Locating Transition States for Biomolecular Dynamics via Invertible Dimensionality Reduction

Locating the transition states (TS) for the conformational changes of biomacromolecules is among the major tasks of biomolecular simulations, as they are the bottlenecks of motion encoding key mechanistic insights. However, identifying the short-lived TSs from (even abundant) simulation data has been a long-standing challenge due to the high dimensionality of the molecules. Gentlest ascent dynamics (GAD) is an effective approach that searches for saddle points but only within spaces of small number of (typically <20) dimensions. Such a restriction of GAD may in principle be relieved by dimensionality reduction (DR) that reduces the high-dimensional configurational space of the molecules to a low-dimensional manifold. However, the vast majority of DR algorithms are built to focus on only high-density regions and have therefore distorted the TS regions, disabling a subsequent GAD search. The recently introduced reaction coordinate flows (RCF) is among the few exceptions. As RCF learns an invertible mapping between the configurational space and the reduced RC space through a loss function incorporating both density and transition pair information, it shall be able to preserve kinetics and therefore TS during DR. GAD can then be readily applied to locate the TS candidates in the RCF-learned RC space, which can be validated rigorously through reverse RCF mapping and committor analysis in the original space. Here, we demonstrate the effectiveness of this RCF-GAD integration through alanine dipeptide and the ground-to-excited transition of the T4 lysozyme L99A variant (T4L-L99A) in explicit solvents. For alanine dipeptide, GAD managed to identify three TSs with the RCF reduction to four RCs, but only two TSs with a reduction to two RCs, due to the merging of two low density stable states in the 2RC representation, indicating the necessity of a priori evaluation of the number of intrinsic dimensions for RCF. For T4L-L99A, the TSs located by GAD in a 4RC RCF reduction successfully resembled those found previously via automated path searching, demonstrating the feasibility of our approach for realistic biomolecular systems.

AMUSET-TICA: A Tensor-Based Approach for Identifying Slow Collective Variables in Biomolecular Dynamics

Elucidating collective variables (CVs) for biomolecular dynamics is crucial for understanding numerous biological processes. By leveraging the tensor-train data structure, a multilinear version of the AMUSE (Algorithm for Multiple Unknown Signals) algorithm for Koopman approximation (AMUSEt) was recently developed to identify CVs for biomolecular dynamics. To find slow CVs, AMUSEt transforms input features (e.g., pairwise atomic distances) into nonlinear basis functions (e.g., Gaussian functions) and encodes these nonlinear basis functions within a tensor-train structure via time-lagged correlation functions. Due to the need to fit these tensor-train data structures into computer memory, AMUSEt can handle only a limited number of input features. Consequently, AMUSEt relies on manually selecting and ranking features based on physical intuition to fully capture the slow dynamics. However, when applied to complex biological systems with numerous features, this selection and ranking process becomes increasingly challenging. To address this challenge, here we present AMUSET-TICA (AMUSEt-based Time-lagged Independent Component Analysis), a CV-identification method using time-structure-independent components (tICs) as the input features for AMUSEt. The key insight of AMUSET-TICA lies in its highly effective embedding of high-dimensional atomistic protein conformations, achieved by expanding orthogonal tICs into overlapping Gaussian basis functions through a tensor-product data structure. This eliminates the need for manually selecting and ranking input features for a wide range of biomolecular systems. We demonstrate that AMUSET-TICA consistently and significantly outperforms AMUSEt and tICA in identifying slow CVs for three different biomolecular systems: alanine dipeptide, the N-terminal domain of L9 (NTL9), and the FIP35 WW domain. For all these systems, the CVs generated by AMUSET-TICA accurately describe the slowest dynamical modes underlying these biological conformational changes. Furthermore, we show that AMUSET-TICA achieves performance comparable to deep-learning approaches like VAMPnets in identifying the slowest dynamical modes, while being significantly more computationally efficient in terms of CPU time. In addition, the CVs yielded by AMUSET-TICA provide insights into the folding mechanisms of NTL9 and the FIP35 WW domain, including CV3 and CV4 of the WW domain, which capture its two parallel folding pathways. We expect AMUSET-TICA can be widely applied to facilitate the investigation of biomolecular dynamics.

Dissecting Large-Scale Structural Transitions in Membrane Transporters Using Advanced Simulation Technologies

Membrane transporters are integral membrane proteins that act as gatekeepers of the cell, controlling fundamental processes such as recruitment of nutrients and expulsion of waste material. At a basic level, transporters operate using the "alternating access model," in which transported substances are accessible from only one side of the membrane at a time. This model usually involves large-scale structural changes in the transporter, which often cannot be captured using unbiased, conventional molecular simulation techniques. In this article, we provide an overview of some of the major simulation techniques that have been applied to characterize the structural dynamics and energetics involved in the transition of membrane transporters between their functional states. After briefly introducing each technique, we discuss some of their advantages and limitations and provide some recent examples of their application to membrane transporters.

Activity Regulation and Conformation Response of Janus Kinase 3 Mediated by Phosphorylation: Exploration from Correlation Network Analysis and Markov Model

The activity of the enzyme JAK3 is modulated by tyrosine phosphorylation, yet the underlying molecular details remain not fully understood. In this study, we employed a GaMD trajectory-based Markov model and correlation network analysis (CNA) to investigate the impact of single phosphorylation (SP) at Y980 (pY980) and double phosphorylation (DP) at Y980/Y981 (pY980/pY981) on the conformational dynamics of JAK3 bound by inhibitors IZA and MI1. The Markov model analysis indicated that both SP and DP result in fewer conformational states and significantly influence the conformational dynamics of the P-loop, αC-helix, and loop1-loop3, while maintaining the hinge region's high rigidity. The CNA findings revealed that phosphorylation alters the communication network among different structural regions of JAK3, providing a rational explanation for how phosphorylation affects the conformational dynamics of the distant P-loop and loop1-loop3. Moreover, the conformational changes mediated by SP and DP further affect the interactions between the inhibitors and the hot spots (L828, V836, E903, Y904, L905, and L956) of JAK3. This work offers valuable theoretical insights into the molecular mechanisms that regulate JAK3 activity.

Conformational and Stability Analysis of SARS-CoV-2 Spike Protein Variants by Molecular Simulation

We performed a comprehensive structural analysis of the conformational space of several spike (S) protein variants using molecular dynamics (MD) simulations. Specifically, we examined four well-known variants (Delta, BA.1, XBB.1.5, and JN.1) alongside the wild-type (WT) form of SARS-CoV-2. The conformational states of each variant were characterized by analyzing their distributions within a selected space of collective variables (CVs), such as inter-domain distances between the receptor-binding domain (RBD) and the N-terminal domain (NTD). Our primary focus was to identify conformational states relevant to potential structural transitions and to determine the set of native contacts (NCs) that stabilize these conformations. The results reveal that genetically more distant variants, such as XBB.1.5, BA.1, and JN.1, tend to adopt more compact conformational states compared to the WT. Additionally, these variants exhibit novel NC profiles, characterized by an increased number of specific contacts distributed among ionic, polar, and nonpolar residues. We further analyzed the impact of specific mutations, including T478K, N500Y, and Y504H. These mutations not only enhance interactions with the human host receptor but also alter inter-chain stability by introducing additional NCs compared to the WT. Consequently, these mutations may influence the accessibility of certain protein regions to neutralizing antibodies. Overall, these findings contribute to a deeper understanding of the structural and functional variations among S protein variants.

