Path Sampling Method Research Articles

Elucidating Dynamical Behaviors in Kinetically Constrained Models via Energy-Activity Double-Biased Matrix Product State Analysis.

We investigate dynamical phase transitions in two representative kinetically constrained models: the 1D Fredrickson-Andersen and East Models. A recently developed energy-activity double-bias approach utilizing both s and g fields, conjugated with dynamical activity and trajectory energy, is combined with matrix product state (MPS) methods. It is demonstrated that MPS methods facilitate the numerical approximation of large-deviation statistics of dynamics by determining the eigenvalues of tilted dynamical generators under the influence of double-biasing fields. Specifically, by focusing on the g-field, a nearly "half-filled" state at moderate negative g values is identified, indicating the potential existence of an anomalous phase. Additionally, dynamical quantities under various s, g, and T conditions are obtained via tensor networks, showing good qualitative consistency with mean-field results and our previous extensive numerical simulation results obtained by the path sampling method. Our study introduces novel methodologies for examining decoupled dynamical behaviors that, although energetically active, remain dynamically inactive within the system. This approach offers a fresh perspective on the theoretical framework and computational strategies for studying dynamical phase transitions in kinetically constrained systems.

The Evolution of the Acylation Mechanism in β-Lactamase and Rapid Protein Dynamics.

β-Lactamases are a class of well-studied enzymes that are known to have existed since billions of years ago, starting as a defense mechanism to stave off competitors and are now enzymes responsible for antibiotic resistance. Using ancestral sequence reconstruction, it is possible to study the crystal structure of a laboratory resurrected 2-3 billion year-old β-lactamase. Comparing the ancestral enzyme to its modern counterpart, a TEM-1 β-lactamase, the structural changes are minor, and it is probable that dynamic effects play an important role in the evolution of function. We used molecular dynamics simulations and employed transition path sampling methods to identify the presence of rate-enhancing dynamics at the femtosecond level in both systems, found that these fast motions are more efficiently coordinated in the modern enzyme, and examined how specific dynamics can pinpoint evolutionary effects that are essential for improving enzymatic catalysis.

WEDAP: A Python Package for Streamlined Plotting of Molecular Simulation Data.

Given the growing interest in path sampling methods for extending the time scales of molecular dynamics (MD) simulations, there has been great interest in software tools that streamline the generation of plots for monitoring the progress of large-scale simulations. Here, we present the WEDAP Python package for simplifying the analysis of data generated from either conventional MD simulations or the weighted ensemble (WE) path sampling method, as implemented in the widely used WESTPA software package. WEDAP facilitates (i) the parsing of WE simulation data stored in highly compressed, hierarchical HDF5 files and (ii) incorporates trajectory weights from WE simulations into all generated plots. Our Python package consists of multiple user-friendly interfaces: a command-line interface, a graphical user interface, and a Python application programming interface. We demonstrate the plotting features of WEDAP through a series of examples using data from WE and conventional MD simulations that focus on the HIV-1 capsid protein's C-terminal domain dimer as a showcase system. The source code for WEDAP is freely available on GitHub at https://github.com/chonglab-pitt/wedap.

Transition Path Sampling Study of Engineered Enzymes That Catalyze the Morita-Baylis-Hillman Reaction: Why Is Enzyme Design so Difficult?

It is hoped that artificial enzymes designed in laboratories can be efficient alternatives to chemical catalysts that have been used to synthesize organic molecules. However, the design of artificial enzymes is challenging and requires a detailed molecular-level analysis to understand the mechanism they promote in order to design efficient variants. In this study, we computationally investigate the mechanism of proficient Morita-Baylis-Hillman enzymes developed using a combination of computational design and directed evolution. The powerful transition path sampling method coupled with in-depth post-processing analysis has been successfully used to elucidate the different chemical pathways, transition states, protein dynamics, and free energy barriers of reactions catalyzed by such laboratory-optimized enzymes. This research provides an explanation for how different chemical modifications in an enzyme affect its catalytic activity in ways that are not predictable by static design algorithms.

Accelerating Kinetics with Time-Reversal Path Sampling.

