Distribution of the energy dissipation rates among degrees of freedom during conformational movements and folding of a macromolecular chain in a viscous medium

The principle of the minimum rate of energy dissipation for conformational movements in a viscous medium formulated in an earlier study (K.V. Shaitan, Biophysics, 2015, Vol. 60, p. 692) has been applied for a theoretical estimation of the distribution functions of the energy dissipation rates for selected elements of a macromolecule. Equipartition of energy dissipation rates among the nodes of the chain upon conformational movements in the statistical limit of the process is obtained. The uniform distribution of energy dissipation rates along the chain does affect on the formation of the collective conformational degrees of freedom and the folding dynamics.     

Relaxation folding and the principle of the minimum rate of energy dissipation for conformational motions in a viscous medium

A numerical simulation of the folding of a model polymer chain of 50 units with valence bonds of a fixed length and fixed valence angle values has been performed using the strong friction approximation. The rate of energy dissipation in the system has been analyzed for conformational motions along a trajectory determined by the equations of mechanics and the trajectories characterized by random and variable deviations from the mechanical path. The validity of the principle of the minimum average rate of the energy dissipation for the conformational relaxation of a macromolecule in a viscous medium has been demonstrated. A profile of the relaxation energy funnel for the folding of a macromolecular chain has been constructed. Slow and rapid stages of folding could be distinguished in the energy funnel profile; the final state was separated from the nearest conformations of the folded chain by an energy gap.     

Coherent conformational degrees of freedom as a structural basis for allosteric communication

Conformational changes in allosteric regulation can to a large extent be described as motion along one or a few coherent degrees of freedom. The states involved are inherent to the protein, in the sense that they are visited by the protein also in the absence of effector ligands. Previously, we developed the measure binding leverage to find sites where ligand binding can shift the conformational equilibrium of a protein. Binding leverage is calculated for a set of motion vectors representing independent conformational degrees of freedom. In this paper, to analyze allosteric communication between binding sites, we introduce the concept of leverage coupling, based on the assumption that only pairs of sites that couple to the same conformational degrees of freedom can be allosterically connected. We demonstrate how leverage coupling can be used to analyze allosteric communication in a range of enzymes (regulated by both ligand binding and post-translational modifications) and huge molecular machines such as chaperones. Leverage coupling can be calculated for any protein structure to analyze both biological and latent catalytic and regulatory sites.     

Lagrangian molecular dynamics using selected conformational degrees of freedom,with application to the pseudorotation dynamics of furanose rings

Using internal conformational degrees of freedom for biopolymers as natural variables, and introducing a Lagrangian dynamics approach, one can simulate time-dependent processes over a much longer time scale than in classical Newtonian molecular dynamics (MD) techniques. Two factors contribute to this: a substantial reduction in the number of degrees of freedom and a very large increase in the size of the time step. We present the Lagrangian equations of motion for repuckering transitions in model furanose ( F ), ribose ( R ), and 2′-deoxyribose ( dR ) ring systems using the pseudorotation phase angle as the single dynamic variable. As in most Lagrangian analyses, the effective masses for the R and dR models are dependent on conformation, and we test the behavior of this variable mass (VM) model. Since the variation in effective mass is small, the VM model is compared with a simplified constant mass (CM) model, which is shown to be an excellent approximation. The equations of motion for the CM and VM models are integrated with the leapfrog and the iterative leapfrog algorithms, respectively. The Lagrangian dynamics approach reduces the number of degrees of freedom from about 40 to 1, and allows the use of time steps on the order of 20 fs, about an order of magnitude greater than is used in conventional MD simulations. © 1994 John Wiley & Sons, Inc.     

