Using Markov state models to study self-assembly

Computational Approaches for Organic Semiconductors: From Chemical and Physical Understanding to Predicting New Materials

While a complete understanding of organic semiconductor (OSC) design principles remains elusive, computational methods─ranging from techniques based in classical and quantum mechanics to more recent data-enabled models─can complement experimental observations and provide deep physicochemical insights into OSC structure-processing-property relationships, offering new capabilities for in silico OSC discovery and design. In this Review, we trace the evolution of these computational methods and their application to OSCs, beginning with early quantum-chemical methods to investigate resonance in benzene and building to recent machine-learning (ML) techniques and their application to ever more sophisticated OSC scientific and engineering challenges. Along the way, we highlight the limitations of the methods and how sophisticated physical and mathematical frameworks have been created to overcome those limitations. We illustrate applications of these methods to a range of specific challenges in OSCs derived from π-conjugated polymers and molecules, including predicting charge-carrier transport, modeling chain conformations and bulk morphology, estimating thermomechanical properties, and describing phonons and thermal transport, to name a few. Through these examples, we demonstrate how advances in computational methods accelerate the deployment of OSCsin wide-ranging technologies, such as organic photovoltaics (OPVs), organic light-emitting diodes (OLEDs), organic thermoelectrics, organic batteries, and organic (bio)sensors. We conclude by providing an outlook for the future development of computational techniques to discover and assess the properties of high-performing OSCs with greater accuracy.

Optimization of non-equilibrium self-assembly protocols using Markov state models

The promise of self-assembly to enable the bottom-up formation of materials with prescribed architectures and functions has driven intensive efforts to uncover rational design principles for maximizing the yield of a target structure. Yet, despite many successful examples of self-assembly, ensuring kinetic accessibility of the target structure remains an unsolved problem in many systems. In particular, long-lived kinetic traps can result in assembly times that vastly exceed experimentally accessible timescales. One proposed solution is to design non-equilibrium assembly protocols in which system parameters change over time to avoid such kinetic traps. Here, we develop a framework to combine Markov state model (MSM) analysis with optimal control theory to compute a time-dependent protocol that maximizes the yield of the target structure at a finite time. We present an adjoint-based gradient descent method that, in conjunction with MSMs for a system as a function of its control parameters, enables efficiently optimizing the assembly protocol. We also describe an interpolation approach to significantly reduce the number of simulations required to construct the MSMs. We demonstrate our approach with two examples; a simple semi-analytic model for the folding of a polymer of colloidal particles, and a more complex model for capsid assembly. Our results show that optimizing time-dependent protocols can achieve significant improvements in the yields of selected structures, including equilibrium free energy minima, long-lived metastable structures, and transient states.

Multiscale Modeling of Hepatitis B Virus Capsid Assembly and Its Dimorphism

Hepatitis B virus (HBV) is an endemic, chronic virus that leads to 800000 deaths per year. Central to the HBV lifecycle, the viral core has a protein capsid assembled from many copies of a single protein. The capsid protein adopts different (quasi-equivalent) conformations to form icosahedral capsids containing 180 or 240 proteins: T = 3 or T = 4, respectively, in Caspar-Klug nomenclature. HBV capsid assembly has become an important target for recently developed antivirals; nonetheless, the assembly pathways and mechanisms that control HBV dimorphism remain unclear. We describe computer simulations of the HBV assembly, using a coarse-grained model that has parameters learned from all-atom molecular dynamics simulations of a complete HBV capsid and yet is computationally tractable. Dynamical simulations with the resulting model reproduce experimental observations of HBV assembly pathways and products. By constructing Markov state models and employing transition path theory, we identify pathways leading to T = 3, T = 4, and other experimentally observed capsid morphologies. The analysis shows that capsid polymorphism is promoted by the low HBV capsid bending modulus, where the key factors controlling polymorphism are the conformational energy landscape and protein-protein binding affinities.

