Eric Vanden-Eijnden

String method for the study of rare events

We present an efficient method for computing the transition pathways, free energy barriers, and transition rates in complex systems with relatively smooth energy landscapes. The method proceeds by evolving strings, i.e., smooth curves with intrinsic parametrization whose dynamics takes them to the most probable transition path between two metastable regions in configuration space. Free energy barriers and transition rates can then be determined by a standard umbrella sampling around the string. Applications to Lennard-Jones cluster rearrangement and thermally induced switching of a magnetic film are presented.

The heterogeneous multiscale method

The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.

Constructing the equilibrium ensemble of folding pathways from short off-equilibrium simulations

Characterizing the equilibrium ensemble of folding pathways, including their relative probability, is one of the major challenges in protein folding theory today. Although this information is in principle accessible via all-atom molecular dynamics simulations, it is difficult to compute in practice because protein folding is a rare event and the affordable simulation length is typically not sufficient to observe an appreciable number of folding events, unless very simplified protein models are used. Here we present an approach that allows for the reconstruction of the full ensemble of folding pathways from simulations that are much shorter than the folding time. This approach can be applied to all-atom protein simulations in explicit solvent. It does not use a predefined reaction coordinate but is based on partitioning the state space into small conformational states and constructing a Markov model between them. A theory is presented that allows for the extraction of the full ensemble of transition pathways from the unfolded to the folded configurations. The approach is applied to the folding of a PinWW domain in explicit solvent where the folding time is two orders of magnitude larger than the length of individual simulations. The results are in good agreement with kinetic experimental data and give detailed insights about the nature of the folding process which is shown to be surprisingly complex and parallel. The analysis reveals the existence of misfolded trap states outside the network of efficient folding intermediates that significantly reduce the folding speed.

String method in collective variables: Minimum free energy paths and isocommittor surfaces

A computational technique is proposed which combines the string method with a sampling technique to determine minimum free energy paths. The technique only requires to compute the mean force and another conditional expectation locally along the string, and therefore can be applied even if the number of collective variables kept in the free energy calculation is large. This is in contrast with other free energy sampling techniques which aim at mapping the full free energy landscape and whose cost increases exponentially with the number of collective variables kept in the free energy. Provided that the number of collective variables is large enough, the new technique captures the mechanism of transition in that it allows to determine the committor function for the reaction and, in particular, the transition state region. The new technique is illustrated on the example of alanine dipeptide, in which we compute the minimum free energy path for the isomerization transition using either two or four dihedral angles as collective variables. It is shown that the mechanism of transition can be captured using the four dihedral angles, but it cannot be captured using only two of them.

Simplified and improved string method for computing the minimum energy paths in barrier-crossing events

We present a simplified and improved version of the string method, originally proposed by E et al. [Phys. Rev. B 66, 052301 (2002)] for identifying the minimum energy paths in barrier-crossing events. In this new version, the step of projecting the potential force to the direction normal to the string is eliminated and the full potential force is used in the evolution of the string. This not only simplifies the numerical procedure, but also makes the method more stable and accurate. We discuss the algorithmic details of the improved string method, analyze its stability, accuracy and efficiency, and illustrate it via numerical examples. We also show how the string method can be combined with the climbing image technique for the accurate calculation of saddle points and we present another algorithm for the accurate calculation of the unstable directions at the saddle points.

Transition-Path Theory and Path-Finding Algorithms for the Study of Rare Events

Transition-path theory is a theoretical framework for describing rare events in complex systems. It can also be used as a starting point for developing efficient numerical algorithms for analyzing such rare events. Here we review the basic components of transition-path theory and path-finding algorithms. We also discuss connections with the classical transition-state theory.

