Modeling the Correlation Functions of Conformational Motions in Proteins

Abstract

A first principles calculation of the correlation function for conformational motion (CM) in proteins is carried out within the framework of a microscopic model of a protein as a heterogeneous system. The fragments of the protein are assumed to be identical hard spheres undergoing the CM within their conformational potentials about some mean equilibrium positions assigned by the tertiary structure. The memory friction function (MFF) for the generalized Langevin equation describing the CM of the particle is obtained on the basis of the direct calculation which is feasible for the present model of the protein due to the existence of a natural large parameter, viz. the ratio of the minimal distance between the mean equilibrium positions of the particles (～7A) to the amplitude of their CM (<1A). A relationship between the MFF and the correlation functions of the CM of the particles is derived which makes their calculation to be a self-consistent mathematical problem. The general analysis of the MFF is exemplified by a simple model case in which the mean equilibrium positions of the particles form a regular lattice so that the correlation functions for all particles are the same. In this case the MFF is shown to be an infinite series of the powers of the auto-correlation function whose coefficients are independent on temperature. The latter is a result of the abstraction of the interaction potential by that of hard spheres which actually corresponds to the high temperature limit. On the examples of cubic and triangular lattices the coefficients are shown to be non-negative values which increase with the increase of the packing density of the particles and quickly tend to zero with the increase of their index. Thus the MFF can be approximated by a polynomial of the correlation function and the resulting mathematical equation is analogous to the one from the dynamic theory of liquids. The correlation function of the CM is obtained by numerical solution of the equation. At realistic packing densities for proteins it exhibits transparent non-exponential decay and includes two relaxation processes: the first one on the intermediate timescale (tens of picoseconds) and the second on the long timescale (its characteristic time is about tens of nanoseconds at small values of the friction coefficient and increases by orders of the magnitude with the increase of the latter). Thus the present approach provides the microscopic basis for previous phenomenologi- cal models of cooperative dynamics in proteins.     

Modeling the "glass" transition in proteins

A model of a protein as a disordered system of identical spherical particles (which imitate protein side chains) interacting with each other via a repulsive soft sphere potential U(r) infinity r(-beta) is constructed. The particles undergo the conformational motion (CM) within their own harmonic conformational potentials around some mean equilibrium positions ascribed by the tertiary structure of the protein. A first principles calculation of the positional correlation functions for the CM is carried out. The general analysis is exemplified by the case in which the mean equilibrium positions of the particles form a cubic tightly-packed (face- centered) lattice (each particle has 12 nearest neighbors) with the step b(hydr) =6.6 A (the average distance between the centers of mass of hydrated protein subunits). The model yields dramatic slowing down of the relaxation with the decrease of temperature followed by a sharp glass transition at some crossover temperature T(c) < 200 K. At the transition the liquid-like dynamic behavior (the correlation functions tend to zero with time) is altered by the glass-like one (the correlation functions tend with time to some non-zero limit). In the liquid-like region above the crossover temperature the relaxation exhibits distinct alpha-process following the beta-one. The glass transition results from the interaction of the particles. Thus the model suggests that namely direct interactions of the fragments of protein structure rather than protein-solvent interactions are the origin of the phenomenon of the glass transition. The known increase of T(c) up to 300 K at dehydration of the protein is attributed to the known concomitant compression of the globule upon drying by about 4-6% so that positions of individual atoms displace by about 0.6 A (modeled by the decrease of the step of the lattice b by 0.6 A so that b(dehydr)=6 A). The model suggests that the solvent influences the phenomenon of the glass transition indirectly determining the tertiary structure of the protein rather than via own freezing. In the model the transition from the liquid-like dynamic behavior to the glass-like one can be obtained even in a cluster containing a few particles. Thus the results of the model can be considered as an argument in favor of the point of view that the transition to the glassy behavior can take place for a very small domains of the protein comprising only several constituting fragments of its structure. The model predicts that for the dehydrated protein the alpha-relaxation process is strongly repressed.     

Osmolyte perturbation reveals conformational equilibria in spin‐labeled proteins

Recent evidence suggests that proteins at equilibrium can exist in a manifold of conformational substates, and that these substates play important roles in protein function. Therefore, there is great interest in identifying regions in proteins that are in conformational exchange. Electron paramagnetic resonance spectra of spin‐labeled proteins containing the nitroxide side chain (R1) often consist of two (or more) components that may arise from slow exchange between conformational substates (lifetimes > 100 ns). However, crystal structures of proteins containing R1 have shown that multicomponent spectra can also arise from equilibria between rotamers of the side chain itself. In this report, it is shown that these scenarios can be distinguished by the response of the system to solvent perturbation with stabilizing osmolytes such as sucrose. Thus, site‐directed spin labeling (SDSL) emerges as a new tool to explore slow conformational exchange in proteins of arbitrary size, including membrane proteins in a native‐like environment. Moreover, equilibrium between substates with even modest differences in conformation is revealed, and the simplicity of the method makes it suitable for facile screening of multiple proteins. Together with previously developed strategies for monitoring picosecond to millisecond backbone dynamics, the results presented here expand the timescale over which SDSL can be used to explore protein flexibility.     

