On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method

Energy landscape analysis of the development of the chromosome structure across the cell cycle

During mitosis, there are significant structural changes in chromosomes. We used a maximum entropy approach to invert experimental Hi-C data to generate effective energy landscapes for chromosomal structures at different stages during the cell cycle. Modeled mitotic structures show a hierarchical organization of helices of helices. High-periodicity loops span hundreds of kilobases or less, while the other low-periodicity ones are larger in genomic separation, spanning several megabases. The structural ensembles reveal a progressive decrease in compartmentalization from interphase to mitosis, accompanied by the appearance of a second diagonal in prometaphase, indicating an organized array of loops. While there is a local tendency to form chiral helices, overall, no preferential left-handed or right-handed chirality appears to develop on the time scale of the cell cycle. Chromatin thus appears to be a liquid crystal containing numerous defects that anneal rather slowly.

The synergy between compartmentalization and motorization in chromatin architecture

High-resolution techniques capable of manipulating from single molecules to millions of cells are combined with three-dimensional modeling followed by simulation to comprehend the specific aspects of chromosomes. From the theoretical perspective, the energy landscape theory from protein folding inspired the development of the minimal chromatin model (MiChroM). In this work, two biologically relevant MiChroM energy terms were minimized under different conditions, revealing a competition between loci compartmentalization and motor-driven activity mechanisms in chromatin folding. Enhancing the motor activity energy baseline increased the lengthwise compaction and reduced the polymer entanglement. Concomitantly, decreasing compartmentalization-related interactions reduced the overall polymer collapse, although compartmentalization given by the microphase separation remained almost intact. For multiple chromosome simulations, increased motorization intensified the territory formation of the different chains and reduced compartmentalization strength lowered the probability of contact formation of different loci between multiple chains, approximating to the experimental inter-contacts of the human chromosomes. These findings have direct implications for experimental data-driven chromosome modeling, specially those involving multiple chromosomes. The interplay between phase-separation and territory formation mechanisms should be properly implemented in order to recover the genome architecture and dynamics, features that might play critical roles in regulating nuclear functions.

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.

Experiment-Guided Refinement of Milestoning Network

Milestoning is an efficient method for calculating rare event kinetics by constructing a continuous-time kinetic network that connects the reactant and product states. Its accuracy depends on both the quality of the underlying force fields and the trajectory sampling. The sampling error can be effectively controlled through various methods. However, the force fields are often not accurate enough, leading to quantitative discrepancies between simulations and experimental data. To address this challenge, we present a refinement approach for Milestoning network based on the maximum caliber (MaxCal), a general variational principle for dynamical systems, to combine simulations and experimental data. The Kullback-Leibler divergence rate between two Milestoning networks is analytically evaluated and minimized as the loss function. Meanwhile, experimental thermodynamic (equilibrium constants) and kinetic (rate constants) data are incorporated as constraints. The use of MaxCal implies that the refined kinetic network is minimally perturbed from the original one while satisfying the experimental constraints. The refined network is expected to align better with available experimental data. The refinement approach is demonstrated using the binding and unbinding dynamics of a series of six small molecule ligands for the model host system, β-cyclodextrin.

Implementation of Time-Averaged Restraints with UNRES Coarse-Grained Model of Polypeptide Chains

Time-averaged restraints from nuclear magnetic resonance (NMR) measurements have been implemented in the UNRES coarse-grained model of polypeptide chains in order to develop a tool for data-assisted modeling of the conformational ensembles of multistate proteins, intrinsically disordered proteins (IDPs) and proteins with intrinsically disordered regions (IDRs), many of which are essential in cell biology. A numerically stable variant of molecular dynamics with time-averaged restraints has been introduced, in which the total energy is conserved in sections of a trajectory in microcanonical runs, the bath temperature is maintained in canonical runs, and the time-average-restraint-force components are scaled up with the length of the memory window so that the restraints affect the simulated structures. The new approach restores the conformational ensembles used to generate ensemble-averaged distances, as demonstrated with synthetic restraints. The approach results in a better fitting of the ensemble-averaged interproton distances to those determined experimentally for multistate proteins and proteins with intrinsically disordered regions, which puts it at an advantage over all-atom approaches with regard to the determination of the conformational ensembles of proteins with diffuse structures, owing to a faster and more robust conformational search.

An integrated machine learning approach delineates an entropic expansion mechanism for the binding of a small molecule to α-synuclein

The mis-folding and aggregation of intrinsically disordered proteins (IDPs) such as α-synuclein (αS) underlie the pathogenesis of various neurodegenerative disorders. However, targeting αS with small molecules faces challenges due to the lack of defined ligand-binding pockets in its disordered structure. Here, we implement a deep artificial neural network-based machine learning approach, which is able to statistically distinguish the fuzzy ensemble of conformational substates of αS in neat water from those in aqueous fasudil (small molecule of interest) solution. In particular, the presence of fasudil in the solvent either modulates pre-existing states of αS or gives rise to new conformational states of αS, akin to an ensemble-expansion mechanism. The ensembles display strong conformation-dependence in residue-wise interaction with the small molecule. A thermodynamic analysis indicates that small-molecule modulates the structural repertoire of αS by tuning protein backbone entropy, however entropy of the water remains unperturbed. Together, this study sheds light on the intricate interplay between small molecules and IDPs, offering insights into entropic modulation and ensemble expansion as key biophysical mechanisms driving potential therapeutics.

Expanded Functionality and Portability for the Colvars Library

Colvars is an open-source C++ library that provides a modular toolkit for collective-variable-based molecular simulations. It allows practitioners to easily create and implement descriptors that best fit a process of interest and to apply a wide range of biasing algorithms in collective variable space. This paper reviews several features and improvements to Colvars that were added since its original introduction. Special attention is given to contributions that significantly expanded the capabilities of this software or its distribution with major MD simulation packages. Collective variables can now be optimized either manually or by machine-learning methods, and the space of descriptors can be explored interactively using the graphical interface included in VMD. Beyond the spatial coordinates of individual molecules, Colvars can now apply biasing forces to mesoscale structures and alchemical degrees of freedom and perform simulations guided by experimental data within ensemble averages or probability distributions. It also features advanced computational schemes to boost the accuracy, robustness, and general applicability of simulation methods, including extended-system and multiple-walker adaptive biasing force, boundary conditions for metadynamics, replica exchange with biasing potentials, and adiabatic bias molecular dynamics. The library is made available directly within the main distributions of the academic software GROMACS, LAMMPS, NAMD, Tinker-HP, and VMD. The robustness of the software and the reliability of the results are ensured through the use of continuous integration with a test suite within the source repository.

