Additive eigenvectors as optimal reaction coordinates, conditioned trajectories, and time-reversible description of stochastic processes.

A fundamental way to analyze complex multidimensional stochastic dynamics is to describe it as diffusion on a free energy landscape-free energy as a function of reaction coordinates (RCs). For such a description to be quantitatively accurate, the RC should be chosen in an optimal way. The committor function is a primary example of an optimal RC for the description of equilibrium reaction dynamics between two states. Here, additive eigenvectors (addevs) are considered as optimal RCs to address the limitations of the committor. An addev master equation for a Markov chain is derived. A stationary solution of the equation describes a sub-ensemble of trajectories conditioned on having the same optimal RC for the forward and time-reversed dynamics in the sub-ensemble. A collection of such sub-ensembles of trajectories, called stochastic eigenmodes, can be used to describe/approximate the stochastic dynamics. A non-stationary solution describes the evolution of the probability distribution. However, in contrast to the standard master equation, it provides a time-reversible description of stochastic dynamics. It can be integrated forward and backward in time. The developed framework is illustrated on two model systems-unidirectional random walk and diffusion.

Nonparametric Analysis of Nonequilibrium Simulations.

We extend the nonparametric framework of reaction coordinate optimization to nonequilibrium ensembles of (short) trajectories. For example, we show how, starting from such an ensemble, one can obtain an equilibrium free-energy profile along the committor, which can be used to determine important properties of the dynamics exactly. A new adaptive sampling approach, the transition-state ensemble enrichment, is suggested, which samples the configuration space by "growing" committor segments toward each other starting from the boundary states. This framework is suggested as a general tool, alternative to the Markov state models, for a rigorous and accurate analysis of simulations of large biomolecular systems, as it has the following attractive properties. It is immune to the curse of dimensionality, does not require system-specific information, can approximate arbitrary reaction coordinates with high accuracy, and has sensitive and rigorous criteria to test optimality and convergence. The approaches are illustrated on a 50-dimensional model system and a realistic protein folding trajectory.

Blind Analysis of Molecular Dynamics.

We describe a nonparametric approach for accurate determination of the slowest relaxation eigenvectors of molecular dynamics. The approach is blind as it uses no system specific information. In particular, it does not require a functional form with many parameters to closely approximate eigenvectors, e.g., linear combinations of molecular descriptors or a deep neural network, and thus no extensive expertise with the system. We suggest a rigorous and sensitive validation/optimality criterion for an eigenvector. The criterion uses only eigenvector time series and can be used to validate eigenvectors computed by other approaches. The power of the approach is illustrated on long atomistic protein folding trajectories. The determined eigenvectors pass the validation test at a time scale of 0.2 ns, much shorter than alternative approaches.

Protein Folding Free Energy Landscape along the Committor - the Optimal Folding Coordinate.

Recent advances in simulation and experiment have led to dramatic increases in the quantity and complexity of produced data, which makes the development of automated analysis tools very important. A powerful approach to analyze dynamics contained in such data sets is to describe/approximate it by diffusion on a free energy landscape - free energy as a function of reaction coordinates (RC). For the description to be quantitatively accurate, RCs should be chosen in an optimal way. Recent theoretical results show that such an optimal RC exists; however, determining it for practical systems is a very difficult unsolved problem. Here we describe a solution to this problem. We describe an adaptive nonparametric approach to accurately determine the optimal RC (the committor) for an equilibrium trajectory of a realistic system. In contrast to alternative approaches, which require a functional form with many parameters to approximate an RC and thus extensive expertise with the system, the suggested approach is nonparametric and can approximate any RC with high accuracy without system specific information. To avoid overfitting for a realistically sampled system, the approach performs RC optimization in an adaptive manner by focusing optimization on less optimized spatiotemporal regions of the RC. The power of the approach is illustrated on a long equilibrium atomistic folding simulation of HP35 protein. We have determined the optimal folding RC - the committor, which was confirmed by passing a stringent committor validation test. It allowed us to determine a first quantitatively accurate protein folding free energy landscape. We have confirmed the recent theoretical results that diffusion on such a free energy profile can be used to compute exactly the equilibrium flux, the mean first passage times, and the mean transition path times between any two points on the profile. We have shown that the mean squared displacement along the optimal RC grows linear with time as for simple diffusion. The free energy profile allowed us to obtain a direct rigorous estimate of the pre-exponential factor for the folding dynamics.

Exploratory search during directed navigation in C. elegans and Drosophila larva.

