A Parallel-pulling Protocol for Free Energy Evaluation INTRODUCTION B A) , p

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1 A Parallel-pulling Protocol for Free Energy Evaluation Van Ngo Collaboratory for Advance Computing and Simulations (CACS, Department of Physics and Astronomy, University of Southern California, 3651 Watt Way, Los Angeles, California , USA. address: Jarzynski s equality (JE allows us to compute free energy differences (FEDs from distributions of work performed on a system. We show that it is possible to generate the work distributions in single step-wise pulling simulations in order to compute FEDs by JE without generating many trajectories using virtual optical traps. We suggest an alternative method for directly computing FEDs in both sequential- and parallel-pulling protocols based on measurements of averaged reaction coordinates along pathways. In comparison with the commonly used Potential of Mean Force method applied to stretching a Deca-Alanine molecule, we show that the parallel-pulling protocol is at least 0 times faster than slow sequential-pulling protocols for producing the same free energy barrier with the uncertainty less than.0 kcal/mol. I. INTRODUCTION In 1997, Jarzynski [1] showed that free energy changes from an initial configuration A could be extracted from finite-time non-equilibrium measurements ΔF A B -β -1 exp(-βw where β1/k B T, T is temperature, k B is Boltzmann constant, W is applied work to bring an interested system to a final configuration B and the bracket denotes the average over all possible trajectories in which work is performed. Those states are associated with external control parameters λ. For simplicity, we assume that our interested system can be characterized by a single control parameter λ that can be used to monitor pathways of a reaction coordinate (RC. If using an optical trap (OT to perform work on a system, parameter λ can be the minimum position of the OT and the center-of-mass position of trapped particles in a direction along λ can be used as an RC. By varying λ from λ 1 (A at time t 1 to λ s (B at time t s, one generate a trajectory or pathway in which work experimentally measured from the resulting force-versus-extension (FVE curve [, 3] is: λ s W exp f λ δλ, (1 λ 1 where f λ is an applied force measured along the pathway at a certain value of control parameter λ whose increment is δλ. Since we vary control parameter λ, the Hamiltonian H(z,λ(t of the system correspondingly changes with time, where z (q, p denotes a point in the phase space of the system. According to Jarzynski, work W can be t s evaluated from the Hamiltonian by W [ H(z,λ(t / t]dt. In experiments of slowly stretching a single RNA molecule [, 3], W exp is measured to test the validity of JE, but not W that is used to prove JE [1, 4]. Even if in slow pulling limits W and W exp coincide, there is a discrepancy between them. Vilar and Rubi [5] theoretically pointed out that the connection of W with the Hamiltonian could be failed due to an introduction of an arbitrary function of time into the Hamiltonian. By contrast f λ, therefore W exp, can be measured without any knowledge of the Hamiltonian. Unfortunately, using W exp for Brownian t 1 dynamics [6] to theoretically investigate the Crooks theorem [7, 8], which is a more generalized version of JE, has not been well accepted [9]. Because of the weak connection between W exp and W, it might give a rise to a doubt about the validity of JE [10]. Although Jarzynski clearly identified which work related to Hamiltonian should be used for JE [11], theoretical considerations of W exp for JE has not been fully investigated. JE was experimentally validated [, 3, 1] by using optical traps to generate all possible FVE curves from which one can construct work distributions. In molecular dynamics (MD simulations [13-18], an important question is how to generate correct work distributions in an efficient way. Such work distributions should contain rare small values of work since the average is taken over exponential functions, which are sensitive with small values of their arguments. In order to ensure rare work distributions there should have a certain overlap of work distributions between forward and reverse processes [4, 19]. To guarantee that condition it is computationally expensive to generate many trajectories in MD simulations. One scheme to overcome this computation difficulty is to use the Potential of Mean Force (PMF method developed by Park et al [13, 18] for Steered Molecular Dynamics (SMD simulations [0] in which W is defined as vk t s t 1 (x λ1 vtdt, where v is a guiding velocity of an OT with spring constant k. In this definition, parameter λ is assumingly linear with time t, i.e., λ~vt. In experiments, this assumption holds if pulling speeds are very slow (~µm/s. In MD simulations, this work is observed to bias resulting free energy changes because MD pulling speeds are thousand times faster than experimental ones. The PMF method remedies the bias problem by utilizing the second order cumulant expansion [13, 18, 1] that can be used to estimate unbiased FEDs with much less cost in comparison with irreversible pulling simulations for acceptable accuracies. Despite of the fact that the PMF method is widely used to study a variety of biological systems [-6], the answer to the 1