Kinetics and thermodynamics of protein adsorption: a generalized molecular theoretical approach

The thermodynamics and kinetics of protein adsorption are studied using a molecular theoretical approach. The cases studied include competitive adsorption from mixtures and the effect of conformational changes upon adsorption. The kinetic theory is based on a generalized diffusion equation in which the driving force for motion is the gradient of chemical potentials of the proteins. The time-dependent chemical potentials, as well as the equilibrium behavior of the system, are obtained using a molecular mean-field theory. The theory provides, within the same theoretical formulation, the diffusion and the kinetic (activated) controlled regimes. By separation of ideal and nonideal contributions to the chemical potential, the equation of motion shows a purely diffusive part and the motion of the particles in the potential of mean force resulting from the intermolecular interactions. The theory enables the calculation of the time-dependent surface coverage of proteins, the dynamic surface tension, and the structure of the adsorbed layer in contact with the approaching proteins. For the case of competitive adsorption from a solution containing a mixture of large and small proteins, a variety of different adsorption patterns are observed depending upon the bulk composition, the strength of the interaction between the particles, and the surface and size of the proteins. It is found that the experimentally observed Vroman sequence is predicted in the case that the bulk solution is at a composition with an excess of the small protein, and that the interaction between the large protein and the surface is much larger than that of the smaller protein. The effect of surface conformational changes of the adsorbed proteins in the time-dependent adsorption is studied in detail. The theory predicts regimes of constant density and dynamic surface tension that are long lived but are only intermediates before the final approach to equilibrium. The implications of the findings to the interpretation of experimental observations is discussed.     

Flap opening and dimer-interface flexibility in the free and inhibitor-bound HIV protease, and their implications for function.

BACKGROUND: (1)H and (15)N transverse relaxation measurements on perdeuterated proteins are ideally suited for detecting backbone conformational fluctuations on the millisecond-microsecond timescale. The identification of conformational exchange on this timescale by measuring the relaxation of both (1)H and (15)N holds great promise for the elucidation of functionally relevant conformational changes in proteins. RESULTS: We measured the transverse (1)H and (15)N relaxation rates of backbone amides of HIV-1 protease in its free and inhibitor-bound forms. An analysis of these rates, obtained as a function of the effective rotating frame field, provided information about the timescale of structural fluctuations in several regions of the protein. The flaps that cover the active site of the inhibitor-bound protein undergo significant changes of backbone (φ,psi) angles, on the 100 micros timescale, in the free protein. In addition, the intermonomer beta-sheet interface of the bound form, which from protease structure studies appears to be rigid, was found to fluctuate on the millisecond timescale. CONCLUSIONS: We present a working model of the flap-opening mechanism in free HIV-1 protease which involves a transition from a semi-open to an open conformation that is facilitated by interaction of the Phe53 ring with the substrate. We also identify a surprising fluctuation of the beta-sheet intermonomer interface that suggests a structural requirement for maturation of the protease. Thus, slow conformational fluctuations identified by (1)H and (15)N transverse relaxation measurements can be related to the biological functions of proteins.     

Evolutionary Conserved Positions Define Protein Conformational Diversity

Conformational diversity of the native state plays a central role in modulating protein function. The selection paradigm sustains that different ligands shift the conformational equilibrium through their binding to highest-affinity conformers. Intramolecular vibrational dynamics associated to each conformation should guarantee conformational transitions, which due to its importance, could possibly be associated with evolutionary conserved traits. Normal mode analysis, based on a coarse-grained model of the protein, can provide the required information to explore these features. Herein, we present a novel procedure to identify key positions sustaining the conformational diversity associated to ligand binding. The method is applied to an adequate refined dataset of 188 paired protein structures in their bound and unbound forms. Firstly, normal modes most involved in the conformational change are selected according to their corresponding overlap with structural distortions introduced by ligand binding. The subspace defined by these modes is used to analyze the effect of simulated point mutations on preserving the conformational diversity of the protein. We find a negative correlation between the effects of mutations on these normal mode subspaces associated to ligand-binding and position-specific evolutionary conservations obtained from multiple sequence-structure alignments. Positions whose mutations are found to alter the most these subspaces are defined as key positions, that is, dynamically important residues that mediate the ligand-binding conformational change. These positions are shown to be evolutionary conserved, mostly buried aliphatic residues localized in regular structural regions of the protein like β-sheets and α-helix.     

