Independent Markov decomposition: Toward modeling kinetics of biomolecular complexes

Integrative spatiotemporal modeling of biomolecular processes: Application to the assembly of the nuclear pore complex

Dynamic processes involving biomolecules are essential for the function of the cell. Here, we introduce an integrative method for computing models of these processes based on multiple heterogeneous sources of information, including time-resolved experimental data and physical models of dynamic processes. First, for each time point, a set of coarse models of compositional and structural heterogeneity is computed (heterogeneity models). Second, for each heterogeneity model, a set of static integrative structure models is computed (a snapshot model). Finally, these snapshot models are selected and connected into a series of trajectories that optimize the likelihood of both the snapshot models and transitions between them (a trajectory model). The method is demonstrated by application to the assembly process of the human nuclear pore complex in the context of the reforming nuclear envelope during mitotic cell division, based on live-cell correlated electron tomography, bulk fluorescence correlation spectroscopy-calibrated quantitative live imaging, and a structural model of the fully assembled nuclear pore complex. Modeling of the assembly process improves the model precision over static integrative structure modeling alone. The method is applicable to a wide range of time-dependent systems in cell biology and is available to the broader scientific community through an implementation in the open source Integrative Modeling Platform (IMP) software.

Discrete state model of a self-aggregating colloidal system with directional interactions

The construction of coarse-grained descriptions of a system's kinetics is well established in biophysics. One prominent example is Markov state models in protein folding dynamics. In this paper, we develop a coarse-grained, discrete state model of a self-aggregating colloidal particle system inspired by the concepts of Markov state modeling. The specific self-aggregating system studied here involves field-responsive colloidal particles in orthogonal electric and magnetic fields. Starting from particle-resolved (Brownian dynamics) simulations, we define the discrete states by categorizing each particle according to its local structure. We then describe the kinetics between these states as a series of stochastic, memoryless jumps. In contrast to other works on colloidal self-assembly, our coarse-grained approach describes the simultaneous formation and evolution of multiple aggregates from single particles. Our discrete model also takes into account the changes in transition dynamics between the discrete states as the size of the largest cluster grows. We validate the coarse-grained model by comparing the predicted population fraction in each of the discrete states with those calculated directly from the particle-resolved simulations as a function of the largest cluster size. We then predict population fractions in the presence of noise-averaging and in a situation where a model parameter is changed instantaneously after a certain time. Finally, we explore the validity of the detailed balance condition in the various stages of aggregation.

RevGraphVAMP: A protein molecular simulation analysis model combining graph convolutional neural networks and physical constraints

Molecular dynamics simulation is a crucial research domain within the life sciences, focusing on comprehending the mechanisms of biomolecular interactions at atomic scales. Protein simulation, as a critical subfield, often utilizes MD for implementation, with trajectory data play a pivotal role in drug discovery. The advancement of high-performance computing and deep learning technology becomes popular and critical to predict protein properties from vast trajectory data, posing challenges regarding data features extraction from the complicated simulation data and dimensionality reduction. Simultaneously, it is essential to provide a meaningful explanation of the biological mechanism behind dimensionality. To tackle this challenge, we propose a new unsupervised model named RevGraphVAMP to intelligently analyze the simulation trajectory. This model is based on the variational approach for Markov processes (VAMP) and integrates graph convolutional neural networks and physical constraint optimization to enhance the learning performance. Additionally, we introduce attention mechanism to assess the importance of key interaction region, facilitating the interpretation of molecular mechanism. In comparison to other VAMPNets models, our model showcases competitive performance, improved accuracy in state transition prediction, as demonstrated through its application to two public datasets and the Shank3-Rap1 complex, which is associated with autism spectrum disorder. Moreover, it enhanced dimensionality reduction discrimination across different substates and provides interpretable results for protein structural characterization.

