Packing regularities in biological structures relate to their dynamics

Entropies Derived from the Packing Geometries within a Single Protein Structure

A fast, simple, yet robust method to calculate protein entropy from a single protein structure is presented here. The focus is on the atomic packing details, which are calculated by combining Voronoi diagrams and Delaunay tessellations. Even though the method is simple, the entropies computed exhibit an extremely high correlation with the entropies previously derived by other methods based on quasi-harmonic motions, quantum mechanics, and molecular dynamics simulations. These packing-based entropies account directly for the local freedom and provide entropy for any individual protein structure that could be used to compute free energies directly during simulations for the generation of more reliable trajectories and also for better evaluations of modeled protein structures. Physico-chemical properties of amino acids are compared with these packing entropies to uncover the relationships with the entropies of different residue types. A public packing entropy web server is provided at packing-entropy.bb.iastate.edu, and the application programing interface is available within the PACKMAN (https://github.com/Pranavkhade/PACKMAN) package.

Insights on Rigidity and Flexibility at the Global and Local Levels of Protein Structures and Their Roles in Homologous Psychrophilic, Mesophilic, and Thermophilic Proteins: A Computational Study

The rigidity and flexibility of homologous psychrophilic (P), mesophilic (M), and thermophilic (T) proteins have been investigated at the global and local levels in terms of "packing factors" and "atomic fluctuations" obtained from B-factors. For comparison of atomic fluctuations, correction of errors by considering errors in B-factors from all sources in a consolidated manner and conversion of the fluctuations to the same temperature have been suggested and validated. The results indicate no differences in the global values like the average packing factor among the three classes of protein homologues, but at local levels there are differences. A comparison of homologous protein triplets show that the average atomic fluctuations at a given temperature mainly obey the order P > M > T. Packing factors and the atomic fluctuations are anti-correlated, suggesting that altering the rigidity of the active site might be a potential strategy to make tailor-made psychrophilic or thermophilic proteins from their mesophilic homologues. The computer codes developed and used in this work are available at the link https://github.com/Munna-Sarkar/proteins-rigidity-flexibility.git.

Computational Ways to Enhance Protein Inhibitor Design

Two new computational approaches are described to aid in the design of new peptide-based drugs by evaluating ensembles of protein structures from their dynamics and through the assessing of structures using empirical contact potential. These approaches build on the concept that conformational variability can aid in the binding process and, for disordered proteins, can even facilitate the binding of more diverse ligands. This latter consideration indicates that such a design process should be less restrictive so that multiple inhibitors might be effective. The example chosen here focuses on proteins/peptides that bind to hemagglutinin (HA) to block the large-scale conformational change for activation. Variability in the conformations is considered from sets of experimental structures, or as an alternative, from their simple computed dynamics; the set of designe peptides/small proteins from the David Baker lab designed to bind to hemagglutinin, is the large set considered and is assessed with the new empirical contact potentials.

Elastic network models capture the motions apparent within ensembles of RNA structures

The role of structure and dynamics in mechanisms for RNA becomes increasingly important. Computational approaches using simple dynamics models have been successful at predicting the motions of proteins and are often applied to ribonucleo-protein complexes but have not been thoroughly tested for well-packed nucleic acid structures. In order to characterize a true set of motions, we investigate the apparent motions from 16 ensembles of experimentally determined RNA structures. These indicate a relatively limited set of motions that are captured by a small set of principal components (PCs). These limited motions closely resemble the motions computed from low frequency normal modes from elastic network models (ENMs), either at atomic or coarse-grained resolution. Various ENM model types, parameters, and structure representations are tested here against the experimental RNA structural ensembles, exposing differences between models for proteins and for folded RNAs. Differences in performance are seen, depending on the structure alignment algorithm used to generate PCs, modulating the apparent utility of ENMs but not significantly impacting their ability to generate functional motions. The loss of dynamical information upon coarse-graining is somewhat larger for RNAs than for globular proteins, indicating, perhaps, the lower cooperativity of the less densely packed RNA. However, the RNA structures show less sensitivity to the elastic network model parameters than do proteins. These findings further demonstrate the utility of ENMs and the appropriateness of their application to well-packed RNA-only structures, justifying their use for studying the dynamics of ribonucleo-proteins, such as the ribosome and regulatory RNAs.

A mechanistic stress model of protein evolution accounts for site-specific evolutionary rates and their relationship with packing density and flexibility

Background

Protein sites evolve at different rates due to functional and biophysical constraints. It is usually considered that the main structural determinant of a site’s rate of evolution is its Relative Solvent Accessibility (RSA). However, a recent comparative study has shown that the main structural determinant is the site’s Local Packing Density (LPD). LPD is related with dynamical flexibility, which has also been shown to correlate with sequence variability. Our purpose is to investigate the mechanism that connects a site’s LPD with its rate of evolution.

