Weighted-ensemble Brownian dynamics simulations for protein association reactions

Inertial effects in diffusion-limited reactions

Diffusion-limited reactions are commonly found in biochemical processes such as enzyme catalysis, colloid and protein aggregation and binding between different macromolecules in cells. Usually, such reactions are modeled within the Smoluchowski framework by considering purely diffusive boundary problems. However, inertial effects are not always negligible in real biological or physical media on typical observation time frames. This is all the more so for non-bulk phenomena involving physical boundaries, that introduce additional time and space constraints. In this paper, we present and test a novel numerical scheme, based on event-driven Brownian dynamics, that allows us to explore a wide range of velocity relaxation times, from the purely diffusive case to the underdamped regime. We show that our algorithm perfectly reproduces the solution of the Fokker-Planck problem with absorbing boundary conditions in all the regimes considered and is thus a good tool for studying diffusion-guided reactions in complex biological environments.

The "weighted ensemble" path sampling method is statistically exact for a broad class of stochastic processes and binning procedures

The "weighted ensemble" method, introduced by Huber and Kim [Biophys. J. 70, 97 (1996)], is one of a handful of rigorous approaches to path sampling of rare events. Expanding earlier discussions, we show that the technique is statistically exact for a wide class of Markovian and non-Markovian dynamics. The derivation is based on standard path-integral (path probability) ideas, but recasts the weighted-ensemble approach as simple "resampling" in path space. Similar reasoning indicates that arbitrary nonstatic binning procedures, which merely guide the resampling process, are also valid. Numerical examples confirm the claims, including the use of bins which can adaptively find the target state in a simple model.

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.

Zipping and unzipping of adenylate kinase: atomistic insights into the ensemble of open<-->closed transitions

Adenylate kinase (AdK), a phosphotransferase enzyme, plays an important role in cellular energy homeostasis. It undergoes a large conformational change between an open and a closed state, even in the absence of substrate. We investigate the apo-AdK transition at the atomic level both with free-energy calculations and with our new dynamic importance sampling (DIMS) molecular dynamics method. DIMS is shown to sample biologically relevant conformations as verified by comparing an ensemble of hundreds of DIMS transitions to AdK crystal structure intermediates. The simulations reveal in atomic detail how hinge regions partially and intermittently unfold during the transition. Conserved salt bridges are seen to have important structural and dynamic roles; in particular, four ionic bonds that open in a sequential, zipper-like fashion and, thus, dominate the free-energy landscape of the transition are identified. Transitions between the closed and open conformations only have to overcome moderate free-energy barriers. Unexpectedly, the closed state and the open state encompass broad free-energy basins that contain conformations differing in domain hinge motions by up to 40 degrees . The significance of these extended states is discussed in relation to recent experimental Förster resonance energy transfer measurements. Taken together, these results demonstrate how a small number of cooperative key interactions can shape the overall dynamics of an enzyme and suggest an "all-or-nothing" mechanism for the opening and closing of AdK. Our efficient DIMS molecular dynamics computer simulation approach can provide a detailed picture of a functionally important macromolecular transition and thus help to interpret and suggest experiments to probe the conformational landscape of dynamic proteins such as AdK.

Transition path sampling and forward flux sampling. Applications to biological systems

The last decade has seen a rapid growth in the number of simulation methods and applications dealing with the sampling of transition pathways of rare nanoscale events. Such studies are crucial, for example, for understanding the mechanism and kinetics of conformational transitions and enzymatic events associated with the function of biomolecules. In this review, a broad account of transition path sampling approaches is provided, starting from the general concepts, progressing to the specific principles that underlie some of the most important methods, and eventually singling out the so-called forward flux sampling method for a more detailed description. This is done because forward flux sampling, despite its appealing simplicity and potential efficiency, has thus far received limited attention from practitioners. While path sampling methods have a widespread application to many types of rare transitional events, here only recent applications involving biomolecules are reviewed, including isomerization, protein folding, and enzyme catalysis.