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

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

Multiscale Approach for Computing Gated Ligand Binding from Molecular Dynamics and Brownian Dynamics Simulations

We develop an approach to characterize the effects of gating by a multiconformation protein consisting of macrostate conformations that are either accessible or inaccessible to ligand binding. We first construct a Markov state model of the apo-protein from atomistic molecular dynamics simulations from which we identify macrostates and their conformations, compute their relative macrostate populations and interchange kinetics, and structurally characterize them in terms of ligand accessibility. We insert the calculated first-order rate constants for conformational transitions into a multistate gating theory from which we derive a gating factor γ that quantifies the degree of conformational gating. Applied to HIV-1 protease, our approach yields a kinetic network of three accessible (semi-open, open, and wide-open) and two inaccessible (closed and a newly identified, "parted") macrostate conformations. The parted conformation sterically partitions the active site, suggesting a possible role in product release. We find that the binding kinetics of drugs and drug-like inhibitors to HIV-1 protease falls in the slow gating regime. However, because γ = 0.75, conformational gating only modestly slows ligand binding. Brownian dynamics simulations of the diffusional association of eight inhibitors to the protease─having a wide range of experimental association constants (∼10⁴-10¹⁰ M^-1 s^-1)─yields gated rate constants in the range of ∼0.5-5.7 × 10⁸ M^-1 s^-1. This indicates that, whereas the association rate of some inhibitors could be described by the model, for many inhibitors either subsequent conformational transitions or alternate binding mechanisms may be rate-limiting. For systems known to be modulated by conformational gating, the approach could be scaled computationally efficiently to screen association kinetics for a large number of ligands.

Roles of the Coupling Agent and Surfactant in Droplet Structure in Sizing Emulsions: A Molecular Dynamics Simulations Study

Sizing emulsions used as glass fiber surface treatments in composites manufacturing are aqueous suspensions of hydrophobic film formers, surface coupling agents, and surfactants. We employ all-atom molecular dynamics simulations to characterize droplet structures in several aqueous blends of the film-former diglycidyl ether of bisphenol A, coupling agent glycidoxypropyl trimethoxysilane, and a triblock copolymer surfactant (Pluronic L35 PEO/PPO copolymer). We show that the quasi-equilibrium states of emulsion droplets are invariant to different initial configurations. We examine the role of the surfactant in determining coupling agent partitioning between the droplet shell and corona and coupling agent cluster size distributions. This work takes a step toward systematic understanding of the sizing chemistry to optimize the interface between the glass and the resin in commercially relevant composites.

Physics of viral dynamics

Viral capsids are often regarded as inert structural units, but in actuality they display fascinating dynamics during different stages of their life cycle. With the advent of single-particle approaches and high-resolution techniques, it is now possible to scrutinize viral dynamics during and after their assembly and during the subsequent development pathway into infectious viruses. In this Review, the focus is on the dynamical properties of viruses, the different physical virology techniques that are being used to study them, and the physical concepts that have been developed to describe viral dynamics.

Should Virus Capsids Assemble Perfectly? Theory and Observation of Defects

Although published structural models of viral capsids generally exhibit a high degree of regularity or symmetry, structural defects might be expected because of the fluctuating environment in which capsids assemble and the requirement of some capsids for disassembly before genome delivery. Defective structures are observed in computer simulations, and are evident in single-particle cryoelectron microscopy studies. Here, we quantify the conditions under which defects might be expected, using a statistical mechanics model allowing for ideal, defective, and vacant sites. The model displays a threshold in affinity parameters below which there is an appreciable population of defective capsids. Even when defective sites are not allowed, there is generally some population of vacancies. Analysis of single particles in cryoelectron microscopy micrographs yields a confirmatory ≳15% of defective particles. Our findings suggest structural heterogeneity in virus capsids may be under-appreciated, and also points to a nontraditional strategy for assembly inhibition.