Many organisms-from bacteria to nematodes to insect larvae-navigate their environments by biasing random movements. In these organisms, navigation in isotropic environments can be characterized as an essentially diffusive and undirected process. In stimulus gradients, movement decisions are biased to drive directed navigation toward favorable environments. How does directed navigation in a gradient modulate random exploration either parallel or orthogonal to the gradient? Here, we introduce methods originally used for analyzing protein folding trajectories to study the trajectories of the nematode Caenorhabditis elegans and the Drosophila larva in isotropic environments, as well as in thermal and chemical gradients. We find that the statistics of random exploration in any direction are little affected by directed movement along a stimulus gradient. A key constraint on the behavioral strategies of these organisms appears to be the preservation of their capacity to continuously explore their environments in all directions even while moving toward favorable conditions.

Nonparametric variational optimization of reaction coordinates.

State of the art realistic simulations of complex atomic processes commonly produce trajectories of large size, making the development of automated analysis tools very important. A popular approach aimed at extracting dynamical information consists of projecting these trajectories into optimally selected reaction coordinates or collective variables. For equilibrium dynamics between any two boundary states, the committor function also known as the folding probability in protein folding studies is often considered as the optimal coordinate. To determine it, one selects a functional form with many parameters and trains it on the trajectories using various criteria. A major problem with such an approach is that a poor initial choice of the functional form may lead to sub-optimal results. Here, we describe an approach which allows one to optimize the reaction coordinate without selecting its functional form and thus avoiding this source of error.

High-resolution free energy landscape analysis of protein folding.

The free energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. The profile together with the optimal coordinate allows one to directly determine such basic properties of folding dynamics as the configurations of the minima and transition states, the heights of the barriers, the value of the pre-exponential factor and its relation to the transition path times. In the present study, we review the framework, in particular, the approach to determine such an optimal coordinate, and its application to the analysis of simulated protein folding dynamics.

Fep1d: a script for the analysis of reaction coordinates.

The dynamics of complex systems with many degrees of freedom can be analyzed by projecting it onto one or few coordinates (collective variables). The dynamics is often described then as diffusion on a free energy landscape associated with the coordinates. Fep1d is a script for the analysis of such one-dimensional coordinates. The script allows one to construct conventional and cut-based free energy profiles, to assess the optimality of a reaction coordinate, to inspect whether the dynamics projected on the coordinate is diffusive, to transform (rescale) the reaction coordinate to more convenient ones, and to compute such quantities as the mean first passage time, the transition path times, the coordinate dependent diffusion coefficient, and so forth. Here, we describe the implemented functionality together with the underlying theoretical framework.

Optimal reaction coordinate as a biomarker for the dynamics of recovery from kidney transplant.

The evolution of disease or the progress of recovery of a patient is a complex process, which depends on many factors. A quantitative description of this process in real-time by a single, clinically measurable parameter (biomarker) would be helpful for early, informed and targeted treatment. Organ transplantation is an eminent case in which the evolution of the post-operative clinical condition is highly dependent on the individual case. The quality of management and monitoring of patients after kidney transplant often determines the long-term outcome of the graft. Using NMR spectra of blood samples, taken at different time points from just before to a week after surgery, we have shown that a biomarker can be found that quantitatively monitors the evolution of a clinical condition. We demonstrate that this is possible if the dynamics of the process is considered explicitly: the biomarker is defined and determined as an optimal reaction coordinate that provides a quantitatively accurate description of the stochastic recovery dynamics. The method, originally developed for the analysis of protein folding dynamics, is rigorous, robust and general, i.e., it can be applied in principle to analyze any type of biological dynamics. Such predictive biomarkers will promote improvement of long-term graft survival after renal transplantation, and have potentially unlimited applications as diagnostic tools.

Robust Estimation of Diffusion-Optimized Ensembles for Enhanced Sampling.

The multicanonical, or flat-histogram, method is a common technique to improve the sampling efficiency of molecular simulations. The idea is that free-energy barriers in a simulation can be removed by simulating from a distribution where all values of a reaction coordinate are equally likely, and subsequently reweight the obtained statistics to recover the Boltzmann distribution at the temperature of interest. While this method has been successful in practice, the choice of a flat distribution is not necessarily optimal. Recently, it was proposed that additional performance gains could be obtained by taking the position-dependent diffusion coefficient into account, thus placing greater emphasis on regions diffusing slowly. Although some promising examples of applications of this approach exist, the practical usefulness of the method has been hindered by the difficulty in obtaining sufficiently accurate estimates of the diffusion coefficient. Here, we present a simple, yet robust solution to this problem. Compared to current state-of-the-art procedures, the new estimation method requires an order of magnitude fewer data to obtain reliable estimates, thus broadening the potential scope in which this technique can be applied in practice.

High-Resolution Free-Energy Landscape Analysis of α-Helical Protein Folding: HP35 and Its Double Mutant.