Structural Heterogeneity of Intermediate States Facilitates CRIPT Peptide Binding to the PDZ3 Domain: Insights from Molecular Dynamics and Markov State Model Analysis

Intrinsically disordered proteins (IDPs), characterized by a lack of defined tertiary structure, are ubiquitous and indispensable components of cellular machinery. These proteins participate in a diverse array of biological processes, often undergoing conformational transitions upon binding to their target, a phenomenon termed "folding-upon-binding." The finding raises the question of how to achieve rapid binding kinetics in the presence of intrinsic disorder, and the underlying molecular mechanism remains elusive. This study investigated the interaction between the C-terminal region of CRIPT and the third PDZ domain of PSD-95, a critical complex in neuronal development. Upon binding, the CRIPT peptide adopts a β-strand conformation, a process meticulously characterized through extensive molecular dynamics simulations totaling 67.7 μs. Our findings reveal a funnel-like binding landscape in which IDPs can adopt multiple conformations prior to binding, forming structurally heterogeneous intermediate complexes and leading to diverse binding pathways. The stabilization of these intermediate complexes necessitates a dynamic interplay of native and non-native interactions. Markov state model analysis underscores the important role of structural heterogeneity as it contributes to accelerated binding. These findings enrich the classical fly-casting mechanism and provide novel insights into the functional advantages conferred by intrinsic disorder.

Markov State Models with Weighted Ensemble Simulation: How to Eliminate the Trajectory Merging Bias

The weighted ensemble (WE) algorithm is gaining popularity as a rare event method for studying long timescale processes with molecular dynamics. WE is particularly useful for determining kinetic properties, such as rates of protein (un)folding and ligand (un)binding, where transition rates can be calculated from the flux of trajectories into a target basin of interest. However, this flux depends exponentially on the number of splitting events that a given trajectory experiences before reaching the target state and can vary by orders of magnitude between WE replicates. Markov state models (MSMs) are helpful tools to aggregate information across multiple WE simulations and have previously been shown to provide more accurate transition rates than WE alone. Discrete-time MSMs are models that coarsely describe the evolution of the system from one discrete state to the next using a discrete lag time, τ. When an MSM is built using conventional MD data, longer values of τ typically provide more accurate results. Combining WE simulations with Markov state modeling presents some additional challenges, especially when using a value of τ that exceeds the lag time between resampling steps in the WE algorithm, τWE. Here, we identify a source of bias that occurs when τ > τWE, which we refer to as "merging bias". We also propose an algorithm to eliminate the merging bias, which results in merging bias-corrected MSMs, or "MBC-MSMs". Using a simple model system, as well as a complex biomolecular example, we show that MBC-MSMs significantly outperform both τ = τWE MSMs and uncorrected MSMs at longer lag times.

Molecular Mechanisms Underlying the Loop-Closing Dynamics of β-1,4 Galactosyltransferase 1

The β-1,4 galactosylation catalyzed by β-1,4 galactosyltransferases (β4Gal-Ts) is not only closely associated with diverse physiological and pathological processes in humans but also widely applied in the N-glycan modification of protein glycoengineering. The loop-closing process of β4Gal-Ts is an essential intermediate step intervening in the binding events of donor substrate (UDP-Gal/Mn2+) and acceptor substrate during its catalytic cycle, with a significant impact on the galactosylation activities. However, the molecular mechanisms in regulating loop-closing dynamics are not entirely clear. Here, we construct Markov state models (MSMs) based on approximately 20 μs of all-atom molecular dynamics simulations to explore the loop-closing dynamics for β-1,4 galactosyltransferase 1 (β4Gal-T1). Our MSM reveals five key metastable states of β4Gal-T1 upon substrate binding, indicating that the entire conformational transition occurs on a time scale of ∼10 μs. Moreover, a regulatory mechanism involving six conserved residues (R187, H190, F222, W310, I341, and D346) among β4Gal-Ts is validated to account for the loop-closing dynamics of the C-loop and W-loop by site-directed mutagenesis and enzymatic activity assays, exhibiting high consistency with our computational predictions. Overall, our research proposes detailed atomic-level insight into the loop-closing dynamics of the C-loop and W-loop on β4Gal-T1, contributing to a deeper understanding of catalytic mechanisms of β-1,4 galactosylation.

Unveiling hidden reaction kinetics of carbon dioxide in supercritical aqueous solutions

Dissolution of CO2 in water followed by the subsequent hydrolysis reactions is of great importance to the global carbon cycle, and carbon capture and storage. Despite numerous previous studies, the reactions are still not fully understood at the atomistic scale. Here, we combined ab initio molecular dynamics (AIMD) simulations with Markov state models to elucidate the reaction mechanisms and kinetics of CO2 in supercritical water both in the bulk and nanoconfined states. The integration of unsupervised learning with first-principles data allows us to identify complex reaction coordinates and pathways automatically instead of a priori human speculation. Interestingly, our unbiased modeling found an unknown pathway of dissolving CO2(aq) under graphene nanoconfinement, involving the pyrocarbonate anion [C2O[Formula: see text](aq)] as an intermediate state. The pyrocarbonate anion was previously hypothesized to have a fleeting existence in water; however, our study reveals that it is a crucial reaction intermediate and stable carbon species in the nanoconfined solutions. We even observed the formation of pyrocarbonic acid [H2C2O5(aq)], which was unknown in water, in our AIMD simulations. The unexpected appearance of pyrocarbonates is related to the superionic behavior of the confined solutions. We also found that carbonation reactions involve collective proton transfer along transient water wires, which exhibits concerted behavior in the bulk solution but proceeds stepwise under nanoconfinement. The first-principles Markov state models show substantial promise for elucidating complex reaction kinetics in aqueous solutions. Our study highlights the importance of large oxocarbons in aqueous carbon reactions, with great implications for the deep carbon cycle and the sequestration of CO2.

Exploring transition states of protein conformational changes via out-of-distribution detection in the hyperspherical latent space

Identifying transitional states is crucial for understanding protein conformational changes that underlie numerous biological processes. Markov state models (MSMs), built from Molecular Dynamics (MD) simulations, capture these dynamics through transitions among metastable conformational states, and have demonstrated success in studying protein conformational changes. However, MSMs face challenges in identifying transition states, as they partition MD conformations into discrete metastable states (or free energy minima), lacking description of transition states located at the free energy barriers. Here, we introduce Transition State identification via Dispersion and vAriational principle Regularized neural networks (TS-DAR), a deep learning framework inspired by out-of-distribution (OOD) detection in trustworthy artificial intelligence (AI). TS-DAR offers an end-to-end pipeline that can simultaneously detect all transition states between multiple free minima from MD simulations using the regularized hyperspherical embeddings in latent space. The key insight of TS-DAR lies in treating transition state structures as OOD data, recognizing that they are sparsely populated and exhibit a distributional shift from metastable states. We demonstrate the power of TS-DAR by applying it to a 2D potential, alanine dipeptide, and the translocation of a DNA motor protein on DNA, where it outperforms previous methods in identifying transition states.

Nucleosome condensate and linker DNA alter chromatin folding pathways and rates

Chromatin organization is essential for DNA packaging and gene regulation in eukaryotic genomes. While significant progresses have been made, the exact atomistic arrangement of nucleosomes remains controversial. Using a well-calibrated residue-level coarse-grained model and advanced dynamics modeling techniques, particularly the non-Markovian dynamics model, we map the free energy landscape of tetra-nucleosome systems, identify both metastable conformations and intermediate states in folding pathways, and quantify the folding kinetics. Our findings show that chromatin with 10 n base pairs (bp) DNA linker lengths favor zigzag fibril structures. However, longer linker lengths destabilize this conformation. When the linker length is 10 n + 5 bp , chromatin loses unique conformations, favoring a dynamic ensemble of structures resembling folding intermediates. Embedding the tetra-nucleosome in a nucleosome condensate similarly shifts stability towards folding intermediates as a result of the competition of inter-nucleosomal contacts. These results suggest that chromatin organization observed in vivo arises from the unfolding of fibril structures due to nucleosome crowding and linker length variation. This perspective aids in unifying experimental studies to develop atomistic models for chromatin.

