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

Computing the committor with the committor to study the transition state ensemble

The study of the kinetic bottlenecks that hinder the rare transitions between long-lived metastable states is a major challenge in atomistic simulations. Here we propose a method to explore the transition state ensemble, which is the distribution of configurations that the system passes through as it translocates from one metastable basin to another. We base our method on the committor function and the variational principle that it obeys. We find its minimum through a self-consistent procedure that starts from information limited to the initial and final states. Right from the start, our procedure allows the sampling of very many transition state configurations. With the help of the variational principle, we perform a detailed analysis of the transition state ensemble, ranking quantitatively the degrees of freedom mostly involved in the transition and enabling a systematic approach for the interpretation of simulation results and the construction of efficient physics-informed collective variables.

Enhanced Sampling for Conformational Changes and Molecular Mechanisms of Human NTHL1

The functionalities of proteins rely on protein conformational changes during many processes. Identification of the protein conformations and capturing transitions among different conformations are important but extremely challenging in both experiments and simulations. In this work, we develop a machine learning based approach to identify a reaction coordinate that accelerates the exploration of protein conformational changes in molecular simulations. We implement our approach to study the conformational changes of human NTHL1 during DNA repair. Our results identified three distinct conformations: open (stable), closed (unstable), and bundle (stable). The existence of the bundle conformation can rationalize recent experimental observations. Comparison with an NTHL1 mutant demonstrates that a closely packed cluster of positively charged residues in the linker could be a factor to search when screening for genetic abnormalities. Results will lead to a better modulation of the DNA repair pathway to protect against carcinogenesis.

On committor functions in milestoning

As an optimal one-dimensional reaction coordinate, the committor function not only describes the probability of a trajectory initiated at a phase space point first reaching the product state before reaching the reactant state but also preserves the kinetics when utilized to run a reduced dynamics model. However, calculating the committor function in high-dimensional systems poses significant challenges. In this paper, within the framework of milestoning, exact expressions for committor functions at two levels of coarse graining are given, including committor functions of phase space point to point (CFPP) and milestone to milestone (CFMM). When combined with transition kernels obtained from trajectory analysis, these expressions can be utilized to accurately and efficiently compute the committor functions. Furthermore, based on the calculated committor functions, an adaptive algorithm is developed to gradually refine the transition state region. Finally, two model examples are employed to assess the accuracy of these different formulations of committor functions.

Predicting the Effect of Single Mutations on Protein Stability and Binding with Respect to Types of Mutations

The development of methods and algorithms to predict the effect of mutations on protein stability, protein-protein interaction, and protein-DNA/RNA binding is necessitated by the needs of protein engineering and for understanding the molecular mechanism of disease-causing variants. The vast majority of the leading methods require a database of experimentally measured folding and binding free energy changes for training. These databases are collections of experimental data taken from scientific investigations typically aimed at probing the role of particular residues on the above-mentioned thermodynamic characteristics, i.e., the mutations are not introduced at random and do not necessarily represent mutations originating from single nucleotide variants (SNV). Thus, the reported performance of the leading algorithms assessed on these databases or other limited cases may not be applicable for predicting the effect of SNVs seen in the human population. Indeed, we demonstrate that the SNVs and non-SNVs are not equally presented in the corresponding databases, and the distribution of the free energy changes is not the same. It is shown that the Pearson correlation coefficients (PCCs) of folding and binding free energy changes obtained in cases involving SNVs are smaller than for non-SNVs, indicating that caution should be used in applying them to reveal the effect of human SNVs. Furthermore, it is demonstrated that some methods are sensitive to the chemical nature of the mutations, resulting in PCCs that differ by a factor of four across chemically different mutations. All methods are found to underestimate the energy changes by roughly a factor of 2.

