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As computational modeling, simulation, and predictions are becoming integral parts of biomedical pipelines, it behooves us to emphasize the reliability of the computational protocol. For any reported quantity of interest (QOI), one must also compute and report a measure of the uncertainty or error associated with the QOI. This is especially important in molecular modeling, since in most practical applications the inputs to the computational protocol are often noisy, incomplete, or low-resolution. Unfortunately, currently available modeling tools do not account for uncertainties and their effect on the final QOIs with sufficient rigor. We have developed a statistical framework that expresses the uncertainty of the QOI as the probability that the reported value deviates from the true value by more than some user-defined threshold. First, we provide a theoretical approach where this probability can be bounded using Azuma-Hoeffding like inequalities. Second, we approximate this probability empirically by sampling the space of uncertainties of the input and provide applications of our framework to bound uncertainties of several QOIs commonly used in molecular modeling. Finally, we also present several visualization techniques to effectively and quantitavely visualize the uncertainties: in the input, final QOIs, and also intermediate states.

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Most of the existing research in assembly pathway prediction/analysis of viral capsids makes the simplifying assumption that the configuration of the intermediate states can be extracted directly from the final configuration of the entire capsid. This assumption does not take into account the conformational changes of the constituent proteins as well as minor changes to the binding interfaces that continue throughout the assembly process until stabilization. This article presents a statistical-ensemble-based approach that samples the configurational space for each monomer with the relative local orientation between monomers, to capture the uncertainties in binding and conformations. Further, instead of using larger capsomers (trimers, pentamers) as building blocks, we allow all possible subassemblies to bind in all possible combinations. We represent the resulting assembly graph in two different ways: First, we use the Wilcoxon signed-rank measure to compare the distributions of binding free energy computed on the sampled conformations to predict likely pathways. Second, we represent chemical equilibrium aspects of the transitions as a Bayesian Factor graph where both associations and dissociations are modeled based on concentrations and the binding free energies. We applied these protocols on the feline panleukopenia virus and the Nudaurelia capensis virus. Results from these experiments showed a significant departure from those that one would obtain if only the static configurations of the proteins were considered. Hence, we establish the importance of an uncertainty-aware protocol for pathway analysis, and we provide a statistical framework as an important first step toward assembly pathway prediction with high statistical confidence.

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We present the data used for an integrative approach to computational modeling of proteins with large variable domains, specifically applied in this context to model HIV Env glycoprotein gp120 in its CD4 and 17b bound state. The initial data involved X-ray structure PDBID:1GC1 and electron microscopy image EMD:5020. Other existing X-ray structures were used as controls to validate and hierarchically refine partial and complete computational models. A summary of the experiment protocol and data was published (Rasheed et al., 2015) [26], along with detailed analysis of the final model (PDBID:3J70) and its implications.

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As computational modeling, simulation, and predictions are becoming integral parts of biomedical pipelines, it behooves us to emphasize the reliability of the computational protocol. For any reported quantity of interest (QOI), one must also compute and report a measure of the uncertainty or error associated with the QOI. This is especially important in molecular modeling, since in most practical applications the inputs to the computational protocol are often noisy, incomplete, or low-resolution. Unfortunately, currently available modeling tools do not account for uncertainties and their effect on the final QOIs with sufficient rigor. We have developed a statistical framework that expresses the uncertainty of the QOI as the probability that the reported value deviates from the true value by more than some user-defined threshold. First, we provide a theoretical approach where this probability can be bounded using Azuma-Hoeffding like inequalities. Second, we approximate this probability empirically by sampling the space of uncertainties of the input and provide applications of our framework to bound uncertainties of several QOIs commonly used in molecular modeling. Finally, we also present several visualization techniques to effectively and quantitavely visualize the uncertainties: in the input, final QOIs, and also intermediate states.

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There continue to be increasing occurrences of both atomistic structure models in the PDB (possibly reconstructed from X-ray diffraction or NMR data), and 3D reconstructed cryo-electron microscopy (3D EM) maps (albeit at coarser resolution) of the same or homologous molecule or molecular assembly, deposited in the EMDB. To obtain the best possible structural model of the molecule at the best achievable resolution, and without any missing gaps, one typically aligns (match and fits) the atomistic structure model with the 3D EM map. We discuss a new algorithm and generalized framework, named PF(2) fit (Polar Fast Fourier Fitting) for the best possible structural alignment of atomistic structures with 3D EM. While PF(2) fit enables only a rigid, six dimensional (6D) alignment method, it augments prior work on 6D X-ray structure and 3D EM alignment in multiple ways: Scoring. PF(2) fit includes a new scoring scheme that, in addition to rewarding overlaps between the volumes occupied by the atomistic structure and 3D EM map, rewards overlaps between the volumes complementary to them. We quantitatively demonstrate how this new complementary scoring scheme improves upon existing approaches. PF(2) fit also includes two scoring functions, the non-uniform exterior penalty and the skeleton-secondary structure score, and implements the scattering potential score as an alternative to traditional Gaussian blurring. Search. PF(2) fit utilizes a fast polar Fourier search scheme, whose main advantage is the ability to search over uniformly and adaptively sampled subsets of the space of rigid-body motions. PF(2) fit also implements a new reranking search and scoring methodology that considerably improves alignment metrics in results obtained from the initial search.