Evolution of Intermediates during Capsid Assembly of Hepatitis B Virus with Phenylpropenamide-Based Antivirals

Self-assembly of virus capsids is a potential target for antivirals due to its importance in the virus lifecycle. Here, we investigate the effect of phenylpropenamide derivatives B-21 and AT-130 on the assembly of hepatitis B virus (HBV) core protein. Phenylpropenamides are widely believed to yield assembly of spherical particles resembling native, empty HBV capsids. Because the details of assembly can be overlooked with ensemble measurements, we performed resistive-pulse sensing on nanofluidic devices with four pores in series to characterize the size distributions of the products in real time. With its single particle sensitivity and compatibility with typical assembly buffers, resistive-pulse sensing is well-suited for analyzing virus assembly in vitro. We observed that assembly with B-21 and AT-130 produced a large fraction of partially complete virus particles that may be on-path, off-path, or trapped. For both B-21 and AT-130, capsid assembly was more sensitive to disruption under conditions where the interprotein association energy was low at lower salt concentrations. Dilution of the reaction solutions led to the rearrangement of the incomplete particles and demonstrated that these large intermediates may be on-path, but are labile, and exist in a frustrated dynamic equilibrium. During capsid assembly, phenylpropenamide molecules modestly increase the association energy of dimers, prevent intermediates from dissociating, and lead to kinetic trapping where the formation of too many capsids has been initiated, which results in both empty and incomplete particles.

Universal folding pathways of polyhedron nets

Low-dimensional objects such as molecular strands, ladders, and sheets have intrinsic features that affect their propensity to fold into 3D objects. Understanding this relationship remains a challenge for de novo design of functional structures. Using molecular dynamics simulations, we investigate the refolding of the 24 possible 2D unfoldings ("nets") of the three simplest Platonic shapes and demonstrate that attributes of a net's topology-net compactness and leaves on the cutting graph-correlate with thermodynamic folding propensity. To explain these correlations we exhaustively enumerate the pathways followed by nets during folding and identify a crossover temperature [Formula: see text] below which nets fold via nonnative contacts (bonds must break before the net can fold completely) and above which nets fold via native contacts (newly formed bonds are also present in the folded structure). Folding above [Formula: see text] shows a universal balance between reduction of entropy via the elimination of internal degrees of freedom when bonds are formed and gain in potential energy via local, cooperative edge binding. Exploiting this universality, we devised a numerical method to efficiently compute all high-temperature folding pathways for any net, allowing us to predict, among the combined 86,760 nets for the remaining Platonic solids, those with highest folding propensity. Our results provide a general heuristic for the design of 2D objects to stochastically fold into target 3D geometries and suggest a mechanism by which geometry and folding propensity are related above [Formula: see text], where native bonds dominate folding.

The role of multivalency in the association kinetics of patchy particle complexes

Association and dissociation of particles are elementary steps in many natural and technological relevant processes. For many such processes, the presence of multiple binding sites is essential. For instance, protein complexes and regular structures such as virus shells are formed from elementary building blocks with multiple binding sites. Here we address a fundamental question concerning the role of multivalency of binding sites in the association kinetics of such complexes. Using single replica transition interface sampling simulations, we investigate the influence of the multivalency on the binding kinetics and the association mechanism of patchy particles that form polyhedral clusters. When the individual bond strength is fixed, the kinetics naturally is very dependent on the multivalency, with dissociation rate constants exponentially decreasing with the number of bonds. In contrast, we find that when the total bond energy per particle is kept constant, association and dissociation rate constants turn out rather independent of multivalency, although of course still very dependent on the total energy. The association and dissociation mechanisms, however, depend on the presence and nature of the intermediate states. For instance, pathways that visit intermediate states are less prevalent for particles with five binding sites compared to the case of particles with only three bonds. The presence of intermediate states can lead to kinetic trapping and malformed aggregates. We discuss implications for natural forming complexes such as virus shells and for the design of artificial colloidal patchy particles.