OpenNucleome for high-resolution nuclear structural and dynamical modeling

The intricate structural organization of the human nucleus is fundamental to cellular function and gene regulation. Recent advancements in experimental techniques, including high-throughput sequencing and microscopy, have provided valuable insights into nuclear organization. Computational modeling has played significant roles in interpreting experimental observations by reconstructing high-resolution structural ensembles and uncovering organization principles. However, the absence of standardized modeling tools poses challenges for furthering nuclear investigations. We present OpenNucleome-an open-source software designed for conducting GPU-accelerated molecular dynamics simulations of the human nucleus. OpenNucleome offers particle-based representations of chromosomes at a resolution of 100 KB, encompassing nuclear lamina, nucleoli, and speckles. This software furnishes highly accurate structural models of nuclear architecture, affording the means for dynamic simulations of condensate formation, fusion, and exploration of non-equilibrium effects. We applied OpenNucleome to uncover the mechanisms driving the emergence of 'fixed points' within the nucleus-signifying genomic loci robustly anchored in proximity to specific nuclear bodies for functional purposes. This anchoring remains resilient even amidst significant fluctuations in chromosome radial positions and nuclear shapes within individual cells. Our findings lend support to a nuclear zoning model that elucidates genome functionality. We anticipate OpenNucleome to serve as a valuable tool for nuclear investigations, streamlining mechanistic explorations and enhancing the interpretation of experimental observations.

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.

Optimizing Force Fields with Experimental Data Using Ensemble Reweighting and Potential Contrasting

Despite force field improvements over the past decades, we still encounter situations where simulation results disagree with experiments due to force field inaccuracies. Such situations provide opportunities to improve force fields. In this study, we introduce a novel framework for optimizing force fields using experimental data. The unique feature of this framework is that it aims to optimize force fields to match experiments while minimizing the perturbation made to the original force field. To achieve this, we combine ensemble reweighting techniques with the potential contrasting method. Ensemble reweighting is used to reweight an ensemble of conformations generated using an existing force field to match experimental data while minimizing the perturbation to the original ensemble. Potential contrasting is then utilized to optimize force field parameters to reproduce the reweighted ensemble. We demonstrate the framework's effectiveness by optimizing a coarse-grained force field for intrinsically disordered proteins using experimental radius of gyration data.

Multiscale Bayesian simulations reveal functional chromatin condensation of gene loci

Chromatin, the complex assembly of DNA and associated proteins, plays a pivotal role in orchestrating various genomic functions. To aid our understanding of the principles underlying chromatin organization, we introduce Hi-C metainference, a Bayesian approach that integrates Hi-C contact frequencies into multiscale prior models of chromatin. This approach combines both bottom-up (the physics-based prior) and top-down (the data-driven posterior) strategies to characterize the 3D organization of a target genomic locus. We first demonstrate the capability of this method to accurately reconstruct the structural ensemble and the dynamics of a system from contact information. We then apply the approach to investigate the Sox2, Pou5f1, and Nanog loci of mouse embryonic stem cells using a bottom-up chromatin model at 1 kb resolution. We observe that the studied loci are conformationally heterogeneous and organized as crumpled globules, favoring contacts between distant enhancers and promoters. Using nucleosome-resolution simulations, we then reveal how the Nanog gene is functionally organized across the multiple scales of chromatin. At the local level, we identify diverse tetranucleosome folding motifs with a characteristic distribution along the genome, predominantly open at cis-regulatory elements and compact in between. At the larger scale, we find that enhancer-promoter contacts are driven by the transient condensation of chromatin into compact domains stabilized by extensive internucleosome interactions. Overall, this work highlights the condensed, but dynamic nature of chromatin in vivo, contributing to a deeper understanding of gene structure-function relationships.

Automatic Forward Model Parameterization with Bayesian Inference of Conformational Populations

To quantify how well theoretical predictions of structural ensembles agree with experimental measurements, we depend on the accuracy of forward models. These models are computational frameworks that generate observable quantities from molecular configurations based on empirical relationships linking specific molecular properties to experimental measurements. Bayesian Inference of Conformational Populations (BICePs) is a reweighting algorithm that reconciles simulated ensembles with ensemble-averaged experimental observations, even when such observations are sparse and/or noisy. This is achieved by sampling the posterior distribution of conformational populations under experimental restraints as well as sampling the posterior distribution of uncertainties due to random and systematic error. In this study, we enhance the algorithm for the refinement of empirical forward model (FM) parameters. We introduce and evaluate two novel methods for optimizing FM parameters. The first method treats FM parameters as nuisance parameters, integrating over them in the full posterior distribution. The second method employs variational minimization of a quantity called the BICePs score that reports the free energy of "turning on" the experimental restraints. This technique, coupled with improved likelihood functions for handling experimental outliers, facilitates force field validation and optimization, as illustrated in recent studies (Raddi et al. 2023, 2024). Using this approach, we refine parameters that modulate the Karplus relation, crucial for accurate predictions of J -coupling constants based on dihedral angles ( ϕ ) between interacting nuclei. We validate this approach first with a toy model system, and then for human ubiquitin, predicting six sets of Karplus parameters forJ H N H α 3 ,J H α C ' 3 ,J H N C β 3 ,J H N C ' 3 ,J C ' C β 3 ,J C ' C ' 3 . This approach, which does not rely on any predetermined parameterization, enhances predictive accuracy and can be used for many applications.