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Envelope glycoprotein gp120 of HIV-1 possesses several variable regions; their precise structure has been difficult to establish. We report a new model of gp120, in complex with antibodies CD4 and 17b, complete with its variable regions. The model was produced by a computational protocol that uses cryo-electron microscopy (EM) maps, atomic-resolution structures of the core, and information on binding interactions. Our model has excellent fit with EMD: 5020, is stereochemically and energetically favorable, and has the expected binding interfaces. Comparison of the ternary arrangement of the loops in this model with those bound to PGT122 and PGV04 suggested a possible motion of the V1V2 away from the CCR5 binding site and toward CD4. Our study also revealed that the CD4-bound state of the V1V2 loop is not optimal for gp120 bound with several neutralizing antibodies.

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We describe the generation of all possible shell and dome shapes that can be uniquely meshed (tiled) using a single type of mesh face (tile), and following a single meshing (tiling) rule that governs the mesh (tile) arrangement with maximal vertex, edge and face symmetries. Such tiling arrangements or congruently tiled meshed shapes, are frequently found in chemical forms (fullerenes or Bucky balls, crystals, quasi-crystals, virus nano shells or capsids), and synthetic shapes (cages, sports domes, modern architectural facades). Congruently tiled meshes are both aesthetic and complete, as they support maximal mesh symmetries with minimal complexity and possess simple generation rules. Here, we generate congruent tilings and meshed shape layouts that satisfy these optimality conditions. Further, the congruent meshes are uniquely mappable to an almost regular 3D polyhedron (or its dual polyhedron) and which exhibits face-transitive (and edge-transitive) congruency with at most two types of vertices (each type transitive to the other). The family of all such congruently meshed polyhedra create a new class of meshed shapes, beyond the well-studied regular, semi-regular and quasi-regular classes, and their duals (platonic, Catalan and Johnson). While our new mesh class is infinite, we prove that there exists a unique mesh parametrization, where each member of the class can be represented by two integer lattice variables, and moreover efficiently constructable.

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MOTIVATION: Computational simulation of protein-protein docking can expedite the process of molecular modeling and drug discovery. This paper reports on our new F(2) Dock protocol which improves the state of the art in initial stage rigid body exhaustive docking search, scoring and ranking by introducing improvements in the shape-complementarity and electrostatics affinity functions, a new knowledge-based interface propensity term with FFT formulation, a set of novel knowledge-based filters and finally a solvation energy (GBSA) based reranking technique. Our algorithms are based on highly efficient data structures including the dynamic packing grids and octrees which significantly speed up the computations and also provide guaranteed bounds on approximation error. RESULTS: The improved affinity functions show superior performance compared to their traditional counterparts in finding correct docking poses at higher ranks. We found that the new filters and the GBSA based reranking individually and in combination significantly improve the accuracy of docking predictions with only minor increase in computation time. We compared F(2) Dock 2.0 with ZDock 3.0.2 and found improvements over it, specifically among 176 complexes in ZLab Benchmark 4.0, F(2) Dock 2.0 finds a near-native solution as the top prediction for 22 complexes; where ZDock 3.0.2 does so for 13 complexes. F(2) Dock 2.0 finds a near-native solution within the top 1000 predictions for 106 complexes as opposed to 104 complexes for ZDock 3.0.2. However, there are 17 and 15 complexes where F(2) Dock 2.0 finds a solution but ZDock 3.0.2 does not and vice versa; which indicates that the two docking protocols can also complement each other. AVAILABILITY: The docking protocol has been implemented as a server with a graphical client (TexMol) which allows the user to manage multiple docking jobs, and visualize the docked poses and interfaces. Both the server and client are available for download. Server: http://www.cs.utexas.edu/~bajaj/cvc/software/f2dock.shtml. Client: http://www.cs.utexas.edu/~bajaj/cvc/software/f2dockclient.shtml.

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MOTIVATION: We present the 'Dynamic Packing Grid' (DPG), a neighborhood data structure for maintaining and manipulating flexible molecules and assemblies, for efficient computation of binding affinities in drug design or in molecular dynamics calculations. RESULTS: DPG can efficiently maintain the molecular surface using only linear space and supports quasi-constant time insertion, deletion and movement (i.e. updates) of atoms or groups of atoms. DPG also supports constant time neighborhood queries from arbitrary points. Our results for maintenance of molecular surface and polarization energy computations using DPG exhibit marked improvement in time and space requirements. AVAILABILITY: http://www.cs.utexas.edu/~bajaj/cvc/software/DPG.shtml.

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