Modeling large-scale dynamics of proteins

Normal-mode driven exploration of protein domain motions

Domain motions involved in the function of proteins can often be well described as a combination of motions along a handfull of low-frequency modes, that is, with the values of a few normal coordinates. This means that, when the functional motion of a protein is unknown, it should prove possible to predict it, since it amounts to guess a few values. However, without the help of additional experimental data, using normal coordinates for generating accurate conformers far away from the initial one is not so straightforward. To do so, a new approach is proposed: instead of building conformers directly with the values of a subset of normal coordinates, they are built in two steps, the conformer built with normal coordinates being just used for defining a set of distance constraints, the final conformer being built so as to match them. Note that this approach amounts to transform the problem of generating accurate protein conformers using normal coordinates into a better known one: the distance-geometry problem, which is herein solved with the help of the ROSETTA software. In the present study, this approach allowed to rebuild accurately six large amplitude conformational changes, using at most six low-frequency normal coordinates. As a consequence of the low-dimensionality of the corresponding subspace, random exploration also proved enough for generating low-energy conformers close to the known end-point of the conformational change of the LAO binding protein, lysozyme T4 and adenylate kinase.

Space warping order parameters and symmetry: application to multiscale simulation of macromolecular assemblies

Coarse-grained features of macromolecular assemblies are understood via a set of order parameters (OPs) constructed in terms of their all-atom configuration. OPs are shown to be slowly changing in time and capture the large-scale spatial features of macromolecular assemblies. The relationship of these variables to the classic notion of OPs based on symmetry breaking phase transitions is discussed. OPs based on space warping transformations are analyzed in detail as they naturally provide a connection between overall structure of an assembly and all-atom configuration. These OPs serve as the basis of a multiscale analysis that yields Langevin equations for OP dynamics. In this context, the characteristics of OPs and PCA modes are compared. The OPs enable efficient all-atom multiscale simulations of the dynamics of macromolecular assemblies in response to changes in microenvironmental conditions, as demonstrated on the structural transitions of cowpea chlorotic mottle virus capsid (CCMV) and RNA of the satellite tobacco mosaic virus (STMV).

Molecular dynamics/order parameter extrapolation for bionanosystem simulations

A multiscale approach, molecular dynamics/order parameter extrapolation (MD/OPX), to the all-atom simulation of large bionanosystems is presented. The approach starts with the introduction of a set of order parameters (OPs) automatically generated with orthogonal polynomials to characterize the nanoscale features of bionanosystems. The OPs are shown to evolve slowly via Newton's equations, and the all-atom multiscale analysis (AMA) developed earlier (Miao and Ortoleva, J Chem Phys 2006, 125, 44901) demonstrates the existence of their stochastic dynamics, which serve as the justification for our MD/OPX approach. In MD/OPX, a short MD run estimates the rate of change of the OPs, which is then used to extrapolate the state of the system over time that is much longer than the 10(-14) second timescale of fast atomic vibrations and collisions. The approach is implemented in NAMD and demonstrated on cowpea chlorotic mottle virus (CCMV) capsid structural transitions (STs). It greatly accelerates the MD code and its underlying all-atom description of the nanosystems enables the use of a universal interatomic force field, avoiding recalibration with each new application as needed for coarse-grained models. The source code of MD/OPX is distributed free of charge at https://simtk.org/home/mdopx and a web portal will be available via http://sysbio.indiana.edu/virusx.

All-atom multiscaling and new ensembles for dynamical nanoparticles

Viruses and other nanoparticles have mixed microscopic/macroscopic character. Thus it is natural to develop an understanding of their dynamics via a multiscale analysis of the Liouville equation following prescriptions introduced for the study of Brownian motion. However, the internal dynamics of the atoms constituting a nanoparticle introduces conceptual and technical difficulties associated with a description involving both the atomistic and nanometer scale properties of these systems and the potential overcounting of degrees of freedom. To overcome these difficulties we introduce a "nanocanonical" ensemble method to facilitate the multiscale analysis of the all-atom Liouville equation. Our approach overcomes technical difficulties associated with the removal of secular behavior, which leads to Fokker-Planck-type equations. Our approach ensures removal of all secular behavior in the N-atom probability density and not just that of a reduced distribution. Being based on a calibrated interatomic force field, our method has the potential to yield parameter-free universal models for nanoparticle dynamics including viral migration in complex media and viral phase transitions and disassembly.

Viral structural transitions: an all-atom multiscale theory

An all-atom theory of viral structural transitions (STs) is developed based on a multiscale analysis of the N-atom Liouville equation. The approach yields an understanding of viral STs from first principles and a calibrated interatomic force field. To carry out the multiscale analysis, we introduce slow variables characterizing the whole-virus dynamics. Use of the "nanocanonical ensemble" technique and the fundamental hypothesis of statistical mechanics (i.e., the equivalence of long-time and ensemble averages) is shown to imply a Fokker-Planck equation yielding the coarse-grained evolution of the slow variables. As viral STs occur on long time scales, transition state theory is used to estimate the energy barrier of transition between free energy wells implied by observed hysteresis in viral STs. Its application to Nudaurelia capensis omega virus provides an upper bound on the free energy barrier when a single dilatational order parameter is used. The long time scale of viral STs is shown to follow from the aggregate effect of inertia, energy barrier, and entropic effects. Our formulation can be generalized for multiple order parameter models to account for lower free energy barrier pathways for transition. The theory with its all-atom description can be applied to nonviral nanoparticles as well.