Nonequilibrium umbrella sampling in spaces of many order parameters

We recently introduced an umbrella sampling method for obtaining nonequilibrium steady-state probability distributions projected onto an arbitrary number of coordinates that characterize a system (order parameters) [A. Warmflash, P. Bhimalapuram, and A. R. Dinner, J. Chem. Phys. 127, 154112 (2007)]. Here, we show how our algorithm can be combined with the image update procedure from the finite-temperature string method for reversible processes [E. Vanden-Eijnden and M. Venturoli, "Revisiting the finite temperature string method for calculation of reaction tubes and free energies," J. Chem. Phys. (in press)] to enable restricted sampling of a nonequilibrium steady state in the vicinity of a path in a many-dimensional space of order parameters. For the study of transitions between stable states, the adapted algorithm results in improved scaling with the number of order parameters and the ability to progressively refine the regions of enforced sampling. We demonstrate the algorithm by applying it to a two-dimensional model of driven Brownian motion and a coarse-grained (Ising) model for nucleation under shear. It is found that the choice of order parameters can significantly affect the convergence of the simulation; local magnetization variables other than those used previously for sampling transition paths in Ising systems are needed to ensure that the reactive flux is primarily contained within a tube in the space of order parameters. The relation of this method to other algorithms that sample the statistics of path ensembles is discussed.

Reaction Brownian dynamics and the effect of spatial fluctuations on the gain of a push-pull network

Brownian Dynamics algorithms have been widely used for simulating systems in soft-condensed matter physics. In recent times, their application has been extended to the simulation of coarse-grained models of biochemical networks. In these models, components move by diffusion and interact with one another upon contact. However, when reactions are incorporated into a Brownian dynamics algorithm, care must be taken to avoid violations of the detailed-balance rule, which would introduce systematic errors in the simulation. We present a Brownian dynamics algorithm for simulating reaction-diffusion systems that rigorously obeys detailed balance for equilibrium reactions. By comparing the simulation results to exact analytical results for a bimolecular reaction, we show that the algorithm correctly reproduces both equilibrium and dynamical quantities. We apply our scheme to a "push-pull" network in which two antagonistic enzymes covalently modify a substrate. Our results highlight that spatial fluctuations of the network components can strongly reduce the gain of the response of a biochemical network.

Efficient and verified simulation of a path ensemble for conformational change in a united-residue model of calmodulin

The computational sampling of rare, large-scale, conformational transitions in proteins is a well appreciated challenge-for which a number of potentially efficient path-sampling methodologies have been proposed. Here, we study a large-scale transition in a united-residue model of calmodulin using the "weighted ensemble" (WE) approach of Huber and Kim. Because of the model's relative simplicity, we are able to compare our results with brute-force simulations. The comparison indicates that the WE approach quantitatively reproduces the brute-force results, as assessed by considering (i) the reaction rate, (ii) the distribution of event durations, and (iii) structural distributions describing the heterogeneity of the paths. Importantly, the WE method is readily applied to more chemically accurate models, and by studying a series of lower temperatures, our results suggest that the WE method can increase efficiency by orders of magnitude in more challenging systems.

A skewed-momenta method to efficiently generate conformational-transition trajectories

We present a novel computational method, the skewed-momenta method (Skew'M), which applies a bias to the Maxwell distribution of initial momenta used to generate ensembles of trajectories. As a result, conformational transitions are accentuated and kinetic properties are calculated more effectively. The connection to the related puddle jumping method is discussed. A reweighting scheme permits the exact calculation of kinetic properties. Applications are presented for the rapid calculation of rate constants for molecular isomerization, and for the efficient reconstruction of free-energy profiles using a straightforward modification of the Jarzynski identity.

Extending molecular dynamics time scales with milestoning: example of complex kinetics in a solvated peptide

A recently introduced computational algorithm to extend time scales of atomically detailed simulations is illustrated. The algorithm, milestoning, is based on partitioning the dynamics to a sequence of trajectories between "milestones" and constructing a non-Markovian model for the motion along a reaction coordinate. The kinetics of a conformational transition in a blocked alanine is computed and shown to be accurate, more efficient than straightforward molecular dynamics by a factor of about 9, and nonexponential. A general scaling argument predicts a linear speedup with the number of milestones for diffusive processes and an exponential speedup for transitions over barriers. The algorithm is also trivial to parallelize. As a side result, milestoning also produces the free energy profile along the reaction coordinate and is able to describe nonequilibrium motions along one (or a few) degrees of freedom.

