Panelling planar graphs

The skeleton of a spherical polyhedron may also be viewed as the skeleton of other panelled structures. We characterize those collections of cycles of a planar graph that bound the panels of hinged-panel structures, and distinguish those that arise from spherical polyhedra

Calculating Ensemble Averaged Descriptions of Protein Rigidity without Sampling

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 properties. This result positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability

A virtual pebble game to ensemble average graph rigidity

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 in- teger Pebble Game (PG) algorithm. However, thermodynamic properties require averaging over many samples from the ensemble of accessible conformations, leading to fluctuations within the network. We have developed a mean field Virtual Pebble Game (VPG) that provides a probabilistic description of the interaction network, meaning that sampling is not required. We extensively test the VPG algorithm over a variety of body-bar networks created on disordered lattices, from these calculations we fully characterize the network conditions under which the performance of the VPG offers the best solution. The VPG provides a satisfactory description of the ensemble averaged PG properties, especially in regions removed from the rigidity transition where ensemble fluctuations are greatest. In further experiments, we characterized the VPG across a structurally nonredundant dataset of 272 proteins. Using quantitative and visual assessments of the rigidity characterizations, the VPG results are shown to accurately reflect the ensemble averaged PG properties. That is, the fluctuating interaction network is well represented by a single calculation that re- places density functions with average values, thus speeding up the desired calculation by several orders of magnitude. Finally, we propose a new algorithm that is based on the combination of PG and VPG to balance the amount of sampling and mean field treatment. While offering interesting results, this approach needs to be further optimized to fully lever- age its utility. All these results positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability

Accurate and Robust Mechanical Modeling of Proteins

Through their motion, proteins perform essential functions in the living cell. Although we cannot observe protein motion directly, over 68,000 crystal structures are freely available from the Protein Data Bank. Computational protein rigidity analysis systems leverage this data, building a mechanical model from atoms and pairwise interactions determined from a static 3D structure. The rigid and flexible components of the model are then calculated with a pebble game algorithm, predicting a protein\u27s flexibility with much more computational efficiency than physical simulation. In prior work with rigidity analysis systems, the available modeling options were hard-coded, and evaluation was limited to case studies. The focus of this thesis is improving accuracy and robustness of rigidity analysis systems. The first contribution is in new approaches to mechanical modeling of noncovalent interactions, namely hydrogen bonds and hydrophobic interactions. Unlike covalent bonds, the behavior of these interactions varies with their energies. I systematically investigate energy-refined modeling of these interactions. Included in this is a method to assign a score to a predicted cluster decomposition, adapted from the B-cubed score from information retrieval. Another contribution of this thesis is in new approaches to measuring the robustness of rigidity analysis results. The protein\u27s fold is held in place by weak, noncovalent interactions, known to break and form during natural fluctuations. Rigidity analysis has been conventionally performed on only a single snapshot, rather than on an entire trajectory, and no information was made available on the sensitivity of the clusters to variations in the interaction network. I propose an approach to measure the robustness of rigidity results, by studying how detrimental the loss of a single interaction may be to a cluster\u27s rigidity. The accompanying study shows that, when present, highly critical interactions are concentrated around the active site, indicating that nature has designed a very versatile system for transitioning between unique conformations. Over the course of this thesis, we develop the KINARI library for experimenting with extensions to rigidity analysis. The modular design of the software allows for easy extensions and tool development. A specific feature is the inclusion of several modeling options, allowing more freedom in exploring biological hypotheses and future benchmarking experiments