2 question asked in MD simulations on large systems remains unsatisfactory. The attractive features of JE are parallel, rateindependent and arbitrarily far-from equilibrium, for instance in ways of pulling or stretching a single biological molecule. The parallel property means that one can carry out all pulling trajectories in parallel (at the same time regardless of pulling speeds (rate-independence [15, 1]. In comparison with the Adaptive Biasing Force (ABF [14, 7] method based on Thermodynamics Integration [1, 8], JE promises a faster method for free energy calculations and is so-called fast-growth method. In the ABF method, forces are adaptive along a pathway and then averaged at a certain number of windows. The integration of the averaged forces along the pathway gives FEDs. As pointed out by Rodriguez-Gomez et al [14], there is a common characteristic between the fast-growth and the ABF method that f λ and adaptive forces must be instantaneously calculated along a pathway. Since by generating work distributions we generate distributions of f λ that can be collected at certain intermediate states or windows. One question may arise is that is it possible to compute FEDs from these non-adaptive force distributions as carried out in the ABF method? If the answer is possible, the parallel pulling might be a way to speed up free energy calculations. In this article we propose a theorem that considers the relationship of the work definitions. We take the definition of W as a starting point but treat control parameter λ and time t in different manners especially aimed from MD simulations, i.e., λ and t are correlated but not simultaneously co-linear. We prove that mechanical work W exp should be the work appearing in JE s expression instead of W for the case of canonical ensembles. By discretizing steps of applying work to unfold a decaalanine molecule, we show that it is possible to construct work distributions in single step-wise pulling simulations. Our theorem suggests a method to compute FEDs from the averaged values of an RC (related to averaged forces along a pathway in sequential- and parallel-pulling protocols (SPPs and PPPs. Based on the theorem and its application s results, we provide a criterion for ensuring that reliable work distributions can be constructed in PPPs that might be a way to significantly speed up free energy calculations. Our article is organized as follows: we present our theorem and propose two methods for free energy calculations in Section II, test them in Section III, and compare our methods with some commonly used methods for free energy calculations in Section IV. II. THEOREM First, we consider a process in which a system of N particles with time-independent Hamiltonian H o ( r 3N 1, x is applied by an OT U(x,λ 1 k(x-λ 1 / in order to bring a set of trapped particle, whose center-of-mass position along x-axis can be defined as a reaction coordinate x, closer to the OT s minimum position λ 1 (control parameter from time t 0 to t 1. Suppose that there is no coupling between the OT with the particles momenta. Then, our Hamiltonian and partition functions can be simply expressed in terms of spatial coordinates ( r 3N 1, x for computing FEDs. The system is in contact with a thermal reservoir at temperature T and assumingly has canonical ensembles for our considerations. The coupling Hamiltonian should have the following formula: H(λ 1, x H o ( r 3N 1, x + k (x θ(t t 0 θ(t 1 t, ( where θ(t is a Heaviside step function. One can in principle take the infinity limit for t 0 and t 1. Time t should be considered as a variable to drive the system to final states, but not λ 1. That is due to the fact that from finite t>t 0, the interested system gradually starts absorbing nothing else but the performance of the potential, which is work, in order to change its internal energy and entropy at constant temperature. The absorption might not be considered as instantaneous conversion of the potential energy into work by using a single Heaviside step function of time [5]. Then, the work driving the system from a specific state at tt 0 to final states should be given by: t 1 H(λ W 1,x dt k ( t (x λ 0 1 (x 1 (3, t 0 where x 0 is the RC s initial position belonging to the ensemble of H ( o r 3N 1, x 0 at time t 0 and unchanged from t t 0, and x 1 is any final positions of the RC at t 1 t 0 belonging to the ensemble of H(λ 1,x 1. If one collects all possible x 1 after turning on the OT, it is possible to take average of exp(-βw over x 1 s ensemble (see Appendix A to get: exp[β k ( x 1 ] exp[βδf(λ 1,k], (4 (x 1,λ 1,k where (x 1,λ 1,k represents all possible points with Hamiltonian H(λ 1,x 1 in phase space, ΔF(λ 1,kF(λ 1,k- F 0 -βlog[z(λ 1,k]+βlog[Z(0], Z(λ 1,k dr 3N 1 dx 1 exp[-β H(λ 1,x 1 ] and Z(0 dr 3N 