Transition-event durations in one-dimensional activated processes

Despite their importance in activated processes, transition-event durations--which are much shorter than first passage times--have not received a complete theoretical treatment. The authors therefore study the distribution rhob(t) of durations of transition events over a barrier in a one-dimensional system undergoing overdamped Langevin dynamics. The authors show that rhob(t) is determined by a Fokker-Planck equation with absorbing boundary conditions and obtain a number of results, including (i) the analytic form of the asymptotic short-time transient behavior, which is universal and independent of the potential function; (ii) the first nonuniversal correction to the short-time behavior leading to an estimate of a key physical time scale; (iii) following previous work, a recursive formulation for calculating, exactly, all moments of rhob based solely on the potential function-along with approximations for the distribution based on a small number of moments; and (iv) a high-barrier approximation to the long-time (t-->infinity) behavior of rhob(t). The authors also find that the mean event duration does not depend simply on the barrier-top frequency (curvature) but is sensitive to details of the potential. All of the analytic results are confirmed by transition-path-sampling simulations implemented in a novel way. Finally, the authors discuss which aspects of the duration distribution are expected to be general for more complex systems.

Introduction to the membrane protein reviews: the interplay of structure, dynamics, and environment in membrane protein function

In our review, we introduce an organizational scheme for membrane protein function. It is the relationship between structure, dynamics, and environment that endows the membrane and its constituents with remarkable sensitivity and robustness. Our understanding begins with landmark advances like those presented in the following chapters. Membrane proteins are notoriously difficult to study, and so the work presented here on the ADP/ATP carrier [Nury et al. ( 2 )], rhodopsin [Palczewski ( 24 )], and the cytochrome b6f complex [Cramer et al. ( 35 )] represents incredible progress in this now blossoming field.

Forward flux sampling-type schemes for simulating rare events: Efficiency analysis

We analyze the efficiency of several simulation methods which we have recently proposed for calculating rate constants for rare events in stochastic dynamical systems in or out of equilibrium. We derive analytical expressions for the computational cost of using these methods and for the statistical error in the final estimate of the rate constant for a given computational cost. These expressions can be used to determine which method to use for a given problem, to optimize the choice of parameters, and to evaluate the significance of the results obtained. We apply the expressions to the two-dimensional nonequilibrium rare event problem proposed by Maier and Stein [Phys. Rev. E 48, 931 (1993)]. For this problem, our analysis gives accurate quantitative predictions for the computational efficiency of the three methods.

Robust biased Brownian dynamics for rate constant calculation

A reaction probability is required to calculate the rate constant of a diffusion-dominated reaction. Due to the complicated geometry and potentially high dimension of the reaction probability problem, it is usually solved by a Brownian dynamics simulation, also known as a random walk or path integral method, instead of solving the equivalent partial differential equation by a discretization method. Building on earlier work, this article completes the development of a robust importance sampling algorithm for Brownian dynamics-i.e., biased Brownian dynamics with weight control-to overcome the high energy and entropy barriers in biomolecular association reactions. The biased Brownian dynamics steers sampling by a bias force, and the weight control algorithm controls sampling by a target weight. This algorithm is optimal if the bias force and the target weight are constructed from the solution of the reaction probability problem. In reality, an approximate reaction probability has to be used to construct the bias force and the target weight. Thus, the performance of the algorithm depends on the quality of the approximation. Given here is a method to calculate a good approximation, which is based on the selection of a reaction coordinate and the variational formulation of the reaction probability problem. The numerically approximated reaction probability is shown by computer experiments to give a factor-of-two speedup over the use of a purely heuristic approximation. Also, the fully developed method is compared to unbiased Brownian dynamics. The tests for human superoxide dismutase, Escherichia coli superoxide dismutase, and antisweetener antibody NC6.8, show speedups of 17, 35, and 39, respectively. The test for reactions between two model proteins with orientations shows speedups of 2578 for one set of configurations and 3341 for another set of configurations.

Fast monte carlo simulations and singularities in the probability distributions of nonequilibrium systems

A numerical technique is introduced that reduces exponentially the time required for Monte Carlo simulations of nonequilibrium systems. Results for the quasistationary probability distribution in two model systems are compared with the asymptotically exact theory in the limit of extremely small noise intensity. Singularities of the nonequilibrium distributions are revealed by the simulations.