Cryo-electron Microscopy and Molecular Modeling Methods to Characterize the Dynamics of Tau Bound to Microtubules

The electron microscopy metainference integrative structural biology method enables the combination of cryo-electron microscopy electron density maps with molecular modeling techniques such as molecular dynamics to unveil the atomistic biomolecular structural ensemble and the error in the map data in an efficient manner. Here we illustrate the electron microscopy metainference protocol and analysis used to elucidate the atomistic structural ensemble of the microtubule-associated protein tau bound to microtubules by using state-of-the-art molecular mechanic force field and the electron density map.

Vendi sampling for molecular simulations: Diversity as a force for faster convergence and better exploration

Molecular dynamics (MD) is the method of choice for understanding the structure, function, and interactions of molecules. However, MD simulations are limited by the strong metastability of many molecules, which traps them in a single conformation basin for an extended amount of time. Enhanced sampling techniques, such as metadynamics and replica exchange, have been developed to overcome this limitation and accelerate the exploration of complex free energy landscapes. In this paper, we propose Vendi Sampling, a replica-based algorithm for increasing the efficiency and efficacy of the exploration of molecular conformation spaces. In Vendi sampling, replicas are simulated in parallel and coupled via a global statistical measure, the Vendi Score, to enhance diversity. Vendi sampling allows for the recovery of unbiased sampling statistics and dramatically improves sampling efficiency. We demonstrate the effectiveness of Vendi sampling in improving molecular dynamics simulations by showing significant improvements in coverage and mixing between metastable states and convergence of free energy estimates for four common benchmarks, including Alanine Dipeptide and Chignolin.

OpenABC enables flexible, simplified, and efficient GPU accelerated simulations of biomolecular condensates

Biomolecular condensates are important structures in various cellular processes but are challenging to study using traditional experimental techniques. In silico simulations with residue-level coarse-grained models strike a balance between computational efficiency and chemical accuracy. They could offer valuable insights by connecting the emergent properties of these complex systems with molecular sequences. However, existing coarse-grained models often lack easy-to-follow tutorials and are implemented in software that is not optimal for condensate simulations. To address these issues, we introduce OpenABC, a software package that greatly simplifies the setup and execution of coarse-grained condensate simulations with multiple force fields using Python scripting. OpenABC seamlessly integrates with the OpenMM molecular dynamics engine, enabling efficient simulations with performance on a single GPU that rivals the speed achieved by hundreds of CPUs. We also provide tools that convert coarse-grained configurations to all-atom structures for atomistic simulations. We anticipate that OpenABC will significantly facilitate the adoption of in silico simulations by a broader community to investigate the structural and dynamical properties of condensates.

Determination of the Structure and Dynamics of the Fuzzy Coat of an Amyloid Fibril of IAPP Using Cryo-Electron Microscopy

In recent years, major advances in cryo-electron microscopy (cryo-EM) have enabled the routine determination of complex biomolecular structures at atomistic resolution. An open challenge for this approach, however, concerns large systems that exhibit continuous dynamics. To address this problem, we developed the metadynamic electron microscopy metainference (MEMMI) method, which incorporates metadynamics, an enhanced conformational sampling approach, into the metainference method of integrative structural biology. MEMMI enables the simultaneous determination of the structure and dynamics of large heterogeneous systems by combining cryo-EM density maps with prior information through molecular dynamics, while at the same time modeling the different sources of error. To illustrate the method, we apply it to elucidate the dynamics of an amyloid fibril of the islet amyloid polypeptide (IAPP). The resulting conformational ensemble provides an accurate description of the structural variability of the disordered region of the amyloid fibril, known as fuzzy coat. The conformational ensemble also reveals that in nearly half of the structural core of this amyloid fibril, the side chains exhibit liquid-like dynamics despite the presence of the highly ordered network backbone of hydrogen bonds characteristic of the cross-β structure of amyloid fibrils.

Efficient Hi-C inversion facilitates chromatin folding mechanism discovery and structure prediction

Genome-wide chromosome conformation capture (Hi-C) experiments have revealed many structural features of chromatin across multiple length scales. Further understanding genome organization requires relating these discoveries to the mechanisms that establish chromatin structures and reconstructing these structures in three dimensions, but both objectives are difficult to achieve with existing algorithms that are often computationally expensive. To alleviate this challenge, we present an algorithm that efficiently converts Hi-C data into contact energies, which measure the interaction strength between genomic loci brought into proximity. Contact energies are local quantities unaffected by the topological constraints that correlate Hi-C contact probabilities. Thus, extracting contact energies from Hi-C contact probabilities distills the biologically unique information contained in the data. We show that contact energies reveal the location of chromatin loop anchors, support a phase separation mechanism for genome compartmentalization, and parameterize polymer simulations that predict three-dimensional chromatin structures. Therefore, we anticipate that contact energy extraction will unleash the full potential of Hi-C data and that our inversion algorithm will facilitate the widespread adoption of contact energy analysis.

BICePs v2.0: Software for Ensemble Reweighting Using Bayesian Inference of Conformational Populations

Bayesian Inference of Conformational Populations (BICePs) version 2.0 (v2.0) is a free, open-source Python package that reweights theoretical predictions of conformational state populations using sparse and/or noisy experimental measurements. In this article, we describe the implementation and usage of the latest version of BICePs (v2.0), a powerful, user-friendly and extensible package which makes several improvements upon the previous version. The algorithm now supports many experimental NMR observables (NOE distances, chemical shifts, J-coupling constants, and hydrogen-deuterium exchange protection factors), and enables convenient data preparation and processing. BICePs v2.0 can perform automatic analysis of the sampled posterior, including visualization, and evaluation of statistical significance and sampling convergence. We provide specific coding examples for these topics, and present a detailed example illustrating how to use BICePs v2.0 to reweight a theoretical ensemble using experimental measurements.