Significance

Atomic structures of chromatin have become increasingly accessible, largely through cryo-EM techniques. Nonetheless, these approaches often face limitations in addressing how intrinsic in vivo factors influence chromatin organization. We present a structural characterization of chromatin under the combined effects of nucleosome condensate crowding and linker DNA length variation-two critical in vivo features that have remained challenging to capture experimentally. This work leverages a novel application of non-Markovian dynamical modeling, providing accurate mapping of chromatin folding kinetics and pathways. Our findings support a hypothesis that in vivo chromatin organization arises from folding intermediates advancing toward a stable fibril configuration, potentially resolving longstanding questions surrounding chromatin atomic structure.

Inhibitor binding and disruption of coupled motions in MmpL3 protein: Unraveling the mechanism of trehalose monomycolate transport

Mycobacterial membrane protein Large 3 (MmpL3) of Mycobacterium tuberculosis (Mtb) is crucial for the translocation of trehalose monomycolate (TMM) across the inner bacterial cell membrane, making it a promising target for anti-tuberculosis (TB) drug development. While several structural, microbiological, and in vitro studies have provided significant insights, the precise mechanisms underlying TMM transport by MmpL3 and its inhibition remain incompletely understood at the atomic level. In this study, molecular dynamic (MD) simulations for the apo form and seven inhibitor-bound forms of Mtb MmpL3 were carried out to obtain a thorough comprehension of the protein's dynamics and function. MD simulations revealed that the seven inhibitors in this work stably bind to the central channel of the transmembrane domain and primarily forming hydrogen bonds with ASP251, ASP640, or both residues. Through dynamical cross-correlation matrix and principal component analysis analyses, several types of coupled motions between different domains were observed in the apo state, and distinct conformational states were identified using Markov state model analysis. These coupled motions and varied conformational states likely contribute to the transport of TMM. However, simulations of inhibitor-bound MmpL3 showed an enlargement of the proton channel, potentially disrupting coupled motions. This indicates that inhibitors may impair MmpL3's transport function by directly blocking the proton channel, thereby hindering coordinated domain movements and indirectly affecting TMM translocation.

Information Bottleneck Approach for Markov Model Construction

Markov state models (MSMs) have proven valuable in studying the dynamics of protein conformational changes via statistical analysis of molecular dynamics simulations. In MSMs, the complex configuration space is coarse-grained into conformational states, with dynamics modeled by a series of Markovian transitions among these states at discrete lag times. Constructing the Markovian model at a specific lag time necessitates defining states that circumvent significant internal energy barriers, enabling internal dynamics relaxation within the lag time. This process effectively coarse-grains time and space, integrating out rapid motions within metastable states. Thus, MSMs possess a multiresolution nature, where the granularity of states can be adjusted according to the time-resolution, offering flexibility in capturing system dynamics. This work introduces a continuous embedding approach for molecular conformations using the state predictive information bottleneck (SPIB), a framework that unifies dimensionality reduction and state space partitioning via a continuous, machine learned basis set. Without explicit optimization of the VAMP-based scores, SPIB demonstrates state-of-the-art performance in identifying slow dynamical processes and constructing predictive multiresolution Markovian models. Through applications to well-validated mini-proteins, SPIB showcases unique advantages compared to competing methods. It autonomously and self-consistently adjusts the number of metastable states based on a specified minimal time resolution, eliminating the need for manual tuning. While maintaining efficacy in dynamical properties, SPIB excels in accurately distinguishing metastable states and capturing numerous well-populated macrostates. This contrasts with existing VAMP-based methods, which often emphasize slow dynamics at the expense of incorporating numerous sparsely populated states. Furthermore, SPIB's ability to learn a low-dimensional continuous embedding of the underlying MSMs enhances the interpretation of dynamic pathways. With these benefits, we propose SPIB as an easy-to-implement methodology for end-to-end MSM construction.

The Arabidopsis AtSWEET13 transporter discriminates sugars by selective facial and positional substrate recognition

Transporters are targeted by endogenous metabolites and exogenous molecules to reach cellular destinations, but it is generally not understood how different substrate classes exploit the same transporter's mechanism. Any disclosure of plasticity in transporter mechanism when treated with different substrates becomes critical for developing general selectivity principles in membrane transport catalysis. Using extensive molecular dynamics simulations with an enhanced sampling approach, we select the Arabidopsis sugar transporter AtSWEET13 as a model system to identify the basis for glucose versus sucrose molecular recognition and transport. Here we find that AtSWEET13 chemical selectivity originates from a conserved substrate facial selectivity demonstrated when committing alternate access, despite mono-/di-saccharides experiencing differing degrees of conformational and positional freedom throughout other stages of transport. However, substrate interactions with structural hallmarks associated with known functional annotations can help reinforce selective preferences in molecular transport.

Advanced computational approaches to understand protein aggregation

Protein aggregation is a widespread phenomenon implicated in debilitating diseases like Alzheimer's, Parkinson's, and cataracts, presenting complex hurdles for the field of molecular biology. In this review, we explore the evolving realm of computational methods and bioinformatics tools that have revolutionized our comprehension of protein aggregation. Beginning with a discussion of the multifaceted challenges associated with understanding this process and emphasizing the critical need for precise predictive tools, we highlight how computational techniques have become indispensable for understanding protein aggregation. We focus on molecular simulations, notably molecular dynamics (MD) simulations, spanning from atomistic to coarse-grained levels, which have emerged as pivotal tools in unraveling the complex dynamics governing protein aggregation in diseases such as cataracts, Alzheimer's, and Parkinson's. MD simulations provide microscopic insights into protein interactions and the subtleties of aggregation pathways, with advanced techniques like replica exchange molecular dynamics, Metadynamics (MetaD), and umbrella sampling enhancing our understanding by probing intricate energy landscapes and transition states. We delve into specific applications of MD simulations, elucidating the chaperone mechanism underlying cataract formation using Markov state modeling and the intricate pathways and interactions driving the toxic aggregate formation in Alzheimer's and Parkinson's disease. Transitioning we highlight how computational techniques, including bioinformatics, sequence analysis, structural data, machine learning algorithms, and artificial intelligence have become indispensable for predicting protein aggregation propensity and locating aggregation-prone regions within protein sequences. Throughout our exploration, we underscore the symbiotic relationship between computational approaches and empirical data, which has paved the way for potential therapeutic strategies against protein aggregation-related diseases. In conclusion, this review offers a comprehensive overview of advanced computational methodologies and bioinformatics tools that have catalyzed breakthroughs in unraveling the molecular basis of protein aggregation, with significant implications for clinical interventions, standing at the intersection of computational biology and experimental research.

Characterization of binding kinetics and intracellular signaling of new psychoactive substances targeting cannabinoid receptor using transition-based reweighting method

New psychoactive substances (NPS) targeting cannabinoid receptor 1 pose a significant threat to society as recreational abusive drugs that have pronounced physiological side effects. These greater adverse effects compared to classical cannabinoids have been linked to the higher downstream β-arrestin signaling. Thus, understanding the mechanism of differential signaling will reveal important structure-activity relationship essential for identifying and potentially regulating NPS molecules. In this study, we simulate the slow (un)binding process of NPS MDMB-Fubinaca and classical cannabinoid HU-210 from CB1 using multi-ensemble simulation to decipher the effects of ligand binding dynamics on downstream signaling. The transition-based reweighing method is used for the estimation of transition rates and underlying thermodynamics of (un)binding processes of ligands with nanomolar affinities. Our analyses reveal major interaction differences with transmembrane TM7 between NPS and classical cannabinoids. A variational autoencoder-based approach, neural relational inference (NRI), is applied to assess the allosteric effects on intracellular regions attributable to variations in binding pocket interactions. NRI analysis indicate a heightened level of allosteric control of NPxxY motif for NPS-bound receptors, which contributes to the higher probability of formation of a crucial triad interaction (Y7.53-Y5.58-T3.46) necessary for stronger β-arrestin signaling. Hence, in this work, MD simulation, data-driven statistical methods, and deep learning point out the structural basis for the heightened physiological side effects associated with NPS, contributing to efforts aimed at mitigating their public health impact.