Integrative spatiotemporal modeling of biomolecular processes: application to the assembly of the Nuclear Pore Complex

Dynamic processes involving biomolecules are essential for the function of the cell. Here, we introduce an integrative method for computing models of these processes based on multiple heterogeneous sources of information, including time-resolved experimental data and physical models of dynamic processes. We first compute integrative structure models at fixed time points and then optimally select and connect these snapshots into a series of trajectories that optimize the likelihood of both the snapshots and transitions between them. The method is demonstrated by application to the assembly process of the human Nuclear Pore Complex in the context of the reforming nuclear envelope during mitotic cell division, based on live-cell correlated electron tomography, bulk fluorescence correlation spectroscopy-calibrated quantitative live imaging, and a structural model of the fully-assembled Nuclear Pore Complex. Modeling of the assembly process improves the model precision over static integrative structure modeling alone. The method is applicable to a wide range of time-dependent systems in cell biology, and is available to the broader scientific community through an implementation in the open source Integrative Modeling Platform software.

Understanding the Fast-Triggering Unfolding Dynamics of FK-11 upon Photoexcitation of Azobenzene

Photoswitchable molecules can control the activity and functions of biomolecules by triggering conformational changes. However, it is still challenging to fully understand such fast-triggering conformational evolution from nonequilibrium to equilibrium distribution at the molecular level. Herein, we successfully simulated the unfolding of the FK-11 peptide upon the photoinduced trans-to-cis isomerization of azobenzene based on the Markov state model. We found that the ensemble of FK-11 contains five conformational states, constituting two unfolding pathways. More intriguingly, we observed the microsecond-scale conformational propagation of the FK-11 peptide from the fully folded state to the equilibrium populations of the five states. The computed CD spectra match well with the experimental data, validating our simulation method. Overall, our study not only offers a protocol to study the photoisomerization-induced conformational changes of enzymes but also could orientate the rational design of a photoswitchable molecule to manipulate biological functions.

Conformational exchange divergence along the evolutionary pathway of eosinophil-associated ribonucleases

The evolutionary role of conformational exchange in the emergence and preservation of function within structural homologs remains elusive. While protein engineering has revealed the importance of flexibility in function, productive modulation of atomic-scale dynamics has only been achieved on a finite number of distinct folds. Allosteric control of unique members within dynamically diverse structural families requires a better appreciation of exchange phenomena. Here, we examined the functional and structural role of conformational exchange within eosinophil-associated ribonucleases. Biological and catalytic activity of various EARs was performed in parallel to mapping their conformational behavior on multiple timescales using NMR and computational analyses. Despite functional conservation and conformational seclusion to a specific domain, we show that EARs can display similar or distinct motional profiles, implying divergence rather than conservation of flexibility. Comparing progressively more distant enzymes should unravel how this subfamily has evolved new functions and/or altered their behavior at the molecular level.

Markov field models: Scaling molecular kinetics approaches to large molecular machines

With recent advances in structural biology, including experimental techniques and deep learning-enabled high-precision structure predictions, molecular dynamics methods that scale up to large biomolecular systems are required. Current state-of-the-art approaches in molecular dynamics modeling focus on encoding global configurations of molecular systems as distinct states. This paradigm commands us to map out all possible structures and sample transitions between them, a task that becomes impossible for large-scale systems such as biomolecular complexes. To arrive at scalable molecular models, we suggest moving away from global state descriptions to a set of coupled models that each describe the dynamics of local domains or sites of the molecular system. We describe limitations in the current state-of-the-art global-state Markovian modeling approaches and then introduce Markov field models as an umbrella term that includes models from various scientific communities, including Independent Markov decomposition, Ising and Potts models, and (dynamic) graphical models, and evaluate their use for computational molecular biology. Finally, we give a few examples of early adoptions of these ideas for modeling molecular kinetics and thermodynamics.