Biophysical and Integrative Characterization of Protein Intrinsic Disorder as a Prime Target for Drug Discovery

Protein intrinsic disorder is increasingly recognized for its biological and disease-driven functions. However, it represents significant challenges for biophysical studies due to its high conformational flexibility. In addressing these challenges, we highlight the complementary and distinct capabilities of a range of experimental and computational methods and further describe integrative strategies available for combining these techniques. Integrative biophysics methods provide valuable insights into the sequence–structure–function relationship of disordered proteins, setting the stage for protein intrinsic disorder to become a promising target for drug discovery. Finally, we briefly summarize recent advances in the development of new small molecule inhibitors targeting the disordered N-terminal domains of three vital transcription factors.

Structure and ensemble refinement against SAXS data: Combining MD simulations with Bayesian inference or with the maximum entropy principle

Small-angle X-ray scattering (SAXS) is a powerful method for tracking conformational transitions of proteins or soft-matter complexes in solution. However, the interpretation of the experimental data is challenged by the low spatial resolution and the low information content of the data, which lead to a high risk of overinterpreting the data. Here, we illustrate how SAXS data can be integrated into all-atom molecular dynamics (MD) simulation to derive atomic structures or heterogeneous ensembles that are compatible with the data. Besides providing atomistic insight, the MD simulation adds physicochemical information, as encoded in the MD force fields, which greatly reduces the risk of overinterpretation. We present an introduction into the theory of SAXS-driven MD simulations as implemented in GROMACS-SWAXS, a modified version of the GROMACS simulation software. We discuss SAXS-driven parallel-replica ensemble refinement with commitment to the maximum entropy principle as well as a Bayesian formulation of SAXS-driven structure refinement. Practical considerations for running and interpreting the simulations are presented. The methods are freely available via GitLab at https://gitlab.com/cbjh/gromacs-swaxs.

Interdomain Dynamics via Paramagnetic NMR on the Highly Flexible Complex Calmodulin/Munc13-1

Paramagnetic NMR constraints are very useful to study protein interdomain motion, but their interpretation is not always straightforward. On the example of the particularly flexible complex Calmodulin/Munc13-1, we present a new approach to characterize this motion with pseudocontact shifts and residual dipolar couplings. Using molecular mechanics, we sampled the conformational space of the complex and used a genetic algorithm to find ensembles that are in agreement with the data. We used the Bayesian information criterion to determine the ideal ensemble size. This way, we were able to make an accurate, unambiguous, reproducible model of the interdomain motion of Calmodulin/Munc13-1 without prior knowledge about the domain orientation from crystallography.

Quantitative prediction of ensemble dynamics, shapes and contact propensities of intrinsically disordered proteins

Intrinsically disordered proteins (IDPs) are highly dynamic systems that play an important role in cell signaling processes and their misfunction often causes human disease. Proper understanding of IDP function not only requires the realistic characterization of their three-dimensional conformational ensembles at atomic-level resolution but also of the time scales of interconversion between their conformational substates. Large sets of experimental data are often used in combination with molecular modeling to restrain or bias models to improve agreement with experiment. It is shown here for the N-terminal transactivation domain of p53 (p53TAD) and Pup, which are two IDPs that fold upon binding to their targets, how the latest advancements in molecular dynamics (MD) simulations methodology produces native conformational ensembles by combining replica exchange with series of microsecond MD simulations. They closely reproduce experimental data at the global conformational ensemble level, in terms of the distribution properties of the radius of gyration tensor, and at the local level, in terms of NMR properties including 15N spin relaxation, without the need for reweighting. Further inspection revealed that 10-20% of the individual MD trajectories display the formation of secondary structures not observed in the experimental NMR data. The IDP ensembles were analyzed by graph theory to identify dominant inter-residue contact clusters and characteristic amino-acid contact propensities. These findings indicate that modern MD force fields with residue-specific backbone potentials can produce highly realistic IDP ensembles sampling a hierarchy of nano- and picosecond time scales providing new insights into their biological function.

Heterogeneous interactions and polymer entropy decide organization and dynamics of chromatin domains

Chromatin is known to be organized into multiple domains of varying sizes and compaction. While these domains are often imagined as static structures, they are highly dynamic and show cell-to-cell variability. Since processes such as gene regulation and DNA replication occur in the context of these domains, it is important to understand their organization, fluctuation, and dynamics. To simulate chromatin domains, one requires knowledge of interaction strengths among chromatin segments. Here, we derive interaction-strength parameters from experimentally known contact maps and use them to predict chromatin organization and dynamics. Taking two domains on the human chromosome as examples, we investigate its three-dimensional organization, size/shape fluctuations, and dynamics of different segments within a domain, accounting for hydrodynamic effects. Considering different cell types, we quantify changes in interaction strengths and chromatin shape fluctuations in different epigenetic states. Perturbing the interaction strengths systematically, we further investigate how epigenetic-like changes can alter the spatio-temporal nature of the domains. Our results show that heterogeneous weak interactions are crucial in determining the organization of the domains. Computing effective stiffness and relaxation times, we investigate how perturbations in interactions affect the solid- and liquid-like nature of chromatin domains. Quantifying dynamics of chromatin segments within a domain, we show how the competition between polymer entropy and interaction energy influence the timescales of loop formation and maintenance of stable loops.

Inferring spatial source of disease outbreaks using maximum entropy

Mathematical modeling of disease outbreaks can infer the future trajectory of an epidemic, allowing for making more informed policy decisions. Another task is inferring the origin of a disease, which is relatively difficult with current mathematical models. Such frameworks, across varying levels of complexity, are typically sensitive to input data on epidemic parameters, case counts, and mortality rates, which are generally noisy and incomplete. To alleviate these limitations, we propose a maximum entropy framework that fits epidemiological models, provides calibrated infection origin probabilities, and is robust to noise due to a prior belief model. Maximum entropy is agnostic to the parameters or model structure used and allows for flexible use when faced with sparse data conditions and incomplete knowledge in the dynamical phase of disease-spread, providing for more reliable modeling at early stages of outbreaks. We evaluate the performance of our model by predicting future disease trajectories based on simulated epidemiological data in synthetic graph networks and the real mobility network of New York State. In addition, unlike existing approaches, we demonstrate that the method can be used to infer the origin of the outbreak with accurate confidence. Indeed, despite the prevalent belief on the feasibility of contact-tracing being limited to the initial stages of an outbreak, we report the possibility of reconstructing early disease dynamics, including the epidemic seed, at advanced stages.