Tutorial on how to build non-Markovian dynamic models from molecular dynamics simulations for studying protein conformational changes

Protein conformational changes play crucial roles in their biological functions. In recent years, the Markov State Model (MSM) constructed from extensive Molecular Dynamics (MD) simulations has emerged as a powerful tool for modeling complex protein conformational changes. In MSMs, dynamics are modeled as a sequence of Markovian transitions among metastable conformational states at discrete time intervals (called lag time). A major challenge for MSMs is that the lag time must be long enough to allow transitions among states to become memoryless (or Markovian). However, this lag time is constrained by the length of individual MD simulations available to track these transitions. To address this challenge, we have recently developed Generalized Master Equation (GME)-based approaches, encoding non-Markovian dynamics using a time-dependent memory kernel. In this Tutorial, we introduce the theory behind two recently developed GME-based non-Markovian dynamic models: the quasi-Markov State Model (qMSM) and the Integrative Generalized Master Equation (IGME). We subsequently outline the procedures for constructing these models and provide a step-by-step tutorial on applying qMSM and IGME to study two peptide systems: alanine dipeptide and villin headpiece. This Tutorial is available at https://github.com/xuhuihuang/GME_tutorials. The protocols detailed in this Tutorial aim to be accessible for non-experts interested in studying the biomolecular dynamics using these non-Markovian dynamic models.

Log-periodic oscillations as real-time signatures of hierarchical dynamics in proteins

The time-dependent relaxation of a dynamical system may exhibit a power-law behavior that is superimposed by log-periodic oscillations. D. Sornette [Phys. Rep. 297, 239 (1998)] showed that this behavior can be explained by a discrete scale invariance of the system, which is associated with discrete and equidistant timescales on a logarithmic scale. Examples include such diverse fields as financial crashes, random diffusion, and quantum topological materials. Recent time-resolved experiments and molecular dynamics simulations suggest that discrete scale invariance may also apply to hierarchical dynamics in proteins, where several fast local conformational changes are a prerequisite for a slow global transition to occur. Employing entropy-based timescale analysis and Markov state modeling to a simple one-dimensional hierarchical model and biomolecular simulation data, it is found that hierarchical systems quite generally give rise to logarithmically spaced discrete timescales. By introducing a one-dimensional reaction coordinate that collectively accounts for the hierarchically coupled degrees of freedom, the free energy landscape exhibits a characteristic staircase shape with two metastable end states, which causes the log-periodic time evolution of the system. The period of the log-oscillations reflects the effective roughness of the energy landscape and can, in simple cases, be interpreted in terms of the barriers of the staircase landscape.

Diffusive dynamics of a model protein chain in solution

A Markov state model is a powerful tool that can be used to track the evolution of populations of configurations in an atomistic representation of a protein. For a coarse-grained linear chain model with discontinuous interactions, the transition rates among states that appear in the Markov model when the monomer dynamics is diffusive can be determined by computing the relative entropy of states and their mean first passage times, quantities that are unchanged by the specification of the energies of the relevant states. In this paper, we verify the folding dynamics described by a diffusive linear chain model of the crambin protein in three distinct solvent systems, each differing in complexity: a hard-sphere solvent, a solvent undergoing multi-particle collision dynamics, and an implicit solvent model. The predicted transition rates among configurations agree quantitatively with those observed in explicit molecular dynamics simulations for all three solvent models. These results suggest that the local monomer-monomer interactions provide sufficient friction for the monomer dynamics to be diffusive on timescales relevant to changes in conformation. Factors such as structural ordering and dynamic hydrodynamic effects appear to have minimal influence on transition rates within the studied solvent densities.

Coenzyme-binding pathway on glutamate dehydrogenase suggested from multiple-binding sites visualized by cryo-electron microscopy

The structure of hexameric glutamate dehydrogenase (GDH) in the presence of the coenzyme nicotinamide adenine dinucleotide phosphate (NADP) was visualized using cryogenic transmission electron microscopy to investigate the ligand-binding pathways to the active site of the enzyme. Each subunit of GDH comprises one hexamer-forming core domain and one nucleotide-binding domain (NAD domain), which spontaneously opens and closes the active-site cleft situated between the two domains. In the presence of NADP, the potential map of GDH hexamer, assuming D3 symmetry, was determined at a resolution of 2.4 Å, but the NAD domain was blurred due to the conformational variety. After focused classification with respect to the NAD domain, the potential maps interpreted as NADP molecules appeared at five different sites in the active-site cleft. The subunits associated with NADP molecules were close to one of the four metastable conformations in the unliganded state. Three of the five binding sites suggested a pathway of NADP molecules to approach the active-site cleft for initiating the enzymatic reaction. The other two binding modes may rarely appear in the presence of glutamate, as demonstrated by the reaction kinetics. Based on the visualized structures and the results from the enzymatic kinetics, we discussed the binding modes of NADP to GDH in the absence and presence of glutamate.

Integrative generalized master equation: A method to study long-timescale biomolecular dynamics via the integrals of memory kernels

The generalized master equation (GME) provides a powerful approach to study biomolecular dynamics via non-Markovian dynamic models built from molecular dynamics (MD) simulations. Previously, we have implemented the GME, namely the quasi Markov State Model (qMSM), where we explicitly calculate the memory kernel and propagate dynamics using a discretized GME. qMSM can be constructed with much shorter MD trajectories than the MSM. However, since qMSM needs to explicitly compute the time-dependent memory kernels, it is heavily affected by the numerical fluctuations of simulation data when applied to study biomolecular conformational changes. This can lead to numerical instability of predicted long-time dynamics, greatly limiting the applicability of qMSM in complicated biomolecules. We present a new method, the Integrative GME (IGME), in which we analytically solve the GME under the condition when the memory kernels have decayed to zero. Our IGME overcomes the challenges of the qMSM by using the time integrations of memory kernels, thereby avoiding the numerical instability caused by explicit computation of time-dependent memory kernels. Using our solutions of the GME, we have developed a new approach to compute long-time dynamics based on MD simulations in a numerically stable, accurate and efficient way. To demonstrate its effectiveness, we have applied the IGME in three biomolecules: the alanine dipeptide, FIP35 WW-domain, and Taq RNA polymerase. In each system, the IGME achieves significantly smaller fluctuations for both memory kernels and long-time dynamics compared to the qMSM. We anticipate that the IGME can be widely applied to investigate biomolecular conformational changes.

Targeted Protein Degradation: Advances, Challenges, and Prospects for Computational Methods

The therapeutic approach of targeted protein degradation (TPD) is gaining momentum due to its potentially superior effects compared with protein inhibition. Recent advancements in the biotech and pharmaceutical sectors have led to the development of compounds that are currently in human trials, with some showing promising clinical results. However, the use of computational tools in TPD is still limited, as it has distinct characteristics compared with traditional computational drug design methods. TPD involves creating a ternary structure (protein-degrader-ligase) responsible for the biological function, such as ubiquitination and subsequent proteasomal degradation, which depends on the spatial orientation of the protein of interest (POI) relative to E2-loaded ubiquitin. Modeling this structure necessitates a unique blend of tools initially developed for small molecules (e.g., docking) and biologics (e.g., protein-protein interaction modeling). Additionally, degrader molecules, particularly heterobifunctional degraders, are generally larger than conventional small molecule drugs, leading to challenges in determining drug-like properties like solubility and permeability. Furthermore, the catalytic nature of TPD makes occupancy-based modeling insufficient. TPD consists of multiple interconnected yet distinct steps, such as POI binding, E3 ligase binding, ternary structure interactions, ubiquitination, and degradation, along with traditional small molecule properties. A comprehensive set of tools is needed to address the dynamic nature of the induced proximity ternary complex and its implications for ubiquitination. In this Perspective, we discuss the current state of computational tools for TPD. We start by describing the series of steps involved in the degradation process and the experimental methods used to characterize them. Then, we delve into a detailed analysis of the computational tools employed in TPD. We also present an integrative approach that has proven successful for degrader design and its impact on project decisions. Finally, we examine the future prospects of computational methods in TPD and the areas with the greatest potential for impact.