Biomolecular diffusional association

The primary method for the computational study of biomolecular diffusional association is Brownian dynamics. Recent work has seen advances in the efficiency of computing association rates and in the accuracy of simulation models. New areas to which Brownian dynamics has been applied include protein polymerisation and protein adsorption to a surface. There has recently been particularly intense study of protein-protein association, and Brownian dynamics, together with other theoretical and experimental approaches, has led to new insights into the determinants of protein-protein binding kinetics.

Open channel block of HERG K(+) channels by vesnarinone

Vesnarinone, a cardiotonic agent, blocks I(Kr) and, unlike other I(Kr) blockers, produces a frequency-dependent prolongation of action potential duration (APD). To elucidate the mechanisms, we studied the effects of vesnarinone on HERG, the cloned human I(Kr) channel, heterologously expressed in Xenopus laevis oocytes. Vesnarinone caused a concentration-dependent inhibition of HERG currents with an IC(50) value of 17.7 +/- 2.5 microM at 0 mV (n = 6). When HERG was coexpressed with the beta-subunit MiRP1, a similar potency for block was measured (IC(50): 15.0 +/- 3.0 microM at 0 mV, n = 5). Tonic block of the HERG channel current was minimal (<5% at 30 microM, n = 5). The rate of onset of block and the steady-state value for block of current were not significantly different for test potentials ranging from -40 to +40 mV [time constant (tau) = 372 +/- 76 ms at +40 mV, n = 4]. Recovery from block at -60, -90, and -120 mV was not significantly different (tau = 8.5 +/- 1.5 s at -90 mV, n = 4). Vesnarinone produced similar effects on inactivation-removed mutant (G628C/S631C) HERG channels. The IC(50) value was 10.7 +/- 3.7 microM at 0 mV (n = 5), and the onset and recovery from block of current findings were similar to those of wild-type HERG. Amino acids important for the binding of vesnarinone were identified using alanine-scanning mutagenesis of residues believed to line the inner cavity of the HERG channel. Six important residues were identified, including G648, F656, and V659 located in the S6 domain and T623, S624, and V625 located at the base of the pore helix. These residues are similar but not identical to those determined previously for MK-499, an antiarrhythmic drug. In conclusion, vesnarinone preferentially blocks open HERG channels, with little effect on channels in the rested or inactivated state. These actions may contribute to the favorable frequency-dependent prolongation in APD.

Effect of anisotropic reactivity on the rate of diffusion-controlled reactions: comparative analysis of the models of patches and hemispheres

A comparative analysis of two models of anisotropic reactivity in bimolecular diffusion-controlled reaction kinetics is presented. One is the conventional model of reactive patches (MRP), where the surface of a molecule is assumed to be reactive over a certain region (circular patch) with the rest of the surface being inert. Another one is the model of reactive hemispheres (MRH), assuming that a molecule is reactive within a certain distance from a point on its surface. The accuracy of the known and newly derived simple analytical expressions for the reaction rate is tested by comparison with the simulation results obtained by the original Brownian dynamics method. These formulas prove to be quite accurate in the practically important limit of strong anisotropy corresponding to small size of the reactive patches or hemispheres. Numerical calculations confirm earlier predictions that the MRP rates are much smaller than the MRH rates for the same radii of the reactive regions, especially in the case where both reacting molecules are anisotropic.

Kinetics of diffusion-assisted reactions in microheterogeneous systems

This review is focused on the basic theory of diffusion-assisted reactions in microheterogeneous systems, from porous solids to self-organized colloids and biomolecules. Rich kinetic behaviors observed experimentally are explained in a unified fashion using simple concepts of competing distance and time scales of the reaction and the embedding structure. We mainly consider pseudo-first-order reactions, such as luminescence quenching, described by the Smoluchowski type of equation for the reactant pair distribution function with a sink term defined by the reaction mechanism. Microheterogeneity can affect the microscopic rate constant. It also enters the evolution equation through various spatial constraints leading to complicated boundary conditions and, possibly, to the reduction of dimensionality of the diffusion space. The reaction coordinate and diffusive motion along this coordinate are understood in a general way, depending on the problem at hand. Thus, the evolution operator can describe translational and rotational diffusion of molecules in a usual sense, it can be a discrete random walk operator when dealing with hopping of adsorbates in solids, or it can correspond to conformational fluctuations in proteins. Mathematical formulation is universal but physical consequences can be different. Understanding the principal features of reaction kinetics in microheterogeneous systems enables one to extract important structural and dynamical information about the host environments by analyzing suitably designed experiments, it helps building effective strategies for computer simulations, and ultimately opens possibilities for designing systems with controllable reactivity properties.

