Diverse classes of proteins function through large-scale conformational changes and various sophisticated computational algorithms have been proposed to enhance sampling of these macromolecular transition paths. Because such paths are curves in a high-dimensional space, it has been difficult to quantitatively compare multiple paths, a necessary prerequisite to, for instance, assess the quality of different algorithms. We introduce a method named Path Similarity Analysis (PSA) that enables us to quantify the similarity between two arbitrary paths and extract the atomic-scale determinants responsible for their differences. PSA utilizes the full information available in 3N-dimensional configuration space trajectories by employing the Hausdorff or Fréchet metrics (adopted from computational geometry) to quantify the degree of similarity between piecewise-linear curves. It thus completely avoids relying on projections into low dimensional spaces, as used in traditional approaches.

To elucidate the principles of PSA, we quantified the effect of path roughness induced by thermal fluctuations using a toy model system. Using, as an example, the closed-to-open transitions of the enzyme adenylate kinase (AdK) in its substrate-free form, we compared a range of protein transition path-generating algorithms. Molecular dynamics-based dynamic importance sampling (DIMS) MD and targeted MD (TMD) and the purely geometric FRODA (Framework Rigidity Optimized Dynamics Algorithm) were tested along with seven other methods publicly available on servers, including several based on the popular elastic network model (ENM). PSA with clustering revealed that paths produced by a given method are more similar to each other than to those from another method and, for instance, that the ENM-based methods produced relatively similar paths. PSA applied to ensembles of DIMS MD and FRODA trajectories of the conformational transition of diphtheria toxin, a particularly challenging example, showed that the geometry-based FRODA occasionally sampled the pathway space of force field-based DIMS MD. For the AdK transition, the new concept of a Hausdorff-pair map enabled us to extract the molecular structural determinants responsible for differences in pathways, namely a set of conserved salt bridges whose charge-charge interactions are fully modelled in DIMS MD but not in FRODA. PSA has the potential to enhance our understanding of transition path sampling methods, validate them, and to provide a new approach to analyzing conformational transitions.

We collect results for bond percolation on various lattices from two to fourteen dimensions that, in the limit of large dimension d or number of neighbors z, smoothly approach a randomly diluted Erdos-Renyi graph. We include results on bond-diluted hypersphere packs in up to nine dimensions, which show the mean coordination, excess kurtosis, and skewness evolving smoothly with dimension towards the Erdos-Renyi limit.

Theoretical modeling is presented for a freestanding vitreous silica bilayer which has recently been synthesized and characterized experimentally in landmark work. While such two-dimensional continuous random covalent networks should likely occur on energetic grounds, no synthetic pathway had been discovered previously. Here the bilayer is modeled using a computer assembly procedure initiated from a single layer of a model of amorphous graphene, generated using a bond-switching algorithm from an initially crystalline graphene structure. Each bond is decorated with an oxygen atom and the carbon atoms are relabeled as silicon, generating a two-dimensional network of corner-sharing triangles. Each triangle is transformed into a tetrahedron, by raising the silicon atom above each triangular base and adding an additional singly coordinated oxygen atom at the apex. The final step in this construction is to mirror-reflect this layer to form a second layer and attach the two layers to form the bilayer. We show that this vitreous silica bilayer has the additional macroscopic degrees of freedom to form easily a network of identical corner-sharing tetrahedra if there is a symmetry plane through the center of the bilayer going through the layer of oxygen ions that join the upper and lower monolayers. This has the consequence that the upper rings lie exactly above the lower rings, which are tilted in general. The assumption of a network of perfect corner-sharing tetrahedra leads to a range of possible densities that we characterize as a flexibility window, with some similarity to flexibility windows in three dimensional zeolites. Finally, using a realistic potential, we have relaxed the bilayer to determine the density and other structural characteristics such as the Si-Si pair distribution functions and the Si-O-Si bond angle distribution, which are compared with experimental results obtained by direct imaging.

The database of molecular motions, MolMovDB (http://molmovdb.org), has been in existence for the past decade. It classifies macromolecular motions and provides tools to interpolate between two conformations (the Morph Server) and predict possible motions in a single structure. In 2005, we expanded the services offered on MolMovDB. In particular, we further developed the Morph Server to produce improved interpolations between two submitted structures. We added support for multiple chains to the original adiabatic mapping interpolation, allowing the analysis of subunit motions. We also added the option of using FRODA interpolation, which allows for more complex pathways, potentially overcoming steric barriers. We added an interface to a hinge prediction service, which acts on single structures and predicts likely residue points for flexibility. We developed tools to relate such points of flexibility in a structure to particular key residue positions, i.e. active sites or highly conserved positions. Lastly, we began relating our motion classification scheme to function using descriptions from the Gene Ontology Consortium.

Protein evolution is most commonly studied by analyzing related protein sequences and generating ancestral sequences through Bayesian and Maximum Likelihood methods, and/or by resurrecting ancestral proteins in the lab and performing ligand binding studies to determine function. Structural and dynamic evolution have largely been left out of molecular evolution studies. Here we incorporate both structure and dynamics to elucidate the molecular principles behind the divergence in the evolutionary path of the steroid receptor proteins. We determine the likely structure of three evolutionarily diverged ancestral steroid receptor proteins using the Zipping and Assembly Method with FRODA (ZAMF). Our predictions are within ∼2.7 Å all-atom RMSD of the respective crystal structures of the ancestral steroid receptors. Beyond static structure prediction, a particular feature of ZAMF is that it generates protein dynamics information. We investigate the differences in conformational dynamics of diverged proteins by obtaining the most collective motion through essential dynamics. Strikingly, our analysis shows that evolutionarily diverged proteins of the same family do not share the same dynamic subspace, while those sharing the same function are simultaneously clustered together and distant from those, that have functionally diverged. Dynamic analysis also enables those mutations that most affect dynamics to be identified. It correctly predicts all mutations (functional and permissive) necessary to evolve new function and ∼60% of permissive mutations necessary to recover ancestral function.