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

Background: Principal component analysis (PCA) is commonly applied to the atomic trajectories of biopolymers to extract essential dynamics that describe biologically relevant motions. Although application of PCA is straightforward, specialized software to facilitate workflows and analysis of molecular dynamics simulation data to fully harness the power of PCA is lacking. The Java Essential Dynamics inspector (JEDi) software is a major upgrade from the previous JED software., Results: Employing multi-threading, JEDi features a user-friendly interface to control rapid workflows for interrogating conformational motions of biopolymers at various spatial resolutions and within subregions, including multiple chain proteins. JEDi has options for Cartesian-based coordinates (cPCA) and internal distance pair coordinates (dpPCA) to construct covariance (Q), correlation (R), and partial correlation (P) matrices. Shrinkage and outlier thresholding are implemented for the accurate estimation of covariance. The effect of rare events is quantified using outlier and inlier filters. Applying sparsity thresholds in statistical models identifies latent correlated motions. Within a hierarchical approach, small-scale atomic motion is first calculated with a separate local cPCA calculation per residue to obtain eigenresidues. Then PCA on the eigenresidues yields rapid and accurate description of large-scale motions. Local cPCA on all residue pairs creates a map of all residue-residue dynamical couplings. Additionally, kernel PCA is implemented. JEDi output gives high quality PNG images by default, with options for text files that include aligned coordinates, several metrics that quantify mobility, PCA modes with their eigenvalues, and displacement vector projections onto the top principal modes. JEDi provides PyMol scripts together with PDB files to visualize individual cPCA modes and the essential dynamics occurring within user-selected time scales. Subspace comparisons performed on the most relevant eigenvectors using several statistical metrics quantify similarity/overlap of high dimensional vector spaces. Free energy landscapes are available for both cPCA and dpPCA., Conclusion: JEDi is a convenient toolkit that applies best practices in multivariate statistics for comparative studies on the essential dynamics of similar biopolymers. JEDi helps identify functional mechanisms through many integrated tools and visual aids for inspecting and quantifying similarity/differences in mobility and dynamic correlations.

Background: Essential Dynamics (ED) is a common application of principal component analysis (PCA) to extract biologically relevant motions from atomic trajectories of proteins. Covariance and correlation based PCA are two common approaches to determine PCA modes (eigenvectors) and their eigenvalues. Protein dynamics can be characterized in terms of Cartesian coordinates or internal distance pairs. In understanding protein dynamics, a comparison of trajectories taken from a set of proteins for similarity assessment provides insight into conserved mechanisms. Comprehensive software is needed to facilitate comparative-analysis with user-friendly features that are rooted in best practices from multivariate statistics., Results: We developed a Java based Essential Dynamics toolkit called JED to compare the ED from multiple protein trajectories. Trajectories from different simulations and different proteins can be pooled for comparative studies. JED implements Cartesian-based coordinates (cPCA) and internal distance pair coordinates (dpPCA) as options to construct covariance (Q) or correlation (R) matrices. Statistical methods are implemented for treating outliers, benchmarking sampling adequacy, characterizing the precision of Q and R, and reporting partial correlations. JED output results as text files that include transformed coordinates for aligned structures, several metrics that quantify protein mobility, PCA modes with their eigenvalues, and displacement vector (DV) projections onto the top principal modes. Pymol scripts together with PDB files allow movies of individual Q- and R-cPCA modes to be visualized, and the essential dynamics occurring within user-selected time scales. Subspaces defined by the top eigenvectors are compared using several statistical metrics to quantify similarity/overlap of high dimensional vector spaces. Free energy landscapes can be generated for both cPCA and dpPCA., Conclusions: JED offers a convenient toolkit that encourages best practices in applying multivariate statistics methods to perform comparative studies of essential dynamics over multiple proteins. For each protein, Cartesian coordinates or internal distance pairs can be employed over the entire structure or user-selected parts to quantify similarity/differences in mobility and correlations in dynamics to develop insight into protein structure/function relationships.

