A Mathematical Framework for Protein Structure Comparison

Comparison of protein structures is important for revealing the evolutionary relationship among proteins, predicting protein functions and predicting protein structures. Many methods have been developed in the past to align two or multiple protein structures. Despite the importance of this problem, rigorous mathematical or statistical frameworks have seldom been pursued for general protein structure comparison. One notable issue in this field is that with many different distances used to measure the similarity between protein structures, none of them are proper distances when protein structures of different sequences are compared. Statistical approaches based on those non-proper distances or similarity scores as random variables are thus not mathematically rigorous. In this work, we develop a mathematical framework for protein structure comparison by treating protein structures as three-dimensional curves. Using an elastic Riemannian metric on spaces of curves, geodesic distance, a proper distance on spaces of curves, can be computed for any two protein structures. In this framework, protein structures can be treated as random variables on the shape manifold, and means and covariance can be computed for populations of protein structures. Furthermore, these moments can be used to build Gaussian-type probability distributions of protein structures for use in hypothesis testing. The covariance of a population of protein structures can reveal the population-specific variations and be helpful in improving structure classification. With curves representing protein structures, the matching is performed using elastic shape analysis of curves, which can effectively model conformational changes and insertions/deletions. We show that our method performs comparably with commonly used methods in protein structure classification on a large manually annotated data set

Understanding diversity of human innate immunity receptors: analysis of surface features of leucine-rich repeat domains in NLRs and TLRs.

BackgroundThe human innate immune system uses a system of extracellular Toll-like receptors (TLRs) and intracellular Nod-like receptors (NLRs) to match the appropriate level of immune response to the level of threat from the current environment. Almost all NLRs and TLRs have a domain consisting of multiple leucine-rich repeats (LRRs), which is believed to be involved in ligand binding. LRRs, found also in thousands of other proteins, form a well-defined "horseshoe"-shaped structural scaffold that can be used for a variety of functions, from binding specific ligands to performing a general structural role. The specific functional roles of LRR domains in NLRs and TLRs are thus defined by their detailed surface features. While experimental crystal structures of four human TLRs have been solved, no structure data are available for NLRs.ResultsWe report a quantitative, comparative analysis of the surface features of LRR domains in human NLRs and TLRs, using predicted three-dimensional structures for NLRs. Specifically, we calculated amino acid hydrophobicity, charge, and glycosylation distributions within LRR domain surfaces and assessed their similarity by clustering. Despite differences in structural and genomic organization, comparison of LRR surface features in NLRs and TLRs allowed us to hypothesize about their possible functional similarities. We find agreement between predicted surface similarities and similar functional roles in NLRs and TLRs with known agonists, and suggest possible binding partners for uncharacterized NLRs.ConclusionDespite its low resolution, our approach permits comparison of molecular surface features in the absence of crystal structure data. Our results illustrate diversity of surface features of innate immunity receptors and provide hints for function of NLRs whose specific role in innate immunity is yet unknown

Spherical harmonics coeffcients for ligand-based virtual screening of cyclooxygenase inhibitors

Background: Molecular descriptors are essential for many applications in computational chemistry, such as ligand-based similarity searching. Spherical harmonics have previously been suggested as comprehensive descriptors of molecular structure and properties. We investigate a spherical harmonics descriptor for shape-based virtual screening. Methodology/Principal Findings: We introduce and validate a partially rotation-invariant three-dimensional molecular shape descriptor based on the norm of spherical harmonics expansion coefficients. Using this molecular representation, we parameterize molecular surfaces, i.e., isosurfaces of spatial molecular property distributions. We validate the shape descriptor in a comprehensive retrospective virtual screening experiment. In a prospective study, we virtually screen a large compound library for cyclooxygenase inhibitors, using a self-organizing map as a pre-filter and the shape descriptor for candidate prioritization. Conclusions/Significance: 12 compounds were tested in vitro for direct enzyme inhibition and in a whole blood assay. Active compounds containing a triazole scaffold were identified as direct cyclooxygenase-1 inhibitors. This outcome corroborates the usefulness of spherical harmonics for representation of molecular shape in virtual screening of large compound collections. The combination of pharmacophore and shape-based filtering of screening candidates proved to be a straightforward approach to finding novel bioactive chemotypes with minimal experimental effort