Deep learning to decompose macromolecules into independent Markovian domains

The increasing interest in modeling the dynamics of ever larger proteins has revealed a fundamental problem with models that describe the molecular system as being in a global configuration state. This notion limits our ability to gather sufficient statistics of state probabilities or state-to-state transitions because for large molecular systems the number of metastable states grows exponentially with size. In this manuscript, we approach this challenge by introducing a method that combines our recent progress on independent Markov decomposition (IMD) with VAMPnets, a deep learning approach to Markov modeling. We establish a training objective that quantifies how well a given decomposition of the molecular system into independent subdomains with Markovian dynamics approximates the overall dynamics. By constructing an end-to-end learning framework, the decomposition into such subdomains and their individual Markov state models are simultaneously learned, providing a data-efficient and easily interpretable summary of the complex system dynamics. While learning the dynamical coupling between Markovian subdomains is still an open issue, the present results are a significant step towards learning Ising models of large molecular complexes from simulation data.

Unifying Single-Channel Permeability From Rare-Event Sampling and Steady-State Flux

Various all-atom molecular dynamics (MD) simulation methods have been developed to compute free energies and crossing rates of ions and small molecules through ion channels. However, a systemic comparison across different methods is scarce. Using a carbon nanotube as a model of small conductance ion channel, we computed the single-channel permeability for potassium ion using umbrella sampling, Markovian milestoning, and steady-state flux under applied voltage. We show that a slightly modified inhomogeneous solubility-diffusion equation yields a single-channel permeability consistent with the mean first passage time (MFPT) based method. For milestoning, applying cylindrical and spherical bulk boundary conditions yield consistent MFPT if factoring in the effective bulk concentration. The sensitivity of the MFPT to the output frequency of collective variables is highlighted using the convergence and symmetricity of the inward and outward MFPT profiles. The consistent transport kinetic results from all three methods demonstrated the robustness of MD-based methods in computing ion channel permeation. The advantages and disadvantages of each technique are discussed, focusing on the future applications of milestoning in more complex systems.

Is There a Need for a More Precise Description of Biomolecule Interactions to Understand Cell Function?

An important goal of biological research is to explain and hopefully predict cell behavior from the molecular properties of cellular components. Accordingly, much work was done to build extensive “omic” datasets and develop theoretical methods, including computer simulation and network analysis to process as quantitatively as possible the parameters contained in these resources. Furthermore, substantial effort was made to standardize data presentation and make experimental results accessible to data scientists. However, the power and complexity of current experimental and theoretical tools make it more and more difficult to assess the capacity of gathered parameters to support optimal progress in our understanding of cell function. The purpose of this review is to focus on biomolecule interactions, the interactome, as a specific and important example, and examine the limitations of the explanatory and predictive power of parameters that are considered as suitable descriptors of molecular interactions. Recent experimental studies on important cell functions, such as adhesion and processing of environmental cues for decision-making, support the suggestion that it should be rewarding to complement standard binding properties such as affinity and kinetic constants, or even force dependence, with less frequently used parameters such as conformational flexibility or size of binding molecules.

Multiscale molecular kinetics by coupling Markov state models and reaction-diffusion dynamics

A novel approach to simulate simple protein-ligand systems at large time and length scales is to couple Markov state models (MSMs) of molecular kinetics with particle-based reaction-diffusion (RD) simulations, MSM/RD. Currently, MSM/RD lacks a mathematical framework to derive coupling schemes, is limited to isotropic ligands in a single conformational state, and lacks multiparticle extensions. In this work, we address these needs by developing a general MSM/RD framework by coarse-graining molecular dynamics into hybrid switching diffusion processes. Given enough data to parameterize the model, it is capable of modeling protein-protein interactions over large time and length scales, and it can be extended to handle multiple molecules. We derive the MSM/RD framework, and we implement and verify it for two protein-protein benchmark systems and one multiparticle implementation to model the formation of pentameric ring molecules. To enable reproducibility, we have published our code in the MSM/RD software package.