Maximum Entropy Technique and Regularization Functional for Determining the Pharmacokinetic Parameters in DCE-MRI

This paper aims to solve the arterial input function (AIF) determination in dynamic contrast-enhanced MRI (DCE-MRI), an important linear ill-posed inverse problem, using the maximum entropy technique (MET) and regularization functionals. In addition, estimating the pharmacokinetic parameters from a DCE-MR image investigations is an urgent need to obtain the precise information about the AIF-the concentration of the contrast agent on the left ventricular blood pool measured over time. For this reason, the main idea is to show how to find a unique solution of linear system of equations generally in the form of [Formula: see text] named an ill-conditioned linear system of equations after discretization of the integral equations, which appear in different tomographic image restoration and reconstruction issues. Here, a new algorithm is described to estimate an appropriate probability distribution function for AIF according to the MET and regularization functionals for the contrast agent concentration when applying Bayesian estimation approach to estimate two different pharmacokinetic parameters. Moreover, by using the proposed approach when analyzing simulated and real datasets of the breast tumors according to pharmacokinetic factors, it indicates that using Bayesian inference-that infer the uncertainties of the computed solutions, and specific knowledge of the noise and errors-combined with the regularization functional of the maximum entropy problem, improved the convergence behavior and led to more consistent morphological and functional statistics and results. Finally, in comparison to the proposed exponential distribution based on MET and Newton's method, or Weibull distribution via the MET and teaching-learning-based optimization (MET/TLBO) in the previous studies, the family of Gamma and Erlang distributions estimated by the new algorithm are more appropriate and robust AIFs.

Simulation-based inference with approximately correct parameters via maximum entropy

Inferring the input parameters of simulators from observations is a crucial challenge with applications from epidemiology to molecular dynamics. Here we show a simple approach in the regime of sparse data and approximately correct models, which is common when trying to use an existing model to infer latent variables with observed data. This approach is based on the principle of maximum entropy (MaxEnt) and provably makes the smallest change in the latent joint distribution to fit new data. This method requires no likelihood or model derivatives and its fit is insensitive to prior strength, removing the need to balance observed data fit with prior belief. The method requires the ansatz that data is fit in expectation, which is true in some settings and may be reasonable in all settings with few data points. The method is based on sample reweighting, so its asymptotic run time is independent of prior distribution dimension. We demonstrate this MaxEnt approach and compare with other likelihood-free inference methods across three systems: a point particle moving in a gravitational field, a compartmental model of epidemic spread and molecular dynamics simulation of a protein.

NMR Provides Unique Insight into the Functional Dynamics and Interactions of Intrinsically Disordered Proteins

Intrinsically disordered proteins are ubiquitous throughout all known proteomes, playing essential roles in all aspects of cellular and extracellular biochemistry. To understand their function, it is necessary to determine their structural and dynamic behavior and to describe the physical chemistry of their interaction trajectories. Nuclear magnetic resonance is perfectly adapted to this task, providing ensemble averaged structural and dynamic parameters that report on each assigned resonance in the molecule, unveiling otherwise inaccessible insight into the reaction kinetics and thermodynamics that are essential for function. In this review, we describe recent applications of NMR-based approaches to understanding the conformational energy landscape, the nature and time scales of local and long-range dynamics and how they depend on the environment, even in the cell. Finally, we illustrate the ability of NMR to uncover the mechanistic basis of functional disordered molecular assemblies that are important for human health.

Generating Ensembles of Dynamic Misfolding Proteins

The early stages of protein misfolding and aggregation involve disordered and partially folded protein conformers that contain a high degree of dynamic disorder. These dynamic species may undergo large-scale intra-molecular motions of intrinsically disordered protein (IDP) precursors, or flexible, low affinity inter-molecular binding in oligomeric assemblies. In both cases, generating atomic level visualization of the interconverting species that captures the conformations explored and their physico-chemical properties remains hugely challenging. How specific sub-ensembles of conformers that are on-pathway to aggregation into amyloid can be identified from their aggregation-resilient counterparts within these large heterogenous pools of rapidly moving molecules represents an additional level of complexity. Here, we describe current experimental and computational approaches designed to capture the dynamic nature of the early stages of protein misfolding and aggregation, and discuss potential challenges in describing these species because of the ensemble averaging of experimental restraints that arise from motions on the millisecond timescale. We give a perspective of how machine learning methods can be used to extract aggregation-relevant sub-ensembles and provide two examples of such an approach in which specific interactions of defined species within the dynamic ensembles of α-synuclein (αSyn) and β2-microgloblulin (β2m) can be captured and investigated.

Visualizing RNA Structures by SAXS-Driven MD Simulations

The biological role of biomolecules is intimately linked to their structural dynamics. Experimental or computational techniques alone are often insufficient to determine accurate structural ensembles in atomic detail. We use all-atom molecular dynamics (MD) simulations and couple it to small-angle X-ray scattering (SAXS) experiments to resolve the structural dynamics of RNA molecules. To accomplish this task, we utilize a set of re-weighting and biasing techniques tailored for RNA molecules. To showcase our approach, we study two RNA molecules: a riboswitch that shows structural variations upon ligand binding, and a two-way junction RNA that displays structural heterogeneity and sensitivity to salt conditions. Integration of MD simulations and experiments allows the accurate construction of conformational ensembles of RNA molecules. We observe a dynamic change of the SAM-I riboswitch conformations depending on its binding partners. The binding of SAM and Mg2+ cations stabilizes the compact state. The absence of Mg2+ or SAM leads to the loss of tertiary contacts, resulting in a dramatic expansion of the riboswitch conformations. The sensitivity of RNA structures to the ionic strength demonstrates itself in the helix junction helix (HJH). The HJH shows non-monotonic compaction as the ionic strength increases. The physics-based picture derived from the experimentally guided MD simulations allows biophysical characterization of RNA molecules. All in all, SAXS-guided MD simulations offer great prospects for studying RNA structural dynamics.