Identifying networks with common organizational principles

Many complex systems can be represented as networks, and the problem of network comparison is becoming increasingly relevant. There are many techniques for network comparison, from simply comparing network summary statistics to sophisticated but computationally costly alignment-based approaches. Yet it remains challenging to accurately cluster networks that are of a different size and density, but hypothesized to be structurally similar. In this paper, we address this problem by introducing a new network comparison methodology that is aimed at identifying common organizational principles in networks. The methodology is simple, intuitive and applicable in a wide variety of settings ranging from the functional classification of proteins to tracking the evolution of a world trade network.Comment: 26 pages, 7 figure

Path Similarity Analysis: a Method for Quantifying Macromolecular Pathways

Diverse classes of proteins function through large-scale conformational changes; sophisticated enhanced sampling methods have been proposed to generate these macromolecular transition paths. As such paths are curves in a high-dimensional space, they have been difficult to compare quantitatively, a prerequisite to, for instance, assess the quality of different sampling algorithms. The Path Similarity Analysis (PSA) approach alleviates these difficulties by utilizing the full information in 3N-dimensional trajectories in configuration space. PSA employs the Hausdorff or Fr\'echet path metrics---adopted from computational geometry---enabling us to quantify path (dis)similarity, while the new concept of a Hausdorff-pair map permits the extraction of atomic-scale determinants responsible for path differences. Combined with clustering techniques, PSA facilitates the comparison of many paths, including collections of transition ensembles. We use the closed-to-open transition of the enzyme adenylate kinase (AdK)---a commonly used testbed for the assessment enhanced sampling algorithms---to examine multiple microsecond equilibrium molecular dynamics (MD) transitions of AdK in its substrate-free form alongside transition ensembles from the MD-based dynamic importance sampling (DIMS-MD) and targeted MD (TMD) methods, and a geometrical targeting algorithm (FRODA). A Hausdorff pairs analysis of these ensembles revealed, for instance, that differences in DIMS-MD and FRODA paths were mediated by a set of conserved salt bridges whose charge-charge interactions are fully modeled in DIMS-MD but not in FRODA. We also demonstrate how existing trajectory analysis methods relying on pre-defined collective variables, such as native contacts or geometric quantities, can be used synergistically with PSA, as well as the application of PSA to more complex systems such as membrane transporter proteins.Comment: 9 figures, 3 tables in the main manuscript; supplementary information includes 7 texts (S1 Text - S7 Text) and 11 figures (S1 Fig - S11 Fig) (also available from journal site

Evolutionary fine-tuning of conformational ensembles in FimH during host-pathogen interactions

Positive selection in the two-domain type 1 pilus adhesin FimH enhances Escherichia coli fitness in urinary tract infection (UTI). We report a comprehensive atomic-level view of FimH in two-state conformational ensembles in solution, composed of one low-affinity tense (T) and multiple high-affinity relaxed (R) conformations. Positively selected residues allosterically modulate the equilibrium between these two conformational states, each of which engages mannose through distinct binding orientations. A FimH variant that only adopts the R state is severely attenuated early in a mouse model of uncomplicated UTI but is proficient at colonizing catheterized bladders in vivo or bladder transitional-like epithelial cells in vitro. Thus, the bladder habitat has barrier(s) to R state–mediated colonization possibly conferred by the terminally differentiated bladder epithelium and/or decoy receptors in urine. Together, our studies reveal the conformational landscape in solution, binding mechanisms, and adhesive strength of an allosteric two-domain adhesin that evolved “moderate” affinity to optimize persistence in the bladder during UTI