1 dx 0 exp(-β H ( o r 3N 1,x 0. ΔF(λ 1,k is the FED between the configurations with and without the OT. There is an interesting point for cancelling the factors related to x 0 in order to arrive at Eq. (4. It should be noted that there is a canonical transformation from x 0 to x 1 with Hamiltonian H(λ 1,x 1 at t t 0. But there is no canonical transformation from the ensemble with H o ( r 3N 1, x 0 to the ensemble with H(λ 1,x 1 vice versa because we simply turn on and off the potential at a certain time t 0. In addition, given the ensemble of x 0 the average can be taken over all x 0 instead of x 1 to compute the same FED: ΔF(λ 1,k β 1 log exp[ β k (x 0 ] (x 0,k 0, (5 where (x 0,k0 represents all possible points with Hamiltonian H o ( r 3N 1, x 0 in phase space. From Eqs. (4, (5 and inequality exp(-βw exp(-β W [9], we have the lower and upper bounds to ΔF(λ 1,k: k (x 1 ΔF(λ 1,k k ( x 1,λ 1,k (x λ (6 0 1 (x 0,k 0

3 Now, we perform a series of pulling in order to pull the center of the OT from λ 1 to λ s by turning on, off and instantaneously moving the OT from λ i-1 to λ i at time t i-1, then the total work (see Appendix B is: s k W total [ (x i 1 λ i (x i λ i ] (7 k [ (x o (x s λ s ] + W mech, where mechanical work W mech for each generated trajectory is defined by: W mech k (λ i+1 λ i ( λ i+1 + λ i f λ δλ, (8 where f λ U(x,λ/ λ for a sufficiently small increment δλ. If s is equal to 1, W mech is zero and the previous consideration is needed. Eq. (7 indicates that in general W total is different from W mech that could be expressed as [ H(λ i+1, x i H(λ i,x i ] that was used to compute FEDs for stochastic processes [30]. It should be noted here that the canonical ensembles with H(λ i,x i are independent of one another. Hence, it is possible to take average of exp(- βw total over all the ensembles of x 0, x 1. x s at the same time to arrive at the following identity: e βw total 1, (9 (x 0,x 1,...,x s or first average exp(-βw total over all x 0, which correspond to H o ( r 3N 1,x 0, and all x s, which correspond to H(λ s,x s H o ( r 3N 1,x 0 +U(x s,λ s, to get: e βw total e β [ F(λ s,k F(λ 1,k ] e βw mech. (10 x 0,x s Then averaging the left hand side of Eq. (10 over all intermediate states (the rest of x i would be exactly equal to averaging its right hand side over all possible W mech, or equivalently all over possible FVE curves. Thus, we recover JE: e βw mech JE FVE e βδf (λ s,λ,k 1, (11 where ΔF JE (λ s,λ 1,kF(λ s,k-f(λ 1,k. The mechanical work W mech defined by Eqs. (1&(8 in principle gives us the desired FED ΔF JE (λ s,λ 1,k, if a correct ensemble of W mech is generated. Eq. (8 indicates that the distributions of W mech are Gaussian, if those of x i are ρ(x i ~exp[-βk(x i - x i /(γ i ], where x i is averaged RC at i th pulling step and γ i are fitting parameters related to the standard deviations of x i s fluctuations. Let s assume that ρ(w mech ~ρ(x 1 ρ(x...ρ(x s-1 for a series of pulling steps with the same increment Δλ(λ s -λ 1 /s, then one can derive a relationship between fitted-gaussian FEDs ΔF G (λ s,λ 1,k and x i : ΔF G (λ s,λ 1,k k(λ λ (1 γ i (λ i s 1 + k(λ s. (1 s s s As the first term vanishes as the number of pulling steps s goes to infinity, i.e., infinitely slow pulling, the FEDs can be determined from the second term that only depends on the average of the differences between λ i and x i. It should λ s λ 1 be emphasized that the index i runs from 1 to s-1 and the sum is divided by s. Let s define: (λ i ΔF fluct (λ s,λ 1,k k(λ s s. (13 Based on our theorem, we propose two methods for free energy calculations: (i using JE and the distributions of mechanical work W mech that defined in Eq. (8; (ii using Eq. (13 and averaged values x i. III. IMPLEMENTATION AND TESTING In this section we apply the two methods to an exemplary system, helix-coil transition of deca-alanine in vacuum (see Ref. [13] for more details of simulation setups. We generate work distributions, free energy profiles by both methods in sequential (a and parallel (b pulling protocols, and investigate the effects of spring constants k (c and relaxation time τ at each pulling step (d. A. Sequential pulling The simulation setups are kept the same as studied by FIG. 1. Distributions of x i (a and mechanical work W mech (b from steps i5 (λ15 Å to i35 (λ30 Å. The inset shows the fitting parameters γ i change as λ increases. Park et al [13]. One end of the molecule is kept fixed at the origin and the other end is sequentially pulled with an OT having the spring constant k7. kcal/mol/å. Instead of continuously varying the center of the OT λ with time, 3

