes longer than in the case of a free Rouse chain, resulting in larger statistical errors given the same trajectory length. In the opposite, high-frequency limit, the friction kernel ̂ approaches another plateau, ξ(∞). In the time domain, this corresponds to a delta-function contribution, implying that the memory kernel has a strictly Markovian component, and the friction force has a component that is proportional to the velocity. This component simply corresponds to the solvent friction force acting on the end monomers, resulting in an effective friction coefficient of γ0/2,6 as opposed to the timedelayed friction force exerted on the end monomers by the rest ̂ of the chain. The high-frequency behavior of the ξ(s) is determined by the short-time behavior of the correlation functions entering eq 10; the accuracy in estimating the plateau value is sensitive to the data sampling rate, which is manifested ̂ by relatively small deviations of the estimated ξ(∞) from γ0/2 in Figure 1. In principle, the Laplace transform of the memory kernel contains all of the physically important information about memory. If the explicit time dependence ξ(t) is desired, the Laplace transform must be inverted numerically, which, in general, is a challenging task. The inset of Figure 1 shows the friction kernel in the time domain obtained via such numerical inversion. The kernel has a delta-function component, which is not shown. Spurious oscillations observed for the high-barrier case, where the statistics is insufficient, illustrate the difficulty in performing an inverse Laplace transform on noisy data. By taking advantage of physical insight into the system, when available, accuracy of estimating ξ(t) may be improved. For example, if one assumes (correctly, for the Rouse model) that the non-Markov part of ξ(t) is a sum of decaying exponentials with positive amplitudes, ξ(t ) = 2γ∞δ(t ) + ∑i ai e−λit , ai > 0 ̂ (cf. eqs 11 and 24), then one can estimate ξ(t) by fitting ξ(s) a i to a sum ξ (̂ s) = γ∞ + ∑i s + λ , thus finding ai and λi. The

Figure 2. Potential of mean force (PMF) on the end-to-end distance of the generalized Rouse model, eq 21. The parameters used here are κ0 = 100, N = 50, ε = 12, σ = 1, and f = 9.2 (same as in Figure 1). Blue dots represent the PMF reconstructed from the equilibrium probability distribution in a simulated trajectory using eq 1. The dashed black line is the exact PMF given by eq 27. A, TS, and B represent the compact chain (folded) basin, the transition state (barrier top), and the expanded chain (unfolded) basin. Orange dashed lines show the transition region boundaries xA and xB used in the trajectory analysis. The dashed blue line is the symmetric harmonic potential used to estimate the transition path times and the transition path velocity using analytical approximations.

i

resulting kernel would then no longer have artifacts such as oscillations. In the absence of insight, however, “blind” inverse Laplace transform methods are preferred. Because of experimental constraints (noise, limited time resolution, etc.), it may be desirable to use an empirical memory kernel, such as the well-studied kernel of the form ξ(t) = 2γ∞δ(t) + ae−λt, which includes a Markovian component and a single term with exponentially decaying memory. Figure 1 illustrates that, in comparison to the Markov ̂ model (where ξ(s) has the same value for any s), such an empirical model provides an improved description of the GRM data, yet it is clearly inadequate, especially considering the behavior of the memory kernel in the time domain.

shape of the PMF is reminiscent to that found in molecular simulations of folding (using different reaction coordinates),65 with a broad unfolded basin and a narrow folded basin. Transition paths are defined as pieces of a trajectory x(t) that enter an interval xA ≤ x ≤ xB (which we call the transition region) through one of its boundaries (say xA) and exit through the other boundary, staying continuously inside this interval between these two events. In what follows, we analyze how various experimentally measurable properties of transition D

DOI: 10.1021/acs.jpcb.8b11137 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

the barrier top, and the activation barrier ΔUA = UTS − UA is the difference between the PMF values at the barrier top and the unfolded minimum. As seen from Table 1, the Kramers theory underestimates the unfolding and refolding rates by a factor of ∼2.5.

paths are affected by the memory effects introduced by the dynamics of the polymer chain. 4.1. Markovianity Test. A recent study48 proposed a simple test allowing one to establish whether the observed dynamics x(t) is Markovian. Consider the probability P(xA → xB|x) that a point x belongs to a transition path from A to B. For a Markov process, the maximum of this probability is always exactly the same

Table 1. Comparison of the Folding and Unfolding Rates from Simulations of the Generalized Rouse Model with the Same Rates Estimated Using the Kramers and Grote−Hynes Approximations

max P(xA → x B|x) = max P(x B → xA|x) = 1/4

xA < x < x B

xA < x < x B

(28)

while this maximum is below 1/4 for a non-Markov process. Indeed, this is what is observed for the unfolding/refolding dynamics of our model (Figure 3): memory effects in the dynamics of this process are readily detected through the deviations from eq 28.

unfolding rate (×10−4 κ0/γ0) folding rate (×10−5 κ0/γ0)

exacta

Grote−Hynes

Kramers

1.40 ± 0.15 1.22 ± 0.13

1.42 1.20

0.57 0.48

a

The rates and error bars were estimated from seven trajectories with a total length of 5 × 106 γ0/κ0.