Roles for holes: are cavities in proteins mere packing defects?

Atomic packing in proteins is not optimized, most structures containing internal cavities, which have been identified by molecular modelling and characterized experimentally. Cavities seem to play a role in assisting conformational changes between domains or subunit interfaces. Comparison between homologous proteins from thermophiles and mesophiles indicates that optimizing packing enhances stabilization at the expense of flexibility. For proteins which interact with small ligands or substrates, cavities seem to play a role in controlling binding and catalysis, rather than being mere "packing defects". We believe that a more complete analysis on the localization, conservation and role of cavities in protein structures (by modelling and site-directed mutagenesis), will reveal that rather than being randomly distributed, they are located in key positions to allow structural dynamics and thereby functional control.     

Structure of the YibK methyltransferase from Haemophilus influenzae (HI0766): a cofactor bound at a site formed by a knot

Despite the suitability of various lattice geometries for coarse-grained modeling of proteins, the actual packing geometry of residues in folded structures has remained largely unexplored. A strong tendency to assume a regular packing geometry is shown here by optimally reorienting and superimposing clusters of neighboring residues from databank structures examined on a coarse-grained (single-site-per-residue) scale. The orientation function (or order parameter) of the examined coordination clusters with respect to fcc lattice directions is found to be 0.82. The observed geometry, which may be termed an incomplete distorted face-centered cubic (fcc) packing, is apparently favored by the drive to maximize packing density, in a fashion analogous to the way identical spheres pack densely and follow fcc geometry. About 2/3 of all residues obey this packing geometry, while the remainder occupy other context-dependent positions. The preferred coordination directions show relatively small variations over the various amino acid types, consistent with uniform residue viewpoint. Both the extremes of solvent-exposed and completely buried residue neighborhoods approximate the same generic packing, the only difference being in the numbers (and not the orientations) of coordination sites that are occupied (or left void for solvent occupancy). We observe the prevalence of a rather uniform (tight) residue packing density throughout the structure, including even the residues packed near solvent-exposed regions. The observed orientation distribution reveals an underlying, intrinsic orientation lattice for proteins.     

Important amino acid properties for enhanced thermostability from mesophilic to thermophilic proteins

Understanding the role of various interactions in enhancing the thermostability of proteins is important not only for clarifying the mechanism of protein stability but also for designing stable proteins. In this work, we have analyzed the thermostability of 16 different families by comparing mesophilic and thermophilic proteins with 48 various physicochemical, energetic and conformational properties. We found that the increase in shape, s (location of branch point in side chain) increases the thermostability, whereas, an opposite trend is observed for Gibbs free energy change of hydration for native proteins, GhN, in 14 families. A good correlation is observed between these two properties and the simultaneous increases of -GhN and s is necessary to enhance the thermostability from mesophile to thermophile. The increase in shape, which tends to increase with increasing number of carbon atoms both for polar and non-polar residues, may generate more packing and compactness, and the position of beta and higher order branches may be important for better packing. On the other hand, the increase in -GhN in thermophilic proteins increases the solubility of the proteins. This tendency counterbalances the increases in insolubility and unfolding heat capacity change due to the increase in the number of carbon atoms. Thus, the present results suggest that the stability of thermophilic proteins may be achieved by a balance between better packing and solubility.     

Fluorescence correlation spectroscopy close to a fluctuating membrane

Compartmentalization of the cytoplasm by membranes should have a strong influence on the diffusion of macromolecules inside a cell, and we have studied how this could be reflected in fluorescence correlation spectroscopy (FCS) experiments. We derived the autocorrelation function measured by FCS for fluorescent particles diffusing close to a soft membrane, and show it to be the sum of two contributions: short timescale correlations come from the diffusion of the particles (differing from free diffusion because of the presence of an obstacle), whereas long timescale correlations arise from fluctuations of the membrane itself (which create intensity fluctuations by modulating the number of detected particles). In the case of thermal fluctuations this second type of correlation depends on the elasticity of the membrane. To illustrate this calculation, we report the results of FCS experiments carried out close to a vesicle membrane. The measured autocorrelation functions display very distinctly the two expected contributions, and allow both to recover the diffusion coefficient of the fluorophore and to characterize the membrane fluctuations in term of a bending rigidity. Our results show that FCS measurements inside cells can lead to erroneous values of the diffusion coefficient if the influence of membranes is not recognized.     