Results

We consider two models: an empirical Flexibility Model and a mechanistic Stress Model. The Flexibility Model postulates a linear increase of site-specific rate of evolution with dynamical flexibility. The Stress Model, introduced here, models mutations as random perturbations of the protein’s potential energy landscape, for which we use simple Elastic Network Models (ENMs). To account for natural selection we assume a single active conformation and use basic statistical physics to derive a linear relationship between site-specific evolutionary rates and the local stress of the mutant’s active conformation.

We compare both models on a large and diverse dataset of enzymes. In a protein-by-protein study we found that the Stress Model outperforms the Flexibility Model for most proteins. Pooling all proteins together we show that the Stress Model is strongly supported by the total weight of evidence. Moreover, it accounts for the observed nonlinear dependence of sequence variability on flexibility. Finally, when mutational stress is controlled for, there is very little remaining correlation between sequence variability and dynamical flexibility.

Conclusions

We developed a mechanistic Stress Model of evolution according to which the rate of evolution of a site is predicted to depend linearly on the local mutational stress of the active conformation. Such local stress is proportional to LPD, so that this model explains the relationship between LPD and evolutionary rate. Moreover, the model also accounts for the nonlinear dependence between evolutionary rate and dynamical flexibility.

Intrinsically disordered regions of p53 family are highly diversified in evolution

Proteins of the p53 family are expressed in vertebrates and in some invertebrate species. The main function of these proteins is to control and regulate cell cycle in response to various cellular signals, and therefore to control the organism's development. The regulatory functions of the p53 family members originate mostly from their highly-conserved and well-structured DNA-binding domains. Many human diseases (including various types of cancer) are related to the missense mutations within this domain. The ordered DNA-binding domains of the p53 family members are surrounded by functionally important intrinsically disordered regions. In this study, substitution rates and propensities in different regions of p53 were analyzed. The analyses revealed that the ordered DNA-binding domain is conserved, whereas disordered regions are characterized by high sequence diversity. This diversity was reflected both in the number of substitutions and in the types of substitutions to which each amino acid was prone. These results support the existence of a positive correlation between protein intrinsic disorder and sequence divergence during the evolutionary process. This higher sequence divergence provides strong support for the existence of disordered regions in p53 in vivo for if they were structured, they would evolve at similar rates as the rest of the protein.

The case for intrinsically disordered proteins playing contributory roles in molecular recognition without a stable 3D structure

The classical 'lock-and-key' and 'induced-fit' mechanisms for binding both originated in attempts to explain features of enzyme catalysis. For both of these mechanisms and for their recent refinements, enzyme catalysis requires exquisite spatial and electronic complementarity between the substrate and the catalyst. Thus, binding models derived from models originally based on catalysis will be highly biased towards mechanisms that utilize structural complementarity. If mere binding without catalysis is the endpoint, then the structural requirements for the interaction become much more relaxed. Recent observations on specific examples suggest that this relaxation can reach an extreme lack of specific 3D structure, leading to molecular recognition with biological consequences that depend not only upon structural and electrostatic complementarity between the binding partners but also upon kinetic, entropic, and generalized electrostatic effects. In addition to this discussion of binding without fixed structure, examples in which unstructured regions carry out important biological functions not involving molecular recognition will also be discussed. Finally, we discuss whether 'intrinsically disordered protein' (IDP) represents a useful new concept.

The importance of slow motions for protein functional loops

Loops in proteins that connect secondary structures such as alpha-helix and beta-sheet, are often on the surface and may play a critical role in some functions of a protein. The mobility of loops is central for the motional freedom and flexibility requirements of active-site loops and may play a critical role for some functions. The structures and behaviors of loops have not been studied much in the context of the whole structure and its overall motions, especially how these might be coupled. Here we investigate loop motions by using coarse-grained structures (C(α) atoms only) to solve the motions of the system by applying Lagrange equations with elastic network models to learn about which loops move in an independent fashion and which move in coordination with domain motions, faster and slower, respectively. The normal modes of the system are calculated using eigen-decomposition of the stiffness matrix. The contribution of individual modes and groups of modes is investigated for their effects on all residues in each loop by using Fourier analyses. Our results indicate overall that the motions of functional sets of loops behave in similar ways as the whole structure. But overall only a relatively few loops move in coordination with the dominant slow modes of motion, and these are often closely related to function.