Escherichia coli adenylate kinase dynamics: comparison of elastic network model modes with mode-coupling (15)N-NMR relaxation data

The dynamics of adenylate kinase of Escherichia coli (AKeco) and its complex with the inhibitor AP(5)A, are characterized by correlating the theoretical results obtained with the Gaussian Network Model (GNM) and the anisotropic network model (ANM) with the order parameters and correlation times obtained with Slowly Relaxing Local Structure (SRLS) analysis of (15)N-NMR relaxation data. The AMPbd and LID domains of AKeco execute in solution large amplitude motions associated with the catalytic reaction Mg(+2)*ATP + AMP --> Mg(+2)*ADP + ADP. Two sets of correlation times and order parameters were determined by NMR/SRLS for AKeco, attributed to slow (nanoseconds) motions with correlation time tau( perpendicular) and low order parameters, and fast (picoseconds) motions with correlation time tau( parallel) and high order parameters. The structural connotation of these patterns is examined herein by subjecting AKeco and AKeco*AP(5)A to GNM analysis, which yields the dynamic spectrum in terms of slow and fast modes. The low/high NMR order parameters correlate with the slow/fast modes of the backbone elucidated with GNM. Likewise, tau( parallel) and tau( perpendicular) are associated with fast and slow GNM modes, respectively. Catalysis-related domain motion of AMPbd and LID in AKeco, occurring per NMR with correlation time tau( perpendicular), is associated with the first and second collective slow (global) GNM modes. The ANM-predicted deformations of the unliganded enzyme conform to the functional reconfiguration induced by ligand-binding, indicating the structural disposition (or potential) of the enzyme to bind its substrates. It is shown that NMR/SRLS and GNM/ANM analyses can be advantageously synthesized to provide insights into the molecular mechanisms that control biological function.

Comparison of full-atomic and coarse-grained models to examine the molecular fluctuations of c-AMP dependent protein kinase

Molecular fluctuations of the native conformation of c-AMP dependent protein kinase (cAPK) have been investigated with three different approaches. The first approach is the full atomic normal mode analysis (NMA) with empirical force fields. The second and third approaches are based on a coarse-grained model with a single single-parameter- harmonic potential between close residues in the crystal structure of the molecule without any residue specificity. The second method calculates only the magnitude of fluctuations whereas the third method is developed to find the directionality of the fluctuations which are essential to understand the functional importance of biological molecules. The aim, in this study, is to determine whether using such coarse-grained models are appropriate for elucidating the global dynamic characteristics of large proteins which reduces the size of the system at least by a factor of ten. The mean-square fluctuations of C(alpha) atoms and the residue cross-correlations are obtained by three approaches. These results are then compared to test the results of coarse grained models on the overall collective motions. All three of the approaches show that highly flexible regions correspond to the activation and solvent exposed loops, whereas the conserved residues (especially in substrate binding regions) exhibit almost no flexibility, adding stability to the structure. The anti-correlated motions of the two lobes of the catalytic core provide flexibility to the molecule. High similarities among the results of these methods indicate that the slowest modes governing the most global motions are preserved in the coarse grained models for proteins. This finding may suggest that the general shapes of the structures are representative of their dynamic characteristics and the dominant motions of protein structures are robust at coarse-grained levels.

New space warping method for the simulation of large-scale macromolecular conformational changes

A space warping method, facilitating the modeling of large-scale conformational changes in mesoscopic systems, is presented. The method uses a set of "global (or collective) coordinates" that capture overall behavior, in conjunction with the set of atomic coordinates. Application of the space warping method to energy minimization is discussed. Several simulations where the method is used to determine the energy minimizing structures of simple central force systems are analyzed. Comparing the results and behavior of the space warping method to simulations involving atomic coordinates only, it is found that the space warping method scales better with system size and also finds lower minima when the potential energy surface has multiple minima. It is shown that the transformation of [Ala16]+ in vacuo from linear to globular is captured efficiently using the space warping method.

Conformational change of proteins arising from normal mode calculations

A normal mode analysis of 20 proteins in 'open' or 'closed' forms was performed using simple potential and protein models. The quality of the results was found to depend upon the form of the protein studied, normal modes obtained with the open form of a given protein comparing better with the conformational change than those obtained with the closed form. Moreover, when the motion of the protein is a highly collective one, then, in all cases considered, there is a single low-frequency normal mode whose direction compares well with the conformational change. When it is not, in most cases there is still a single low-frequency normal mode giving a good description of the pattern of the atomic displacements, as they are observed experimentally during the conformational change. Hence a lot of information on the nature of the conformational change of a protein is often found in a single low-frequency normal mode of its open form. Since this information can be obtained through the normal mode analysis of a model as simple as that used in the present study, it is likely that the property captured by such an analysis is for the most part a property of the shape of the protein itself. One of the points that has to be clarified now is whether or not amino acid sequences have been selected in order to allow proteins to follow a single normal mode direction, as least at the very beginning of their conformational change.

Building-block approach for determining low-frequency normal modes of macromolecules

Normal mode analysis of proteins of various sizes, ranging from 46 (crambin) up to 858 residues (dimeric citrate synthase) were performed, by using standard approaches, as well as a recently proposed method that rests on the hypothesis that low-frequency normal modes of proteins can be described as pure rigid-body motions of blocks of consecutive amino-acid residues. Such a hypothesis is strongly supported by our results, because we show that the latter method, named RTB, yields very accurate approximations for the low-frequency normal modes of all proteins considered. Moreover, the quality of the normal modes thus obtained depends very little on the way the polypeptidic chain is split into blocks. Noteworthy, with six amino-acids per block, the normal modes are almost as accurate as with a single amino-acid per block. In this case, for a protein of n residues and N atoms, the RTB method requires the diagonalization of an n x n matrix, whereas standard procedures require the diagonalization of a 3N x 3N matrix. Being a fast method, our approach can be useful for normal mode analyses of large systems, paving the way for further developments and applications in contexts for which the normal modes are needed frequently, as for example during molecular dynamics calculations.