The Distance Constraint Model (DCM) is a computational modeling scheme that uniquely integrates thermodynamic and mechanical descriptions of protein structure. As such, quantitative stability-flexibility relationships (QSFR) that describe the interrelationships of thermodynamics and mechanics can be quickly computed. Using comparative QSFR analyses, we have previously investigated these relationships across a small number of protein orthologs, ranging from two to a dozen [1, 2]. However, our ultimate goal is provide a comprehensive analysis of whole protein families, which requires consideration of many more structures. To that end, we have developed homology modeling and assessment protocols so that we can robustly calculate QSFR properties for proteins without experimentally derived structures. The approach, which is presented here, starts from a large ensemble of potential homology models and uses a clustering algorithm to identify the best models, thus paving the way for a comprehensive QSFR analysis across hundreds of proteins in a protein family.

It has become commonplace to employ principal component analysis to reveal the most important motions in proteins. This method is more commonly known by its acronym, PCA. While most popular molecular dynamics packages inevitably provide PCA tools to analyze protein trajectories, researchers often make inferences of their results without having insight into how to make interpretations, and they are often unaware of limitations and generalizations of such analysis. Here we review best practices for applying standard PCA, describe useful variants, discuss why one may wish to make comparison studies, and describe a set of metrics that make comparisons possible. In practice, one will be forced to make inferences about the essential dynamics of a protein without having the desired amount of samples. Therefore, considerable time is spent on describing how to judge the significance of results, highlighting pitfalls. The topic of PCA is reviewed from the perspective of many practical considerations, and useful recipes are provided.

Previous works have demonstrated that protein rigidity is related to thermodynamic stability, especially under conditions that favor formation of native structure. Mechanical network rigidity properties of a single conformation are efficiently calculated using the integer body-bar Pebble Game (PG) algorithm. However, thermodynamic properties require averaging over many samples from the ensemble of accessible conformations to accurately account for fluctuations in network topology. We have developed a mean field Virtual Pebble Game (VPG) that represents the ensemble of networks by a single effective network. That is, all possible number of distance constraints (or bars) that can form between a pair of rigid bodies is replaced by the average number. The resulting effective network is viewed as having weighted edges, where the weight of an edge quantifies its capacity to absorb degrees of freedom. The VPG is interpreted as a flow problem on this effective network, which eliminates the need to sample. Across a nonredundant dataset of 272 protein structures, we apply the VPG to proteins for the first time. Our results show numerically and visually that the rigidity characterizations of the VPG accurately reflect the ensemble averaged [Formula: see text] properties. This result positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability.

The distance constraint model (DCM) is a unique computational modeling paradigm that integrates mechanical and thermodynamic descriptions of macromolecular structure. That is, network rigidity calculations are used to account for nonadditivity within entropy components, thus restoring the utility of free-energy decomposition. The DCM outputs a large number of structural characterizations that collectively allow for quantified stability-flexibility relationships (QSFR) to be identified. In this review, we describe the theoretical underpinnings of the DCM and introduce several common QSFR metrics. Application of the DCM across protein families highlights the sensitivity within the set of protein structure residue-to-residue couplings. Further, we have developed a perturbation method to identify putative allosteric sites, where large changes in QSFR upon rigidification (mimicking ligand-binding) detect sites likely to invoke allosteric changes.

To clarify the extent structure plays in determining protein dynamics, a comparative study is made using three models that characterize native state dynamics of single domain proteins starting from known structures taken from four distinct SCOP classifications. A geometrical simulation using the framework rigidity optimized dynamics algorithm (FRODA) based on rigid cluster decomposition is compared to the commonly employed elastic network model (specifically the Anisotropic Network Model ANM) and molecular dynamics (MD) simulation. The essential dynamics are quantified by a mode subspace constructed from ANM and a principal component analysis (PCA) on FRODA and MD trajectories. Aggregate conformational ensembles are constructed to provide a basis for quantitative comparisons between FRODA runs using different parameter settings to critically assess how the predictions of essential dynamics depend on a priori arbitrary user-defined distance constraint rules. We established a range of physicality for these parameters. Surprisingly, FRODA maintains greater intra-consistent results than obtained from MD trajectories, comparable to ANM. Additionally, a mode subspace is constructed from PCA on an exemplar set of myoglobin structures from the Protein Data Bank. Significant overlap across the three model subspaces and the experimentally derived subspace is found. While FRODA provides the most robust sampling and characterization of the native basin, all three models give similar dynamical information of a native state, further demonstrating that structure is the key determinant of dynamics., (Copyright © 2011 Elsevier Inc. All rights reserved.)