4 we increase λ by Δλ0.5 Å from λ 1 13 to λ s 33 Å (s40. At each step, λ i, the system is relaxed for τ10 ns. Hence, the averaged pulling speed is 5.0 mm/s. We record the positions of the pulled end x i (considered as RC at each relaxation step (RS every 0.1 ps to construct their distributions as briefly shown in Fig. 1a. We use Least Square Fitting method to obtain parameters γ i as shown by the inset in Fig. 1b. Dimensionless fitting parameters γ i fall in the range of 1.1 to 1.4 and are peaked at λ5 Å. The values of these parameters indicate how the approximated widths of x i s distributions change along our pathway. The fitting parameters are related to the strength of our OT, kσ / (σ is an averaged standard deviation of the RC in comparison with thermal energy k B T/. If kσ / is larger than k B T/, γ i is larger than unity, vice versa. We divide the ranges of x i and W mech into small bins and generate work distributions (see Appendix C at every RS as briefly shown in Fig. 1b. The work distributions are moving with their peaks lower and widths more broaden as λ increases. Plugging these distributions and measured x i into Eqs. (8, (11, (1 and (13, we compute applied mechanical work W mech, FEDs ΔF JE, ΔF G and ΔF fluct as functions of λ, respectively. The values of W mech are located at the centers of the corresponding distributions shown in Fig. 1b and noticeably higher than ΔF JE whose values are in a good agreement with those of the PMF method (the PMF pulling speed v0.1 Å/ns and ΔF fluct as shown in Fig. a. We observe that at the minimum position (λ~15å ΔF JE is the same as the PMF free energy (~-.1 kcal/mol that is lower than ΔF fluct by 0.5 kcal/mol. At λ 5 Å (barrier position, the PMF values are larger than ΔF fluct by 1.0 kcal/mol, but only differ from ΔF JE by 0.5 kcal/mol. Free energies ΔF G are lower than the others since by fitting x i s distributions to find parameters γ i we actually throw some important data contributing to the FEDs and the effects of finite s and τ cause a significant contribution to ΔF G. The evolution of parameters γ i a long our pathway indicates the transition occurring at which the FEDs are observed to be a plateau because the highest value of γ i indicates the largest standard deviation of x i at a transition state. B. Parallel pulling The coincidence between ΔF fluct (Eq. (13 and ΔF JE (Eq. (11 might provide a much easier way to compute FEDs. These equations suggest us that one can compute FEDs if the values of x i and their fluctuations are known in any pulling protocols. In order to verify this observation, we perform 1 10ns-simulations in parallel of stretching the molecule from the same initial state by varying parameter λ from 13 Å to 33 Å (Δλ1 Å and record all values of x i (s1 corresponding to λ i. One can reduce the amount of relaxation time for small λ in order to speed up the simulations. For the purpose of comparison with the previous pulling, we keep the relaxation time the same. The resulting free energy profile of ΔF fluct is plotted in Fig. b together with the profile obtained in the previous sequential pulling. Fig. b shows that both parallel and sequential ΔF fluct have the same minimum FED. For larger values of λ, the parallel curve is shifted below the sequential curve by an amount of 1.0 kcal/mol. Interestingly, with these parallel data we are able to generate work distributions and compute ΔF JE as shown in Fig. b (JE parallel curve. The values of ΔF JE are only.0 kcal/mol larger than those of the sequential ΔF fluct for λ 5 Å. The sequential free energy profile lying between the parallel curves suggests that both ΔF JE and ΔF fluct can be combined to estimate the exact FIG.. (a Mechanical work (boxes, free energies and PMF energy (crosses versus λ. ΔF JE (empty circles and ΔF G (triangles are free energies computed by Eq. (11 and by Eq. (1 in the sequential pulling, respectively. ΔF fluct (full circles are computed by Eq. (13. (b Free energy ΔF fluct (dot dashed line in the sequential pulling with Δλ0.5 Å, ΔF fluct (triangles and ΔF JE (boxes in the parallel pulling with Δλ1.0 Å. The inset shows the accumulating forces ΔF fluct /(λ s -λ 1 k (λ i - x i /s for i from 1 to s-1versus λ for both cases. free energy profile of our system. Subsequently, the uncertainty of the estimated exact profile would be half of the difference between ΔF JE and ΔF fluct that is at most 1.5 kcal/mol. In terms of computational cost, the PPP is definitely at least 0 times faster than the SPP. It is worth to compare the averaged applied forces in both the SPP and PPP along our pathway. The factor k (λ i - x i /s in Eq. (13 can be interpreted as averaged 4

5 accumulating forces along our pathway. The inset in Fig. shows how the accumulating forces are changing with λ. In both pulling protocols, the behaviors of the forces are almost the same. Both of them are saturated after λ equal to 5Å. The saturation indicates that the linear regime of ΔF fluct as the function of extension λ s -λ 1 could be used for λ>5. It is possible to collect 11 x i s distributions (s11 out of the 1 simulations with Δλ Å in order to construct corresponding work distributions. As a result, the FEDs at λ5 Å computed from this data set using JE and Eq. (13 are about 31.8 and 11.4 kcal/mol, respectively. In other words, with these data we overestimate the FED if using JE and underestimate it if using Eq. (13. These numbers clearly indicate that the data set with Δλ Å cannot be used to construct correct work distributions and measure a sufficient