Survival strategies of artificial active agents

Artificial cells can be engineered to display dynamics sharing remarkable features in common with the survival behavior of living organisms. In particular, such active systems can respond to stimuli provided by the environment and undertake specific displacements to remain out of equilibrium, e.g. by moving towards regions with higher fuel concentration. In spite of the intense experimental activity aiming at investigating this fascinating behavior, a rigorous definition and characterization of such "survival strategies" from a statistical physics perspective is still missing. In this work, we take a first step in this direction by adapting and applying to active systems the theoretical framework of Transition Path Theory, which was originally introduced to investigate rare thermally activated transitions in passive systems. We perform experiments on camphor disks navigating Petri dishes and perform simulations in the paradigmatic active Brownian particle model to show how the notions of transition probability density and committor function provide the pivotal concepts to identify survival strategies, improve modeling, and obtain and validate experimentally testable predictions. The definition of survival in these artificial systems paves the way to move beyond simple observation and to formally characterize, design and predict complex life-like behaviors.

Revealing Free Energy Landscape From MD Data via Conditional Angle Partition Tree

Deciphering the free energy landscape of biomolecular structure space is crucial for understanding many complex molecular processes, such as protein-protein interaction, RNA folding, and protein folding. A major source of current dynamic structure data is Molecular Dynamics (MD) simulations. Several methods have been proposed to investigate the free energy landscape from MD data, but all of them rely on the assumption that kinetic similarity is associated with global geometric similarity, which may lead to unsatisfactory results. In this paper, we proposed a new method called Conditional Angle Partition Tree to reveal the hierarchical free energy landscape by correlating local geometric similarity with kinetic similarity. Its application on the benchmark alanine dipeptide MD data showed a much better performance than existing methods in exploring and understanding the free energy landscape. We also applied it to the MD data of Villin HP35. Our results are more reasonable on various aspects than those from other methods and very informative on the hierarchical structure of its energy landscape.

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.

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.

Out-of-Equilibrium Biophysical Chemistry: The Case for Multidimensional, Integrated Single-Molecule Approaches

Out-of-equilibrium processes are ubiquitous across living organisms and all structural hierarchies of life. At the molecular scale, out-of-equilibrium processes (for example, enzyme catalysis, gene regulation, and motor protein functions) cause biological macromolecules to sample an ensemble of conformations over a wide range of time scales. Quantifying and conceptualizing the structure–dynamics to function relationship is challenging because continuously evolving multidimensional energy landscapes are necessary to describe nonequilibrium biological processes in biological macromolecules. In this perspective, we explore the challenges associated with state-of-the-art experimental techniques to understanding biological macromolecular function. We argue that it is time to revisit how we probe and model functional out-of-equilibrium biomolecular dynamics. We suggest that developing integrated single-molecule multiparametric force–fluorescence instruments and using advanced molecular dynamics simulations to study out-of-equilibrium biomolecules will provide a path towards understanding the principles of and mechanisms behind the structure–dynamics to function paradigm in biological macromolecules.

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.

String Method with Swarms-of-Trajectories, Mean Drifts, Lag Time, and Committor

The kinetics of a dynamical system comprising two metastable states is formulated in terms of a finite-time propagator in phase space (position and velocity) adapted to the underdamped Langevin equation. Dimensionality reduction to a subspace of collective variables yields familiar expressions for the propagator, committor, and steady-state flux. A quadratic expression for the steady-state flux between the two metastable states can serve as a robust variational principle to determine an optimal approximate committor expressed in terms of a set of collective variables. The theoretical formulation is exploited to clarify the foundation of the string method with swarms-of-trajectories, which relies on the mean drift of short trajectories to determine the optimal transition pathway. It is argued that the conditions for Markovity within a subspace of collective variables may not be satisfied with an arbitrary short time-step and that proper kinetic behaviors appear only when considering the effective propagator for longer lag times. The effective propagator with finite lag time is amenable to an eigenvalue-eigenvector spectral analysis, as elaborated previously in the context of position-based Markov models. The time-correlation functions calculated by swarms-of-trajectories along the string pathway constitutes a natural extension of these developments. The present formulation provides a powerful theoretical framework to characterize the optimal pathway between two metastable states of a system.