The Lifson-Roig Model (LRM) and all its variants describe the α-helix to coil transition in terms of additive component-free energies within a free energy decomposition scheme, and these contributions are interpreted through sequence-context dependent nucleation and propagation parameters. Although this phenomenological approach is able to adequately fit experimental data on helix content and heat capacity, the number of required parameters increases dramatically with additional sequence variation. Moreover, due to nonadditive competing microscopic effects that are difficult to disentangle within a LRM, large uncertainties within the parameters emerge. We offer an alternative view that removes the need for sequence-context parameterization by focusing on individual microsopic interactions within a free energy decomposition and explicitly account for nonadditivity in conformational entropy through network rigidity using a Distance Constraint Model (DCM). We apply a LRM and a DCM to previously published experimental heat capacity and helix content data for a series of heterogeneous polypeptides. Both models describe the experimental data well, and the parameters from both models are consistent with prior work. However, the number of DCM parameters is independent of sequence-variability, the parameter values exhibit better transferability, and the helix nucleation is predicted by the DCM explicitly through the nonadditive nature of conformational entropy. The importance of these results is that the DCM offers a system-independent approach for modeling stability within polypeptides and proteins, where the demonstrated accuracy for the α-helix to coil transition over a series of heterogeneous polypeptides described here is one case in point., (Copyright © 2010 Wiley Periodicals, Inc.)

Previously, we employed a Maxwell counting distance constraint model (McDCM) to describe α-helix formation in polypeptides. Unlike classical helix-coil transition theories, the folding mechanism derives from nonadditivity in conformational entropy caused by rigidification of molecular structure as intramolecular cross-linking interactions form along the backbone. For example, when a hydrogen bond forms within a flexible region, both energy and conformational entropy decrease. However, no conformational entropy is lost when the region is already rigid because atomic motions are not constrained further. Unlike classical zipper models, the same mechanism also describes a coil-to-β-hairpin transition. Special topological features of the helix and hairpin structures allow the McDCM to be solved exactly. Taking full advantage of the fact that Maxwell constraint counting is a mean field approximation applied to the distribution of cross-linking interactions, we present an exact transfer matrix method that does not require any special topological feature. Upon application of the model to proteins, cooperativity within the folding transition is yet again appropriately described. Notwithstanding other contributing factors such as the hydrophobic effect, this simple model identifies a universal mechanism for cooperativity within polypeptide and protein-folding transitions, and it elucidates scaling laws describing hydrogen-bond patterns observed in secondary structure. In particular, the native state should have roughly twice as many constraints as there are degrees of freedom in the coil state to ensure high fidelity in two-state folding cooperativity, which is empirically observed., (Copyright © 2011 Biophysical Society. Published by Elsevier Inc. All rights reserved.)

Molecular dynamics (MD) simulation is a natural approach for studying protein dynamics, and coupled with the ideas of multiscale modeling, MD proves to be the gold standard in computational biology to investigate mechanistic details related to protein function. In principle, if MD trajectories are long enough, the ensemble of protein conformations generated allows thermodynamic and kinetic properties to be predicted. We know from experiments that proteins exhibit a high degree of fidelity in function, and that empirical kinetic models are successful in describing kinetics, suggesting that the ensemble of conformations cluster into well-defined thermodynamic states, which are frequently metastable. The experimental evidence suggest that more efficient computational models that retain only essential properties of the protein can be constructed to faithfully reproduce the relatively few observed thermodynamic states, and perhaps describe transition states if the model is sufficiently detailed. Indeed, there are many so-called ensemble-based methods that attempt to generate more complete ensembles than MD can provide by focusing on the most important driving forces through simplified representations of how elements within the protein interact. Although coarse-graining is employed in MD and other approaches, such as in elastic network models, the key distinguishing factor of ensemble-based methods is that they are meant to efficiently generate a large ensemble of conformations without solving explicit equations of motion. This review highlights three types of ensemble-based methods, illustrated by 'COREX' and the Wako-Saito-Munoz-Eaton (WSME) model, the Framework Rigidity Optimized Dynamic Algorithm (FRODA) and the distance constraint model (DCM)., (Copyright © 2010 Elsevier Ltd. All rights reserved.)