Harmonic analysis of DNA dynamics in a viscous medium

The harmonic dynamics of normal modes of double-stranded DNA in a viscous fluid are investigated. The model DNA consists of two backbone-supported DNA strands coiling around a common helix axis with base stacking, sugar puckering, interstrand hydrogen bonding, and intrastrand sugar-base interactions assigned values based on published data. Assuming that the DNA bases are shielded from direct bombardment by the solvent, analytical solutions are obtained. The dissipation and fluctuation of the normal modes of the bases moving along the spirals display the effect of the medium indirectly through interactions with the backbone. The dynamics of the backbone are found to be overdamped with the characteristic damping times extending to the picosecond region for disturbance in position and to the sub-picosecond region for disturbance in velocity. In addition to the dynamic mode of a rigid rod, the motions of the bases are coupled to the motions of the backbone with comparable amplitudes for disturbance in position. For disturbance in velocity, however, the bases are effectively at rest, not being able to follow the motions of the backbone. The angular frequencies of the underdamped vibrational modes, identified as the ringing modes of the bases with the backbone effectively at rest, are insensitive to the viscosity and lie in the low frequency region of the Raman spectrum. These findings indicate that the backbone of DNA plays a significant role in modulating the dynamics of double-stranded DNA in an overdamping environment. This modulation of the dynamics of the motions of the bases in DNA by environmental impediments to molecular motion is briefly discussed in connection with protein- and drug- DNA interactions as well as gene regulation.     

Essential dynamics of reversible peptide folding: memory-free conformational dynamics governed by internal hydrogen bonds

A principal component analysis has been applied on equilibrium simulations of a beta-heptapeptide that shows reversible folding in a methanol solution. The analysis shows that the configurational space contains only three dense sub-states. These states of relatively low free energy correspond to the "native" left-handed helix, a partly helical intermediate, and a hairpin-like structure. The collection of unfolded conformations form a relatively diffuse cloud with little substructure. Internal hydrogen-bonding energies were found to correlate well with the degree of folding. The native helical structure folds from the N terminus; the transition from the major folding intermediate to the native helical structure involves the formation of the two most C-terminal backbone hydrogen bonds. A four-state Markov model was found to describe transition frequencies between the conformational states within error limits, indicating that memory-effects are negligible beyond the nanosecond time-scale. The dominant native state fluctuations were found to be very similar to unfolding motions, suggesting that unfolding pathways can be inferred from fluctuations in the native state. The low-dimensional essential subspace, describing 69% of the collective atomic fluctuations, was found to converge at time-scales of the order of one nanosecond at all temperatures investigated, whereas folding/unfolding takes place at significantly longer time-scales, even above the melting temperature.     

Water as a conformational editor in protein folding

As molecules approach one another in aqueous solution, desolvation free energy barriers to association are encountered. Experiments suggest these (de)solvation effects contribute to the free energy barriers separating the folded and unfolded states of protein molecules. To explore their influence on the energy landscapes of protein folding reactions, we have incorporated desolvation barriers into a semi-realistic, off-lattice protein model that uses a simplified physico-chemical force-field determined solely by the sequence of amino acids. Monte Carlo sampling techniques were used to study the effects on the thermodynamics and kinetics of folding of a number of systems, diverse in structure and sequence. In each case, desolvation barriers increase the stability of the native conformation and the cooperativity of the major folding/unfolding transition. The folding times of these systems are reduced significantly upon inclusion of desolvation barriers, demonstrating that the particulate nature of the solvent engenders a more defined route to the native fold.     

Simulation of protein conformational freedom as a function of pH: constant-pH molecular dynamics using implicit titration

Solution pH is a determinant parameter on protein function and stability, and its inclusion in molecular dynamics simulations is attractive for studies at the molecular level. Current molecular dynamics simulations can consider pH only in a very limited way, through a somewhat arbitrary choice of a set of fixed charges on the titrable sites. Conversely, continuum electrostatic methods that explicitly treat pH effects assume a single protein conformation whose choice is not clearly defined. In this paper we describe a general method that combines both titration and conformational freedom. The method is based on a potential of mean force for implicit titration and combines both usual molecular dynamics and pH-dependent calculations based on continuum methods. A simple implementation of the method, using a mean field approximation, is presented and applied to the bovine pancreatic trypsin inhibitor. We believe that this constant-pH molecular dynamics method, by correctly sampling both charges and conformation, can become a valuable help in the understanding of the dependence of protein function and stability on pH. © 1997 Wiley-Liss Inc.     

Analysis of conformational variation in macromolecular structural models

Experimental conditions or the presence of interacting components can lead to variations in the structural models of macromolecules. However, the role of these factors in conformational selection is often omitted by in silico methods to extract dynamic information from protein structural models. Structures of small peptides, considered building blocks for larger macromolecular structural models, can substantially differ in the context of a larger protein. This limitation is more evident in the case of modeling large multi-subunit macromolecular complexes using structures of the individual protein components. Here we report an analysis of variations in structural models of proteins with high sequence similarity. These models were analyzed for sequence features of the protein, the role of scaffolding segments including interacting proteins or affinity tags and the chemical components in the experimental conditions. Conformational features in these structural models could be rationalized by conformational selection events, perhaps induced by experimental conditions. This analysis was performed on a non-redundant dataset of protein structures from different SCOP classes. The sequence-conformation correlations that we note here suggest additional features that could be incorporated by in silico methods to extract dynamic information from protein structural models.     