In comparison to numerous enhanced sampling methods for equilibrium thermodynamics, accelerating simulations for kinetics and nonequilibrium statistics are relatively rare and less effective. Here, we derive a time-reversal path sampling (tRPS) method based on time reversibility to accelerate simulations for determining the transition rates between free-energy basins. It converts the difficult uphill path sampling into an easy downhill problem. This method is easy to implement, i.e., forward and backward shooting simulations with opposite initial velocities are conducted from random initial conformations within a transition-state region until they reach the basin minima, which are then assembled to give the distribution of transition paths efficiently. The effects of tRPS are demonstrated using a comparison with direct simulations of protein folding and unfolding, where tRPS is shown to give results consistent with direct simulations and increase the efficiency by up to five orders of magnitude. This approach is generally applicable to stochastic processes with microscopic reversibility, regardless of whether the variables are continuous or discrete.

Characterization of geological uncertainties from limited boreholes using copula-based coupled Markov chains for underground construction

This paper proposes an efficient method for quantifying the stratigraphic uncertainties and modeling the geological formations based on boreholes. Two Markov chains are used to describe the soil transitions along different directions, and the transition probability matrices (TPMs) of the Markov chains are analytically expressed by copulas. This copula expression is efficient since it can represent a large TPM by a few unknown parameters. Due to the analytical expression of the TPMs, the likelihood function of the Markov chain model is given in an explicit form. The estimation of the TPMs is then re-casted as a multi-objective constrained optimization problem that aims to maximize the likelihoods of two independent Markov chains subject to a set of parameter constraints. Unlike the method which determines the TPMs by counting the number of transitions between soil types, the proposed method is more statistically sound. Moreover, a random path sampling method is presented to avoid the directional effect problem in simulations. The soil type at a location is inferred from its nearest known neighbors along the cardinal directions. A general form of the conditional probability, based on Pickard's theorem and Bayes rule, is presented for the soil type generation. The proposed stratigraphic characterization and simulation method is applied to real borehole data collected from a construction site in Wuhan, China. It is illustrated that the proposed method is accurate in prediction and does not show an inclination during simulation.

Adaptive Sampling Methods for Molecular Dynamics in the Era of Machine Learning.

Molecular dynamics (MD) simulations are fundamental computational tools for the study of proteins and their free energy landscapes. However, sampling protein conformational changes through MD simulations is challenging due to the relatively long time scales of these processes. Many enhanced sampling approaches have emerged to tackle this problem, including biased sampling and path-sampling methods. In this Perspective, we focus on adaptive sampling algorithms. These techniques differ from other approaches because the thermodynamic ensemble is preserved and the sampling is enhanced solely by restarting MD trajectories at particularly chosen seeds rather than introducing biasing forces. We begin our treatment with an overview of theoretically transparent methods, where we discuss principles and guidelines for adaptive sampling. Then, we present a brief summary of select methods that have been applied to realistic systems in the past. Finally, we discuss recent advances in adaptive sampling methodology powered by deep learning techniques, as well as their shortcomings.

World‐Space Spatiotemporal Path Resampling for Path Tracing

AbstractWith the advent of hardware‐accelerated ray tracing, more and more real‐time rendering applications tend to render images with ray‐traced global illumination (GI). However, the low sample counts at real‐time framerates bring enormous challenges to existing path sampling methods. Recent work (ReSTIR GI) samples indirect illumination effectively with a dramatic bias reduction. However, as a screen‐space based path resampling approach, it can only reuse the path at the first bounce and brings subtle benefits for complex scenes. To this end, we propose a world‐space based spatiotemporal path resampling approach. Our approach caches more path samples into a world‐space grid, which allows reusing sub‐path starting from non‐primary path vertices. Furthermore, we introduce a practical normal‐aware hash grid construction approach, providing more efficient candidate samples for path resampling. Eventually, our method achieves improvements ranging from 16.6% to 41.9% in terms of mean squared errors (MSE) compared against the previous method with only 4.4% ∼ 8.4% extra time cost.

How far can we stretch the timescale with RETIS?