Conformational ensembles of intrinsically disordered proteins and flexible multidomain proteins

Intrinsically disordered proteins (IDPs) and multidomain proteins with flexible linkers show a high level of structural heterogeneity and are best described by ensembles consisting of multiple conformations with associated thermodynamic weights. Determining conformational ensembles usually involves the integration of biophysical experiments and computational models. In this review, we discuss current approaches to determine conformational ensembles of IDPs and multidomain proteins, including the choice of biophysical experiments, computational models used to sample protein conformations, models to calculate experimental observables from protein structure, and methods to refine ensembles against experimental data. We also provide examples of recent applications of integrative conformational ensemble determination to study IDPs and multidomain proteins and suggest future directions for research in the field.

Characterizing conformational ensembles of multi-domain proteins using anisotropic paramagnetic NMR restraints

It has been over two decades since paramagnetic NMR started to form part of the essential techniques for structural analysis of proteins under physiological conditions. Paramagnetic NMR has significantly expanded our understanding of the inherent flexibility of proteins, in particular, those that are formed by combinations of two or more domains. Here, we present a brief overview of techniques to characterize conformational ensembles of such multi-domain proteins using paramagnetic NMR restraints produced through anisotropic metals, with a focus on the basics of anisotropic paramagnetic effects, the general procedures of conformational ensemble reconstruction, and some representative reweighting approaches.

Modified Maximum Entropy Method and Estimating the AIF via DCE-MRI Data Analysis

Background: For the kinetic models used in contrast-based medical imaging, the assignment of the arterial input function named AIF is essential for the estimation of the physiological parameters of the tissue via solving an optimization problem. Objective: In the current study, we estimate the AIF relayed on the modified maximum entropy method. The effectiveness of several numerical methods to determine kinetic parameters and the AIF is evaluated—in situations where enough information about the AIF is not available. The purpose of this study is to identify an appropriate method for estimating this function. Materials and Methods: The modified algorithm is a mixture of the maximum entropy approach with an optimization method, named the teaching-learning method. In here, we applied this algorithm in a Bayesian framework to estimate the kinetic parameters when specifying the unique form of the AIF by the maximum entropy method. We assessed the proficiency of the proposed method for assigning the kinetic parameters in the dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI), when determining AIF with some other parameter-estimation methods and a standard fixed AIF method. A previously analyzed dataset consisting of contrast agent concentrations in tissue and plasma was used. Results and Conclusions: We compared the accuracy of the results for the estimated parameters obtained from the MMEM with those of the empirical method, maximum likelihood method, moment matching (“method of moments”), the least-square method, the modified maximum likelihood approach, and our previous work. Since the current algorithm does not have the problem of starting point in the parameter estimation phase, it could find the best and nearest model to the empirical model of data, and therefore, the results indicated the Weibull distribution as an appropriate and robust AIF and also illustrated the power and effectiveness of the proposed method to estimate the kinetic parameters.

Recent Developments in Data-Assisted Modeling of Flexible Proteins

Many proteins can fold into well-defined conformations. However, intrinsically-disordered proteins (IDPs) do not possess a defined structure. Moreover, folded multi-domain proteins often digress into alternative conformations. Collectively, the conformational dynamics enables these proteins to fulfill specific functions. Thus, most experimental observables are averaged over the conformations that constitute an ensemble. In this article, we review the recent developments in the concept and methods for the determination of the dynamic structures of flexible peptides and proteins. In particular, we describe ways to extract information from nuclear magnetic resonance small-angle X-ray scattering (SAXS), and chemical cross-linking coupled with mass spectroscopy (XL-MS) measurements. All these techniques can be used to obtain ensemble-averaged restraints or to re-weight the simulated conformational ensembles.

Criteria ruling particle agglomeration

Most of the technically important properties of nanomaterials, such as superparamagnetism or luminescence, depend on the particle size. During synthesis and handling of nanoparticles, agglomeration may occur. Agglomeration of nanoparticles may be controlled by different mechanisms. During synthesis one observes agglomeration controlled by the geometry and electrical charges of the particles. Additionally, one may find agglomeration controlled by thermodynamic interaction of the particles in the direction of a minimum of the free enthalpy. In this context, one may observe mechanisms leading to a reduction of the surface energy or controlled by the van der Waals interaction. Additionally, the ensemble may arrange in the direction of a maximum of the entropy. Simulations based on Monte Carlo methods teach that, in case of any energetic interaction of the particles, the influence of the entropy is minor or even negligible. Complementary to the simulations, the extremum of the entropy was determined using the Lagrange method. Both approaches yielded identical result for the particle size distribution of an agglomerated ensemble, that is, an exponential function characterized by two parameters. In this context, it is important to realize that one has to take care of fluctuations of the entropy.

Chromosome Modeling on Downsampled Hi-C Maps Enhances the Compartmentalization Signal

The human genome is organized within a nucleus where chromosomes fold into an ensemble of different conformations. Chromosome conformation capture techniques such as Hi-C provide information about the genome architecture by creating a 2D heat map. Initially, Hi-C map experiments were performed in human interphase cell lines. Recently, efforts were expanded to several different organisms, cell lines, tissues, and cell cycle phases where obtaining high-quality maps is challenging. Poor sampled Hi-C maps present high sparse matrices where compartments located far from the main diagonal are difficult to observe. Aided by recently developed models for chromatin folding and dynamics investigation, we introduce a framework to enhance the compartments' information far from the diagonal observed in experimental sparse matrices. The simulations were performed using the Open-MiChroM platform aided by new trained parameters in the minimal chromatin model (MiChroM) energy function. The simulations optimized on a downsampled experimental map (10% of the original data) allow the prediction of a contact frequency similar to that of the complete (100%) experimental Hi-C. The modeling results open a discussion on how simulations and modeling can increase the statistics and help fill in some Hi-C regions not captured by poor sampling experiments. Open-MiChroM simulations allow us to explore the 3D genome organization of different organisms, cell lines, and cell phases that often do not produce high-quality Hi-C maps.