The kinetics of side chain stabilization during protein folding

The rate of stabilization of side chains during protein folding has never been carefully studied. Recent developments in labeling proteins with (19)F-labeled amino acids coupled with real-time NMR measurements have allowed such measurements to be made. This paper describes the application of this method to the study of several proteins using 6-(19)F-tryptophan as the reporting group. It is found that these side chains adopt their final stable state at the last stages of the folding process and that the stabilization of side chains into their final conformation is a highly cooperative process. It is also possible to show the presence of intermediates in which the side chains are not correctly packed. The technique should be applicable to many systems.     

Coupling of conformational folding and disulfide-bond reactions in oxidative folding of proteins

The oxidative folding of proteins consists of conformational folding and disulfide-bond reactions. These two processes are coupled significantly in folding-coupled regeneration steps, in which a single chemical reaction (the "forward" reaction) converts a conformationally unstable precursor species into a conformationally stable, disulfide-protected successor species. Two limiting-case mechanisms for folding-coupled regeneration steps are described. In the folded-precursor mechanism, the precursor species is preferentially folded at the moment of the forward reaction. The (transient) native structure increases the effective concentrations of the reactive thiol and disulfide groups, thus favoring the forward reaction. By contrast, in the quasi-stochastic mechanism, the forward reaction occurs quasi-stochastically in an unfolded precursor; i.e., reactive groups encounter each other with a probability determined primarily by loop entropy, albeit modified by conformational biases in the unfolded state. The resulting successor species is initially unfolded, and its folding competes with backward chemical reactions to the unfolded precursors. The folded-precursor and quasi-stochastic mechanisms may be distinguished experimentally by the dependence of their kinetics on factors affecting the rates of thiol--disulfide exchange and conformational (un)folding. Experimental data and structural and biochemical arguments suggest that the quasi-stochastic mechanism is more plausible than the folded-precursor mechanism for most proteins.     

Autocatalytic peptide cyclization during chain folding of histidine ammonia-lyase.

Histidine ammonia-lyase requires a 4-methylidene-imidazole-5-one group (MIO) that is produced autocatalytically by a cyclization and dehydration step in a 3-residue loop of the polypeptide. The crystal structures of three mutants have been established. Two mutants were inactive and failed to form MIO, but remained unchanged elsewhere. The third mutant showed very low activity and formed MIO, although it differed from an MIO-less mutant only by an additional 329-C(beta) atom. This atom forms one constraint during MIO formation, the other being the strongly connected Asp145. An exploration of the conformational space of the MIO-forming loop showed that the cyclization is probably enforced by a mechanic compression in a late stage of chain folding and is catalyzed by a well-connected internal water molecule. The cyclization of the respective 3-residue loop of green fluorescent protein is likely to occur in a similar reaction.     

Increasing stability reduces conformational heterogeneity in a protein folding intermediate ensemble

A multi-site, time-resolved fluorescence resonance energy transfer methodology has been used to study structural heterogeneity in a late folding intermediate ensemble, IL, of the small protein barstar. Four different intra-molecular distances have been measured within the structural components of IL. The IL ensemble is shown to consist of different sub-populations of molecules, in each of which one or more of the four distances are native-like and the remaining distances are unfolded-like. In very stable conditions that favor formation of IL, all four distances are native-like in most molecules. In less stable conditions, one or more distances are unfolded-like. As stability is decreased, the proportion of molecules with unfolded-like distances increases. Thus, the results show that protein folding intermediates are ensembles of different structural forms, and they demonstrate that conformational entropy increases as structures become less stable. These observations provide direct experimental evidence in support of a basic tenet of energy landscape theory for protein folding, that available conformational space, as represented by structural heterogeneity in IL, becomes restricted as the stability is increased. The results also vindicate an important prediction of energy landscape theory, that different folding pathways may become dominant under different folding conditions. In more stable folding conditions, uniformly native-like compactness is achieved during folding to IL, whereas in less stable conditions, uniformly native-like compactness is achieved only later during the folding of IL to N.     

