Variational calculation of macrostate transition rates

Perspective: Computer simulations of long time dynamics

Atomically detailed computer simulations of complex molecular events attracted the imagination of many researchers in the field as providing comprehensive information on chemical, biological, and physical processes. However, one of the greatest limitations of these simulations is of time scales. The physical time scales accessible to straightforward simulations are too short to address many interesting and important molecular events. In the last decade significant advances were made in different directions (theory, software, and hardware) that significantly expand the capabilities and accuracies of these techniques. This perspective describes and critically examines some of these advances.

On the assumptions underlying milestoning

Milestoning is a procedure to compute the time evolution of complicated processes such as barrier crossing events or long diffusive transitions between predefined states. Milestoning reduces the dynamics to transition events between intermediates (the milestones) and computes the local kinetic information to describe these transitions via short molecular dynamics (MD) runs between the milestones. The procedure relies on the ability to reinitialize MD trajectories on the milestones to get the right kinetic information about the transitions. It also rests on the assumptions that the transition events between successive milestones and the time lags between these transitions are statistically independent. In this paper, we analyze the validity of these assumptions. We show that sets of optimal milestones exist, i.e., sets such that successive transitions are indeed statistically independent. The proof of this claim relies on the results of transition path theory and uses the isocommittor surfaces of the reaction as milestones. For systems in the overdamped limit, we also obtain the probability distribution to reinitialize the MD trajectories on the milestones, and we discuss why this distribution is not available in closed form for systems with inertia. We explain why the time lags between transitions are not statistically independent even for optimal milestones, but we show that working with such milestones allows one to compute mean first passage times between milestones exactly. Finally, we discuss some practical implications of our results and we compare milestoning with Markov state models in view of our findings.

Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics

To meet the challenge of modeling the conformational dynamics of biological macromolecules over long time scales, much recent effort has been devoted to constructing stochastic kinetic models, often in the form of discrete-state Markov models, from short molecular dynamics simulations. To construct useful models that faithfully represent dynamics at the time scales of interest, it is necessary to decompose configuration space into a set of kinetically metastable states. Previous attempts to define these states have relied upon either prior knowledge of the slow degrees of freedom or on the application of conformational clustering techniques which assume that conformationally distinct clusters are also kinetically distinct. Here, we present a first version of an automatic algorithm for the discovery of kinetically metastable states that is generally applicable to solvated macromolecules. Given molecular dynamics trajectories initiated from a well-defined starting distribution, the algorithm discovers long lived, kinetically metastable states through successive iterations of partitioning and aggregating conformation space into kinetically related regions. The authors apply this method to three peptides in explicit solvent-terminally blocked alanine, the 21-residue helical F(s) peptide, and the engineered 12-residue beta-hairpin trpzip2-to assess its ability to generate physically meaningful states and faithful kinetic models.

Exploring the complex folding kinetics of RNA hairpins: I. General folding kinetics analysis

Depending on the nucleotide sequence, the temperature, and other conditions, RNA hairpin-folding kinetics can be very complex. The complexity with a wide range of cooperative and noncooperative kinetic behaviors arises from the interplay between the formation of the loops, the disruption of the misfolded states, and the formation of the rate-limiting base stacks. With a rate constant model and a kinetic-cluster theory, we explore the broad landscape for RNA hairpin-folding kinetics. The model is validated through direct tests against several experimental measurements. The general kinetic folding mechanisms and the predicted great variety of folding kinetics are directly applicable and quantitatively testable in experiments. The results from this study suggest that 1), previous experimental findings based on the individual hairpins revealed only a small fraction of much broader and more complex RNA hairpin-folding landscapes; 2), even for structures as simple as hairpins, universal folding timescales and pathways do not exist; and 3), to treat the loop size as the sole factor to determine the hairpin-folding rate is an oversimplification.

Analyzing the biopolymer folding rates and pathways using kinetic cluster method

A kinetic cluster method enables us to analyze biopolymer folding kinetics with discrete rate-limiting steps by classifying biopolymer conformations into pre-equilibrated clusters. The overall folding kinetics is determined by the intercluster transitions. Due to the complex energy landscapes of biopolymers, the intercluster transitions have multiple pathways and can have kinetic intermediates (local free-energy minima) distributed on the intercluster pathways. We focus on the RNA secondary structure folding kinetics. The dominant folding pathways and the kinetic partitioning mechanism can be identified and quantified from the rate constants for different intercluster pathways. Moreover, the temperature dependence of the folding rate can be analyzed from the interplay between the stabilities of the on-pathway (nativelike) and off-pathway (misfolded) conformations and from the kinetic partitioning between different intercluster pathways. The predicted folding kinetics can be directly tested against experiments.

Macrostate data clustering

We develop an effective nonhierarchical data clustering method using an analogy to the dynamic coarse graining of a stochastic system. Analyzing the eigensystem of an interitem transition matrix identifies fuzzy clusters corresponding to the metastable macroscopic states (macrostates) of a diffusive system. A "minimum uncertainty criterion" determines the linear transformation from eigenvectors to cluster-defining window functions. Eigenspectrum gap and cluster certainty conditions identify the proper number of clusters. The physically motivated fuzzy representation and associated uncertainty analysis distinguishes macrostate clustering from spectral partitioning methods. Macrostate data clustering solves a variety of test cases that challenge other methods.

Top-down free-energy minimization on protein potential energy landscapes

The hierarchical properties of potential energy landscapes have been used to gain insight into thermodynamic and kinetic properties of protein ensembles. It also may be possible to use them to direct computational searches for thermodynamically stable macroscopic states, i.e., computational protein folding. To this end, we have developed a top-down search procedure in which conformation space is recursively dissected according to the intrinsic hierarchical structure of a landscape's effective-energy barriers. This procedure generates an inverted tree similar to the disconnectivity graphs generated by local minima-clustering methods, but it fundamentally differs in the manner in which the portion of the tree that is to be computationally explored is selected. A key ingredient is a branch-selection algorithm that takes advantage of statistically predictive properties of the landscape to guide searches down the tree branches that are most likely to lead to the physically relevant macroscopic states. Using the computational folding of a beta-hairpin-forming peptide as an example, we show that such predictive properties indeed exist and can be used for structure prediction by free-energy global minimization.

Efficient dynamic importance sampling of rare events in one dimension

Exploiting stochastic path-integral theory, we obtain by simulation substantial gains in efficiency for the computation of reaction rates in one-dimensional, bistable, overdamped stochastic systems. Using a well-defined measure of efficiency, we compare implementations of "dynamic importance sampling" (DIMS) methods to unbiased simulation. The best DIMS algorithms are shown to increase efficiency by factors of approximately 20 for a 5k(B)T barrier height and 300 for 9k(B)T, compared to unbiased simulation. The gains result from close emulation of natural (unbiased), instantonlike crossing events with artificially decreased waiting times between events that are corrected for in rate calculations. The artificial crossing events are generated using the closed-form solution to the most probable crossing event described by the Onsager-Machlup action. While the best biasing methods require the second derivative of the potential (resulting from the "Jacobian" term in the action, which is discussed at length), algorithms employing solely the first derivative do nearly as well. We discuss the importance of one-dimensional models to larger systems, and suggest extensions to higher-dimensional systems.