A rigid network of long-range contacts increases thermostability in a mutant endoglucanase.

Thermodynamic stability of a protein at elevated temperatures is a key factor for thermostable enzymes to catalyze their specific reactions. Yet our understanding of biological determinants of thermostability is far from complete. Many different atomistic factors have been suggested as possible means for such proteins to preserve their activity at high temperatures. Among these factors are specific local interatomic interactions or enrichment of specific amino acid types. The case of glycosyl hydrolase family endoglucanase of Trichoderma reesei defies current hypotheses for thermostability because a single mutation far from the active site (A35 V) converts this mesostable protein into a thermostable protein without significant change in the protein structure. This substantial change in enzymatic activity cannot be explained on the basis of local intramolecular interactions alone. Here we present a more global view of the induced thermostability and show that the A35 V mutation affects the underlying structural rigidity of the whole protein via a number of long-range, non-local interactions. Our analysis of this structure reveals a precisely tuned, rigid network of atomic interactions. This cooperative, allosteric effect promotes the transformation of this mesostable protein into a thermostable one.

Correlating allostery with rigidity.

Allosteric proteins demonstrate the phenomenon of a ligand binding to a protein at a regulatory or effector site and thereby changing the chemical affinity of the catalytic site. As such, allostery is extremely important biologically as a regulatory mechanism for molecular concentrations in many cellular processes. One particularly interesting feature of allostery is that often the catalytic and effector sites are separated by a large distance. Structural comparisons of allosteric proteins resolved in both inactive and active states indicate that a variety of structural rearrangement and changes in motions may contribute to general allosteric behavior. In general it is expected that the coupling of catalytic and regulatory sites is responsible for allosteric behavior. We utilize a novel examination of allostery using rigidity analysis of the underlying graph of the protein structures. Our results indicate a general global change in rigidity associated with allosteric transitions where the R state is more rigid than the T state. A set of allosteric proteins with heterotropic interactions is used to test the hypothesis that catalytic and effector sites are structurally coupled. Observation of a rigid path connecting the effector and catalytic sites in 68.75% of the structures points to rigidity as a means by which the distal sites communicate with each other and so contribute to allosteric regulation. Thus structural rigidity is shown to be a fundamental underlying property that promotes cooperativity and non-locality seen in allostery.

Coarse-grained models: getting more with less.

There has recently been a proliferation of simplified, coarse-grained models to study aspects of biomolecular dynamics, binding, assembly and folding. Despite differences in construction these various coarse-grained models share a common underlying desire to identify the minimal set of variables required to realistically describe the essence of these molecules. Recent results emphasizing common and distinctive features are highlighted. For someone not involved in developing such models it is a daunting task to decide which, if any, coarse-grained model would be appropriate for a given system. Although this decision ultimately depends upon what kinds of questions one is probing, suggestions about reaching a conclusion are provided.

Thermostability in rubredoxin and its relationship to mechanical rigidity.

The source of increased stability in proteins from organisms that thrive in extreme thermal environments is not well understood. Previous experimental and theoretical studies have suggested many different features possibly responsible for such thermostability. Many of these thermostabilizing mechanisms can be accounted for in terms of structural rigidity. Thus a plausible hypothesis accounting for this remarkable stability in thermophilic enzymes states that these enzymes have enhanced conformational rigidity at temperatures below their native, functioning temperature. Experimental evidence exists to both support and contradict this supposition. We computationally investigate the relationship between thermostability and rigidity using rubredoxin as a case study. The mechanical rigidity is calculated using atomic models of homologous rubredoxin structures from the hyperthermophile Pyrococcus furiosus and mesophile Clostridium pasteurianum using the FIRST software. A global increase in structural rigidity (equivalently a decrease in flexibility) corresponds to an increase in thermostability. Locally, rigidity differences (between mesophilic and thermophilic structures) agree with differences in protection factors.

Identification of putative, stable binding regions through flexibility analysis of HIV-1 gp120.

The acquired-immunodeficiency syndrome has evolved into a major worldwide epidemic. Significant effort has been made in the development of antiviral therapies. A new strategy for vaccine and drug design that complements the existing cocktail therapy is to target entry of the human immunodeficiency virus (HIV). Such an approach provides the advantage of interfering with multiple intermediates in this multi-step process. The extraordinary conformational flexibility, glycosylation, and strain variations of viral glycoprotein gp120 cause general viral evasion of humoral immune response and thus complicate the development of an effective vaccine. Especially difficult to define are the conformation of gp120 before CD4 engagement as well as the relative orientations of the V1/V2 and V3 loops with respect to the inner and outer domains. In this study, we used Floppy Inclusion and Rigid Substructure Topography (FIRST), a program based on graph theory, to analyze the flexibility and rigidity of all known HIV-1 gp120 structures. A flexibility index is used to describe and compare the spatial distribution of protein flexibility and rigidity of these structures in isolation and in complex with CD4, CD4-mimics, and neutralizing antibodies. Using this flexibility analysis, we identified a universal rigid region (the alpha2 helix) as well as the consensus largest rigid cluster involving a beta-sheet located on the coreceptor binding face. Both of these regions may serve as stable targets for vaccine design and drug discovery. Detailed comparisons of the changes in flexibility based on strain variations, stabilizing mutations, binding features of CD4 mimics, and impact of b12 binding are reported.