Role of surface hydrophobic residues in the conformational stability of human lysozyme at three different positions

To evaluate the contribution of the amino acid residues on the surface of a protein to its stability, a series of hydrophobic mutant human lysozymes (Val to Gly, Ala, Leu, Ile, Met, and Phe) modified at three different positions on the surface, which are located in the alpha-helix (Val 110), the beta-sheet (Val 2), and the loop (Val 74), were constructed. Their thermodynamic parameters of denaturation and crystal structures were examined by calorimetry and by X-ray crystallography at 100 K, respectively. Differences in the denaturation Gibbs energy change between the wild-type and the hydrophobic mutant proteins ranged from 4.6 to -9.6 kJ/mol, 2.7 to -1.5 kJ/mol, and 3.6 to -0.2 kJ/mol at positions 2, 74, and 110, respectively. The identical substitution at different positions and different substitutions at the same position resulted in different degrees of stabilization. Changes in the stability of the mutant proteins could be evaluated by a unique equation considering the conformational changes due to the substitutions [Funahashi et al. (1999) Protein Eng. 12, 841-850]. For this calculation, secondary structural propensities were newly considered. However, some mutant proteins were not adapted to the equation. The hydration structures around the mutation sites of the exceptional mutant proteins were affected due to the substitutions. The stability changes in the exceptional mutant proteins could be explained by the formation or destruction of the hydration structures. These results suggest that the hydration structure mediated via hydrogen bonds covering the protein surface plays an important role in the conformational stability of the protein.     

Dynamics of folding and unfolding transition in a globular protein studied by time correlation functions from computer simulation

A method of calculating time correlation functions from records of computer simulated equilibrium conformational fluctuations in a globular protein is discussed. Use of the calculated time correlation function for discussions of dynamics of folding and unfolding transition in the two-dimensional lattice model of proteins. The time correlation functions can be approximated in general by a sum of two simple exponential terms. The relaxation time of the slower mode does not depend on the nature of the physical quantity with respect to which the time correlation function is calculated. This time characterizes the overall folding and unfolding transition. The relaxation time of the faster mode depends on the nature of the physical quantity and characterizes conformational fluctuations within each of the native and denatured states. The mechanism of a previously observed phenomenon of the acceleration of the folding and unfolding transition by short-range interactions is discussed.     

Evaluating Molecular Mechanical Potentials for Helical Peptides and Proteins

Multiple variants of the AMBER all-atom force field were quantitatively evaluated with respect to their ability to accurately characterize helix-coil equilibria in explicit solvent simulations. Using a global distributed computing network, absolute conformational convergence was achieved for large ensembles of the capped A₂₁ and F_s helical peptides. Further assessment of these AMBER variants was conducted via simulations of a flexible 164-residue five-helix-bundle protein, apolipophorin-III, on the 100 ns timescale. Of the contemporary potentials that had not been assessed previously, the AMBER-99SB force field showed significant helix-destabilizing tendencies, with beta bridge formation occurring in helical peptides, and unfolding of apolipophorin-III occurring on the tens of nanoseconds timescale. The AMBER-03 force field, while showing adequate helical propensities for both peptides and stabilizing apolipophorin-III, (i) predicts an unexpected decrease in helicity with ALA→ARG⁺ substitution, (ii) lacks experimentally observed 3₁₀ helical content, and (iii) deviates strongly from average apolipophorin-III NMR structural properties. As is observed for AMBER-99SB, AMBER-03 significantly overweighs the contribution of extended and polyproline backbone configurations to the conformational equilibrium. In contrast, the AMBER-99φ force field, which was previously shown to best reproduce experimental measurements of the helix-coil transition in model helical peptides, adequately stabilizes apolipophorin-III and yields both an average gyration radius and polar solvent exposed surface area that are in excellent agreement with the NMR ensemble.     