Systematic Strategies for Stochastic Mode Reduction in Climate

A systematic strategy for stochastic mode reduction is applied here to three prototype ‘‘toy’’ models with nonlinear behavior mimicking several features of low-frequency variability in the extratropical atmosphere. Two of the models involve explicit stable periodic orbits and multiple equilibria in the projected nonlinear climate dynamics. The systematic strategy has two steps: stochastic consistency and stochastic mode elimination. Both aspects of the mode reduction strategy are tested in an a priori fashion in the paper. In all three models the stochastic mode elimination procedure applies in a quantitative fashion for moderately large values of « 0.5 or even « 1, where the parameter « roughly measures the ratio of correlation times of unresolved variables to resolved climate variables, even though the procedure is only justified mathematically for « K 1. The results developed here provide some new perspectives on both the role of stable nonlinear structures in projected nonlinear climate dynamics and the regression fitting strategies for stochastic climate modeling. In one example, a deterministic system with 102 degrees of freedom has an explicit stable periodic orbit for the projected climate dynamics in two variables; however, the complete deterministic system has instead a probability density function with two large isolated peaks on the ‘‘ghost’’ of this periodic orbit, and correlation functions that only weakly ‘‘shadow’’ this periodic orbit. Furthermore, all of these features are predicted in a quantitative fashion by the reduced stochastic model in two variables derived from the systematic theory; this reduced model has multiplicative noise and augmented nonlinearity. In a second deterministic model with 101 degrees of freedom, it is established that stable multiple equilibria in the projected climate dynamics can be either relevant or completely irrelevant in the actual dynamics for the climate variable depending on the strength of nonlinearity and the coupling to the unresolved variables. Furthermore, all this behavior is predicted in a quantitative fashion by a reduced nonlinear stochastic model for a single climate variable with additive noise, which is derived from the systematic mode reduction procedure. Finally, the systematic mode reduction strategy is applied in an idealized context to the stochastic modeling of the effect of mountain torque on the angular momentum budget. Surprisingly, the strategy yields a nonlinear stochastic equation for the large-scale fluctuations, and numerical simulations confirm significantly improved predicted correlation functions from this model compared with a standard linear model with damping and white noise forcing.

Transition Path Theory for Markov Jump Processes

The framework of transition path theory (TPT) is developed in the context of continuous-time Markov chains on discrete state-spaces. Under assumption of ergodicity, TPT singles out any two subsets in the state-space and analyzes the statistical properties of the associated reactive trajectories, i.e., those trajectories by which the random walker transits from one subset to another. TPT gives properties such as the probability distribution of the reactive trajectories, their probability current and flux, and their rate of occurrence and the dominant reaction pathways. In this paper the framework of TPT for Markov chains is developed in detail, and the relation of the theory to electric resistor network theory and data analysis tools such as Laplacian eigenmaps and diffusion maps is discussed as well. Various algorithms for the numerical calculation of the various objects in TPT are also introduced. Finally, the theory and the algorithms are illustrated in several examples.

Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates

An efficient simulation algorithm for chemical kinetic systems with disparate rates is proposed. This new algorithm is quite general, and it amounts to a simple and seamless modification of the classical stochastic simulation algorithm (SSA), also known as the Gillespie [J. Comput. Phys. 22, 403 (1976); J. Phys. Chem. 81, 2340 (1977)] algorithm. The basic idea is to use an outer SSA to simulate the slow processes with rates computed from an inner SSA which simulates the fast reactions. Averaging theorems for Markov processes can be used to identify the fast and slow variables in the system as well as the effective dynamics over the slow time scale, even though the algorithm itself does not rely on such information. This nested SSA can be easily generalized to systems with more than two separated time scales. Convergence and efficiency of the algorithm are discussed using the established error estimates and illustrated through examples.

Revisiting the finite temperature string method for the calculation of reaction tubes and free energies

An improved and simplified version of the finite temperature string (FTS) method [W. E, W. Ren, and E. Vanden-Eijnden, J. Phys. Chem. B 109, 6688 (2005)] is proposed. Like the original approach, the new method is a scheme to calculate the principal curves associated with the Boltzmann-Gibbs probability distribution of the system, i.e., the curves which are such that their intersection with the hyperplanes perpendicular to themselves coincides with the expected position of the system in these planes (where perpendicular is understood with respect to the appropriate metric). Unlike more standard paths such as the minimum energy path or the minimum free energy path, the location of the principal curve depends on global features of the energy or the free energy landscapes and thereby may remain appropriate in situations where the landscape is rough on the thermal energy scale and/or entropic effects related to the width of the reaction channels matter. Instead of using constrained sampling in hyperplanes as in the original FTS, the new method calculates the principal curve via sampling in the Voronoi tessellation whose generating points are the discretization points along this curve. As shown here, this modification results in greater algorithmic simplicity. As a by-product, it also gives the free energy associated with the Voronoi tessellation. The new method can be applied both in the original Cartesian space of the system or in a set of collective variables. We illustrate FTS on test-case examples and apply it to the study of conformational transitions of the nitrogen regulatory protein C receiver domain using an elastic network model and to the isomerization of solvated alanine dipeptide.