Reduced temperature dependence of collective conformational opening in a hyperthermophile rubredoxin.

Spatially localized differences in the conformational dynamics of the rubredoxins from the hyperthermophile Pyrococcus furiosus (Pf) and the mesophile Clostridium pasteurianum (Cp) are monitored via amide exchange measurements. As shown previously for the hyperthermophile protein, nearly all backbone amides of the Cp rubredoxin exhibit EX(2) hydrogen exchange kinetics with conformational opening rates of >1 s(-)(1). Significantly slower amide exchange is observed for Pf rubredoxin in the region surrounding the metal site and the proximal end of the three-stranded beta-sheet, while for the rest of the structure, the exchange rates at 23 degrees C are similar for both proteins. For the multiple-turn region comprising residues 14-32 in both rubredoxins, the uniformity of both the exchange rate constants and the values of the activation energy at the slowly exchanging sites is consistent with a model of solvent exposure via a subglobal cooperative conformational opening. In contrast to the common expectation of increased rigidity in the hyperthermophile proteins, below room temperature Pf rubredoxin exhibits a larger apparent flexibility in this multiple-turn region. The smaller enthalpy for the conformational opening process of this region in Pf rubredoxin reflects the much weaker temperature dependence of the underlying conformational equilibrium in the hyperthermophile protein.     

Rigid-cluster models of conformational transitions in macromolecular machines and assemblies

We present a rigid-body-based technique (called rigid-cluster elastic network interpolation) to generate feasible transition pathways between two distinct conformations of a macromolecular assembly. Many biological molecules and assemblies consist of domains which act more or less as rigid bodies during large conformational changes. These collective motions are thought to be strongly related with the functions of a system. This fact encourages us to simply model a macromolecule or assembly as a set of rigid bodies which are interconnected with distance constraints. In previous articles, we developed coarse-grained elastic network interpolation (ENI) in which, for example, only Calpha atoms are selected as representatives in each residue of a protein. We interpolate distance differences of two conformations in ENI by using a simple quadratic cost function, and the feasible conformations are generated without steric conflicts. Rigid-cluster interpolation is an extension of the ENI method with rigid-clusters replacing point masses. Now the intermediate conformations in an anharmonic pathway can be determined by the translational and rotational displacements of large clusters in such a way that distance constraints are observed. We present the derivation of the rigid-cluster model and apply it to a variety of macromolecular assemblies. Rigid-cluster ENI is then modified for a hybrid model represented by a mixture of rigid clusters and point masses. Simulation results show that both rigid-cluster and hybrid ENI methods generate sterically feasible pathways of large systems in a very short time. For example, the HK97 virus capsid is an icosahedral symmetric assembly composed of 60 identical asymmetric units. Its original Hessian matrix size for a Calpha coarse-grained model is >(300,000)(2). However, it reduces to (84)(2) when we apply the rigid-cluster model with icosahedral symmetry constraints. The computational cost of the interpolation no longer scales heavily with the size of structures; instead, it depends strongly on the minimal number of rigid clusters into which the system can be decomposed.     

Evaluation of conformational changes in diabetes‐associated mutation in insulin a chain: A molecular dynamics study

Insulin plays a central role in the regulation of metabolism in humans. Mutations in the insulin gene can impair the folding of its precursor protein, proinsulin, and cause permanent neonatal‐onset diabetes mellitus known as Mutant INS‐gene induced Diabetes of Youth (MIDY) with insulin deficiency. To gain insights into the molecular basis of this diabetes‐associated mutation, we perform molecular dynamics simulations in wild‐type and mutant (Cys^A7 to Tyr or C(A7)Y) insulin A chain in aqueous solutions. The C(A7)Y mutation is one of the identified mutations that impairs the protein folding by substituting the cysteine residue which is required for the disulfide bond formation. A comparative analysis reveals structural differences between the wild‐type and the mutant conformations. The analyzed mutant insulin A chain forms a metastable state with major effects on its N‐terminal region. This suggests that MIDY mutant involves formation of a partially folded intermediate with conformational change in N‐terminal region in A chain that generates flexible N‐terminal domain. This may lead to the abnormal interactions with other proinsulins in the aggregation process. Proteins 2015; 83:662–669. © 2015 Wiley Periodicals, Inc.