MAVENs: motion analysis and visualization of elastic networks and structural ensembles

Background

The ability to generate, visualize, and analyze motions of biomolecules has made a significant impact upon modern biology. Molecular Dynamics has gained substantial use, but remains computationally demanding and difficult to setup for many biologists. Elastic network models (ENMs) are an alternative and have been shown to generate the dominant equilibrium motions of biomolecules quickly and efficiently. These dominant motions have been shown to be functionally relevant and also to indicate the likely direction of conformational changes. Most structures have a small number of dominant motions. Comparing computed motions to the structure's conformational ensemble derived from a collection of static structures or frames from an MD trajectory is an important way to understand functional motions as well as evaluate the models. Modes of motion computed from ENMs can be visualized to gain functional and mechanistic understanding and to compute useful quantities such as average positional fluctuations, internal distance changes, collectiveness of motions, and directional correlations within the structure.

Results

Our new software, MAVEN, aims to bring ENMs and their analysis to a broader audience by integrating methods for their generation and analysis into a user friendly environment that automates many of the steps. Models can be constructed from raw PDB files or density maps, using all available atomic coordinates or by employing various coarse-graining procedures. Visualization can be performed either with our software or exported to molecular viewers. Mixed resolution models allow one to study atomic effects on the system while retaining much of the computational speed of the coarse-grained ENMs. Analysis options are available to further aid the user in understanding the computed motions and their importance for its function.

Conclusion

MAVEN has been developed to simplify ENM generation, allow for diverse models to be used, and facilitate useful analyses, all on the same platform. This represents an integrated approach that incorporates all four levels of the modeling process - generation, evaluation, analysis, visualization - and also brings to bear multiple ENM types. The intension is to provide a versatile modular suite of programs to a broader audience. MAVEN is available for download at http://maven.sourceforge.net.

Free energies for coarse-grained proteins by integrating multibody statistical contact potentials with entropies from elastic network models

We propose a novel method of calculation of free energy for coarse grained models of proteins by combining our newly developed multibody potentials with entropies computed from elastic network models of proteins. Multi-body potentials have been of much interest recently because they take into account three dimensional interactions related to residue packing and capture the cooperativity of these interactions in protein structures. Combining four-body non-sequential, four-body sequential and pairwise short range potentials with optimized weights for each term, our coarse-grained potential improved recognition of native structure among misfolded decoys, outperforming all other contact potentials for CASP8 decoy sets and performance comparable to the fully atomic empirical DFIRE potentials. By combing statistical contact potentials with entropies from elastic network models of the same structures we can compute free energy changes and improve coarse-grained modeling of protein structure and dynamics. The consideration of protein flexibility and dynamics should improve protein structure prediction and refinement of computational models. This work is the first to combine coarse-grained multibody potentials with an entropic model that takes into account contributions of the entire structure, investigating native-like decoy selection.

Immunoglobulin Structure Exhibits Control over CDR Motion

Motions of the IgG structure are evaluated using normal mode analysis of an elastic network model to detect hinges, the dominance of low frequency modes, and the most important internal motions. One question we seek to answer is whether or not IgG hinge motions facilitate antigen binding. We also evaluate the protein crystal and packing effects on the experimental temperature factors and disorder predictions. We find that the effects of the protein environment on the crystallographic temperature factors may be misleading for evaluating specific functional motions of IgG. The extent of motion of the antigen binding domains is computed to show their large spatial sampling. We conclude that the IgG structure is specifically designed to facilitate large excursions of the antigen binding domains. Normal modes are shown as capable of computationally evaluating the hinge motions and the spatial sampling by the structure. The antigen binding loops and the major hinge appear to behave similarly to the rest of the structure when we consider the dominance of the low frequency modes and the extent of internal motion. The full IgG structure has a lower spectral dimension than individual F_ab domains, pointing to more efficient information transfer through the antibody than through each domain. This supports the claim that the IgG structure is specifically constructed to facilitate antigen binding by coupling motion of the antigen binding loops with the large scale hinge motions.

Voronoia: analyzing packing in protein structures

The packing of protein atoms is an indicator for their stability and functionality, and applied in determining thermostability, in protein design, ligand binding and to identify flexible regions in proteins. Here, we present Voronoia, a database of atomic-scale packing data for protein 3D structures. It is based on an improved Voronoi Cell algorithm using hyperboloid interfaces to construct atomic volumes, and to resolve solvent-accessible and -inaccessible regions of atoms. The database contains atomic volumes, local packing densities and interior cavities calculated for 61 318 biological units from the PDB. A report for each structure summarizes the packing by residue and atom types, and lists the environment of interior cavities. The packing data are compared to a nonredundant set of structures from SCOP superfamilies. Both packing densities and cavities can be visualized in the 3D structures by the Jmol plugin. Additionally, PDB files can be submitted to the Voronoia server for calculation. This service performs calculations for most full-atomic protein structures within a few minutes. For batch jobs, a standalone version of the program with an optional PyMOL plugin is available for download. The database can be freely accessed at: http://bioinformatics.charite.de/voronoia.