The seven deadly sins: When computing crystal nucleation rates, the devil is in the details

The formation of crystals has proven to be one of the most challenging phase transformations to quantitatively model-let alone to actually understand-be it by means of the latest experimental technique or the full arsenal of enhanced sampling approaches at our disposal. One of the most crucial quantities involved with the crystallization process is the nucleation rate, a single elusive number that is supposed to quantify the average probability for a nucleus of critical size to occur within a certain volume and time span. A substantial amount of effort has been devoted to attempt a connection between the crystal nucleation rates computed by means of atomistic simulations and their experimentally measured counterparts. Sadly, this endeavor almost invariably fails to some extent, with the venerable classical nucleation theory typically blamed as the main culprit. Here, we review some of the recent advances in the field, focusing on a number of perhaps more subtle details that are sometimes overlooked when computing nucleation rates. We believe it is important for the community to be aware of the full impact of aspects, such as finite size effects and slow dynamics, that often introduce inconspicuous and yet non-negligible sources of uncertainty into our simulations. In fact, it is key to obtain robust and reproducible trends to be leveraged so as to shed new light on the kinetics of a process, that of crystal nucleation, which is involved into countless practical applications, from the formulation of pharmaceutical drugs to the manufacturing of nano-electronic devices.

The Two-Step Clustering Approach for Metastable States Learning

Understanding the energy landscape and the conformational dynamics is crucial for studying many biological or chemical processes, such as protein-protein interaction and RNA folding. Molecular Dynamics (MD) simulations have been a major source of dynamic structure. Although many methods were proposed for learning metastable states from MD data, some key problems are still in need of further investigation. Here, we give a brief review on recent progresses in this field, with an emphasis on some popular methods belonging to a two-step clustering framework, and hope to draw more researchers to contribute to this area.

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.

An effective one-dimensional approach to calculating mean first passage time in multi-dimensional potentials

Thermally activated escape processes in multi-dimensional potentials are of interest to a variety of fields, so being able to calculate the rate of escape-or the mean first-passage time (MFPT)-is important. Unlike in one dimension, there is no general, exact formula for the MFPT. However, Langer's formula, a multi-dimensional generalization of Kramers's one-dimensional formula, provides an approximate result when the barrier to escape is large. Kramers's and Langer's formulas are related to one another by the potential of mean force (PMF): when calculated along a particular direction (the unstable mode at the saddle point) and substituted into Kramers's formula, the result is Langer's formula. We build on this result by using the PMF in the exact, one-dimensional expression for the MFPT. Our model offers better agreement with Brownian dynamics simulations than Langer's formula, although discrepancies arise when the potential becomes less confining along the direction of escape. When the energy barrier is small our model offers significant improvements upon Langer's theory. Finally, the optimal direction along which to evaluate the PMF no longer corresponds to the unstable mode at the saddle point.

Assembly of heteropolymers via a network of reaction coordinates

In biochemistry, heteropolymers encoding biological information are assembled out of equilibrium by sequentially incorporating available monomers found in the environment. Current models of polymerization treat monomer incorporation as a sequence of discrete chemical reactions between intermediate metastable states. In this paper, we use ideas from reaction rate theory and describe nonequilibrium assembly of a heteropolymer via a continuous reaction coordinate. Our approach allows for estimating the copy error and incorporation speed from the Gibbs free energy landscape of the process. We apply our theory to several examples from a simple reaction characterized by a free energy barrier to more complex cases incorporating error correction mechanisms, such as kinetic proofreading.

Oscillations in the mean transition time of a particle scattered on a double slit potential

Scattering through a double slit potential is one of the most fundamental problems in quantum mechanics. It is well understood that due to the superposition of amplitudes, one observes a spatial interference pattern in the scattered wavefunction reflecting the superposition of amplitudes coming from both slits. However, the effect of the double slit on the mean time it takes to traverse the slit has not been considered previously. Using a transition path time formalism, we show that when a single Gaussian wavepacket is scattered through a double slit potential, one finds not only oscillations in the scattered density resulting from the spatial interference created by the splitting of the wavepacket but also an oscillatory pattern in the mean scattering time. Long times are associated with low values of a suitably defined momentum, and short times with higher values. The double slit thus serves as a momentum filtering device. We also find an interference pattern in the time averaged momentum weak value profile of the scattered particle implying that the double slit also acts as a weak momentum filter. These results not only demonstrate the value of considering transition path time distributions in their quantum mechanical context but also present a challenge to semiclassical approximations-can they account for temporal interference?