Predicting Rare Events in Molecular Dynamics

Nuclear Quantum Effects on the Nature of Hydroboration Selectivity: Experimental Effects of First-Collision Tunneling

The understanding of selectivity in reactions exhibiting nonstatistical dynamics is impeded by the limitations of trajectory studies with regard to nuclear quantum effects, especially tunneling. We described here the use of ring-polymer molecular dynamics (RPMD) to account for an unusual regiochemical isotope effect on the regioselectivity of hydroborations of alkenes with BH3/BD3. RPMD is able to account for the experimental observation, while statistical approaches and classical trajectories fail. The combination of experiment and RPMD trajectories suggests that tunneling in the initial collision of reactants is the major source of the nonstatistical selectivity.

DeepWEST: Deep Learning of Kinetic Models with the Weighted Ensemble Simulation Toolkit for Enhanced Sampling

Recent advances in computational power and algorithms have enabled molecular dynamics (MD) simulations to reach greater time scales. However, for observing conformational transitions associated with biomolecular processes, MD simulations still have limitations. Several enhanced sampling techniques seek to address this challenge, including the weighted ensemble (WE) method, which samples transitions between metastable states using many weighted trajectories to estimate kinetic rate constants. However, initial sampling of the potential energy surface has a significant impact on the performance of WE, i.e., convergence and efficiency. We therefore introduce deep-learned kinetic modeling approaches that extract statistically relevant information from short MD trajectories to provide a well-sampled initial state distribution for WE simulations. This hybrid approach overcomes any statistical bias to the system, as it runs short unbiased MD trajectories and identifies meaningful metastable states of the system. It is shown to provide a more refined free energy landscape closer to the steady state that could efficiently sample kinetic properties such as rate constants.

Transition rate theory, spectral analysis, and reactive paths

The kinetics of a dynamical system dominated by two metastable states is examined from the perspective of the activated-dynamics reactive flux formalism, Markov state eigenvalue spectral decomposition, and committor-based transition path theory. Analysis shows that the different theoretical formulations are consistent, clarifying the significance of the inherent microscopic lag-times that are implicated, and that the most meaningful one-dimensional reaction coordinate in the region of the transition state is along the gradient of the committor in the multidimensional subspace of collective variables. It is shown that the familiar reactive flux activated dynamics formalism provides an effective route to calculate the transition rate in the case of a narrow sharp barrier but much less so in the case of a broad flat barrier. In this case, the standard reactive flux correlation function decays very slowly to the plateau value that corresponds to the transmission coefficient. Treating the committor function as a reaction coordinate does not alleviate all issues caused by the slow relaxation of the reactive flux correlation function. A more efficient activated dynamics simulation algorithm may be achieved from a modified reactive flux weighted by the committor. Simulation results on simple systems are used to illustrate the various conceptual points.

Perspective: Defining and quantifying the role of dynamics in enzyme catalysis

Enzymes control chemical reactions that are key to life processes, and allow them to take place on the time scale needed for synchronization between the relevant reaction cycles. In addition to general interest in their biological roles, these proteins present a fundamental scientific puzzle, since the origin of their tremendous catalytic power is still unclear. While many different hypotheses have been put forward to rationalize this, one of the proposals that has become particularly popular in recent years is the idea that dynamical effects contribute to catalysis. Here, we present a critical review of the dynamical idea, considering all reasonable definitions of what does and does not qualify as a dynamical effect. We demonstrate that no dynamical effect (according to these definitions) has ever been experimentally shown to contribute to catalysis. Furthermore, the existence of non-negligible dynamical contributions to catalysis is not supported by consistent theoretical studies. Our review is aimed, in part, at readers with a background in chemical physics and biophysics, and illustrates that despite a substantial body of experimental effort, there has not yet been any study that consistently established a connection between an enzyme's conformational dynamics and a significant increase in the catalytic contribution of the chemical step. We also make the point that the dynamical proposal is not a semantic issue but a well-defined scientific hypothesis with well-defined conclusions.

Monte Carlo simulations of single- and multistep enzyme-catalyzed reaction sequences: effects of diffusion, cell size, enzyme fluctuations, colocalization, and segregation

Following the discovery of slow fluctuations in the catalytic activity of an enzyme in single-molecule experiments, it has been shown that the classical Michaelis-Menten (MM) equation relating the average enzymatic velocity and the substrate concentration may hold even for slowly fluctuating enzymes. In many cases, the average velocity is that given by the MM equation with time-averaged values of the fluctuating rate constants and the effect of enzyme fluctuations is simply averaged out. The situation is quite different for a sequence of reactions. For colocalization of a pair of enzymes in a sequence to be effective in promoting reaction, the second must be active when the first is active or soon after. If the enzymes are slowly varying and only rarely active, the product of the first reaction may diffuse away before the second enzyme is active, and colocalization may have little value. Even for single-step reactions the interplay of reaction and diffusion with enzyme fluctuations leads to added complexities, but for multistep reactions the interplay of reaction and diffusion, cell size, compartmentalization, enzyme fluctuations, colocalization, and segregation is far more complex than for single-step reactions. In this paper, we report the use of stochastic simulations at the level of whole cells to explore, understand, and predict the behavior of single- and multistep enzyme-catalyzed reaction systems exhibiting some of these complexities. Results for single-step reactions confirm several earlier observations by others. The MM relationship, with altered constants, is found to hold for single-step reactions slowed by diffusion. For single-step reactions, the distribution of enzymes in a regular grid is slightly more effective than a random distribution. Fluctuations of enzyme activity, with average activity fixed, have no observed effects for simple single-step reactions slowed by diffusion. Two-step sequential reactions are seen to be slowed by segregation of the enzymes for each step, and results of the calculations suggest limits for cell size. Colocalization of enzymes for a two-step sequence is seen to promote reaction, and rates fall rapidly with increasing distance between enzymes. Low frequency fluctuations of the activities of colocalized enzymes, with average activities fixed, can greatly reduce reaction rates for sequential reactions.