In single-molecule force spectroscopy studies, the PMF can be measured,67,68 and one usually uses eq 29 to estimate the value of the friction coefficient γ. To obtain agreement between the Kramers formula and simulated dynamics, then the value of the friction coefficient should be a factor of ∼2.5 lower and the value of the diffusion coefficient a factor of ∼2.5 higher than its true value in the Markov limit. Memory effects can, however, be treated using the more general Grote−Hynes theory, which predicts the following expression for the transition rate69−71 kA → B = Figure 3. Markovianity test48 applied to the generalized Rouse model of unfolding/refolding under a stretching force, with the same parameters as those in Figure 2. The probability P(xA → xB|x) is shown in black, the orange dashed lines represent the transition region boundaries, and the dashed blue line here represents the expected maximum value if the trajectory were Markovian.

″ λξ (̂ λ) = UTS

(31)

In particular, the Kramers formula, eq 29, is recovered from eqs 30 and 31 when ξ̂(λ) can be approximated by ∞ ̂ ξ (0) = ∫ ξ(t ) dt = γ . 0

As seen from Table 1, the Grote−Hynes theory, unlike the Kramers formula, predicts rates that are very close to the “exact” transition rates obtained directly from the simulated trajectories. 4.3. Distribution of Transition Path Times. Perhaps the most (experimentally) accessible property of transition paths is their temporal duration, or transition path time. As such, this property has received considerable experimental11−14,72,73 and theoretical16−18,20−22,26,55,56,74−76 attention. In the analysis of experimental13 and, sometimes, simulated76 distributions of transition path times, one often takes advantage of analytic approximations available when the barrier is approximated by an inverted parabola. For a parabolic barrier with a curvature κb and a width 2a (cf. Figure 2) whose height is much greater than the thermal energy, the following result for the distribution of transition path times tTP was recently obtained55,56 in the case of GLE dynamics:

A

reactive flux across the barrier with nTP being the number of transition paths in a trajectory of total time T, and x TS PA = ∫ peq (x) dx is the equilibrium probability to be in −∞

the initial basin A. Note that this result is insensitive to the choice of the boundaries of the transition region as long as those boundaries are chosen far enough (within a few kBT) from the barrier top and the PMF minima. According to the Kramers theory in the overdamped limit, the transition (unfolding) rate is given by ″ |1/2 −ΔUA / kBT |UA″|1/2 |UTS e 2πγ

(30)

where λ is the reactive frequency satisfying the equation

4.2. Folding and Unfolding Rates. Protein folding kinetics is frequently analyzed using the model of Brownian motion over a barrier due to Kramers.2 To assess the applicability of this model to our system, we have computed the folding and unfolding rates in the following way: Assuming the validity of first order kinetics, the rate of transitions from F 1n state A to B is66 kA → B = AP→ B , where FA → B = 2 TTP is the

kA → B =

1/2 λ |UA″| e−ΔUA / kBT ″ |1/2 2π |UTS

(29)

where γ is the friction coefficient given by eq 14, UA″ and UTS ″ are the curvatures of the PMF at the unfolded minimum and at E

DOI: 10.1021/acs.jpcb.8b11137 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B p(t TP) =

κba2/2 = UTS − U(xA) = UTS − U(xB), as shown in Figure 2. Despite the asymmetry of the actual PMF and the relatively low height of the barrier traversed by transition paths (∼2.5kBT), the non-Markov prediction, eq 33, agrees well with the actual distribution, while the Markov approximation, eq 34, significantly overestimates transition path times, a result consistent with the above observation that the apparent diffusion coefficient should be greater than kBT/γ to match the observed time scales. 4.4. Transition Path Velocity Profile. Other experimentally accessible properties of transition paths include their average shape23,24,26 and velocity profile25 vTP(x). We focus on the latter, as it has recently been measured experimentally,28,63 and as a corresponding shape of the transition path, xTP(t), can be determined from its velocity profile through the implicit equation

2κba 2 πkBT

2 y i χ 2 (t TP) − 1 (χ (t TP) − 1) erfcjjj κba zzz k 2kBT { ÄÅ ÉÑ 2 ÅÅ κ a χ (t ) + 1 ÑÑ TP ÑÑ expÅÅÅÅ− b ÑÑ ÅÅÇÅ 2kBT χ (t TP) − 1 ÑÑÖÑ (32) Here the function χ(t) has the following Laplace transform