history of x i. This observation is consistent with the implication of W mech s expression that the increments λ i -λ i-1 should not exceed the magnitudes of x i s fluctuations; otherwise small and rare values of W mech cannot be sampled. Fig. 1a shows that the magnitudes of x i s fluctuations are around 1.0 Å that is observed in the parallel pulling (s1. One can reduce the strength of the OT, i.e., spring constant k, in order to enlarge x i s fluctuations, however that would require more relaxation time. Hence, there is a tradeoff between k and relaxation time τ that depends upon systems properties. C. Effects of spring constant k In order to investigate how the FEDs vary in respond to different spring constants k, we perform two sequential pulling simulations in which k1.0 (~1.6 k B T/Å and 50.0 kcal/mol/å are used for Δλ0.5 Å and τ10 ns. It is more convenient to do analysis in these sequential pulling simulations than in similar parallel pulling simulations since one does not need to add extra time to τ for intermediate paths to x i in parallel pulling for small k. We observe that the sets of x i in all three cases are not the same and in the case of k1 kcal/mol/å x i are not as perfectly linear with λ as in the other cases. However, the free energy profiles of ΔF fluct as plotted in Fig. 3a are in a good agreement with one another. The minimum energies are -0.9, -1.6 and -1.9 kcal/mol corresponding to k1.0, 7. and 50.0 kcal/mol/å, respectively. At λ 5 Å, the three curves are merging and shifted by about 1.0 kcal/mol from each other with the same order as appearing at the minimum position (see the insets in Fig. 3a. The smallest spring constant gives us higher minimum and barrier FEDs, while the strongest one gives rise to the smaller values. With the uncertainty of.0 kcal/mol, we can confirm that those important values are independent of spring constants in the range from 1.0 to 50.0 kcal/mol/å, even if there is noticeable lowering of the FEDs at k1.0 kcal/mol/å along the pathway from the minimum to the barrier. However, as computed by JE, the free energy profiles are distinguishable as seen in Fig. 3b. The FEDs at both smaller and larger values of k are clearly higher than that in the case of k7. kcal/mol/å for λ>5 Å. In order to explain the significant distinction among the profiles, we look at the distributions of x i at step i0 and 1 (λ~3 Å where the transition is about to occur. Fig. 3c shows that the two x i s distributions at k1 kcal/mol/å are lagging behind the others, whereas at k50 kcal/mol/å they have much less overlapping compared to the others. Moreover, FIG. 3. Free energy profiles of (a ΔF fluct and (b ΔF JE for different spring contants k1.0 (dots, 7. (empty boxes and 50.0 (triangles kcal/mol/å. (c Distributions of x i at step i0 (gray and 1 (black in the three cases. The insets in (a are zoomed at minimum and barrier positions. The inset in (c shows fitting parameters γ i versus λ for different spring constants. 5

6 the fitting parameters γ i in the inset of Fig. (3c indicate how x i s distributions relatively change along our pathway. The smaller values of γ i indicate less overlapping between the successive distributions since the same increment Δλ0.5 Å is used. The parameters in the case of k1 kcal/mol/å reached the highest value indicate that x i is likely still strapped in some previous pulling steps before the transition occurs. These suggest a possible explanation based on the expression of W mech in Eq. (8. For small spring constants, it is possible that one has to tune λ i quite far from x i in order to overcome a free energy barrier. This will increase rare values of work at the transition state if relaxation time is not long enough, since the RC might be trapped longer before the transition occurs. For large spring constants the RC is trapped by the OT in very narrow regions so that its distributions have less mutual overlaps. Consequently, large k likely reduces number of rare values of work. D. Effects of relaxation time τ We repeat the parallel pulling protocol by reducing relaxation time τ from 10 ns down to 0.01 ns in order to optimize required time to perform such simulations, particularly for our test case. Spring constant k7. kcal/mol/å and increment Δλ1.0 Å are used for all cases here. We record all FEDs ΔF fluct and ΔF JE at λ6 Å (barrier position corresponding to relaxation time τ. Fig. FIG. 4. (a Free energy ΔF fluct (dots and ΔF JE (triangles versus relaxation time τ, (b rare work distribution in a range from 0 to 5 kcal/mol at τ0.4 (dots, 0.6 (boxes and 4.0 (triangles ns and (c work distribution at τ0.01 ns at λ6 Å. The inset in (b shows the overall work distributions of the three values of τ. The inset in (c is x 14 s distribution. 