Toward a Benchmark for Markov State Models: The Folding of HP35

Adopting a 300 μs long MD trajectory of the folding of villin headpiece (HP35) by D. E. Shaw Research, we recently constructed a Markov state model (MSM) based on inter-residue contacts. The model reproduces the folding time and predicts that the native basin and unfolded region consist of metastable substates that are structurally well-characterized. Recognizing the need to establish well-defined benchmark problems, we study to what extent and in what sense this MSM can be employed as a reference model. Hence, we test the robustness of the MSM by comparing it to models that use alternative combinations of features, dimensionality reduction methods, and clustering schemes. The study suggests some main characteristics of the folding of HP35 that should be reproduced by other competitive models. Moreover, the discussion reveals which parts of the MSM workflow matter most for the considered problem and illustrates the promises and pitfalls of state-based models for the interpretation of biomolecular simulations.

An Efficient Path Classification Algorithm Based on Variational Autoencoder to Identify Metastable Path Channels for Complex Conformational Changes

Conformational changes (i.e., dynamic transitions between pairs of conformational states) play important roles in many chemical and biological processes. Constructing the Markov state model (MSM) from extensive molecular dynamics (MD) simulations is an effective approach to dissect the mechanism of conformational changes. When combined with transition path theory (TPT), MSM can be applied to elucidate the ensemble of kinetic pathways connecting pairs of conformational states. However, the application of TPT to analyze complex conformational changes often results in a vast number of kinetic pathways with comparable fluxes. This obstacle is particularly pronounced in heterogeneous self-assembly and aggregation processes. The large number of kinetic pathways makes it challenging to comprehend the molecular mechanisms underlying conformational changes of interest. To address this challenge, we have developed a path classification algorithm named latent-space path clustering (LPC) that efficiently lumps parallel kinetic pathways into distinct metastable path channels, making them easier to comprehend. In our algorithm, MD conformations are first projected onto a low-dimensional space containing a small set of collective variables (CVs) by time-structure-based independent component analysis (tICA) with kinetic mapping. Then, MSM and TPT are constructed to obtain the ensemble of pathways, and a deep learning architecture named the variational autoencoder (VAE) is used to learn the spatial distributions of kinetic pathways in the continuous CV space. Based on the trained VAE model, the TPT-generated ensemble of kinetic pathways can be embedded into a latent space, where the classification becomes clear. We show that LPC can efficiently and accurately identify the metastable path channels in three systems: a 2D potential, the aggregation of two hydrophobic particles in water, and the folding of the Fip35 WW domain. Using the 2D potential, we further demonstrate that our LPC algorithm outperforms the previous path-lumping algorithms by making substantially fewer incorrect assignments of individual pathways to four path channels. We expect that LPC can be widely applied to identify the dominant kinetic pathways underlying complex conformational changes.

Selecting Features for Markov Modeling: A Case Study on HP35

Markov state models represent a popular means to interpret molecular dynamics trajectories in terms of memoryless transitions between metastable conformational states. To provide a mechanistic understanding of the considered biomolecular process, these states should reflect structurally distinct conformations and ensure a time scale separation between fast intrastate and slow interstate dynamics. Adopting the folding of villin headpiece (HP35) as a well-established model problem, here we discuss the selection of suitable input coordinates or "features", such as backbone dihedral angles and interresidue distances. We show that dihedral angles account accurately for the structure of the native energy basin of HP35, while the unfolded region of the free energy landscape and the folding process are best described by tertiary contacts of the protein. To construct a contact-based model, we consider various ways to define and select contact distances and introduce a low-pass filtering of the feature trajectory as well as a correlation-based characterization of states. Relying on input data that faithfully account for the mechanistic origin of the studied process, the states of the resulting Markov model are clearly discriminated by the features, describe consistently the hierarchical structure of the free energy landscape, and─as a consequence─correctly reproduce the slow time scales of the process.

Hybrid Quantum Mechanical/Molecular Mechanical Methods For Studying Energy Transduction in Biomolecular Machines

Hybrid quantum mechanical/molecular mechanical (QM/MM) methods have become indispensable tools for the study of biomolecules. In this article, we briefly review the basic methodological details of QM/MM approaches and discuss their applications to various energy transduction problems in biomolecular machines, such as long-range proton transports, fast electron transfers, and mechanochemical coupling. We highlight the particular importance for these applications of balancing computational efficiency and accuracy. Using several recent examples, we illustrate the value and limitations of QM/MM methodologies for both ground and excited states, as well as strategies for calibrating them in specific applications. We conclude with brief comments on several areas that can benefit from further efforts to make QM/MM analyses more quantitative and applicable to increasingly complex biological problems.

Loading dynamics of one SARS-CoV-2-derived peptide into MHC-II revealed by kinetic models

Major histocompatibility complex class II (MHC-II) plays an indispensable role in activating CD4+ T cell immune responses by presenting antigenic peptides on the cell surface for recognition by T cell receptors. The assembly of MHC-II and antigenic peptide is therefore a prerequisite for the antigen presentation. To date, however, the atomic-level mechanism underlying the peptide-loading dynamics for MHC-II is still elusive. Here, by constructing Markov state models based on extensive all-atom molecular dynamics simulations, we reveal the complete peptide-loading dynamics into MHC-II for one SARS-CoV-2 S-protein-derived antigenic peptide (235ITRFQTLLALHRSYL249). Our Markov state model identifies six metastable states (S1-S6) during the peptide-loading process and determines two dominant loading pathways. The peptide could potentially approach the antigen-binding groove via either its N- or C-terminus. Then, the consecutive insertion of several anchor residues into the binding pockets profoundly dictates the peptide-loading dynamics. Notably, the MHC-II αA52-E55 motif could guide the peptide loading into the antigen-binding groove via forming β-sheets conformation with the incoming peptide. The rate-limiting step, namely S5→S6, is mainly attributed to a considerable desolvation penalty triggered by the binding of the peptide C-terminus. Moreover, we further examined the conformational changes associated with the peptide exchange process catalyzed by the chaperon protein HLA-DM. A flipped-out conformation of MHC-II αW43 captured in S1-S3 is considered a critical anchor point for HLA-DM to modulate the structural dynamics. Our work provides deep structural insights into the key regulatory factors in MHC-II responsible for peptide recognition and guides future design for peptide vaccines against SARS-CoV-2.

Memory Unlocks the Future of Biomolecular Dynamics: Transformative Tools to Uncover Physical Insights Accurately and Efficiently

Conformational changes underpin function and encode complex biomolecular mechanisms. Gaining atomic-level detail of how such changes occur has the potential to reveal these mechanisms and is of critical importance in identifying drug targets, facilitating rational drug design, and enabling bioengineering applications. While the past two decades have brought Markov state model techniques to the point where practitioners can regularly use them to glimpse the long-time dynamics of slow conformations in complex systems, many systems are still beyond their reach. In this Perspective, we discuss how including memory (i.e., non-Markovian effects) can reduce the computational cost to predict the long-time dynamics in these complex systems by orders of magnitude and with greater accuracy and resolution than state-of-the-art Markov state models. We illustrate how memory lies at the heart of successful and promising techniques, ranging from the Fokker-Planck and generalized Langevin equations to deep-learning recurrent neural networks and generalized master equations. We delineate how these techniques work, identify insights that they can offer in biomolecular systems, and discuss their advantages and disadvantages in practical settings. We show how generalized master equations can enable the investigation of, for example, the gate-opening process in RNA polymerase II and demonstrate how our recent advances tame the deleterious influence of statistical underconvergence of the molecular dynamics simulations used to parameterize these techniques. This represents a significant leap forward that will enable our memory-based techniques to interrogate systems that are currently beyond the reach of even the best Markov state models. We conclude by discussing some current challenges and future prospects for how exploiting memory will open the door to many exciting opportunities.