The free-energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. Here, we construct the optimal coordinate and the associated free-energy profile for all-helical proteins HP35 and its norleucine (Nle/Nle) double mutant, based on realistic equilibrium folding simulations [Piana et al. Proc. Natl. Acad. Sci. U.S.A.2012, 109, 17845]. From the obtained profiles, we directly determine such basic properties of folding dynamics as the configurations of the minima and transition states (TS), the formation of secondary structure and hydrophobic core during the folding process, the value of the pre-exponential factor and its relation to the transition path times, the relation between the autocorrelation times in TS and minima. We also present an investigation of the accuracy of the pre-exponential factor estimation based on the transition-path times. Four different estimations of the pre-exponential factor for both proteins give k0-1 values of approximately a few tens of nanoseconds. Our analysis gives detailed information about folding of the proteins and can serve as a rigorous common language for extensive comparison between experiment and simulation.

On Reaction Coordinate Optimality.

The following question is addressed: how to establish that a constructed reaction coordinate is optimal, i.e., that it provides an accurate description of dynamics. It is shown that the reaction coordinate is optimal if its cut free energy profile, determined using length-weighted transitions, is constant, i.e., it is position and sampling interval independent. The observation leads to a number of interesting results. In particular, the equilibrium flux between two boundary states can be computed exactly as diffusion on a free energy profile associated with the coordinate. The mean square displacement, for the trajectory projected onto the coordinate, grows linear with time. That for the same trajectory projected onto a suboptimal coordinate grows slower than linear with time. The results are illustrated on a number of model systems, Sierpinski gasket, FIP35 protein, and beta3s peptide.

Method to describe stochastic dynamics using an optimal coordinate.

A general method to describe the stochastic dynamics of Markov processes is suggested. The method aims to solve three related problems: the determination of an optimal coordinate for the description of stochastic dynamics; the reconstruction of time from an ensemble of stochastic trajectories; and the decomposition of stationary stochastic dynamics into eigenmodes which do not decay exponentially with time. The problems are solved by introducing additive eigenvectors which are transformed by a stochastic matrix in a simple way - every component is translated by a constant distance. Such solutions have peculiar properties. For example, an optimal coordinate for stochastic dynamics with detailed balance is a multivalued function. An optimal coordinate for a random walk on a line corresponds to the conventional eigenvector of the one-dimensional Dirac equation. The equation for the optimal coordinate in a slowly varying potential reduces to the Hamilton-Jacobi equation for the action function.

The free energy landscape analysis of protein (FIP35) folding dynamics.

A fundamental problem in the analysis of protein folding and other complex reactions is the determination of the reaction free energy landscape. The current experimental techniques lack the necessary spatial and temporal resolution to construct such landscapes. The properties of the landscapes can be probed only indirectly. Simulation, assuming that it reproduces the experimental dynamics, can provide the necessary spatial and temporal resolution. It is, arguably, the only way for direct rigorous construction of the quantitatively accurate free energy landscapes. Here, such landscape is constructed from the equilibrium folding simulation of FIP35 protein reported by Shaw et al. Science 2010, 330, 341-346. For the dynamics to be accurately described as diffusion on the free energy landscape, the choice of reaction coordinates is crucial. The reaction coordinate used here is such that the dynamics projected on it is diffusive, so the description is consistent and accurate. The obtained landscape suggests an alternative interpretation of the simulation, markedly different from that of Shaw et al. In particular, FIP35 is not an incipient downhill folder, it folds via a populated on-pathway intermediate separated by high free energy barriers; the high free energy barriers rather than landscape roughness are a major determinant of the rates for conformational transitions; the preexponential factor of folding kinetics 1/k(0) ∼ 10 ns rather than 1 µs.

Optimal dimensionality reduction of complex dynamics: the chess game as diffusion on a free-energy landscape.

Dimensionality reduction is ubiquitous in the analysis of complex dynamics. The conventional dimensionality reduction techniques, however, focus on reproducing the underlying configuration space, rather than the dynamics itself. The constructed low-dimensional space does not provide a complete and accurate description of the dynamics. Here I describe how to perform dimensionality reduction while preserving the essential properties of the dynamics. The approach is illustrated by analyzing the chess game--the archetype of complex dynamics. A variable that provides complete and accurate description of chess dynamics is constructed. The winning probability is predicted by describing the game as a random walk on the free-energy landscape associated with the variable. The approach suggests a possible way of obtaining a simple yet accurate description of many important complex phenomena. The analysis of the chess game shows that the approach can quantitatively describe the dynamics of processes where human decision-making plays a central role, e.g., financial and social dynamics.

Numerical construction of the p(fold) (committor) reaction coordinate for a Markov process.

To simplify the description of a complex multidimensional dynamical process, one often projects it onto a single reaction coordinate. In protein folding studies, the folding probability p(fold) is an optimal reaction coordinate which preserves many important properties of the dynamics. The construction of the coordinate is difficult. Here, an efficient numerical approach to construct the p(fold) reaction coordinate for a Markov process (satisfying the detailed balance) is described. The coordinate is obtained by optimizing parameters of a chosen functional form to make a generalized cut-based free energy profile the highest. The approach is illustrated by constructing the p(fold) reaction coordinate for the equilibrium folding simulation of FIP35 protein reported by Shaw et al. (Science 2010, 330, 341-346).