4a shows how our FEDs change in respond to the variation of τ. ΔF JE curve is noticeably higher than ΔF fluct curve that starts saturated at τ0.4 ns. The convergence of ΔF fluct is due to the strong applied OT that makes the corresponding sets of x i not essentially change even at τ~0.4 ns. Surprisingly, ΔF JE curve does not as smoothly converge as ΔF fluct curve does. The values of ΔF JE have suddenly clear jumps at τ~0.6 and 6.0 ns. These jumps indicate the changes in work distributions, for example as seen in Fig. 4b the rare work distribution at τ~0.6 ns is lower than those at τ~0.4 and 4.0 ns. Nevertheless, the inset of Fig. 4b shows no visible deviation among their overall distributions. These rare work distributions have major contributions to ΔF JE. Work having lower distributions gives a rise to larger ΔF JE. It should be noted that at τ0.01 ns work distributions are still well-defined Gaussian as shown in Fig. 4c in spite of the fact that x i s distributions are more spiky (see the inset in Fig. 4c. IV. DISCUSSION AND CONCLUSION Our theorem illustrates a consideration that the correlation between control parameter λ and time t should be treated carefully. We suggest that control parameter λ should not be considered as a deterministic function of time t. Parameter λ should be a time-independent parameter characterizing perturbed states and time t should be the one driving systems into those perturbed states. The introduction of the double Heaviside function of time t does describe the correct physics of how gradually systems absorb energies from applied potentials over time to evolve into those states. For that reason, their internal energies and entropies are not changed as instantaneously as by a single Heaviside function of time t [5]. The resulting relationship between Hamiltonian-related work W (or W total and W mech is as clear as the expression in Eq. (7. As indicated by the identity Eq. (9 W should not be used to compute FEDs by means of JE. Importantly, our theorem provides a simple proof that mechanical work W mech (or W exp does appear in JE if one can completely control λ in a time-independent manner. The expression of W mech Eq. (8 agrees with the one for stochastic processes [4, 30, 31] in which JE clearly holds. Therefore, the concerns of time-dependent Hamiltonian [5] or heat baths [3] having influences on the validity of JE might be less severe. We have shown that it is possible to construct work distributions from an RC s distributions in both sequential and parallel pulling simulations. An acceptable agreement between the sequential and parallel FEDs in Section II part B suggests that the construction of work distributions can have wider applications in spite of the fact that our theorem has been derived for sequential pulling processes. In terms of computational expense, the parallel pulling simulations are at least 0 times faster than the sequential ones and have the cost equal to performing 100 trajectories with pulling speed v10 Å/ns using the PMF method [13] in which the attractive parallel feature of JE is utilized. In the PMF method a finite number of the original parallel 6

7 pulling from configuration A to configuration B, as illustrated in Fig. 5a, would produce FEDs that are likely pulling-speed dependence [13, 14, 3, 4]. It has been not clear how to guarantee that the pulling protocol passes through rare states that have a major contribution to work distributions in MD simulations, while in experiments the Crooks theorem [7, 8] can provide an important framework for extracting free energies by performing work in forward and reverse processes [3]. Our approach suggests a protocol that requires appropriate overlaps (~1.0 Å among intermediate x i s distributions, as schematically described in Fig. 5b, so that one can carry out pulling simulations in parallel in order to produce trustworthy work distributions. The overlapping requirement resembles the one suggested by Kofke [19, 33] for the initial and final configurations A and B so that the separation of the work distributions in forward and reverse processes is adequate. Our step-wise pulling protocols are similar to the concepts of the Umbrella Sampling method [34-36] in the sense that distributions of x i must be well-defined Gaussian in every window. FIG. 5. Schematic illustration for (a original parallel pulling protocol, (b our proposed parallel pulling protocol. An RC, x o, represents any state without virtual optical traps. A and B are initial and final configurations, respectively. The circles represent sets of states. N is a number of trajectories. Shaded areas are mutual overlapping states. Index i denotes ordered number of pulling simulations from 1 to s, where indices 1 and s also represent the initial and final configurations, respectively. In the limit of very slow pulling, i.e., infinitely small Δλ, Eq. (13 becomes well-known Thermodynamic Integration (TI [1, 8] and Adaptive Biasing Force (ABF equations [14, 7]: λ s λ H(λ,k s ΔF fluct (λ s,λ 1,k dλ f (x λ λ λ dλ, λ 1 where f(x λ λ are averaged forces along a pathway. In the ABF method one has to collect all possible external forces from s windows in comparison with only s-1 windows in our methods. The difference between ΔF fluct (λ s,λ 1,k and ΔF ABF for finite increment Δλ is Δλ f s, where f s is the final force at λλ s. The error of the ABF method