Improving the productivity of monodisperse polyhedral cages by the rational design of kinetic self-assembly pathways

Hollow polyhedral cages hold great potential for application in nanotechnological and biomedical fields. Understanding the formation mechanism of these self-assembled structures could provide guidance for the rational design of the desired polyhedral cages. Here, by constructing kinetic network models from extensive coarse-grained molecular dynamics simulations, we elucidated the formation mechanism of the dodecahedral cage, which is formed by the self-assembly of patchy particles. We found that the dodecahedral cage is formed through increasing the aggregate size followed by structure rearrangement. Based on this mechanistic understanding, we improved the productivity of the dodecahedral cage through the rational design of the patch arrangement of patchy particles, which promotes the structural rearrangement process. Our results demonstrate that it should be a feasible strategy to achieve the rational design of the desired nanostructures via the kinetic analysis. We anticipate that this methodology could be extended to other self-assembly systems for the fabrication of functional nanomaterials.

Harnessing complexity in molecular self-assembly using computer simulations

In molecular self-assembly, hundreds of thousands of freely-diffusing molecules associate to form ordered and functional architectures in the absence of an actuator. This intriguing phenomenon plays a critical role in biology and has become a powerful tool for the fabrication of advanced nanomaterials. Due to the limited spatial and temporal resolutions of current experimental techniques, computer simulations offer a complementary strategy to explore self-assembly with atomic resolution. Here, we review recent computational studies focusing on both thermodynamic and kinetic aspects. As we shall see, thermodynamic approaches based on modeling and statistical mechanics offer initial guidelines to design nanostructures with modest computational effort. Computationally more intensive analyses based on molecular dynamics simulations and kinetic network models (KNMs) reach beyond it, opening the door to the rational design of self-assembly pathways. Current limitations of these methodologies are discussed. We anticipate that the synergistic use of thermodynamic and kinetic analyses based on computer simulations will provide an important contribution to the de novo design of self-assembly.

Elucidation of the conformational dynamics of multi-body systems by construction of Markov state models

Constructing Markov State Models (MSMs) based on short molecular dynamics simulations is a powerful computational technique to complement experiments in predicting long-time kinetics of biomolecular processes at atomic resolution. Even though the MSM approach has been widely applied to study one-body processes such as protein folding and enzyme conformational changes, the majority of biological processes, e.g. protein-ligand recognition, signal transduction, and protein aggregation, essentially involve multiple entities. Here we review the attempts at constructing MSMs for multi-body systems, point out the challenges therein and discuss recent algorithmic progresses that alleviate these challenges. In particular, we describe an automatic kinetics based partitioning method that achieves optimal definition of the conformational states in a multi-body system, and discuss a novel maximum-likelihood approach that efficiently estimates the slow uphill kinetics utilizing pre-computed equilibrium populations of all states. We expect that these new algorithms and their combinations may boost investigations of important multi-body biological processes via the efficient construction of MSMs.

Kinetics-Controlled Amphiphile Self-Assembly Processes

Amphiphile self-assembly is an essential bottom-up approach of fabricating advanced functional materials. Self-assembled materials with desired structures are often obtained through thermodynamic control. Here, we demonstrate that the selection of kinetic pathways can lead to drastically different self-assembled structures, underlining the significance of kinetic control in self-assembly. By constructing kinetic network models from large-scale molecular dynamics simulations, we show that two largely similar amphiphiles, 1-[11-oxo-11-(pyren-1-ylmethoxy)-undecyl]pyridinium bromide (PYR) and 1-(11-((5a1,8a-dihydropyren-1-yl)methylamino)-11-oxoundecyl)pyridinium bromide (PYN), prefer distinct kinetic assembly pathways. While PYR prefers an incremental growth mechanism and forms a nanotube, PYN favors a hopping growth pathway leading to a vesicle. Such preference was found to originate from the subtle difference in the distributions of hydrophobic and hydrophilic groups in their chemical structures, which leads to different rates of the adhesion process among the aggregating micelles. Our results are in good agreement with experimental results, and accentuate the role of kinetics in the rational design of amphiphile self-assembly.