Relating protein conformational changes to packing efficiency and disorder

Changes in protein conformation play key roles in facilitating various biochemical processes, ranging from signaling and phosphorylation to transport and catalysis. While various factors that drive these motions such as environmental changes and binding of small molecules are well understood, specific causative effects on the structural features of the protein due to these conformational changes have not been studied on a large scale. Here, we study protein conformational changes in relation to two key structural metrics: packing efficiency and disorder. Packing has been shown to be crucial for protein stability and function by many protein design and engineering studies. We study changes in packing efficiency during conformational changes, thus extending the analysis from a static context to a dynamic perspective and report some interesting observations. First, we study various proteins that adopt alternate conformations and find that tendencies to show motion and change in packing efficiency are correlated: residues that change their packing efficiency show larger motions. Second, our results suggest that residues that show higher changes in packing during motion are located on the changing interfaces which are formed during these conformational changes. These changing interfaces are slightly different from shear or static interfaces that have been analyzed in previous studies. Third, analysis of packing efficiency changes in the context of secondary structure shows that, as expected, residues buried in helices show the least change in packing efficiency, whereas those embedded in bends are most likely to change packing. Finally, by relating protein disorder to motions, we show that marginally disordered residues which are ordered enough to be crystallized but have sequence patterns indicative of disorder show higher dislocation and a higher change in packing than ordered ones and are located mostly on the changing interfaces. Overall, our results demonstrate that between the two conformations, the cores of the proteins remain mostly intact, whereas the interfaces display the most elasticity, both in terms of disorder and change in packing efficiency. By doing a variety of tests, we also show that our observations are robust to the solvation state of the proteins.     

Packing of the extracellular domain hydrophobic core has evolved to facilitate pentameric ligand-gated ion channel function

Protein function depends on conformational flexibility and folding stability. Loose packing of hydrophobic cores is not infrequent in proteins, as the enhanced flexibility likely contributes to their biological function. Here, using experimental and computational approaches, we show that eukaryotic pentameric ligand-gated ion channels are characterized by loose packing of their extracellular domain β-sandwich cores, and that loose packing contributes to their ability to rapidly switch from closed to open channel states in the presence of ligand. Functional analyses of GABA(A) receptors show that increasing the β-core packing disrupted GABA-mediated currents, with impaired GABA efficacy and slowed GABA current activation and desensitization. We propose that loose packing of the hydrophobic β-core developed as an evolutionary strategy aimed to facilitate the allosteric mechanisms of eukaryotic pentameric ligand-gated ion channels.     

Prediction of amyloidogenic and disordered regions in protein chains

The determination of factors that influence protein conformational changes is very important for the identification of potentially amyloidogenic and disordered regions in polypeptide chains. In our work we introduce a new parameter, mean packing density, to detect both amyloidogenic and disordered regions in a protein sequence. It has been shown that regions with strong expected packing density are responsible for amyloid formation. Our predictions are consistent with known disease-related amyloidogenic regions for eight of 12 amyloid-forming proteins and peptides in which the positions of amyloidogenic regions have been revealed experimentally. Our findings support the concept that the mechanism of amyloid fibril formation is similar for different peptides and proteins. Moreover, we have demonstrated that regions with weak expected packing density are responsible for the appearance of disordered regions. Our method has been tested on datasets of globular proteins and long disordered protein segments, and it shows improved performance over other widely used methods. Thus, we demonstrate that the expected packing density is a useful value with which one can predict both intrinsically disordered and amyloidogenic regions of a protein based on sequence alone. Our results are important for understanding the structural characteristics of protein folding and misfolding.     

Modeling the step of endosomal escape during cell infection by a nonenveloped virus

Widely disparate viruses enter the host cell through an endocytic pathway and travel the cytoplasm inside an endosome. For the viral genetic material to be delivered into the cytoplasm, these viruses have to escape the endosomal compartment, an event triggered by the conformational changes of viral endosomolytic proteins. We focus here on small nonenveloped viruses such as adeno-associated viruses, which contain few penetration proteins. The first time a penetration protein changes conformation defines the slowest timescale responsible for the escape. To evaluate this time, we construct what to our knowledge is a novel biophysical model based on a stochastic approach that accounts for the small number of proteins, the endosomal maturation, and the protease activation dynamics. We show that the escape time increases with the endosomal size, whereas decreasing with the number of viral particles inside the endosome. We predict that the optimal escape probability is achieved when the number of proteases in the endosome is in the range of 250-350, achieved for three viral particles.     

Optimization of the gbeta1 domain by computational design and by in vitro evolution: structural and energetic basis of stabilization

Computational design and in vitro evolution are major strategies for stabilizing proteins. For the four critical positions 16, 18, 25, and 29 of the B domain of the streptococcal protein G (Gbeta1), they identified the same optimal residues at positions 16 and 25, but not at 18 and 29. Here we analyzed the energetic contributions of the residues from these two approaches by single and double mutant analyses and determined crystal structures for a variant from the calculation (I16/L18/E25/K29) and from the selection (I16/I18/E25/F29). The structural analysis explains the observed differences in stabilization. Residues 16, 18, and 29 line an invagination, which results from a packing defect between the helix and the beta-sheet of Gbeta1. In all stabilized variants, residues with larger side-chains occur at these positions and packing is improved. In the selected variant, packing is better optimized than in the computed variant. Such differences in side-chain packing strongly affect stability but are difficult to evaluate by computation.