χ ̇ (t TP)

χ ̂ (s ) =

ξ (̂ s) ̂ sξ (s) − κb

(33)

Note that eq 33 is identical to eq 16, where κ = −κb. According to eq 32, the properties of the distribution of transition path times are determined by the function χ(t). For a parabolic barrier, this function describes how the (initially small) departure from the barrier top grows, on average, as a function of time.55,56 Importantly, χ(t) is a “slow” function, with a characteristic time scale being the “barrier relaxation time”, which, in turn, is comparable to the mean transition path time (unlike the characteristic time scale over which the memory kernel ξ(t) decays, which may be much shorter). ̂ ̂ In the Markov case, ξ(s) = ξ(0) = γ, eq 33 becomes 1 κbt/γ χ̂ (s) = s − κ / γ or χ(t) = e and eq 32 reduces to16,18,74,75

t=

κb κba 2 2 y i πkBT 2γ erfcjjj κba zzz 2kBT k { ÄÅ ÉÑ ÅÅ κba2 Ñ expÅÅÅ− 2k T coth(κbt TP/2γ )ÑÑÑÑ ÅÇ B ÑÖ sinh[κbt TP/2γ ] sinh[κbt TP/γ ]

(35)

The transition path velocity vTP(x) is the ratio of a short distance Δx over the average cumulative time ⟨Δt(x − Δx/2, x + Δx/2)⟩ spent by a transition path in the interval (x − Δx/2, x + Δx/2). This velocity is related to the probability distribution of all points belonging to transition paths, p(x| TP), by25 1 vTP(x) = p(x|TP)⟨t TP⟩ (36)

b

p(t TP) =

∫ dxTP/vTP(xTP)

where ⟨tTP⟩ is the mean transition path time. In Figure 5, the transition path velocity profile obtained directly from the simulated trajectories is compared to that

(34)

We test the distribution of transition path times obtained from simulations against the parabolic barrier approximation with (eq 33) and without (eq 34) memory in Figure 4. The width of the parabolic barrier was chosen to match that of the actual transition region, 2a = xB − xA, and the curvature κb was chosen such that its height (measured relative to the entrance/ exit of the transition region) matches the actual barrier, i.e., Figure 5. Transition path velocity profile for the generalized Rouse model of folding/unfolding under mechanical force with the same parameters as those in Figure 2 (black). Blue dashes represent the corresponding transition path velocity profile for Markovian dynamics in the same potential of mean force (eq 6). The blue solid line is the theoretical prediction for diffusion in a parabolic barrier, eq 37.

obtained using the Markovian approximation to GLE, as well as to the analytic result for diffusion in a parabolic barrier25 (with the same parameters as above) ÄÅ ÉÑ−1 ÅÅ ÑÑ κbx 2 y 2i j z ÅÅ ÑÑ j z erf j z − 1/2 Å ÅÅ ÑÑ 2kBT Ñ kBTκb i π y k { Å ÑÑ j z ÅÅ1 − jj zz vTP(x) = Å ÑÑÑ 2 y γ ÅÅ κ a k8{ 2i ÅÅ erf jjj b zzz ÑÑÑ ÅÅÅÇ k 2kBT { ÑÑÑÖ ij κ a 2 yz ij κba 2 x 2 yzzz j zz j b erf −1jjj zz expjjj− z jj 2kBT zz jj 2kBT a 2 zzz (37) k { k {

Figure 4. Distribution of the transition path times for the end-to-end distance of the generalized Rouse chain with the same parameters as those in Figure 2. The green curve is the prediction of eq 33 using the memory kernel shown in Figure 1. The blue curve is using the Markov approximation for a parabolic barrier, eq 34. The inset shows the same plot on a log scale. F