Molecular dynamics (MD) and Monte Carlo (MC) have long coexisted as two main independent branches of molecular simulation. In the late eighties, however, algorithms based on the combination of both were created such as hybrid Monte Carlo which uses large MD steps as MC moves. An entirely different kind of combination emerged a decade later via the transition path sampling (TPS) method in which MD trajectories are not just part of the MC move, but also form the state space being sampled. Algorithms like replica exchange transition interface sampling (RETIS) exploit this idea to compute reaction rates via a series of TPS simulations. RETIS yields results identical to hypothetical long MD runs, but with exponentially reduced computation time. This perspective describes the RETIS method and discusses recent and future advancements that will enable the study of even longer molecular timescales with reasonable computational resources.

RNA folding pathways from all-atom simulations with a variationally improved history-dependent bias

Atomically detailed simulations of RNA folding have proven very challenging in view of the difficulties of developing realistic force fields and the intrinsic computational complexity of sampling rare conformational transitions. As a step forward in tackling these issues, we extend to RNA an enhanced path-sampling method previously successfully applied to proteins. In this scheme, the information about the RNA’s native structure is harnessed by a soft history-dependent biasing force promoting the generation of productive folding trajectories in an all-atom force field with explicit solvent. A rigorous variational principle is then applied to minimize the effect of the bias. Here, we report on an application of this method to RNA molecules from 20 to 47 nucleotides long and increasing topological complexity. By comparison with analog simulations performed on small proteins with similar size and architecture, we show that the RNA folding landscape is significantly more frustrated, even for relatively small chains with a simple topology. The predicted RNA folding mechanisms are found to be consistent with the available experiments and some of the existing coarse-grained models. Due to its computational performance, this scheme provides a promising platform to efficiently gather atomistic RNA folding trajectories, thus retain the information about the chemical composition of the sequence.

Path sampling and integration method to calculate speckle patterns.

A stable speckle pattern is generated when a coherent beam illuminates a stationary scattering medium that contains numerous scatterers with fixed positions. To date, there has been no valid method to the best of our knowledge for calculating the speckle pattern of a macro medium with a large number of scatterers. Here, a new method based on possible path sampling with corresponding weights and coherent superposition is presented for the simulation of optical field propagation in a scattering medium and output speckle patterns. In this method, a photon is launched onto a medium with fixed scatterers. It propagates in one direction; upon collision with a scatterer, its direction is updated. The procedure is repeated until it exits the medium. A sampled path is obtained in this manner. By repeatedly launching photons, numerous independent optical paths can be sampled. A speckle pattern, corresponding to the probability density of the photon, is formed by the coherent superposition of sufficiently sampled path lengths ending on a receiving screen. This method can be used in sophisticated studies of the influences of medium parameters, motion of scatterers, sample distortions on speckle distributions, and morphological appearances. It can be used for micro-examination of optical fields in scattering media and may inspire new methods and techniques for non-invasive precision detection and diagnosis of scattering media.

ATESA: An Automated Aimless Shooting Workflow.

Transition path sampling methods are powerful tools for studying the dynamics of rare events in molecular simulations. However, these methods are generally restricted to experts with the knowledge and resources to properly set up and analyze the often hundreds of thousands of simulations that constitute a complete study. Aimless Transition Ensemble Sampling and Analysis (ATESA) is a new open-source software program written in Python that automates a full transition path sampling workflow based on the aimless shooting algorithm, streamlining the process and reducing the barrier to use for researchers new to this approach. This introduction to ATESA includes a demonstration of a complete transition path sampling process flow for an example reaction, including finding an initial transition state, sampling with aimless shooting, building a reaction coordinate with inertial likelihood maximization, verifying that coordinate with committor analysis, and measuring the reaction energy profile with umbrella sampling. We also describe our implementation of a termination criterion for aimless shooting based on the Godambe information calculated during model building with likelihood maximization as well as a novel approach to constraining simulations to the desired rare event pathway during umbrella sampling.