Bimolecular reaction simulation using Weighted Ensemble Brownian dynamics and the University of Houston Brownian Dynamics program

We discuss here the implementation of the Weighted Ensemble Brownian (WEB) dynamics algorithm of Huber and Kim in the University of Houston Brownian Dynamics (UHBD) suite of programs and its application to bimolecular association problems. WEB dynamics is a biased Brownian dynamics (BD) algorithm that is more efficient than the standard Northrup-Allison-McCammon (NAM) method in cases where reaction events are infrequent because of intervening free energy barriers. Test cases reported here include the Smoluchowski rate for association of spheres, the association of the enzyme copper-zinc superoxide dismutase with superoxide anion, and the binding of the superpotent sweetener N-(p-cyanophenyl)-N'-(diphenylmethyl)-guanidinium acetic acid to a monoclonal antibody fragment, NC6.8. Our results show that the WEB dynamics algorithm is a superior simulation method for enzyme-substrate reaction encounters with large free energy barriers.

Biased Brownian dynamics for rate constant calculation

An enhanced sampling method-biased Brownian dynamics-is developed for the calculation of diffusion-limited biomolecular association reaction rates with high energy or entropy barriers. Biased Brownian dynamics introduces a biasing force in addition to the electrostatic force between the reactants, and it associates a probability weight with each trajectory. A simulation loses weight when movement is along the biasing force and gains weight when movement is against the biasing force. The sampling of trajectories is then biased, but the sampling is unbiased when the trajectory outcomes are multiplied by their weights. With a suitable choice of the biasing force, more reacted trajectories are sampled. As a consequence, the variance of the estimate is reduced. In our test case, biased Brownian dynamics gives a sevenfold improvement in central processing unit (CPU) time with the choice of a simple centripetal biasing force.

Regulation of milk protein gene expression

Studies using both transgenic mice and transfected mammary epithelial cells have established that composite response elements containing multiple binding sites for several transcription factors mediate the hormonal and developmental regulation of milk protein gene expression. Activation of signal transduction pathways by lactogenic hormones and cell-substratum interactions activate transcription factors and change chromatin structure and milk protein gene expression. The casein promoters have binding sites for signal transducers and activators of transcription 5, Yin Yang 1, CCAAT/enhancer binding protein, and the glucocorticoid receptor. The whey protein gene promoters have binding sites for nuclear factor I, as well as the glucocorticoid receptor and the signal transducers and activators of transcription 5. The functional importance of some of these factors in mammary gland development and milk protein gene expression has been elucidated by studying mice in which some of these factors have been deleted.

Brownian dynamics simulations of protein folding: access to milliseconds time scale and beyond

Protein folding occurs on a time scale ranging from milliseconds to minutes for a majority of proteins. Computer simulation of protein folding, from a random configuration to the native structure, is nontrivial owing to the large disparity between the simulation and folding time scales. As an effort to overcome this limitation, simple models with idealized protein subdomains, e.g., the diffusion-collision model of Karplus and Weaver, have gained some popularity. We present here new results for the folding of a four-helix bundle within the framework of the diffusion-collision model. Even with such simplifying assumptions, a direct application of standard Brownian dynamics methods would consume 10,000 processor-years on current supercomputers. We circumvent this difficulty by invoking a special Brownian dynamics simulation. The method features the calculation of the mean passage time of an event from the flux overpopulation method and the sampling of events that lead to productive collisions even if their probability is extremely small (because of large free-energy barriers that separate them from the higher probability events). Using these developments, we demonstrate that a coarse-grained model of the four-helix bundle can be simulated in several days on current supercomputers. Furthermore, such simulations yield folding times that are in the range of time scales observed in experiments.

Theory of biomolecular recognition

Specific, noncovalent binding of biomolecules can only be understood by considering structural, thermodynamic, and kinetic issues. The theoretical foundations for such analyses have been clarified in the past year. Computational techniques for both particle-based and continuum models continue to improve and to yield useful insights into an ever wider range of biomolecular systems.