is proportional to square root of Δλ f s. Accordingly, at the lowest order of error for a sufficiently large number of available data, ΔF fluct (λ s,λ 1,k and ΔF ABF would be the same. The ABF method is more general than our proposed method (Eq. (13 because one does not need to specify λ λ 1 any forms of applied forces that are adaptive or constrained along pathways. In our method (Eq. (13, no constraint is imposed and the behavior of the averaged accumulating forces (see the inset in Fig. b in SPPs and PPPs are almost the same. Moreover, with appropriate OTs (strong enough string constant k it does not take long relaxation time τ to collect reasonable averaged values of an RC x i along a pathway for free energy calculations (see Fig. 4a. These two results suggest that FEDs can be computed by performing such PPPs to measure x i or averaged accumulating forces along pathways. The disadvantage of our methods is that there is no systematic way of choosing parameters Δλ, τ and k that are correlated and have some effects on the accuracy of FEDs. A strategy to extract the accuracy from our methods is to compare FEDs computed by JE and by Eq. (13. Since JE usually gives overestimated FEDs and Eq. (13 gives underestimated FEDs (see Figs. 3a & 4a. It might not be sufficient to just rely on accuracy of either JE or Eq. (13, although Eq. (13 can be used for acceptable accuracies for many values of spring constant k and relaxation time τ as small as 0.4 ns. Using both equations, one can estimate how many pulling steps needed to have free energy profiles with an accuracy equal to half of the deviation between ΔF fluct and ΔF JE as illustrated in Section II part A and B (see Fig.. For purposes of direct and controllable pulling, our methods might provide a simple strategy for studying local free energy changes. In conclusion, we suggest that mechanical work W mech should be the work used for JE but probably not W[ H/ t]dt in general, even though they become identical in slow pulling protocols. We have shown that it is possible to generate work distributions in step-wise pulling protocols in order to speed up free energy calculations by JE. Our theorem differs from previous studies on JE in a way that we construct work distributions from an RC s distributions. The idea of discretizing pulling processes using a harmonic OT leads to an interesting formula, Eq. (13, that enables us to directly compute FEDs in both parallel and sequential pulling protocols from measurements of x i. We show in a test case that with appropriate choices of parameters Δλ, τ and k, our methods can be used to investigate free energy landscapes of interested systems by various pulling protocols. We acknowledge Dr. Nakano and Dr. Vemparala for valuable discussions. APPENDIX A: SINGLE PULLING STEP Here we explicitly derive Eqs. (4&(5 with the assumption that different canonical ensembles of x 0 and x 1 exist. Averaging of exp(-βw over x s ensemble can be computed as: 7

8 d ( e βw x 1,λ 1,k r 3N 1 dx 1 e β (W + H o ( r 3N 1,x 1 + k (x 1 λ 1 dr 3N 1 dx 1 e β ( H o ( r 3N 1,x 1 + k (x 1 λ 1 k (x 0 λ 1 + H o ( r 3N 1,x 1 dr 3N 1 dx 1 e β Z(λ 1,k e β k (x 0 λ 1 d Z(λ 1,k exp β k (x λ [ 0 1 ( x 1 ] r 3N 1 dx 1 e βh o ( r 3N 1,x 1 ( x1,λ 1,k exp β k (x 0 exp [ βδf(λ 1,k ]. Since a single value of x 0 at time t 0 is constant in the average, we can cancel both sides the factors related to x 0. Thus, we arrive at Eq. (4: exp β k x λ ( 1 1 exp[ βδf(λ 1,k]. (x 1,λ 1,k A similar step can be carried out to derive Eq. (5: dr 3N 1 dx 0 e β (W + H o ( r 3N 1,x 0 ( dr 3N 1 dx 0 e βh o ( r 3N 1,x 0 e βw x 0,k 0 e β k (x 1 λ 1 dr 3N 1 dx 0 e β Z(0 k (x 0 λ 1 + H o ( r 3N 1,x 0 ΔF(λ 1,k β 1 log exp[ β k (x 0 ] (x 0,k 0 APPENDIX B: SERIES OF PULLING STEPS Since Hamiltonian H o ( r 3N 1,x is time-independent, the total work absorbed by the system in a series of pulling steps can be given by: t s H(λ W total 1,λ...λ s, x dt t t s t 1 t 1 t H ( o r 3N 1, x + k s (x λ i θ(t t i 1 θ(t i t dt k [ (x λ 0 1 (x s λ s ] + k [(λ i+1 (x i λ i ] [ ] k (x 0 (x s λ s + [ H(λ i+1,x i H(λ i,x i ]. (B.1. With the assumption that all x i s ensembles are canonical and independent, the average of exp(-βw total can be computed as: e βw total dr 3N 1 dx 0 e βh o ( r 3N 1,x 0 e β k (x 0 λ 1 (x 0,x 1,...,x s Z(0 dr 3N 1 dx i e β ( H o ( Z(λ 1,k...Z(λ,k d [ ] r 3N 1,x i + k (x i λ i e β k (x i λ i+1 (x i λ i r 3N 1 dx s e β ( H o ( r 3N 1,x s + k (x s λ s +β k e (x s λ s Z(λ s,k 1 (B. By definition W mech does not contain x 0 and x s, we can first take the average over these variables: dr 3N 1 dx e βw total e βw mech 0 e β ( H o ( r 3N 1,x 0 + k (x 0 λ 1 x 0,x s Z(0 d r 3N 1 dx s e βh o ( r 3N 1,x 0 Z(λ s,k e β [ F(λ s,k F(λ 1,k ] e βw mech. If we have all possible values of W mech that can be constructed from either FVE curves or the ensembles of x i with i from 1 to s-1, we recover JE by means of the identity (B.: ΔF JE (λ s,λ 1,k F(λ s,k F(λ 1,k β ln e βw mech FVE (B.3 Furthermore, if x i s distributions are perfectly Gaussian (since they belong to the canonical ensembles, we can derive a relationship between FEDs and x i s distributions with the assumption ρ(w mech ~ρ(x 1 ρ(x ρ(x s-1 : dw mech ρ( W mech exp( βw mech exp[ βw mech ] FVE dw mech ρ( W mech (B.4 dx i exp βk (x i /γ i + Δλ(λ i + Δλ dx i exp[ βk (x i ( /γ i ] { [ ] /} exp βkδλ [(1 γ i + λ x i i ]. Δλ ΔF G (λ s,λ 1,k kδλ (1 γ i + kδλ (λ i. APPENDIX C: WORK DISTRIBUTION CONSTRUCTION (B.5 8