Chemical and entropic control on the molecular self-assembly process

Molecular self-assembly refers to the spontaneous assembly of molecules into larger structures. In order to exploit molecular self-assembly for the bottom-up synthesis of nanomaterials, the effects of chemical control (strength of the directionality in the intermolecular interaction) and entropic control (temperature) on the self-assembly process should be clarified. Here we present a theoretical methodology that unambiguously distinguishes the effects of chemical and entropic control on the self-assembly of molecules adsorbed to metal surfaces. While chemical control simply increases the formation probability of ordered structures, entropic control induces a variety of effects. These effects range from fine structure modulation of ordered structures, through to degrading large, amorphous structures into short, chain-shaped structures. Counterintuitively, the latter effect shows that entropic control can improve molecular ordering. By identifying appropriate levels of chemical and entropic control, our methodology can, therefore, identify strategies for optimizing the yield of desired nanostructures from the molecular self-assembly process.

Reaction-diffusion basis of retroviral infectivity

Retrovirus particle (virion) infectivity requires diffusion and clustering of multiple transmembrane envelope proteins (Env₃) on the virion exterior, yet is triggered by protease-dependent degradation of a partially occluding, membrane-bound Gag polyprotein lattice on the virion interior. The physical mechanism underlying such coupling is unclear and only indirectly accessible via experiment. Modelling stands to provide insight but the required spatio-temporal range far exceeds current accessibility by all-atom or even coarse-grained molecular dynamics simulations. Nor do such approaches account for chemical reactions, while conversely, reaction kinetics approaches handle neither diffusion nor clustering. Here, a recently developed multiscale approach is considered that applies an ultra-coarse-graining scheme to treat entire proteins at near-single particle resolution, but which also couples chemical reactions with diffusion and interactions. A model is developed of Env₃ molecules embedded in a truncated Gag lattice composed of membrane-bound matrix proteins linked to capsid subunits, with freely diffusing protease molecules. Simulations suggest that in the presence of Gag but in the absence of lateral lattice-forming interactions, Env₃ diffuses comparably to Gag-absent Env₃ Initial immobility of Env₃ is conferred through lateral caging by matrix trimers vertically coupled to the underlying hexameric capsid layer. Gag cleavage by protease vertically decouples the matrix and capsid layers, induces both matrix and Env₃ diffusion, and permits Env₃ clustering. Spreading across the entire membrane surface reduces crowding, in turn, enhancing the effect and promoting infectivity.This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.

Nanoscale Engineering of Designer Cellulosomes

Biocatalysts showcase the upper limit obtainable for high-speed molecular processing and transformation. Efforts to engineer functionality in synthetic nanostructured materials are guided by the increasing knowledge of evolving architectures, which enable controlled molecular motion and precise molecular recognition. The cellulosome is a biological nanomachine, which, as a fundamental component of the plant-digestion machinery from bacterial cells, has a key potential role in the successful development of environmentally-friendly processes to produce biofuels and fine chemicals from the breakdown of biomass waste. Here, the progress toward so-called "designer cellulosomes", which provide an elegant alternative to enzyme cocktails for lignocellulose breakdown, is reviewed. Particular attention is paid to rational design via computational modeling coupled with nanoscale characterization and engineering tools. Remaining challenges and potential routes to industrial application are put forward.

State-space reduction and equivalence class sampling for a molecular self-assembly model

Direct simulation of a model with a large state space will generate enormous volumes of data, much of which is not relevant to the questions under study. In this paper, we consider a molecular self-assembly model as a typical example of a large state-space model, and present a method for selectively retrieving 'target information' from this model. This method partitions the state space into equivalence classes, as identified by an appropriate equivalence relation. The set of equivalence classes H, which serves as a reduced state space, contains none of the superfluous information of the original model. After construction and characterization of a Markov chain with state space H, the target information is efficiently retrieved via Markov chain Monte Carlo sampling. This approach represents a new breed of simulation techniques which are highly optimized for studying molecular self-assembly and, moreover, serves as a valuable guideline for analysis of other large state-space models.