Phosphorylation Regulation Mechanism of β2 Integrin for the Binding of Filamin Revealed by Markov State Model

Leukocyte adhesion deficiency-1 (LAD-1) disorder is a severe immunodeficiency syndrome caused by deficiency or mutation of β2 integrin. The phosphorylation on threonine 758 of β2 integrin acts as a molecular switch inhibiting the binding of filamin. However, the switch mechanism of site-specific phosphorylation at the atom level is still poorly understood. To resolve the regulation mechanism, all-atom molecular dynamics simulation and Markov state model were used to study the dynamic regulation pathway of phosphorylation. Wild type system possessed lower binding free energy and fewer number of states than the phosphorylated system. Both systems underwent local disorder-to-order conformation conversion when achieving steady states. To reach steady states, wild type adopted less number of transition paths/shortest path according to the transition path theory than the phosphorylated system. The underlying phosphorylated regulation pathway was from P1 to P0 and then P4 state, and the main driving force should be hydrogen bond and hydrophobic interaction disturbing the secondary structure of phosphorylated states. These studies will shed light on the pathogenesis of LAD-1 disease and lay a foundation for drug development.

Nucleotide addition and cleavage by RNA polymerase II: Coordination of two catalytic reactions using a single active site

RNA polymerase II (Pol II) incorporates complementary ribonucleotides into the growing RNA chain one at a time via the nucleotide addition cycle. The nucleotide addition cycle, however, is prone to misincorporation of noncomplementary nucleotides. Thus, to ensure transcriptional fidelity, Pol II backtracks and then cleaves the misincorporated nucleotides. These two reverse reactions, nucleotide addition and cleavage, are catalyzed in the same active site of Pol II, which is different from DNA polymerases or other endonucleases. Recently, substantial progress has been made to understand how Pol II effectively performs its dual role in the same active site. Our review highlights these recent studies and provides an overall model of the catalytic mechanisms of Pol II. In particular, RNA extension follows the two-metal-ion mechanism, and several Pol II residues play important roles to facilitate the catalysis. In sharp contrast, the cleavage reaction is independent of any Pol II residues. Interestingly, Pol II relies on its residues to recognize the misincorporated nucleotides during the backtracking process, prior to cleavage. In this way, Pol II efficiently compartmentalizes its two distinct catalytic functions using the same active site. Lastly, we also discuss a new perspective on the potential third Mg²⁺ in the nucleotide addition and intrinsic cleavage reactions.

Markov State Models Underlying the N-Terminal Premodel of TOPK/PBK

Lymphokine-activated killer T-cell-originated protein kinase (TOPK) is a potential target for cancer therapy. To explore the micromechanism, we proposed the N-terminal premodel (NTPM) of the TOPK monomer via homology modeling and molecular dynamic simulations and analyzed the conformational dynamics by Markov state model analysis. The electronegative insert (ENI) motif of the NTPM can be opened with a small probability under wild type, regulated by the so-called "N-C" interaction zone consisting of the N-terminal head, the coil between β₃-strand and αC-helix, and the ENI motif. Glutamate substitution at threonine residue 9 or tyrosine residue 74 promotes the closed-open transition, revealing the details of phosphorylation. Allosteric effects induce functionally relevant structural changes, such as increased structural flexibility and active sites, which are thought to be necessary for further activation or binding. These findings provide rational structural templates for designing state-dependent inhibitors and give insight into the molecular regulatory mechanisms of TOPK monomers.

Automated Path Searching Reveals the Mechanism of Hydrolysis Enhancement by T4 Lysozyme Mutants

Bacteriophage T4 lysozyme (T4L) is a glycosidase that is widely applied as a natural antimicrobial agent in the food industry. Due to its wide applications and small size, T4L has been regarded as a model system for understanding protein dynamics and for large-scale protein engineering. Through structural insights from the single conformation of T4L, a series of mutations (L99A,G113A,R119P) have been introduced, which have successfully raised the fractional population of its only hydrolysis-competent excited state to 96%. However, the actual impact of these substitutions on its dynamics remains unclear, largely due to the lack of highly efficient sampling algorithms. Here, using our recently developed travelling-salesman-based automated path searching (TAPS), we located the minimum-free-energy path (MFEP) for the transition of three T4L mutants from their ground states to their excited states. All three mutants share a three-step transition: the flipping of F114, the rearrangement of α0/α1 helices, and final refinement. Remarkably, the MFEP revealed that the effects of the mutations are drastically beyond the expectations of their original design: (a) the G113A substitution not only enhances helicity but also fills the hydrophobic Cavity I and reduces the free energy barrier for flipping F114; (b) R119P barely changes the stability of the ground state but stabilizes the excited state through rarely reported polar contacts S117_OG:N132_ND2, E11_OE1:R145_NH1, and E11_OE2:Q105_NE2; (c) the residue W138 flips into Cavity I and further stabilizes the excited state for the triple mutant L99A,G113A,R119P. These novel insights that were unexpected in the original mutant design indicated the necessity of incorporating path searching into the workflow of rational protein engineering.

Protein Function Analysis through Machine Learning

Machine learning (ML) has been an important arsenal in computational biology used to elucidate protein function for decades. With the recent burgeoning of novel ML methods and applications, new ML approaches have been incorporated into many areas of computational biology dealing with protein function. We examine how ML has been integrated into a wide range of computational models to improve prediction accuracy and gain a better understanding of protein function. The applications discussed are protein structure prediction, protein engineering using sequence modifications to achieve stability and druggability characteristics, molecular docking in terms of protein-ligand binding, including allosteric effects, protein-protein interactions and protein-centric drug discovery. To quantify the mechanisms underlying protein function, a holistic approach that takes structure, flexibility, stability, and dynamics into account is required, as these aspects become inseparable through their interdependence. Another key component of protein function is conformational dynamics, which often manifest as protein kinetics. Computational methods that use ML to generate representative conformational ensembles and quantify differences in conformational ensembles important for function are included in this review. Future opportunities are highlighted for each of these topics.

Nanoparticle cluster formation mechanisms elucidated via Markov state modeling: Attraction range effects, aggregation pathways, and counterintuitive transition rates

Nanoparticle clusters are promising candidates for developing functional materials. However, it is still a challenging task to fabricate them in a predictable and controllable way, which requires investigation of the possible mechanisms underlying cluster formation at the nanoscale. By constructing Markov state models (MSMs) at the microstate level, we find that for highly dispersed particles to form a highly aggregated cluster, there are multiple coexisting pathways, which correspond to direct aggregation, or pathways that need to pass through partially aggregated, intermediate states. Varying the range of attraction between nanoparticles is found to significantly affect pathways. As the attraction range becomes narrower, compared to direct aggregation, some pathways that need to pass through partially aggregated intermediate states become more competitive. In addition, from MSMs constructed at the macrostate level, the aggregation rate is found to be counterintuitively lower with a lower free-energy barrier, which is also discussed.

Community Reaction Network Reduction for Constructing a Coarse-Grained Representation of Combustion Reaction Mechanisms

A community-reaction network reduction (CNR) approach is presented for mechanism reduction on the basis of a network-based community detection technique, a concept related to pre-equilibrium in chemical kinetics. In this method, the detailed combustion mechanism is first transformed into a weighted network, in which communities of species that have dense inner connections under the critical ignition conditions are identified. By analyzing the community partitions in different regions, we determine the effective functional groups and driving processes. Then, a skeletal model for the overall mechanism is deduced according to the network centrality data, including transition pathway identification and reaction-path flux. The CNR method is illustrated on the hydrogen autoignition system which has been extensively investigated, and a new reduced mechanism involving seven processes is proposed. Dynamics simulations employing the present CNR model show that the computed ignition time and distribution of major species on a wide range of temperature and pressure conditions are in accord with the experiments and results from other methods.