A complete description of protein structure and function must include a proper treatment of mechanisms that lead to cooperativity. The helix/coil transition serves as a simple example of a cooperative folding process, commonly described by a nucleation-propagation mechanism. The prevalent view is that coil structure must first form a short segment of helix in a localized region despite paying a free energy cost (nucleation). Afterward, helical structure propagates outward from the nucleation site. Both processes entail enthalpy-entropy compensation that derives from the loss in conformational entropy on helix formation with concomitant gain in favorable interactions. Nucleation-propagation models inherently assume that cooperativity arises from a sequential series of local events. An alternative distance constraint model asserts there is a direct link between available degrees of freedom and cooperativity through the nonadditivity in conformational entropy. That is, helix nucleation is a concerted manifestation of rigidity propagating through atomic structure. The link between network rigidity and nonadditivity of conformational entropy is shown in this study by solving the distance constraint model using a simple global constraint counting approximation. Cooperativity arises from competition between excess and deficiency in available degrees of freedom in the coil and helix states respectively.

Additivity principles in chemistry, biochemistry, and biophysics have been used extensively for decades. Nevertheless, it is well known that additivity frequently breaks down in complex biomacromolecules. Nonadditivity within protein double mutant free energy cycles of spatially close residue pairs is a generally well-understood phenomenon, whereas a robust description of nonadditivity extending over large distances remains to be developed. Here, we test the hypothesis that the mutational effects tend to be nonadditive if two structurally well-separated mutated residues belong to the same rigid cluster within the wild type protein, and additive if they are located within different clusters. We find the hypothesis to be statistically significant with P-values that range from 10(-5) to 10(-6). To the best of our knowledge, this result represents the first demonstration of a statistically significant preponderance for nonadditivity over long distances. These findings provide new insight into the origins of long-range nonadditivity in double mutant cycles, which complements the conventional wisdom that nonadditivity arises in double mutations involving contacting residues. Consequently, these results should have far-reaching implications for a proper understanding of protein stability, structure/function analyses, and protein design., ((c) 2007 Wiley-Liss, Inc.)

Le Châtelier’s principle is the cornerstone of our understanding of chemical equilibria. When a system at equilibrium undergoes a change in concentration or thermodynamic state (i.e., temperature, pressure, etc.), La Châtelier’s principle states that an equilibrium shift will occur to offset the perturbation and a new equilibrium is established. We demonstrate that the effects of stabilizing mutations on the rigidity ⇔ flexibility equilibrium within the native state ensemble manifest themselves through enthalpy-entropy compensation as the protein structure adjusts to restore the global balance between the two. Specifically, we characterize the effects of mutation to single chain fragments of the anti-lymphotoxin-β receptor antibody using a computational Distance Constraint Model. Statistically significant changes in the distribution of both rigidity and flexibility within the molecular structure is typically observed, where the local perturbations often lead to distal shifts in flexibility and rigidity profiles. Nevertheless, the net gain or loss in flexibility of individual mutants can be skewed. Despite all mutants being exclusively stabilizing in this dataset, increased flexibility is slightly more common than increased rigidity. Mechanistically the redistribution of flexibility is largely controlled by changes in the H-bond network. For example, a stabilizing mutation can induce an increase in rigidity locally due to the formation of new H-bonds, and simultaneously break H-bonds elsewhere leading to increased flexibility distant from the mutation site via Le Châtelier. Increased flexibility within the VH β4/β5 loop is a noteworthy illustration of this long-range effect. [ABSTRACT FROM AUTHOR]