Comparisons of protein family dynamics.

Similarities between different protein structures have led to the identification of protein families based upon some measure of structural similarity. Using these similarities one can classify proteins into structural families and higher-order groupings from which inferred function can be transferred. When taken for a large number of proteins, these schemes point to evolutionary relationships between organisms. We propose a novel classification scheme based upon the structurally-inspired dynamics of each protein. This classification scheme has the advantages of being quantitative, automatically assigned, and able to also make distinctions within protein families. Results are presented for five protein families illustrating the correct identification of previously un-classified structures and sources of intrafamily distinctions.

Comparison of stability predictions and simulated unfolding of rhodopsin structures.

Developing a better mechanistic understanding of membrane protein folding is urgently needed because of the discovery of an increasing number of human diseases, where membrane protein instability and misfolding is involved. Towards this goal, we investigated folding and stability of 7-transmembrane (TM) helical bundles by computational methods. We compared the results of three different algorithms for predicting changes in stability of proteins against an experimental mutation dataset obtained for bacteriorhodopsin (BR) and mammalian rhodopsin and find that 61.6% and 70.6% of the mutation results can potentially be explained by known local contributors to the stability of the folded state of BR and mammalian rhodopsin, respectively. To obtain further information on the predicted folding pathway of 7-TM proteins, we conducted simulated thermal unfolding experiments of all available rhodopsin structures with resolution better than 3 angstroms using the Floppy Inclusions and Rigid Substructure Topography (FIRST) method (Jacobs, D. J., A. J. Rader, L. A. Kuhn and M. F. Thorpe [2001] Proteins 44, 150) described previously for a single mammalian rhodopsin structure (Rader et al. [2004] PNAS 101, 7246). In statistical comparison we found that structures of mammalian rhodopsin have a stability core that is characterized by long-range interactions involving amino acids close in space but distant in sequence comprising positions from both extracellular loop and TM regions. In contrast, BR-simulated unfolding does not reveal such a core but is dominated by interactions within individual and groups of TM helices, consistent with the two-stage hypothesis of membrane protein folding. Similar results were obtained for halo- and sensory rhodopsins as for BRs. However, the average folding core energies of sensory rhodopsins were in between those observed for mammalian rhodopsins and BRs hinting at a possible evolution of these structures toward a rhodopsin-like behavior. These results support the conclusion that although the two-stage model can explain the mechanisms of folding and stability of BR, it fails to account for the folding and stability of mammalian rhodopsin, even though the two proteins are structurally related.

Predisposition of the dark state of rhodopsin to functional changes in structure.

As the only member of the family of G-protein-coupled receptors for which atomic coordinates are available, rhodopsin is widely studied for insight into the molecular mechanism of G-protein-coupled receptor activation. The currently available structures refer to the inactive, dark state, of rhodopsin, rather than the light-activated metarhodopsin II (Meta II) state. A model for the Meta II state is proposed here by analyzing elastic network normal modes in conjunction with experimental data. Key mechanical features and interactions broken/formed in the proposed model are found to be consistent with the experimental data. The model is further tested by using a set of Meta II fluorescence decay rates measured to empirically characterize the deactivation of rhodopsin mutants. The model is found to correctly predict 93% of the experimentally observed effects in 119 rhodopsin mutants for which the decay rates and misfolding data have been measured, including a systematic analysis of Cys-->Ser replacements reported here. Based on the detailed comparison between model and experiments, a cooperative activation mechanism is deduced that couples retinal isomerization to concerted changes in conformation, facilitated by the intrinsic dynamics of rhodopsin. A global hinge site is identified near the retinal-binding pocket that ensures the efficient propagation of signals from the central transmembrane region to both cytoplasmic and extracellular ends. The predicted activation mechanism opens the transmembrane helices at the critical G-protein binding cytoplasmic domain. This model provides a detailed, mechanistic description of the activation process, extending experimental observations and yielding new insights for further tests.

oGNM: online computation of structural dynamics using the Gaussian Network Model.