Conserve Water: A Method for the Analysis of Solvent in Molecular Dynamics

Molecular dynamics with explicit solvent is favored for its ability to more correctly simulate aqueous biological processes and has become routine thanks to increasingly powerful computational resources. However, analysis techniques including Markov state models (MSMs) ignore solvent atoms and focus solely on solute coordinates despite solvent being implicated in myriad biological phenomena. We present a unified framework called "solvent-shells featurization" for including solvent degrees of freedom in analysis and show that this method produces better models. We apply this method to simulations of dewetting in the two-domain protein BphC to generate a predictive MSM and identify functional water molecules. Furthermore, the proposed methodology could be easily extended for building MSMs of any systems with indistinguishable components.

Allosteric Control of Icosahedral Capsid Assembly

During the life cycle of a virus, viral proteins and other components self-assemble to form an ordered protein shell called a capsid. This assembly process is subject to multiple competing constraints, including the need to form a thermostable shell while avoiding kinetic traps. It has been proposed that viral assembly satisfies these constraints through allosteric regulation, including the interconversion of capsid proteins among conformations with different propensities for assembly. In this article, we use computational and theoretical modeling to explore how such allostery affects the assembly of icosahedral shells. We simulate assembly under a wide range of protein concentrations, protein binding affinities, and two different mechanisms of allosteric control. We find that above a threshold strength of allosteric control, assembly becomes robust over a broad range of subunit binding affinities and concentrations, allowing the formation of highly thermostable capsids. Our results suggest that allostery can significantly shift the range of protein binding affinities that lead to successful assembly and thus should be taken into account in models that are used to estimate interaction parameters from experimental data.

Advances in the Simulation of Protein Aggregation at the Atomistic Scale

Protein aggregation into highly structured amyloid fibrils is associated with various diseases including Alzheimer's disease, Parkinson's disease, and type II diabetes. Amyloids can also have normal biological functions and, in the future, could be used as the basis for novel nanoscale materials. However, a full understanding of the physicochemical forces that drive protein aggregation is still lacking. Such understanding is crucial for the development of drugs that can effectively inhibit aberrant amyloid aggregation and for the directed design of functional amyloids. Atomistic simulations can help understand protein aggregation. In particular, atomistic simulations can be used to study the initial formation of toxic oligomers which are hard to characterize experimentally and to understand the difference in aggregation behavior between different amyloidogenic peptides. Here, we review the latest atomistic simulations of protein aggregation, concentrating on amyloidogenic protein fragments, and provide an outlook for the future in this field.

Elucidating dominant pathways of the nano-particle self-assembly process

Self-assembly processes play a key role in the fabrication of functional nano-structures with wide application in drug delivery and micro-reactors.

Tabulation as a high-resolution alternative to coarse-graining protein interactions: Initial application to virus capsid subunits

Traditional coarse-graining based on a reduced number of interaction sites often entails a significant sacrifice of chemical accuracy. As an alternative, we present a method for simulating large systems composed of interacting macromolecules using an energy tabulation strategy previously devised for small rigid molecules or molecular fragments [S. Lettieri and D. M. Zuckerman, J. Comput. Chem. 33, 268-275 (2012); J. Spiriti and D. M. Zuckerman, J. Chem. Theory Comput. 10, 5161-5177 (2014)]. We treat proteins as rigid and construct distance and orientation-dependent tables of the interaction energy between them. Arbitrarily detailed interactions may be incorporated into the tables, but as a proof-of-principle, we tabulate a simple α-carbon Gō-like model for interactions between dimeric subunits of the hepatitis B viral capsid. This model is significantly more structurally realistic than previous models used in capsid assembly studies. We are able to increase the speed of Monte Carlo simulations by a factor of up to 6700 compared to simulations without tables, with only minimal further loss in accuracy. To obtain further enhancement of sampling, we combine tabulation with the weighted ensemble (WE) method, in which multiple parallel simulations are occasionally replicated or pruned in order to sample targeted regions of a reaction coordinate space. In the initial study reported here, WE is able to yield pathways of the final ∼25% of the assembly process.

PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models

Markov (state) models (MSMs) and related models of molecular kinetics have recently received a surge of interest as they can systematically reconcile simulation data from either a few long or many short simulations and allow us to analyze the essential metastable structures, thermodynamics, and kinetics of the molecular system under investigation. However, the estimation, validation, and analysis of such models is far from trivial and involves sophisticated and often numerically sensitive methods. In this work we present the open-source Python package PyEMMA ( http://pyemma.org ) that provides accurate and efficient algorithms for kinetic model construction. PyEMMA can read all common molecular dynamics data formats, helps in the selection of input features, provides easy access to dimension reduction algorithms such as principal component analysis (PCA) and time-lagged independent component analysis (TICA) and clustering algorithms such as k-means, and contains estimators for MSMs, hidden Markov models, and several other models. Systematic model validation and error calculation methods are provided. PyEMMA offers a wealth of analysis functions such that the user can conveniently compute molecular observables of interest. We have derived a systematic and accurate way to coarse-grain MSMs to few states and to illustrate the structures of the metastable states of the system. Plotting functions to produce a manuscript-ready presentation of the results are available. In this work, we demonstrate the features of the software and show new methodological concepts and results produced by PyEMMA.

Mechanisms of virus assembly

Viruses are nanoscale entities containing a nucleic acid genome encased in a protein shell called a capsid and in some cases are surrounded by a lipid bilayer membrane. This review summarizes the physics that govern the processes by which capsids assemble within their host cells and in vitro. We describe the thermodynamics and kinetics for the assembly of protein subunits into icosahedral capsid shells and how these are modified in cases in which the capsid assembles around a nucleic acid or on a lipid bilayer. We present experimental and theoretical techniques used to characterize capsid assembly, and we highlight aspects of virus assembly that are likely to receive significant attention in the near future.

Biomolecular engineering of virus-like particles aided by computational chemistry methods

Multi-scale investigation of VLP self-assembly aided by computational methods is facilitating the design, redesign, and modification of functionalized VLPs.

Automatic state partitioning for multibody systems (APM): an efficient algorithm for constructing Markov state models to elucidate conformational dynamics of multibody systems

The conformational dynamics of multibody systems plays crucial roles in many important problems. Markov state models (MSMs) are powerful kinetic network models that can predict long-time-scale dynamics using many short molecular dynamics simulations. Although MSMs have been successfully applied to conformational changes of individual proteins, the analysis of multibody systems is still a challenge because of the complexity of the dynamics that occur on a mixture of drastically different time scales. In this work, we have developed a new algorithm, automatic state partitioning for multibody systems (APM), for constructing MSMs to elucidate the conformational dynamics of multibody systems. The APM algorithm effectively addresses different time scales in the multibody systems by directly incorporating dynamics into geometric clustering when identifying the metastable conformational states. We have applied the APM algorithm to a 2D potential that can mimic a protein-ligand binding system and the aggregation of two hydrophobic particles in water and have shown that it can yield tremendous enhancements in the computational efficiency of MSM construction and the accuracy of the models.

Pathways for virus assembly around nucleic acids

Understanding the pathways by which viral capsid proteins assemble around their genomes could identify key intermediates as potential drug targets. In this work, we use computer simulations to characterize assembly over a wide range of capsid protein-protein interaction strengths and solution ionic strengths. We find that assembly pathways can be categorized into two classes, in which intermediates are either predominantly ordered or disordered. Our results suggest that estimating the protein-protein and the protein-genome binding affinities may be sufficient to predict which pathway occurs. Furthermore, the calculated phase diagrams suggest that knowledge of the dominant assembly pathway and its relationship to control parameters could identify optimal strategies to thwart or redirect assembly to block infection. Finally, analysis of simulation trajectories suggests that the two classes of assembly pathways can be distinguished in single-molecule fluorescence correlation spectroscopy or bulk time-resolved small-angle X-ray scattering experiments.