Efficient Differentiation of Pixel Reconstruction Filters for Path-Space Differentiable Rendering

Pixel reconstruction filters play an important role in physics-based rendering and have been thoroughly studied. In physics-based differentiable rendering, however, the proper treatment of pixel filters remains largely under-explored. We present a new technique to efficiently differentiate pixel reconstruction filters based on the path-space formulation. Specifically, we formulate the pixel boundary integral that models discontinuities in pixel filters and introduce new antithetic sampling methods that support differentiable path sampling methods, such as adjoint particle tracing and bidirectional path tracing. We demonstrate both the need and efficacy of antithetic sampling when estimating this integral, and we evaluate its effectiveness across several differentiable- and inverse-rendering settings.

Path Sampling Simulations Reveal How the Q61L Mutation Alters the Dynamics of KRas.

Flexibility is essential for many proteins to function, but can be difficult to characterize. Experiments lack resolution in space and time, while the time scales involved are prohibitively long for straightforward molecular dynamics simulations. In this work, we present a multiple state transition path sampling simulation study of a protein that has been notoriously difficult to characterize in its active state. The GTPase enzyme KRas is a signal transduction protein in pathways for cell differentiation, growth, and division. When active, KRas tightly binds guanosine triphosphate (GTP) in a rigid state. The protein-GTP complex can also visit more flexible states, in which it is not active. KRas mutations can affect the conversion between these rigid and flexible states, thus prolonging the activation of signal transduction pathways, which may result in tumor formation. In this work, we apply path sampling simulations to investigate the dynamic behavior of KRas-4B (wild type, WT) and the oncogenic mutant Q61L (Q61L). Our results show that KRas visits several conformational states, which are the same for WT and Q61L. The multiple state transition path sampling (MSTPS) method samples transitions between the different states in a single calculation. Tracking which transitions occur shows large differences between WT and Q61L. The MSTPS results further reveal that for Q61L, a route to a more flexible state is inaccessible, thus shifting the equilibrium to more rigid states. The methodology presented here enables a detailed characterization of protein flexibility on time scales not accessible with brute-force molecular dynamics simulations.

Exchanging Replicas with Unequal Cost, Infinitely and Permanently.

We developed a replica exchange method that is effectively parallelizable even if the computational cost of the Monte Carlo moves in the parallel replicas are considerably different, for instance, because the replicas run on different types of processor units or because of the algorithmic complexity. To prove detailed-balance, we make a paradigm shift from the common conceptual viewpoint in which the set of parallel replicas represents a high-dimensional superstate, to an ensemble-based criterion in which the other ensembles represent an environment that might or might not participate in the Monte Carlo move. In addition, based on a recent algorithm for computing permanents, we effectively increase the exchange rate to infinite without the steep factorial scaling as a function of the number of replicas. We illustrate the effectiveness of this replica exchange methodology by combining it with a quantitative path sampling method, replica exchange transition interface sampling (RETIS), in which the costs for a Monte Carlo move can vary enormously as paths in a RETIS algorithm do not have the same length and the average path lengths tend to vary considerably for the different path ensembles that run in parallel. This combination, coined ∞RETIS, was tested on three model systems.

Perspective: Path Sampling Methods Applied to Enzymatic Catalysis.

This Perspective reviews the use of Transition Path Sampling methods to study enzymatically catalyzed chemical reactions. First applied by our group to an enzymatic reaction over 15 years ago, the method has uncovered basic principles in enzymatic catalysis such as the protein promoting vibration, and it has also helped harmonize such ideas as electrostatic preorganization with dynamic views of enzyme function. It is now being used to help uncover principles of protein design necessary to artificial enzyme creation.