An assessment of the equilibrium dynamics of biomolecular systems, and in particular their most cooperative fluctuations accessible under native state conditions, is a first step towards understanding molecular mechanisms relevant to biological function. We present a web-based system, oGNM that enables users to calculate online the shape and dispersion of normal modes of motion for proteins, oligonucleotides and their complexes, or associated biological units, using the Gaussian Network Model (GNM). Computations with the new engine are 5-6 orders of magnitude faster than those using conventional normal mode analyses. Two cases studies illustrate the utility of oGNM. The first shows that the thermal fluctuations predicted for 1250 non-homologous proteins correlate well with X-ray crystallographic data over a broad range [7.3-15 A] of inter-residue interaction cutoff distances and the correlations improve with increasing observation temperatures. The second study, focused on 64 oligonucleotides and oligonucleotide-protein complexes, shows that good agreement with experiments is achieved by representing each nucleotide by three GNM nodes (as opposed to one-node-per-residue in proteins) along with uniform interaction ranges for all components of the complexes. These results open the way to a rapid assessment of the dynamics of DNA/RNA-containing complexes. The server can be accessed at http://ignm.ccbb.pitt.edu/GNM_Online_Calculation.htm.

Elastic network models for understanding biomolecular machinery: from enzymes to supramolecular assemblies.

With advances in structure genomics, it is now recognized that knowledge of structure alone is insufficient to understand and control the mechanisms of biomolecular function. Additional information in the form of dynamics is needed. As demonstrated in a large number of studies, the machinery of proteins and their complexes can be understood to a good approximation by adopting Gaussian (or elastic) network models (GNM) for simplified normal mode analyses. While this approximation lacks chemical details, it provides us with a means for assessing the collective motions of large structures/assemblies and perform a comparative analysis of a series of proteins, thus providing insights into the mechanical aspects of biomolecular dynamics. In this paper, we discuss recent applications of GNM to a series of enzymes as well as large structures such as the HK97 bacteriophage viral capsids. Understanding the dynamics of large protein structures can be computationally challenging. To this end, we introduce a new approach for building a hierarchical, reduced rank representation of the protein topology and consequently the fluctuation dynamics.

Coarse-grained normal mode analysis in structural biology.

The realization that experimentally observed functional motions of proteins can be predicted by coarse-grained normal mode analysis has renewed interest in applications to structural biology. Notable applications include the prediction of biologically relevant motions of proteins and supramolecular structures driven by their structure-encoded collective dynamics; the refinement of low-resolution structures, including those determined by cryo-electron microscopy; and the identification of conserved dynamic patterns and mechanically key regions within protein families. Additionally, hybrid methods that couple atomic simulations with deformations derived from coarse-grained normal mode analysis are able to sample collective motions beyond the range of conventional molecular dynamics simulations. Such applications have provided great insight into the underlying principles linking protein structures to their dynamics and their dynamics to their functions.

iGNM: a database of protein functional motions based on Gaussian Network Model.

MOTIVATION: The knowledge of protein structure is not sufficient for understanding and controlling its function. Function is a dynamic property. Although protein structural information has been rapidly accumulating in databases, little effort has been invested to date toward systematically characterizing protein dynamics. The recent success of analytical methods based on elastic network models, and in particular the Gaussian Network Model (GNM), permits us to perform a high-throughput analysis of the collective dynamics of proteins. RESULTS: We computed the GNM dynamics for 20 058 structures from the Protein Data Bank, and generated information on the equilibrium dynamics at the level of individual residues. The results are stored on a web-based system called iGNM and configured so as to permit the users to visualize or download the results through a standard web browser using a simple search engine. Static and animated images for describing the conformational mobility of proteins over a broad range of normal modes are accessible, along with an online calculation engine available for newly deposited structures. A case study of the dynamics of 20 non-homologous hydrolases is presented to illustrate the utility of the iGNM database for identifying key residues that control the cooperative motions and revealing the connection between collective dynamics and catalytic activity.

Maturation dynamics of bacteriophage HK97 capsid.

Maturation of the bacteriophage HK97 capsid requires a large conformational change of the virus capsid. Experimental studies have identified several intermediates along this maturation pathway. To gain insights into the molecular mechanisms of capsid maturation, we examined the fluctuation dynamics of the procapsid and mature capsid using a residue-level computational approach. The most cooperative motions of the procapsid are found to be consistent with the observed change in configuration that takes place during maturation. A few dominant modes of motion are sufficient to describe the anisotropic expansion that accompanies maturation. Based upon these modes, maturation is proposed to occur via an overall expansion and reconfiguration of the capsid initiated by puckering of the pentamers, followed by flattening and crosslinking of the hexameric subunits, and finally crosslinking of the pentameric subunits. The highly mobile E loops are stabilized by anchoring to highly stable residues belonging to neighboring subunits.

Global ribosome motions revealed with elastic network model.