Prediction of Protein–Protein Interactions Between Alsin DH/PH and Rac1 and Resulting Protein Dynamics

Alsin is a protein of 1,657 amino acids known for its crucial role in vesicular trafficking in neurons thanks to its ability to interact with two guanosine triphosphatases, Rac1 and Rab5. Evidence suggests that Rac1 can bind Alsin central region, composed by a Dbl Homology (DH) domain followed by a Pleckstrin Homology (PH) domain, leading to Alsin relocalization. However, Alsin three-dimensional structure and its relationship with known biological functions of this protein are still unknown. In this work, a homology model of the Alsin DH/PH domain was developed and studied through molecular dynamics both in the presence and in the absence of its binding partner, Rac1. Due to different conformations of DH domain, the presence of Rac1 seems to stabilize an open state of the protein, while the absence of its binding partner results in closed conformations. Furthermore, Rac1 interaction was able to reduce the fluctuations in the second conserved region of DH motif, which may be involved in the formation of a homodimer. Moreover, the dynamics of DH/PH was described through a Markov State Model to study the pathways linking the open and closed states. In conclusion, this work provided an all-atom model for the DH/PH domain of Alsin protein; moreover, molecular dynamics investigations suggested underlying molecular mechanisms in the signal transduction between Rac1 and Alsin, providing the basis for a deeper understanding of the whole structure–function relationship for Alsin protein.

Markov State Models to Study the Functional Dynamics of Proteins in the Wake of Machine Learning

Markov state models (MSMs) based on molecular dynamics (MD) simulations are routinely employed to study protein folding, however, their application to functional conformational changes of biomolecules is still limited. In the past few years, the field of computational chemistry has experienced a surge of advancements stemming from machine learning algorithms, and MSMs have not been left out. Unlike global processes, such as protein folding, the application of MSMs to functional conformational changes is challenging because they mostly consist of localized structural transitions. Therefore, it is critical to properly select a subset of structural features that can describe the slowest dynamics of these functional conformational changes. To address this challenge, we recommend several automatic feature selection methods such as Spectral-OASIS. To identify states in MSMs, the chosen features can be subject to dimensionality reduction methods such as TICA or deep learning based VAMPNets to project MD conformations onto a few collective variables for subsequent clustering. Another challenge for the application of MSMs to the study of functional conformational changes is the ability to comprehend their biophysical mechanisms, as MSMs built for these processes often require a large number of states. We recommend the recently developed quasi-MSMs (qMSMs) to address this issue. Compared to MSMs, qMSMs encode the non-Markovian dynamics via the generalized master equation and can significantly reduce the number of states. As a result, qMSMs can be built with a handful of states to facilitate the interpretation of functional conformational changes. In the wake of machine learning, we believe that the rapid advancement in the MSM methodology will lead to their wider application in studying functional conformational changes of biomolecules.

Assessing the Performance of Traveling-salesman based Automated Path Searching (TAPS) on Complex Biomolecular Systems

Though crucial for understanding the function of large biomolecular systems, locating the minimum free energy paths (MFEPs) between their key conformational states is far from trivial due to their high-dimensional nature. Most existing path-searching methods require a static collective variable space as input, encoding intuition or prior knowledge of the transition mechanism. Such information is, however, hardly available a priori and expensive to validate. To alleviate this issue, we have previously introduced a Traveling-salesman based Automated Path Searching method (TAPS) and demonstrated its efficiency on simple peptide systems. Having implemented a parallel version of this method, here we assess the performance of TAPS on three realistic systems (tens to hundreds of residues) in explicit solvents. We show that TAPS successfully located the MFEP for the ground/excited state transition of the T4 lysozyme L99A variant, consistent with previous findings. TAPS also helped identifying the important role of the two polar contacts in directing the loop-in/loop-out transition of the mitogen-activated protein kinase kinase (MEK1), which explained previous mutant experiments. Remarkably, at a minimal cost of 126 ns sampling, TAPS revealed that the Ltn40/Ltn10 transition of lymphotactin needs no complete unfolding/refolding of its β-sheets and that five polar contacts are sufficient to stabilize the various partially unfolded intermediates along the MFEP. These results present TAPS as a general and promising tool for studying the functional dynamics of complex biomolecular systems.

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.

CLoNe: automated clustering based on local density neighborhoods for application to biomolecular structural ensembles

Motivation

Proteins are intrinsically dynamic entities. Flexibility sampling methods, such as molecular dynamics or those arising from integrative modeling strategies, are now commonplace and enable the study of molecular conformational landscapes in many contexts. Resulting structural ensembles increase in size as technological and algorithmic advancements take place, making their analysis increasingly demanding. In this regard, cluster analysis remains a go-to approach for their classification. However, many state-of-the-art algorithms are restricted to specific cluster properties. Combined with tedious parameter fine-tuning, cluster analysis of protein structural ensembles suffers from the lack of a generally applicable and easy to use clustering scheme.

Results

We present CLoNe, an original Python-based clustering scheme that builds on the Density Peaks algorithm of Rodriguez and Laio. CLoNe relies on a probabilistic analysis of local density distributions derived from nearest neighbors to find relevant clusters regardless of cluster shape, size, distribution and amount. We show its capabilities on many toy datasets with properties otherwise dividing state-of-the-art approaches and improves on the original algorithm in key aspects. Applied to structural ensembles, CLoNe was able to extract meaningful conformations from membrane binding events and ligand-binding pocket opening as well as identify dominant dimerization motifs or inter-domain organization. CLoNe additionally saves clusters as individual trajectories for further analysis and provides scripts for automated use with molecular visualization software.

Availability and implementation

www.epfl.ch/labs/lbm/resources, github.com/LBM-EPFL/CLoNe.

Supplementary information

Supplementary data are available at Bioinformatics online.

A comprehensive mechanism for 5-carboxylcytosine-induced transcriptional pausing revealed by Markov state models

RNA polymerase II (Pol II) surveils the genome, pausing as it encounters DNA lesions and base modifications and initiating signals for DNA repair among other important regulatory events. Recent work suggests that Pol II pauses at 5-carboxycytosine (5caC), an epigenetic modification of cytosine, because of a specific hydrogen bond between the carboxyl group of 5caC and a specific residue in fork loop 3 of Pol II. This hydrogen bond compromises productive NTP binding and slows down elongation. Apart from this specific interaction, the carboxyl group of 5caC can potentially interact with numerous charged residues in the cleft of Pol II. However, it is not clear how other interactions between Pol II and 5caC contribute to pausing. In this study, we use Markov state models (a type of kinetic network models) built from extensive molecular dynamics simulations to comprehensively study the impact of 5caC on Pol II translocation. We describe two translocation intermediates with specific interactions that prevent the template base from loading into the Pol II active site. In addition to the previously observed state with 5caC constrained by fork loop 3, we discovered a new intermediate state with a hydrogen bond between 5caC and fork loop 2. Surprisingly, we find that 5caC may curb translocation by suppressing kinking of the helix bordering the active site (the bridge helix) because its high flexibility is critical to translocation. Our work provides new insights into how epigenetic modifications of genomic DNA can modulate Pol II translocation, inducing pauses in transcription.

