Donald J. Jacobs

Protein flexibility predictions using graph theory

Techniques from graph theory are applied to analyze the bond networks in proteins and identify the flexible and rigid regions. The bond network consists of distance constraints defined by the covalent and hydrogen bonds and salt bridges in the protein, identified by geometric and energetic criteria. We use an algorithm that counts the degrees of freedom within this constraint network and that identifies all the rigid and flexible substructures in the protein, including overconstrained regions (with more crosslinking bonds than are needed to rigidify the region) and underconstrained or flexible regions, in which dihedral bond rotations can occur. The number of extra constraints or remaining degrees of bond‐rotational freedom within a substructure quantifies its relative rigidity/flexibility and provides a flexibility index for each bond in the structure. This novel computational procedure, first used in the analysis of glassy materials, is approximately a million times faster than molecular dynamics simulations and captures the essential conformational flexibility of the protein main and side‐chains from analysis of a single, static three‐dimensional structure. This approach is demonstrated by comparison with experimental measures of flexibility for three proteins in which hinge and loop motion are essential for biological function: HIV protease, adenylate kinase, and dihydrofolate reductase. Proteins 2001;44:150–165.

Self-organization in network glasses

Abstract The continuous random network model is widely used as a realistic description of the structure of covalent glasses and amorphous solids. We point out that in real glasses and amorphous materials, there are non-random structural elements that go beyond just simple chemical ordering. We propose that the network can self-organize at its formation or fictive temperature, and examine some of the possible consequences of such self-organization. We find that the absence of small rings can cause the mechanical threshold to change from a second order to a first order transition. We show that if stressed regions are inhibited in the network, then there are two-phase transitions and an intermediate phase that is rigid but stress-free. This intermediate phase is bounded by a second order transition on one side and a first order transition on the other. Recent experiments in chalcogenide glasses give evidence for this intermediate phase.

Protein flexibility and dynamics using constraint theory.

A new approach is presented for determining the rigid regions in proteins and the flexible joints between them. The short-range forces in proteins are modeled as constraints and we use a recently developed formalism from graph theory to analyze flexibility in the bond network. Forces included in the analysis are the covalent bond-stretching and bond-bending forces, salt bridges, and hydrogen bonds. We use a local function to associate an energy with individual hydrogen bonds, which then can be included or excluded depending on the bond strength. Colored maps of the rigid and flexible regions provide a direct visualization of where the motion of the protein can take place, consistent with these distance constraints. We also define a flexibility index that quantifies the local density of flexible or floppy modes, in terms of the dihedral angles that remain free to rotate in each flexible region. A negative flexibility index provides a measure of the density of redundant bonds in rigid regions. A new application of this approach is to simulate the maximal range of possible motions of the flexible regions by introducing Monte Carlo changes in the free dihedral angles, subject to the distance constraints. This is done using a method that maintains closure of the rings formed by covalent and hydrogen bonds in the flexible parts of the protein, and van der Waals overlaps between atoms are avoided. We use the locus of the possible motions of HIV protease as an example: movies of its motion can be seen at http://www.pa.msu.edu/~lei.

Principal Component Analysis: A Method for Determining the Essential Dynamics of Proteins

It has become commonplace to employ principal component analysis to reveal the most important motions in proteins. This method is more commonly known by its acronym, PCA. While most popular molecular dynamics packages inevitably provide PCA tools to analyze protein trajectories, researchers often make inferences of their results without having insight into how to make interpretations, and they are often unaware of limitations and generalizations of such analysis. Here we review best practices for applying standard PCA, describe useful variants, discuss why one may wish to make comparison studies, and describe a set of metrics that make comparisons possible. In practice, one will be forced to make inferences about the essential dynamics of a protein without having the desired amount of samples. Therefore, considerable time is spent on describing how to judge the significance of results, highlighting pitfalls. The topic of PCA is reviewed from the perspective of many practical considerations, and useful recipes are provided.

Generic rigidity in three-dimensional bond-bending networks

Much progress can be made in studying the mechanical stability of frameworks when they are treated as generic, which lack any special symmetries. This is because testing for rigidity becomes topological in nature rather than geometrical. Generic rigidity, synonymous with graph rigidity, depends only on the connectivity of the network, making it a simpler problem to deal with in principle. A complete combinatorial constraint counting characterization of graph rigidity is given by Lamans theorem for two dimensions. Unfortunately there is no known corresponding theorem for three dimensions. Herein it is proposed that the theorem of Laman generalizes to three dimensions for bar-joint networks that have no implied-hinge joints. Particular attention is given to bond-bending networks, having a truss structure with constraints between nearest and next-nearest neighbours, that are suitable for modelling many covalent network glasses and macromolecules. It is shown that implied-hinge joints do not exist in bond-bending networks. Based on the proposition that an all subgraph constraint counting characterization of generic rigidity is recovered in three-dimensional bar-joint networks having no implied hinge joints, an efficient combinatorial algorithm is constructed for bond-bending networks. Complete agreement is found with exact calculations involving diagonalization of dynamical matrices, for systems up to degrees of freedom.