Is protein folding sub-diffusive?

Protein folding dynamics is often described as diffusion on a free energy surface considered as a function of one or few reaction coordinates. However, a growing number of experiments and models show that, when projected onto a reaction coordinate, protein dynamics is sub-diffusive. This raises the question as to whether the conventionally used diffusive description of the dynamics is adequate. Here, we numerically construct the optimum reaction coordinate for a long equilibrium folding trajectory of a Go model of a -repressor protein. The trajectory projected onto this coordinate exhibits diffusive dynamics, while the dynamics of the same trajectory projected onto a sub-optimal reaction coordinate is sub-diffusive. We show that the higher the (cut-based) free energy profile for the putative reaction coordinate, the more diffusive the dynamics become when projected on this coordinate. The results suggest that whether the projected dynamics is diffusive or sub-diffusive depends on the chosen reaction coordinate. Protein folding can be described as diffusion on the free energy surface as function of the optimum reaction coordinate. And conversely, the conventional reaction coordinates, even though they might be based on physical intuition, are often sub-optimal and, hence, show sub-diffusive dynamics.

Analysis of the free-energy surface of proteins from reversible folding simulations.

Computer generated trajectories can, in principle, reveal the folding pathways of a protein at atomic resolution and possibly suggest general and simple rules for predicting the folded structure of a given sequence. While such reversible folding trajectories can only be determined ab initio using all-atom transferable force-fields for a few small proteins, they can be determined for a large number of proteins using coarse-grained and structure-based force-fields, in which a known folded structure is by construction the absolute energy and free-energy minimum. Here we use a model of the fast folding helical lambda-repressor protein to generate trajectories in which native and non-native states are in equilibrium and transitions are accurately sampled. Yet, representation of the free-energy surface, which underlies the thermodynamic and dynamic properties of the protein model, from such a trajectory remains a challenge. Projections over one or a small number of arbitrarily chosen progress variables often hide the most important features of such surfaces. The results unequivocally show that an unprojected representation of the free-energy surface provides important and unbiased information and allows a simple and meaningful description of many-dimensional, heterogeneous trajectories, providing new insight into the possible mechanisms of fast-folding proteins.

Diffusive reaction dynamics on invariant free energy profiles.

A fundamental problem in the analysis of protein folding and other complex reactions in which the entropy plays an important role is the determination of the activation free energy from experimental measurements or computer simulations. This article shows how to combine minimum-cut-based free energy profiles (F(C)), obtained from equilibrium molecular dynamics simulations, with conventional histogram-based free energy profiles (F(H)) to extract the coordinate-dependent diffusion coefficient on the F(C) (i.e., the method determines free energies and a diffusive preexponential factor along an appropriate reaction coordinate). The F(C), in contrast to the F(H), is shown to be invariant with respect to arbitrary transformations of the reaction coordinate, which makes possible partition of configuration space into basins in an invariant way. A "natural coordinate," for which F(H) and F(C) differ by a multiplicative constant (constant diffusion coefficient), is introduced. The approach is illustrated by a model one-dimensional system, the alanine dipeptide, and the folding reaction of a double beta-hairpin miniprotein. It is shown how the results can be used to test whether the putative reaction coordinate is a good reaction coordinate.

One-dimensional barrier-preserving free-energy projections of a beta-sheet miniprotein: new insights into the folding process.

The conformational space of a 20-residue three-stranded antiparallel beta-sheet peptide (double hairpin) was sampled by equilibrium folding/unfolding molecular dynamics simulations for a total of 20 micros. The resulting one-dimensional free-energy profiles (FEPs) provide a detailed description of the free-energy basins and barriers for the folding reaction. The similarity of the FEPs obtained using the probability of folding before unfolding (pfold) or the mean first passage time supports the robustness of the procedure. The folded state and the most populated free-energy basins in the denatured state are described by the one-dimensional FEPs, which avoid the overlap of states present in the usual one- or two-dimensional projections. Within the denatured state, a basin with fluctuating helical conformations and a heterogeneous entropic state are populated near the melting temperature at about 11% and 33%, respectively. Folding pathways from the helical basin or enthalpic traps (with only one of the two hairpins formed) reach the native state through the entropic state, which is on-pathway and is separated by a low barrier from the folded state. A simplified equilibrium kinetic network based on the FEPs shows the complexity of the folding reaction and indicates, as augmented by additional analyses, that the basins in the denatured state are connected primarily by the native state. The overall folding kinetics shows single-exponential behavior because barriers between the non-native basins and the folded state have similar heights.