Forward flux sampling for rare event simulations

Rare events are ubiquitous in many different fields, yet they are notoriously difficult to simulate because few, if any, events are observed in a conventional simulation run. Over the past several decades, specialized simulation methods have been developed to overcome this problem. We review one recently developed class of such methods, known as forward flux sampling. Forward flux sampling uses a series of interfaces between the initial and final states to calculate rate constants and generate transition paths for rare events in equilibrium or nonequilibrium systems with stochastic dynamics. This review draws together a number of recent advances, summarizes several applications of the method and highlights challenges that remain to be overcome.

Efficiency analysis of reaction rate calculation methods using analytical models I: The two-dimensional sharp barrier

We analyze the efficiency of different methods for the calculation of reaction rates in the case of a simple two-dimensional analytical benchmark system. Two classes of methods are considered: the first is based on the free energy calculation along a reaction coordinate and the calculation of the transmission coefficient, the second on the sampling of dynamical pathways. We give scaling rules for how this efficiency depends on barrier height and width, and we hand out simple optimization rules for the method-specific parameters. We show that the path sampling methods, using the transition interface sampling technique, become exceedingly more efficient than the others when the reaction coordinate is not the optimal one.

Army ants algorithm for rare event sampling of delocalized nonadiabatic transitions by trajectory surface hopping and the estimation of sampling errors by the bootstrap method

The most widely used algorithm for Monte Carlo sampling of electronic transitions in trajectory surface hopping (TSH) calculations is the so-called anteater algorithm, which is inefficient for sampling low-probability nonadiabatic events. We present a new sampling scheme (called the army ants algorithm) for carrying out TSH calculations that is applicable to systems with any strength of coupling. The army ants algorithm is a form of rare event sampling whose efficiency is controlled by an input parameter. By choosing a suitable value of the input parameter the army ants algorithm can be reduced to the anteater algorithm (which is efficient for strongly coupled cases), and by optimizing the parameter the army ants algorithm may be efficiently applied to systems with low-probability events. To demonstrate the efficiency of the army ants algorithm, we performed atom-diatom scattering calculations on a model system involving weakly coupled electronic states. Fully converged quantum mechanical calculations were performed, and the probabilities for nonadiabatic reaction and nonreactive deexcitation (quenching) were found to be on the order of 10(-8). For such low-probability events the anteater sampling scheme requires a large number of trajectories ( approximately 10(10)) to obtain good statistics and converged semiclassical results. In contrast by using the new army ants algorithm converged results were obtained by running 10(5) trajectories. Furthermore, the results were found to be in excellent agreement with the quantum mechanical results. Sampling errors were estimated using the bootstrap method, which is validated for use with the army ants algorithm.

Transition rate prefactors for systems of many degrees of freedom

When a minimum on the potential energy surface is surrounded by multiple saddle points with similar energy barriers, the transition pathways with greater prefactors are more important than those that have similar energy barriers but smaller prefactors. In this paper, we present a theoretical formulation for the prefactors, computing the probabilities for transition paths from a minimum to its surrounding saddle points. We apply this formulation to a system of 2 degrees of freedom and a system of 14 degrees of freedom. The first is Brownian motion in a two-dimensional potential whose global anharmonicities play a dominant role in determining the transition rates. The second is a Lennard-Jones (LJ) cluster of seven particles in two dimensions. Low lying transition states of the LJ cluster, which can be reached directly from a minimum without passing through another minimum, are identified without any presumption of their characteristics nor of the product states they lead to. The probabilities are computed for paths going from an equilibrium ensemble of states near a given minimum to the surrounding transition states. These probabilities are directly related to the prefactors in the rate formula. This determination of the rate prefactors includes all anharmonicities, near or far from transition states, which are pertinent in the very sophisticated energy landscape of LJ clusters and in many other complex systems.

Transition state theory: Variational formulation, dynamical corrections, and error estimates

Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient kappaS of the TST dividing surface are given, and it is shown that the relative error on kappaS scales as 1/square root(kappaS) when kappaS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient kappaS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise kappaS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum kappaS, but this coefficient may still be very small).

Evaluation of canonical and microcanonical nonadiabatic reaction rate constants by using the Zhu–Nakamura formulas

We consider a problem of calculating both thermal and microcanonical rate constants for nonadiabatic chemical reactions. Instead of using the conventional transition state theory, we use a generalized seam surface and introduce a concept of a coordinate dependent effective nonadiabatic transition probability based on the Zhu-Nakamura theory which can treat the nonadiabatic tunneling properly. The present approach can be combined with Monte Carlo method so as to be applicable to chemical reactions in complicated systems. The method is demonstrated to work well in wide energy and temperature range. Numerical tests also show that it is very essential for accurate evaluation of the thermal rate constant to use the generalized seam surface and take into account the nonadiabatic tunneling effect.

Transition path sampling: throwing ropes over rough mountain passes, in the dark

This article reviews the concepts and methods of transition path sampling. These methods allow computational studies of rare events without requiring prior knowledge of mechanisms, reaction coordinates, and transition states. Based upon a statistical mechanics of trajectory space, they provide a perspective with which time dependent phenomena, even for systems driven far from equilibrium, can be examined with the same types of importance sampling tools that in the past have been applied so successfully to static equilibrium properties.