Multiscale modeling of genome organization with maximum entropy optimization

Three-dimensional (3D) organization of the human genome plays an essential role in all DNA-templated processes, including gene transcription, gene regulation, and DNA replication. Computational modeling can be an effective way of building high-resolution genome structures and improving our understanding of these molecular processes. However, it faces significant challenges as the human genome consists of over 6 × 109 base pairs, a system size that exceeds the capacity of traditional modeling approaches. In this perspective, we review the progress that has been made in modeling the human genome. Coarse-grained models parameterized to reproduce experimental data via the maximum entropy optimization algorithm serve as effective means to study genome organization at various length scales. They have provided insight into the principles of whole-genome organization and enabled de novo predictions of chromosome structures from epigenetic modifications. Applications of these models at a near-atomistic resolution further revealed physicochemical interactions that drive the phase separation of disordered proteins and dictate chromatin stability in situ. We conclude with an outlook on the opportunities and challenges in studying chromosome dynamics.

The diversity of molecular interactions involving intrinsically disordered proteins: A molecular modeling perspective

Intrinsically Disordered Proteins and Regions (IDPs/IDRs) are key components of a multitude of biological processes. Conformational malleability enables IDPs/IDRs to perform very specialized functions that cannot be accomplished by globular proteins. The functional role for most of these proteins is related to the recognition of other biomolecules to regulate biological processes or as a part of signaling pathways. Depending on the extent of disorder, the number of interacting sites and the type of partner, very different architectures for the resulting assemblies are possible. More recently, molecular condensates with liquid-like properties composed of multiple copies of IDPs and nucleic acids have been proven to regulate key processes in eukaryotic cells. The structural and kinetic details of disordered biomolecular complexes are difficult to unveil experimentally due to their inherent conformational heterogeneity. Computational approaches, alone or in combination with experimental data, have emerged as unavoidable tools to understand the functional mechanisms of this elusive type of assemblies. The level of description used, all-atom or coarse-grained, strongly depends on the size of the molecular systems and on the timescale of the investigated mechanism. In this mini-review, we describe the most relevant architectures found for molecular interactions involving IDPs/IDRs and the computational strategies applied for their investigation.

Halting the Spread of Herpes Simplex Virus-1: The Discovery of an Effective Dual αvβ6/αvβ8 Integrin Ligand

Over recent years, αvβ6 and αvβ8 Arg-Gly-Asp (RGD) integrins have risen to prominence as interchangeable co-receptors for the cellular entry of herpes simplex virus-1 (HSV-1). In fact, the employment of subtype-specific integrin-neutralizing antibodies or gene-silencing siRNAs has emerged as a valuable strategy for impairing HSV infectivity. Here, we shift the focus to a more affordable pharmaceutical approach based on small RGD-containing cyclic pentapeptides. Starting from our recently developed αvβ6-preferential peptide [RGD-Chg-E]-CONH2 (1), a small library of N-methylated derivatives (2-6) was indeed synthesized in the attempt to increase its affinity toward αvβ8. Among the novel compounds, [RGD-Chg-(NMe)E]-CONH2 (6) turned out to be a potent αvβ6/αvβ8 binder and a promising inhibitor of HSV entry through an integrin-dependent mechanism. Furthermore, the renewed selectivity profile of 6 was fully rationalized by a NMR/molecular modeling combined approach, providing novel valuable hints for the design of RGD integrin ligands with the desired specificity profile.

Reconciling Simulations and Experiments With BICePs: A Review

Bayesian Inference of Conformational Populations (BICePs) is an algorithm developed to reconcile simulated ensembles with sparse experimental measurements. The Bayesian framework of BICePs enables population reweighting as a post-simulation processing step, with several advantages over existing methods, including the proper use of reference potentials, and the estimation of a Bayes factor-like quantity called the BICePs score for model selection. Here, we summarize the theory underlying this method in context with related algorithms, review the history of BICePs applications to date, and discuss current shortcomings along with future plans for improvement.

Consistent Force Field Captures Homologue-Resolved HP1 Phase Separation

Many proteins have been shown to function via liquid-liquid phase separation. Computational modeling could offer much needed structural details of protein condensates and reveal the set of molecular interactions that dictate their stability. However, the presence of both ordered and disordered domains in these proteins places a high demand on the model accuracy. Here, we present an algorithm to derive a coarse-grained force field, MOFF, which can model both ordered and disordered proteins with consistent accuracy. It combines maximum entropy biasing, least-squares fitting, and basic principles of energy landscape theory to ensure that MOFF recreates experimental radii of gyration while predicting the folded structures for globular proteins with lower energy. The theta temperature determined from MOFF separates ordered and disordered proteins at 300 K and exhibits a strikingly linear relationship with amino acid sequence composition. We further applied MOFF to study the phase behavior of HP1, an essential protein for post-translational modification and spatial organization of chromatin. The force field successfully resolved the structural difference of two HP1 homologues despite their high sequence similarity. We carried out large-scale simulations with hundreds of proteins to determine the critical temperature of phase separation and uncover multivalent interactions that stabilize higher-order assemblies. In all, our work makes significant methodological strides to connect theories of ordered and disordered proteins and provides a powerful tool for studying liquid-liquid phase separation with near-atomistic details.

Full structural ensembles of intrinsically disordered proteins from unbiased molecular dynamics simulations

Molecular dynamics (MD) simulation is widely used to complement ensemble-averaged experiments of intrinsically disordered proteins (IDPs). However, MD often suffers from limitations of inaccuracy. Here, we show that enhancing the sampling using Hamiltonian replica-exchange MD (HREMD) led to unbiased and accurate ensembles, reproducing small-angle scattering and NMR chemical shift experiments, for three IDPs of varying sequence properties using two recently optimized force fields, indicating the general applicability of HREMD for IDPs. We further demonstrate that, unlike HREMD, standard MD can reproduce experimental NMR chemical shifts, but not small-angle scattering data, suggesting chemical shifts are insufficient for testing the validity of IDP ensembles. Surprisingly, we reveal that despite differences in their sequence, the inter-chain statistics of all three IDPs are similar for short contour lengths (< 10 residues). The results suggest that the major hurdle of generating an accurate unbiased ensemble for IDPs has now been largely overcome.