Brownian dynamics simulation of protein-protein diffusional encounter

Protein association events are ubiquitous in biological systems. Some protein associations and subsequent responses are diffusion controlled in vivo. Hence, it is important to be able to compute bimolecular diffusional association rates for proteins. The Brownian dynamics simulation methodology may be used to simulate protein-protein encounter, compute association rates, and examine their dependence on protein mutation and the nature of the physical environment (e.g., as a function of ionic strength or viscosity). Here, the theory for Brownian dynamics simulations is described, and important methodological aspects, particularly pertaining to the correct modeling of electrostatic forces and definition of encounter complex formation, are highlighted. To illustrate application of the method, simulations of the diffusional encounter of the extracellular ribonuclease, barnase, and its intracellular inhibitor, barstar, are described. This shows how experimental rates for a series of mutants and the dependence of rates on ionic strength can be reproduced well by Brownian dynamics simulations. Potential future uses of the Brownian dynamics method for investigating protein-protein association are discussed.

Rapid inactivation determines the rectification and [K+]o dependence of the rapid component of the delayed rectifier K+ current in cardiac cells

Two characteristic features of the rapid component of the cardiac delayed rectifier current (IKr) are prominent inward rectification and an unexpected reduction in activating current with decreased [K+]o. Similar features are observed with heterologous expression of HERG, the gene thought to encode the channel carrying IKr, moreover, recent studies indicate that the mechanism underlying rectification of HERG current is the inactivation that channels rapidly undergo during depolarizing pulses. The present studies were designed to determine the mechanism of IKr rectification and [K+]o sensitivity in the mouse atrial myocyte cell line, AT-1 cells. Reducing [Mg2+]i to 0, which reverses inward rectification of some K+ channels, did not alter IKr current-voltage relationships, although it did decrease sensitivity to the IKr blockers dofetilide and quinidine 2- to 5-fold. To determine the presence and extent of fast inactivation of IKr in AT-1 cells, a brief hyperpolarizing pulse (20 ms to -120 mV) was applied during long depolarizations. Immediately after this pulse, a very large outward current that decayed rapidly to the previous activating current baseline was observed. This outward current component was blocked by the IKr-specific inhibitor dofetilide, indicating that it represented recovery from fast inactivation during the hyperpolarizing step, with fast reinactivation during the return to depolarized potential. With removal of inactivation using this approach, current-voltage relationships for IKr ([K+]o, 1 to 20 mmol/L) were linar and reversed close to the predicted Nernst potential for K+. In addition, decreased [K+]o decreased the time constants for open-->inactivated and inactivated-->open transitions. Thus, in these cardiac myocytes, as with heterologously expressed HERG, IKr undergoes fast inactivation that determines its characteristic inward rectification. These studies demonstrate that the mechanism underlying decreased activating current observed at low [K+]o is more extensive fast inactivation.

Theory and simulation of the time-dependent rate coefficients of diffusion-influenced reactions

A general formalism is developed for calculating the time-dependent rate coefficient k(t) of an irreversible diffusion-influenced reaction. This formalism allows one to treat most factors that affect k(t), including rotational Brownian motion and conformational gating of reactant molecules and orientation constraint for product formation. At long times k(t) is shown to have the asymptotic expansion k(infinity)[1 + k(infinity) (pie Dt)-1/2 /4 pie D + ...], where D is the relative translational diffusion constant. An approximate analytical method for calculating k(t) is presented. This is based on the approximation that the probability density of the reactant pair in the reactive region keeps the equilibrium distribution but with a decreasing amplitude. The rate coefficient then is determined by the Green function in the absence of chemical reaction. Within the framework of this approximation, two general relations are obtained. The first relation allows the rate coefficient for an arbitrary amplitude of the reactivity to be found if the rate coefficient for one amplitude of the reactivity is known. The second relation allows the rate coefficient in the presence of conformational gating to be found from that in the absence of conformational gating. The ratio k(t)/k(0) is shown to be the survival probability of the reactant pair at time t starting from an initial distribution that is localized in the reactive region. This relation forms the basis of the calculation of k(t) through Brownian dynamics simulations. Two simulation procedures involving the propagation of nonreactive trajectories initiated only from the reactive region are described and illustrated on a model system. Both analytical and simulation results demonstrate the accuracy of the equilibrium-distribution approximation method.