A flexible approach for understanding protein stability

A distance constraint model (DCM) is presented that identifies flexible regions within protein structure consistent with specified thermodynamic condition. The DCM is based on a rigorous free energy decomposition scheme representing structure as fluctuating constraint topologies. Entropy non‐additivity is problematic for naive decompositions, limiting the success of heat capacity predictions. The DCM resolves non‐additivity by summing over independent entropic components determined by an efficient network‐rigidity algorithm. A minimal 3‐parameter DCM is demonstrated to accurately reproduce experimental heat capacity curves. Free energy landscapes and quantitative stability‐flexibility relationships are obtained in terms of global flexibility. Several connections to experiment are made.

Extending the fast multipole method to charges inside or outside a dielectric sphere

In this paper, we propose a novel method to extend the fast multipole method (FMM) to calculate the electrostatic potential due to charges inside or outside a dielectric sphere. The key result which allows such an extension is the construction of a small number (two for a 10^-^2 relative error in reaction potentials inside the sphere) of image point charges for source point charges inside or outside the dielectric sphere. Numerical results validate the accuracy and high efficiency of the resulting algorithm.

Network rigidity at finite temperature: relationships between thermodynamic stability, the nonadditivity of entropy, and cooperativity in molecular systems.

A statistical mechanical distance constraint model (DCM) is presented that explicitly accounts for network rigidity among constraints present within a system. Constraints are characterized by local microscopic free-energy functions. Topological rearrangements of thermally fluctuating constraints are permitted. The partition function is obtained by combining microscopic free energies of individual constraints using network rigidity as an underlying long-range mechanical interaction, giving a quantitative explanation for the nonadditivity in component entropies exhibited in molecular systems. Two exactly solved two-dimensional toy models representing flexible molecules that can undergo conformational change are presented to elucidate concepts, and to outline a DCM calculation scheme applicable to many types of physical systems. It is proposed that network rigidity plays a central role in balancing the energetic and entropic contributions to the free energy of biopolymers, such as proteins. As a demonstration, the distance constraint model is solved exactly for the alpha-helix to coil transition in homogeneous peptides. Temperature and size independent model parameters are fitted to Monte Carlo simulation data, which includes peptides of length 10 for gas phase, and lengths 10, 15, 20, and 30 in water. The DCM is compared to the Lifson-Roig model. It is found that network rigidity provides a mechanism for cooperativity in molecular structures including their ability to spontaneously self-organize. In particular, the formation of a characteristic topological arrangement of constraints is associated with the most probable microstates changing under different thermodynamic conditions.

Conserved quantitative stability/flexibility relationships (QSFR) in an orthologous RNase H pair

Many reports qualitatively describe conserved stability and flexibility profiles across protein families, but biophysical modeling schemes have not been available to robustly quantify both. Here we investigate an orthologous RNase H pair by using a minimal distance constraint model (DCM). The DCM is an all atom microscopic model [Jacobs and Dallakyan, Biophys J 2005;88(2):903–915] that accurately reproduces heat capacity measurements [Livesay et al., FEBS Lett 2004;576(3):468–476], and is unique in its ability to harmoniously calculate thermodynamic stability and flexibility in practical computing times. Consequently, quantified stability/flexibility relationships (QSFR) can be determined using the DCM. For the first time, a comparative QSFR analysis is performed, serving as a paradigm study to illustrate the utility of a QSFR analysis for elucidating evolutionarily conserved stability and flexibility profiles. Despite global conservation of QSFR profiles, distinct enthalpy‐entropy compensation mechanisms are identified between the RNase H pair. In both cases, local flexibility metrics parallel H/D exchange experiments by correctly identifying the folding core and several flexible regions. Remarkably, at appropriately shifted temperatures (e.g., melting temperature), these differences lead to a global conservation in Landau free energy landscapes, which directly relate thermodynamic stability to global flexibility. Using ensemble‐based sampling within free energy basins, rigidly, and flexibly correlated regions are quantified through cooperativity correlation plots. Five conserved flexible regions are identified within the structures of the orthologous pair. Evolutionary conservation of these flexibly correlated regions is strongly suggestive of their catalytic importance. Conclusions made herein are demonstrated to be robust with respect to the DCM parameterization. Proteins 2006.

Structural rigidity in the capsid assembly of cowpea chlorotic mottle virus

The cowpea chlorotic mottle virus (CCMV) has a protein cage, or capsid, which encloses its genetic material. The structure of the capsid consists of 180 copies of a single protein that self-assemble inside a cell to form a complete capsid with icosahedral symmetry. The icosahedral surface can be naturally divided into pentagonal and hexagonal faces, and the formation of either of these faces has been proposed to be the first step in the capsid assembly process. We have used the software FIRST to analyse the rigidity of pentameric and hexameric substructures of the complete capsid to explore the viability of certain capsid assembly pathways. FIRST uses the 3D pebble game to determine structural rigidity, and a brief description of this algorithm, as applied to body–bar networks, is given here. We find that the pentameric substructure, which corresponds to a pentagonal face on the icosahedral surface, provides the best structural properties for nucleating the capsid assembly process, consistent with experimental observations.