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time behavior of the function χ(t) (which is related to the short-time behavior of the friction kernel through eq 11 and which determines the distribution of transition path times through eq 32) is given by χ(t) ≈ 1 − κt/γ∞ (see section 2). To estimate this behavior accurately, the time resolution of the data should thus be much shorter than the high-frequency relaxation time τ∞ = γ∞/κ. Experimentally, therefore, one can verify whether the sampling rate of the data is sufficient given the estimated values of γ∞ and κ (with the latter estimated as a typical curvature of the PMF obtained using eq 1). Importantly, the relaxation time scale τ∞ does not correspond to the time scale over which the memory kernel decays itself. For example, in the Markovian case (eq 6), the memory kernel decays infinitely fast, yet this does not cause any time resolution issues, as this behavior is readily obtained from eq 11 with exponentially decaying correlation functions, Cxx(t) = (kBT/κ)e−tκ/γ, as long as the time resolution is better than γ/κ. Fast decaying memory kernels can be estimated with modest time resolution. It is tempting to extend the above arguments and claim that the long-time/low-frequency limit of the memory kernel can be estimated reliably whenever the trajectory’s temporal length ̂ exceeds the relaxation time ξ(0)/κ, with κ estimated, again, as a typical PMF curvature. The results shown in Figure 1, however, suggest that this is not the case: instead, the length of the trajectory must exceed the longest relaxation time of the system. In cases where the dynamics of the system involves the crossing of a free energy barrier, this relaxation time is related to the barrier crossing time (as opposed to a local relaxation time within a single potential well), and it increases exponentially with the barrier height. As a practical rule, then, the experimental trajectory must display a sufficiently large number of barrier crossing events. Of course, estimation of the memory kernel from experimental data is only meaningful if the overdamped GLE description of the experimental signal, eq 7, is an adequate one. In this regard, the Markovianity test proposed in ref 48 and illustrated in Figure 3 may be a more general one, as it does not rely on a specific model of non-Markovian dynamics; the caveat of using the latter test is that it provides no information about the physical nature of non-Markovian effects. Testing whether GLE is an adequate model of folding dynamics as observed in molecular simulations is an important direction that we plan to pursue in the future.

While use of the parabolic approximation does not introduce significant errors to the velocity profile (data shown in blue in Figure 5), the memory effects change the velocity profile significantly.

5. CONCLUSIONS Although the picture of diffusive barrier crossing provides a simple and often accurate description of folding kinetics, the emerging experimental studies of microscopic transition paths are poised to reveal finer features of barrier crossing dynamics. In particular, memory effects are expected to be a universal feature when multidimensional molecular dynamics is projected onto a single coordinate,77 as is inevitably done in experimental studies. The study of a toy model described here suggests a roadmap for experimental data analysis that could lead to more accurate low-dimensional models of folding dynamics. Indeed, most of the analysis presented here for simulated data can be (and, in the case of transition path times and velocities, has already been13,28,63) equally applied to experimental signals. Since the underlying dynamics of the model studied here is known exactly, we can also assess the effect of various approximations commonly made in data analysis. In particular, we find that the velocity profiles (Figure 5), and hence “mean” transition path shapes (eq 35), are sensitive to memory effects (Figure 5); at the same time, the neglect of anharmonicity of the underlying potential of mean force has surprisingly little effect on the transition path velocity profile (Figure 5, data shown in blue), the transition path time distributions (Figure 4), and the transition rates (see Table 1, which shows that the Grote−Hynes theory predicts the rate accurately). This supports the use of the analytic approximations (eqs 30−32, 37) that take advantage of the parabolic barrier approximation in data analysis. At the same time, this observation also suggests that the deviations of, for example, velocity profiles observed experimentally28 from theoretical predictions24,25 may not be attributable to anharmonicity: rather, effects of memory, coordinate dependence of the diffusion coefficient, or, possibly, experimental artifacts should be examined. In addition to the indirect signatures of non-Markovian effects that can be observed in transition path time distributions and velocity profiles, our results suggest a simple and direct test: estimate the memory kernel directly from the experimental trajectory x(t) using eq 10. This should be possible because experimental trajectories, often in contrast to simulated ones, can be long. As a result, the long-time (small s) behavior can be obtained accurately; the short-time behavior of the estimated memory kernel, however, may be affected by the experimental time resolution/data sampling rate or by experimental noise. Indeed, Figure 1 illustrates that the high ̂ frequency behavior of ξ(s) is sensitive to the sampling rate. However, because the correlation functions entering eq 10 are “slow” ones, with a characteristic time scale comparable to that of the relaxation time of the reaction coordinate x, estimating the memory kernel should be within reach of modern single molecule force spectroscopy techniques.78 Specific requirements for the length and for the time resolution of an experimental trajectory required to obtain the short-time and long-time behavior of the system accurately can be estimated as follows. Consider the dynamics in a parabolic potential with a positive or negative spring constant κ (the latter case corresponding to a barrier). Assuming that the higĥ frequency friction coefficient ξ(∞) = γ∞ is nonzero, the short-

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: (512) 471-4575. ORCID

Dmitrii E. Makarov: 0000-0002-8421-1846 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Mona Ozmaian for providing her Brownian dynamics simulation code and to Dave Thirumalai for helpful discussions. This work was supported by the Robert A. Welch Foundation (Grant No. F-1514) and the National Science Foundation (Grant No. CHE 1566001). G

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