Experimentally Consistent Simulation of Aβ21-30 Peptides with a Minimal NMR Bias

Misfolded amyloid peptides are neurotoxic molecules associated with Alzheimer's disease. The Aβ_21-30 peptide fragment is a decapeptide fragment of the complete Aβ42 peptide which is a hypothesized cause of Alzheimer's disease via amyloid fibrillogenesis. Aβ_21-30 is investigated here with a combination of NMR (nuclear magnetic resonance) spectroscopy experiments and molecular dynamics simulations with experiment directed simulation (EDS). EDS is a maximum entropy biasing method that augments a molecular dynamics simulation with experimental data (NMR chemical shifts) to improve agreement with experiments and thus accuracy. EDS molecular dynamics shows that the Aβ_21-30 monomer has a β turn stabilized by the following interactions: S26-K28, D23-S26, and D23-K28. NMR, total correlation spectroscopy, and rotating frame Overhauser effect spectroscopy experiments provide independent agreement. Subsequent two- and four-monomer EDS simulations show aggregation. Diffusion coefficients calculated from molecular simulation also agreed with experimentally measured values only after using EDS, providing independent assessment of accuracy. This work demonstrates how accuracy can be improved by directly using experimental data in molecular dynamics of complex processes like self-assembly.

Data-Driven Molecular Dynamics: A Multifaceted Challenge

The big data concept is currently revolutionizing several fields of science including drug discovery and development. While opening up new perspectives for better drug design and related strategies, big data analysis strongly challenges our current ability to manage and exploit an extraordinarily large and possibly diverse amount of information. The recent renewal of machine learning (ML)-based algorithms is key in providing the proper framework for addressing this issue. In this respect, the impact on the exploitation of molecular dynamics (MD) simulations, which have recently reached mainstream status in computational drug discovery, can be remarkable. Here, we review the recent progress in the use of ML methods coupled to biomolecular simulations with potentially relevant implications for drug design. Specifically, we show how different ML-based strategies can be applied to the outcome of MD simulations for gaining knowledge and enhancing sampling. Finally, we discuss how intrinsic limitations of MD in accurately modeling biomolecular systems can be alleviated by including information coming from experimental data.

Multi-state recognition pathway of the intrinsically disordered protein kinase inhibitor by protein kinase A

In the nucleus, the spatiotemporal regulation of the catalytic subunit of cAMP-dependent protein kinase A (PKA-C) is orchestrated by an intrinsically disordered protein kinase inhibitor, PKI, which recruits the CRM1/RanGTP nuclear exporting complex. How the PKA-C/PKI complex assembles and recognizes CRM1/RanGTP is not well understood. Using NMR, SAXS, fluorescence, metadynamics, and Markov model analysis, we determined the multi-state recognition pathway for PKI. After a fast binding step in which PKA-C selects PKI's most competent conformations, PKI folds upon binding through a slow conformational rearrangement within the enzyme's binding pocket. The high-affinity and pseudo-substrate regions of PKI become more structured and the transient interactions with the kinase augment the helical content of the nuclear export sequence, which is then poised to recruit the CRM1/RanGTP complex for nuclear translocation. The multistate binding mechanism featured by PKA-C/PKI complex represents a paradigm on how disordered, ancillary proteins (or protein domains) are able to operate multiple functions such as inhibiting the kinase while recruiting other regulatory proteins for nuclear export.

Bayesian inference of chromatin structure ensembles from population-averaged contact data

Mounting experimental evidence suggests a role for the spatial organization of chromatin in crucial processes of the cell nucleus such as transcription regulation. Chromosome conformation capture techniques allow us to characterize chromatin structure by mapping contacts between chromosomal loci on a genome-wide scale. The most widespread modality is to measure contact frequencies averaged over a population of cells. Single-cell variants exist, but suffer from low contact numbers and have not yet gained the same resolution as population methods. While intriguing biological insights have already been garnered from ensemble-averaged data, information about three-dimensional (3D) genome organization in the underlying individual cells remains largely obscured because the contact maps show only an average over a huge population of cells. Moreover, computational methods for structure modeling of chromatin have mostly focused on fitting a single consensus structure, thereby ignoring any cell-to-cell variability in the model itself. Here, we propose a fully Bayesian method to infer ensembles of chromatin structures and to determine the optimal number of states in a principled, objective way. We illustrate our approach on simulated data and compute multistate models of chromatin from chromosome conformation capture carbon copy (5C) data. Comparison with independent data suggests that the inferred ensembles represent the underlying sample population faithfully. Harnessing the rich information contained in multistate models, we investigate cell-to-cell variability of chromatin organization into topologically associating domains, thus highlighting the ability of our approach to deliver insights into chromatin organization of great biological relevance.

Small-Angle X-ray Scattering Curves of Detergent Micelles: Effects of Asymmetry, Shape Fluctuations, Disorder, and Atomic Details

Small-angle X-ray scattering (SAXS) is a widely used experimental technique, providing structural and dynamic insight into soft-matter complexes and biomolecules under near-native conditions. However, interpreting the one-dimensional scattering profiles in terms of three-dimensional structures and ensembles remains challenging, partly because it is poorly understood how structural information is encoded along the measured scattering angle. We combined all-atom SAXS-restrained ensemble simulations, simplified continuum models, and SAXS experiments of a n-dodecyl-β-d-maltoside (DDM) micelle to decipher the effects of model asymmetry, shape fluctuations, atomic disorder, and atomic details on SAXS curves. Upon interpreting the small-angle regime, we find remarkable agreement between (i) a two-component triaxial ellipsoid model fitted against the data and (ii) a SAXS-refined all-atom ensemble. However, continuum models fail at wider angles, even if they account for shape fluctuations, disorder, and asymmetry of the micelle. We conclude that modeling atomic details is mandatory for explaining SAXS curves at wider angles.