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

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.

Non-Markovian Dynamic Models Identify Non-Canonical KRAS-VHL Encounter Complex Conformations for Novel PROTAC Design

Targeted protein degradation (TPD) is emerging as a promising therapeutic approach for cancer and other diseases, with an increasing number of programs demonstrating its efficacy in human clinical trials. One notable method for TPD is Proteolysis Targeting Chimeras (PROTACs) that selectively degrade a protein of interest (POI) through E3-ligase induced ubiquitination followed by proteasomal degradation. PROTACs utilize a warhead-linker-ligand architecture to bring the POI (bound to the warhead) and the E3 ligase (bound to the ligand) into proximity. The resulting non-native protein-protein interactions (PPIs) formed between the POI and E3 ligase lead to the formation of a stable ternary complex, enhancing cooperativity for TPD. A significant challenge in PROTAC design is the screening of the linkers to induce favorable non-native PPIs between POI and E3 ligase. Here, we present a physics-based computational protocol to predict noncanonical and metastable PPI interfaces between an E3 ligase and a given POI, aiding in the design of linkers to stabilize the ternary complex and enhance degradation. Specifically, we build the non-Markovian dynamic model using the Integrative Generalized Master equation (IGME) method from ∼1.5 ms all-atom molecular dynamics simulations of linker-less encounter complex, to systematically explore the inherent PPIs between the oncogene homologue protein and the von Hippel-Lindau E3 ligase. Our protocol revealed six metastable states each containing a different PPI interface. We selected three of these metastable states containing promising PPIs for linker design. Our selection criterion included thermodynamic and kinetic stabilities of PPIs and the accessibility between the solvent-exposed sites on the warheads and E3 ligand. One selected PPIs closely matches a recent cocrystal PPI interface structure induced by an experimentally designed PROTAC with potent degradation efficacy. We anticipate that our protocol has significant potential for widespread application in predicting metastable POI-ligase interfaces that can enable rational design of PROTACs.

Markov State Models Reveal How Folding Kinetics Influence Absorption Spectra of Foldamers

Self-assembly of platinum(II) complex foldamers is an essential approach to fabricate advanced luminescent materials. However, a comprehensive understanding of folding kinetics and their absorption spectra remains elusive. By constructing Markov state models (MSMs) from large-scale molecular dynamics simulations, we reveal that two largely similar dinuclear alknylplatinum(II) terpyridine foldamers, Pt-PEG and Pt-PE with slightly different bridges, exhibit distinctive folding kinetics. Particularly, Pt-PEG bears bridge-dominant, plane-dominant, and cooperative pathways, while Pt-PE only prefers the plane-dominant pathway. Such preference originates from their difference in intrabridge electrostatic interactions, leading to contrastive distributions of metastable states. We also found that the bridge-dominant pathway for Pt-PEG becomes more favorable when lowering the temperature. Interestingly, based on the comprehensive conformation ensembles from our MSMs, we reveal the conformation-dependent absorption spectra of Pt-PEG and Pt-PE. Our theoretical spectra not only align with experimental results but also reveal the contributions of diverse conformations to the overall absorption bands explicitly, facilitating the rational design of stimuli-responsive smart luminescent molecules.

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.

An Information Bottleneck Approach for Markov Model Construction

Markov state models (MSMs) have proven valuable in studying dynamics of protein conformational changes via statistical analysis of molecular dynamics (MD) 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 multi-resolution 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 multi-resolution 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 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 MSMs construction.

Accurate Memory Kernel Extraction from Discretized Time-Series Data

Memory effects emerge as a fundamental consequence of dimensionality reduction when low-dimensional observables are used to describe the dynamics of complex many-body systems. In the context of molecular dynamics (MD) data analysis, accounting for memory effects using the framework of the generalized Langevin equation (GLE) has proven efficient, accurate, and insightful, particularly when working with high-resolution time series data. However, in experimental systems, high-resolution data are often unavailable, raising questions about the impact of the data resolution on the estimated GLE parameters. This study demonstrates that direct memory extraction from time series data remains accurate when the discretization time is below the memory time. To obtain memory functions reliably, even when the discretization time exceeds the memory time, we introduce a Gaussian Process Optimization (GPO) scheme. This scheme minimizes the deviation of discretized two-point correlation functions between time series data and GLE simulations and is able to estimate accurate memory kernels as long as the discretization time stays below the longest time scale in the data, typically the barrier crossing time.

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.

Submillisecond Atomistic Molecular Dynamics Simulations Reveal Hydrogen Bond-Driven Diffusion of a Guest Peptide in Protein-RNA Condensate

Liquid-liquid phase separation mediated by proteins and/or nucleic acids is believed to underlie the formation of many distinct condensed phases, or membraneless organelles, within living cells. These condensates have been proposed to orchestrate a variety of important processes. Despite recent advances, the interactions that regulate the dynamics of molecules within a condensate remain poorly understood. We performed accumulated 564.7 μs all-atom molecular dynamics (MD) simulations (system size ∼200k atoms) of model condensates formed by a scaffold RNA oligomer and a scaffold peptide rich in arginine (Arg). These model condensates contained one of three possible guest peptides: the scaffold peptide itself or a variant in which six Arg residues were replaced by lysine (Lys) or asymmetric dimethyl arginine (ADMA). We found that the Arg-rich peptide can form the largest number of hydrogen bonds and bind the strongest to the scaffold RNA in the condensate, relative to the Lys- and ADMA-rich peptides. Our MD simulations also showed that the Arg-rich peptide diffused more slowly in the condensate relative to the other two guest peptides, which is consistent with a recent fluorescence microscopy study. There was no significant increase in the number of cation-π interactions between the Arg-rich peptide and the scaffold RNA compared to the Lys-rich and ADMA-rich peptides. Our results indicate that hydrogen bonds between the peptides and the RNA backbone, rather than cation-π interactions, play a major role in regulating peptide diffusion in the condensate.

Accurate estimates of dynamical statistics using memory

Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis. Because the projected dynamics are generally not memoryless, the Markov approximation can result in significant systematic errors. Inspired by quasi-Markov state models, which employ the generalized master equation to encode memory resulting from the projection, we reformulate DGA to account for memory and analyze its performance on two systems: a two-dimensional triple well and the AIB9 peptide. We demonstrate that our method is robust to the choice of basis and can decrease the time series length required to obtain accurate kinetics by an order of magnitude.

Reaction coordinate flows for model reduction of molecular kinetics

In this work, we introduce a flow based machine learning approach called reaction coordinate (RC) flow for the discovery of low-dimensional kinetic models of molecular systems. The RC flow utilizes a normalizing flow to design the coordinate transformation and a Brownian dynamics model to approximate the kinetics of RC, where all model parameters can be estimated in a data-driven manner. In contrast to existing model reduction methods for molecular kinetics, RC flow offers a trainable and tractable model of reduced kinetics in continuous time and space due to the invertibility of the normalizing flow. Furthermore, the Brownian dynamics-based reduced kinetic model investigated in this work yields a readily discernible representation of metastable states within the phase space of the molecular system. Numerical experiments demonstrate how effectively the proposed method discovers interpretable and accurate low-dimensional representations of given full-state kinetics from simulations.