What Markov State Models Can and Cannot Do: Correlation versus Path-Based Observables in Protein-Folding Models

Markov state models (MSMs) have been widely applied to study the kinetics and pathways of protein conformational dynamics based on statistical analysis of molecular dynamics (MD) simulations. These MSMs coarse-grain both configuration space and time in ways that limit what kinds of observables they can reproduce with high fidelity over different spatial and temporal resolutions. Despite their popularity, there is still limited understanding of which biophysical observables can be computed from these MSMs in a robust and unbiased manner, and which suffer from the space-time coarse-graining intrinsic in the MSM model. Most theoretical arguments and practical validity tests for MSMs rely on long-time equilibrium kinetics, such as the slowest relaxation time scales and experimentally observable time-correlation functions. Here, we perform an extensive assessment of the ability of well-validated protein folding MSMs to accurately reproduce path-based observable such as mean first-passage times (MFPTs) and transition path mechanisms compared to a direct trajectory analysis. We also assess a recently proposed class of history-augmented MSMs (haMSMs) that exploit additional information not accounted for in standard MSMs. We conclude with some practical guidance on the use of MSMs to study various problems in conformational dynamics of biomolecules. In brief, MSMs can accurately reproduce correlation functions slower than the lag time, but path-based observables can only be reliably reproduced if the lifetimes of states exceed the lag time, which is a much stricter requirement. Even in the presence of short-lived states, we find that haMSMs reproduce path-based observables more reliably.

Role of bacterial RNA polymerase gate opening dynamics in DNA loading and antibiotics inhibition elucidated by quasi-Markov State Model

To initiate transcription, the holoenzyme (RNA polymerase [RNAP] in complex with σ factor) loads the promoter DNA via the flexible loading gate created by the clamp and β-lobe, yet their roles in DNA loading have not been characterized. We used a quasi-Markov State Model (qMSM) built from extensive molecular dynamics simulations to elucidate the dynamics of Thermus aquaticus holoenzyme's gate opening. We showed that during gate opening, β-lobe oscillates four orders of magnitude faster than the clamp, whose opening depends on the Switch 2's structure. Myxopyronin, an antibiotic that binds to Switch 2, was shown to undergo a conformational selection mechanism to inhibit clamp opening. Importantly, we reveal a critical but undiscovered role of β-lobe, whose opening is sufficient for DNA loading even when the clamp is partially closed. These findings open the opportunity for the development of antibiotics targeting β-lobe of RNAP. Finally, we have shown that our qMSMs, which encode non-Markovian dynamics based on the generalized master equation formalism, hold great potential to be widely applied to study biomolecular dynamics.

Markov models from the square root approximation of the Fokker-Planck equation: calculating the grid-dependent flux

Molecular dynamics (MD) are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the MD as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker-Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive MD simulations. The square-root approximation of the Fokker-Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used MD simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to [Formula: see text] states can be obtained within seconds on a single high-performance compute server if regular grids are used.

On the advantages of exploiting memory in Markov state models for biomolecular dynamics

Biomolecular dynamics play an important role in numerous biological processes. Markov State Models (MSMs) provide a powerful approach to study these dynamic processes by predicting long time scale dynamics based on many short molecular dynamics (MD) simulations. In an MSM, protein dynamics are modeled as a kinetic process consisting of a series of Markovian transitions between different conformational states at discrete time intervals (called "lag time"). To achieve this, a master equation must be constructed with a sufficiently long lag time to allow interstate transitions to become truly Markovian. This imposes a major challenge for MSM studies of proteins since the lag time is bound by the length of relatively short MD simulations available to estimate the frequency of transitions. Here, we show how one can employ the generalized master equation formalism to obtain an exact description of protein conformational dynamics both at short and long time scales without the time resolution restrictions imposed by the MSM lag time. Using a simple kinetic model, alanine dipeptide, and WW domain, we demonstrate that it is possible to construct these quasi-Markov State Models (qMSMs) using MD simulations that are 5-10 times shorter than those required by MSMs. These qMSMs only contain a handful of metastable states and, thus, can greatly facilitate the interpretation of mechanisms associated with protein dynamics. A qMSM opens the door to the study of conformational changes of complex biomolecules where a Markovian model with a few states is often difficult to construct due to the limited length of available MD simulations.

Connecting dynamic reweighting Algorithms: Derivation of the dynamic reweighting family tree

Thermally driven processes of molecular systems include transitions of energy barriers on the microsecond timescales and higher. Sufficient sampling of such processes with molecular dynamics simulations is challenging and often requires accelerating slow transitions using external biasing potentials. Different dynamic reweighting algorithms have been proposed in the past few years to recover the unbiased kinetics from biased systems. However, it remains an open question if and how these dynamic reweighting approaches are connected. In this work, we establish the link between the two main reweighting types, i.e., path-based and energy-based reweighting. We derive a path-based correction factor for the energy-based dynamic histogram analysis method, thus connecting the previously separate reweighting types. We show that the correction factor can be used to combine the advantages of path-based and energy-based reweighting algorithms: it is integrator independent, more robust, and at the same time able to reweight time-dependent biases. We can furthermore demonstrate the relationship between two independently derived path-based reweighting algorithms. Our theoretical findings are verified on a one-dimensional four-well system. By connecting different dynamic reweighting algorithms, this work helps to clarify the strengths and limitations of the different methods and enables a more robust usage of the combined types.

EspcTM: Kinetic Transition Network Based on Trajectory Mapping in Effective Energy Rescaling Space

The transition network provides a key to reveal the thermodynamic and kinetic properties of biomolecular systems. In this paper, we introduce a new method, named effective energy rescaling space trajectory mapping (EspcTM), to detect metastable states and construct transition networks based on the simulation trajectories of the complex biomolecular system. It mapped simulation trajectories into an orthogonal function space, whose bases were rescaled by effective energy, and clustered the interrelation between these trajectories to locate metastable states. By using the EspcTM method, we identified the metastable states and elucidated interstate transition kinetics of a Brownian particle and a dodecapeptide. It was found that the scaling parameters of effective energy also provided a clue to the dominating factors in dynamics. We believe that the EspcTM method is a useful tool for the studies of dynamics of the complex system and may provide new insight into the understanding of thermodynamics and kinetics of biomolecular systems.

MSMPathfinder: Identification of Pathways in Markov State Models

Markov state models represent a popular means to interpret biomolecular processes in terms of memoryless transitions between metastable conformational states. To gain insight into the underlying mechanism, it is instructive to determine all relevant pathways between initial and final states of the process. Currently available methods, such as Markov chain Monte Carlo and transition path theory, are convenient for identifying the most frequented pathways. They are less suited to account for the typically huge amount of pathways with low probability which, though, may dominate the cumulative flux of the reaction. On the basis of a systematic construction of all possible pathways, the here proposed method MSMPathfinder is able to characterize the multitude of unique pathways (say, up to 10¹⁰) in a complex system and to quantitatively calculate their correct weights and associated waiting times with predefined accuracy. Adopting the chiral transitions of a peptide helix and the folding of the villin headpiece as model problems, mechanisms and associated waiting times of these processes are discussed using a kinetic network representation. The analysis reveals that the waiting time distribution may yield only little insight into the diversity of pathways, because the measured folding times do typically not reflect the most probable path lengths but rather the cumulative effect of many different pathways.

Elucidating molecular mechanisms of functional conformational changes of proteins via Markov state models

Functional conformational changes of proteins can facilitate numerous biological events in cells. The Markov state model (MSM) built from molecular dynamics simulations provide a powerful approach to study them. We here introduce a protocol that is tailor-made for constructing MSMs to study the functional conformational changes of proteins. In this protocol, one of the important steps is to select proper molecular features that can collectively describe the slowest timescales of conformational changes of interest. We recommend spectral oASIS, the modified version of oASIS, as a promising approach for automatic feature selection. Recently developed deep learning methods could also serve efficient approaches for selecting features and finding collective variables. Using DNA repair enzymes and RNA polymerases as examples, we review recent applications of MSMs to elucidate molecular mechanisms of functional conformational changes. Finally, we discuss remaining challenges and future perspectives for constructing MSMs to study functional conformational changes of proteins.