The motions of large systems such as the ribosome are not fully accessible with conventional molecular simulations. A coarse-grained, less-than-atomic-detail model such as the anisotropic network model (ANM) is a convenient informative tool to study the cooperative motions of the ribosome. The motions of the small 30S subunit, the larger 50S subunit, and the entire 70S assembly of the two subunits have been analyzed using ANM. The lowest frequency collective modes predicted by ANM show that the 50S subunit and 30S subunit are strongly anti-correlated in the motion of the 70S assembly. A ratchet-like motion is observed that corresponds well to the experimentally reported ratchet motion. Other slow modes are also examined because of their potential links to the translocation steps in the ribosome. We identify several modes that may facilitate the E-tRNA exiting from the assembly. The A-site t-RNA and P-site t-RNA are found to be strongly coupled and positively correlated in these slow modes, suggesting that the translocations of these two t-RNAs occur simultaneously, while the motions of the E-site t-RNA are less correlated, and thus less likely to occur simultaneously. Overall the t-RNAs exhibit relatively large deformations. Animations of these slow modes of motion can be viewed at.

Identification of core amino acids stabilizing rhodopsin.

Rhodopsin is the only G protein-coupled receptor (GPCR) whose 3D structure is known; therefore, it serves as a prototype for studies of the GPCR family of proteins. Rhodopsin dysfunction has been linked to misfolding, caused by chemical modifications that affect the naturally occurring disulfide bond between C110 and C187. Here, we identify the structural elements that stabilize rhodopsin by computational analysis of the rhodopsin structure and comparison with data from previous in vitro mutational studies. We simulate the thermal unfolding of rhodopsin by breaking the native-state hydrogen bonds sequentially in the order of their relative strength, using the recently developed Floppy Inclusion and Rigid Substructure Topography (FIRST) method [Jacobs, D. J., Rader, A. J., Kuhn, L. A. & Thorpe, M. F. (2001) Proteins 44, 150-165]. Residues most stable under thermal denaturation are part of a core, which is assumed to be important for the formation and stability of folded rhodopsin. This core includes the C110-C187 disulfide bond at the center of residues forming the interface between the transmembrane and the extracellular domains near the retinal binding pocket. Fast mode analysis of rhodopsin using the Gaussian network model also identifies the disulfide bond and the retinal ligand binding pocket to be the most rigid region in rhodopsin. Experiments confirm that 90% of the amino acids predicted by the FIRST method to be part of the core cause misfolding upon mutation. The observed high degree of conservation (78.9%) of this disulfide bond across all GPCR classes suggests that it is critical for the stability and function of GPCRs.

Identifying protein folding cores from the evolution of flexible regions during unfolding.

The unfolding of a protein can be described as a transition from a predominantly rigid, folded structure to an ensemble of denatured states. During unfolding, the hydrogen bonds and salt bridges break, destabilizing the secondary and tertiary structure. Our previous work shows that the network of covalent bonds, salt bridges, hydrogen bonds, and hydrophobic interactions forms constraints that define which regions of the native protein are flexible or rigid (structurally stable). Here, we test the hypothesis that information about the folding pathway is encoded in the energetic hierarchy of non-covalent interactions in the native-state structure. The incremental thermal denaturation of protein structures is simulated by diluting the network of salt bridges and hydrogen bonds, breaking them one by one, from weakest to strongest. The structurally stable and flexible regions are identified at each step, providing information about the evolution of flexible regions during denaturation. The folding core, or center of structure formation during folding, is predicted as the region formed by two or more secondary structures having the greatest stability against denaturation. For 10 proteins with different architectures, we show that the predicted folding cores from this flexibility/stability analysis are in good agreement with those identified by native-state hydrogen-deuterium exchange experiments.

Protein unfolding: rigidity lost.

We relate the unfolding of a protein to its loss of structural stability or rigidity. Rigidity and flexibility are well defined concepts in mathematics and physics, with a body of theorems and algorithms that have been applied successfully to materials, allowing the constraints in a network to be related to its deformability. Here we simulate the weakening or dilution of the noncovalent bonds during protein unfolding, and identify the emergence of flexible regions as unfolding proceeds. The transition state is determined from the inflection point in the change in the number of independent bond-rotational degrees of freedom (floppy modes) of the protein as its mean atomic coordination decreases. The first derivative of the fraction of floppy modes as a function of mean coordination is similar to the fraction-folded curve for a protein as a function of denaturant concentration or temperature. The second derivative, a specific heat-like quantity, shows a peak around a mean coordination of = 2.41 for the 26 diverse proteins we have studied. As the protein denatures, it loses rigidity at the transition state, proceeds to a state where just the initial folding core remains stable, then becomes entirely denatured or flexible. This universal behavior for proteins of diverse architecture, including monomers and oligomers, is analogous to the rigid to floppy phase transition in network glasses. This approach provides a unifying view of the phase transitions of proteins and glasses, and identifies the mean coordination as the relevant structural variable, or reaction coordinate, along the unfolding pathway.

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.

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.