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We describe efficient methods for consistently coloring and visualizing collections of rigid cluster decompositions obtained from variations of a protein structure, and lay the foundation for more complex setups, that may involve different computational and experimental methods. The focus here is on three biological applications: the conceptually simpler problems of visualizing results of dilution and mutation analyses, and the more complex task of matching decompositions of multiple Nucleic Magnetic Resonance (NMR) models of the same protein. Implemented into the KINematics And RIgidity (KINARI) web server application, the improved visualization techniques give useful information about protein folding cores, help examining the effect of mutations on protein flexibility and function, and provide insights into the structural motions of Protein Data Bank proteins solved with solution NMR. These tools have been developed with the goal of improving and validating rigidity analysis as a credible coarse-grained model capturing essential information about a protein's slow motions near the native state.

Background: We initiate in silico rigidity-theoretical studies of biological assemblies and small crystals for protein structures. The goal is to determine if, and how, the interactions among neighboring cells and subchains affect the flexibility of a molecule in its crystallized state. We use experimental X-ray crystallography data from the Protein Data Bank (PDB). The analysis relies on an effcient graph-based algorithm. Computational experiments were performed using new protein rigidity analysis tools available in the new release of our KINARI-Web server http://kinari.cs.umass.edu., Results: We provide two types of results: on biological assemblies and on crystals. We found that when only isolated subchains are considered, structural and functional information may be missed. Indeed, the rigidity of biological assemblies is sometimes dependent on the count and placement of hydrogen bonds and other interactions among the individual subchains of the biological unit. Similarly, the rigidity of small crystals may be affected by the interactions between atoms belonging to different unit cells., Conclusion: The rigidity analysis of a single asymmetric unit may not accurately reflect the protein's behavior in the tightly packed crystal environment. Using our KINARI software, we demonstrated that additional functional and rigidity information can be gained by analyzing a protein's biological assembly and/or crystal structure. However, performing a larger scale study would be computationally expensive (due to the size of the molecules involved). Overcoming this limitation will require novel mathematical and computational extensions to our software.

Background: Protein rigidity analysis is an efficient computational method for extracting flexibility information from static, X-ray crystallography protein data. Atoms and bonds are modeled as a mechanical structure and analyzed with a fast graph-based algorithm, producing a decomposition of the flexible molecule into interconnected rigid clusters. The result depends critically on noncovalent atomic interactions, primarily on how hydrogen bonds and hydrophobic interactions are computed and modeled. Ongoing research points to the stringent need for benchmarking rigidity analysis software systems, towards the goal of increasing their accuracy and validating their results, either against each other and against biologically relevant (functional) parameters. We propose two new methods for modeling hydrogen bonds and hydrophobic interactions that more accurately reflect a mechanical model, without being computationally more intensive. We evaluate them using a novel scoring method, based on the B-cubed score from the information retrieval literature, which measures how well two cluster decompositions match., Results: To evaluate the modeling accuracy of KINARI, our pebble-game rigidity analysis system, we use a benchmark data set of 20 proteins, each with multiple distinct conformations deposited in the Protein Data Bank. Cluster decompositions for them were previously determined with the RigidFinder method from Gerstein's lab and validated against experimental data. When KINARI's default tuning parameters are used, an improvement of the B-cubed score over a crude baseline is observed in 30% of this data. With our new modeling options, improvements were observed in over 70% of the proteins in this data set. We investigate the sensitivity of the cluster decomposition score with case studies on pyruvate phosphate dikinase and calmodulin., Conclusion: To substantially improve the accuracy of protein rigidity analysis systems, thorough benchmarking must be performed on all current systems and future extensions. We have measured the gain in performance by comparing different modeling methods for noncovalent interactions. We showed that new criteria for modeling hydrogen bonds and hydrophobic interactions can significantly improve the results. The two new methods proposed here have been implemented and made publicly available in the current version of KINARI (v1.3), together with the benchmarking tools, which can be downloaded from our software's website, http://kinari.cs.umass.edu.

Predicting the effect of a single amino acid substitution on the stability of a protein structure is a fundamental task in macromolecular modeling. It has relevance to drug design and understanding of disease-causing protein variants. We present KINARI-Mutagen, a web server for performing in silico mutation experiments on protein structures from the Protein Data Bank. Our rigidity-theoretical approach permits fast evaluation of the effects of mutations that may not be easy to perform in vitro, because it is not always possible to express a protein with a specific amino acid substitution. We use KINARI-Mutagen to identify critical residues, and we show that our predictions correlate with destabilizing mutations to glycine. In two in-depth case studies we show that the mutated residues identified by KINARI-Mutagen as critical correlate with experimental data, and would not have been identified by other methods such as Solvent Accessible Surface Area measurements or residue ranking by contributions to stabilizing interactions. We also generate 48 mutants for 14 proteins, and compare our rigidity-based results against experimental mutation stability data. KINARI-Mutagen is available at http://kinari.cs.umass.edu.

Motivated by recently developed computational techniques for studying protein flexibility, and their potential applications in docking, we propose an efficient method for sampling the conformational space of complex molecular structures. We focus on the loop closure problem, identified in the work of Thorpe and Lei (2004 Phil. Mag. 84 1323-31) as a primary bottleneck in the fast simulation of molecular motions. By modeling a molecular structure as a branching robot, we use an intuitive method in which the robot holds onto itself for maintaining loop constraints. New conformations are generated by applying random external forces, while internal, attractive forces pull the loops closed. Our implementation, tested on several model molecules with low number of degrees of freedom but many interconnected loops, gives promising results that show an almost four times speed-up on the benchmark cube-molecule of Thorpe and Lei.