9 Given λ 1, λ s, Δλ and all x i s distributions ρ i (x i, we divide a sufficiently large interval, which can contain all values of x i, into K bins with a width of δx. For example, we choose the interval to be 50.0 Å with δx0.001 Å. Similarly, we estimate a range of all mechanical work W mech (W 1, W W s-1 can fall into, for instance, 00.0 kcal/mol with a bin width δw0.01 kcal/mol and a number of bins M are used. As pulling step i is equal to 1, W 1 is zero (see our theorem. As pulling step i is equal to, we construct the distribution of W, here denoted by Ω (W, as the following: for (j1; j<k; j++ { if (ρ 1 (j!0 { // ρ 1 (x 1 are non-zero in small regions around x 1. W kδλ(λ 1 -jδx +Δλ/; w INT(W /δw ; Ω (wρ 1 (j; // all Ω i are initialized to be zero. }} As pulling steps i are greater than, the work distributions for these pulling steps are accumulated as the following: for (i3; i<s-1; i++ { // i should not exceed s-1 because we don t want to compute FED for λλ s +Δλ. for (j1; j<k; j++ { if (ρ i-1 (j! 0 { for (w 1 -M ;w 1 <M; w 1 ++ { if(ω i-1 (w 1!0 { W i w 1 δw + kδλ(λ i-1 -jδx +Δλ/; w INT(W i /δw; Ω i (w + ρ i-1 (j Ω i-1 (w 1 ; }}}}} We observed that INT function gives an unwanted spike appearing in work distributions at W i ~0.0. We smoothed work distributions at this value by assigning Ω i (0 (Ω i (-1+ Ω i (1/. From the distributions of work Ω i, it is straightforward to compute FEDs based on Eq. (11 or Eq. (B.5. The error analysis of these numerical calculations can be found elsewhere [14, 37]. The variance of FEDs can be estimated by σ W Q + β σ W (Q 1, where σ W are the standard deviations of work distributions Ω i (W i and Q is a number of bins which have non-zero Ω i (W i. For example, at the sequential pulling step i35 (see Fig. 1b σ W of the corresponding work distribution is about 10 kcal/mol with Q~0000, then the variance is about 0.7 kcal/mol for temperature T300 K. Hence, the standard deviation of the corresponding FED is about 0.8 kcal/mol. We found that the choices of δx and δw give reasonable estimates for work distributions and their FEDs in all cases of investigated spring constants and relaxation time. [1] C. Jarzynski, Physical Review Letters 78, 690 (1997. [] J. T. Liphardt et al., Biophysical Journal 8, 193A (00. [3] D. Collin et al., Nature 437, 31 ( [4] C. Jarzynskia, The European Physical Journal B - Condensed Matter and Complex Systems 64, 331 (008. [5] J. M. G. Vilar, and J. M. Rubi, Physical Review Letters 100 (008. [6] L. Y. Chen, Journal of Chemical Physics 19 (008. [7] G. E. Crooks, Physical Review E 60, 71 (1999. [8] G. E. Crooks, Physical Review E 61, 361 (000. [9] G. E. Crooks, The Journal of Chemical Physics 130, (009. [10] E. G. D. Cohen, and D. Mauzerall, Journal of Statistical Mechanics-Theory and Experiment (004. [11] C. Jarzynski, Comptes Rendus Physique 8, 495. [1] F. Douarche, and et al., EPL (Europhysics Letters 70, 593 (005. [13] S. Park et al., Journal of Chemical Physics 119, 3559 (003. [14] D. Rodriguez-Gomez, E. Darve, and A. Pohorille, Journal of Chemical Physics 10, 3563 (004. [15] H. Oberhofer, C. Dellago, and P. L. Geissler, Journal of Physical Chemistry B 109, 690 (005. [16] D. A. Hendrix, and C. Jarzynski, The Journal of Chemical Physics 114, 5974 (001. [17] D. K. West, P. D. Olmsted, and E. Paci, The Journal of Chemical Physics 15, (006. [18] S. Park, and K. Schulten, The Journal of Chemical Physics 10, 5946 (004. [19] D. A. Kofke, Molecular Physics 104, 3701 (006. [0] B. Isralewitz, M. Gao, and K. Schulten, Current Opinion in Structural Biology 11, 4 (001. [1] G. Hummer, Journal of Chemical Physics 114, 7330 (001. [] B. Ilan et al., Proteins: Structure, Function, and Bioinformatics 55, 3 (004. [3] T. Bastug et al., The Journal of Chemical Physics 18, (008. [4] T. Bastug, and S. Kuyucak, Chemical Physics Letters 436, 383 (007. [5] I. Kosztin, B. Barz, and L. Janosi, The Journal of Chemical Physics 14, (006. [6] M. ò. Jensen et al., Proceedings of the National Academy of Sciences 99, 6731 (00. [7] E. Darve, and A. Pohorille, Journal of Chemical Physics 115, 9169 (001. [8] P. Kollman, Chemical Reviews 93, 395 (1993. [9] R. W. Zwanzig, The Journal of Chemical Physics, 140 (1954. [30] G. Crooks, Journal of Statistical Physics 90, 1481 (1998. [31] C. Jarzynski, Physical Review E 56, 5018 (1997. [3] D. A. Cohen, and T. Z. Lys, Journal of Accounting & Economics 36, 147 (003. [33] N. D. Lu, D. A. Kofke, and T. B. Woolf, Journal of Computational Chemistry 5, 8 (004. [34] M. Souaille, and B. Roux, Computer Physics Communications 135, 40 (001. [35] G. M. Torrie, and J. P. Valleau, Journal of Computational Physics 3, 187 (

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