Sampling rare conformational transitions with a quantum computer

Structural rearrangements play a central role in the organization and function of complex biomolecular systems. In principle, Molecular Dynamics (MD) simulations enable us to investigate these thermally activated processes with an atomic level of resolution. In practice, an exponentially large fraction of computational resources must be invested to simulate thermal fluctuations in metastable states. Path sampling methods focus the computational power on sampling the rare transitions between states. One of their outstanding limitations is to efficiently generate paths that visit significantly different regions of the conformational space. To overcome this issue, we introduce a new algorithm for MD simulations that integrates machine learning and quantum computing. First, using functional integral methods, we derive a rigorous low-resolution spatially coarse-grained representation of the system’s dynamics, based on a small set of molecular configurations explored with machine learning. Then, we use a quantum annealer to sample the transition paths of this low-resolution theory. We provide a proof-of-concept application by simulating a benchmark conformational transition with all-atom resolution on the D-Wave quantum computer. By exploiting the unique features of quantum annealing, we generate uncorrelated trajectories at every iteration, thus addressing one of the challenges of path sampling. Once larger quantum machines will be available, the interplay between quantum and classical resources may emerge as a new paradigm of high-performance scientific computing. In this work, we provide a platform to implement this integrated scheme in the field of molecular simulations.

Non-Markov-Type Analysis and Diffusion Map Analysis for Molecular Dynamics Trajectory of Chignolin at a High Temperature.

We apply the non-Markov-type analysis of state-to-state transitions to nearly microsecond molecular dynamics (MD) simulation data at a folding temperature of a small artificial protein, chignolin, and we found that the time scales obtained are consistent with our previous result using the weighted ensemble simulations, which is a general path-sampling method to extract the kinetic properties of molecules. Previously, we also applied diffusion map (DM) analysis, which is one of a manifold of learning techniques, to the same trajectory of chignolin in order to cluster the conformational states and found that DM and relaxation mode analysis give similar results for the eigenvectors. In this paper, we divide the same trajectory into shorter pieces and further apply DM to such short-length trajectories to investigate how the obtained eigenvectors are useful to characterize the conformational change of chignolin.

Reaction Coordinate and Thermodynamics of Base Flipping in RNA.

Base flipping is a key biophysical event involved in recognition of various ligands by ribonucleic acid (RNA) molecules. However, the mechanism of base flipping in RNA remains poorly understood, in part due to the lack of atomistic details on complex rearrangements in neighboring bases. In this work, we applied transition path sampling (TPS) methods to study base flipping in a double-stranded RNA (dsRNA) molecule that is known to interact with RNA-editing enzymes through this mechanism. We obtained an ensemble of 1000 transition trajectories to describe the base-flipping process. We used the likelihood maximization method to determine the refined reaction coordinate (RC) consisting of two collective variables (CVs), a distance and a dihedral angle between nucleotides that form stacking interactions with the flipping base. The free energy profile projected along the refined RC revealed three minima, two corresponding to the initial and final states and one for a metastable state. We suggest that the metastable state likely represents a wobbled conformation of nucleobases observed in NMR studies that is often characterized as the flipped state. The analyses of reactive trajectories further revealed that the base flipping is coupled to a global conformational change in a stem-loop of dsRNA.

Catalytic Mechanism of Processive GlfT2: Transition Path Sampling Investigation of Substrate Translocation.

We applied the transition path sampling (TPS) method to study the translocation step of the catalytic mechanism of galactofuranosyl transferase 2 (GlfT2). Using TPS in the field of enzymatic reactions is still relatively rare, and we show its effectiveness on this enzymatic system. We decipher an unknown mechanism of the translocation step and, thus, provide a complete understanding of the catalytic mechanism of GlfT2 at the atomistic level. The GlfT2 enzyme is involved in the formation of the mycobacterial cell wall and transfers galactofuranose (Galf) from UDP-Galf onto a growing acceptor Galf chain. The biosynthesis of the galactan chain is accomplished in a processive manner, with the growing acceptor substrate remaining bound to GlfT2. The glycosidic bond formed by GlfT2 between the two Galf residues alternates between β-(1–6) and β-(1–5) linkages. The translocation of the growing galactan between individual additions of Galf residues is crucial for the function of GlfT2. Analysis of unbiased trajectory ensembles revealed that the translocation proceeds differently depending on the glycosidic linkage between the last two Galf residues. We also showed that the protonation state of the catalytic residue Asp372 significantly influences the translocation. Approximate transition state structures and potential energy reaction barriers of the translocation process were determined. The calculated potential reaction barriers in the range of 6–14 kcal/mol show that the translocation process is not the rate-limiting step in galactan biosynthesis.