Sculpting proteins interactively: continual energy minimization embedded in a graphical modeling system.

We describe a new paradigm for modeling proteins in interactive computer graphics systems--continual maintenance of a physically valid representation, combined with direct user control and visualization. This is achieved by a fast algorithm for energy minimization, capable of real-time performance on all atoms of a small protein, plus graphically specified user tugs. The modeling system, called Sculpt, rigidly constrains bond lengths, bond angles, and planar groups (similar to existing interactive modeling programs), while it applies elastic restraints to minimize the potential energy due to torsions, hydrogen bonds, and van der Waals and electrostatic interactions (similar to existing batch minimization programs), and user-specified springs. The graphical interface can show bad and/or favorable contacts, and individual energy terms can be turned on or off to determine their effects and interactions. Sculpt finds a local minimum of the total energy that satisfies all the constraints using an augmented Lagrange-multiplier method; calculation time increases only linearly with the number of atoms because the matrix of constraint gradients is sparse and banded. On a 100-MHz MIPS R4000 processor (Silicon Graphics Indigo), Sculpt achieves 11 updates per second on a 20-residue fragment and 2 updates per second on an 80-residue protein, using all atoms except non-H-bonding hydrogens, and without electrostatic interactions. Applications of Sculpt are described: to reverse the direction of bundle packing in a designed 4-helix bundle protein, to fold up a 2-stranded beta-ribbon into an approximate beta-barrel, and to design the sequence and conformation of a 30-residue peptide that mimics one partner of a protein subunit interaction. Computer models that are both interactive and physically realistic (within the limitations of a given force field) have 2 significant advantages: (1) they make feasible the modeling of very large changes (such as needed for de novo design), and (2) they help the user understand how different energy terms interact to stabilize a given conformation. The Sculpt paradigm combines many of the best features of interactive graphical modeling, energy minimization, and actual physical models, and we propose it as an especially productive way to use current and future increases in computer speed.     

Inclusion of solvation free energy with molecular mechanics energy: alanyl dipeptide as a test case.

A combined force field of molecular mechanics and solvation free energy is tested by carrying out energy minimization and molecular dynamics on several conformations of the alanyl dipeptide. Our results are qualitatively consistent with previous experimental and computational studies, in that the addition of solvation energy stabilizes the C5 conformation of the alanyl dipeptide relative to the C7.     

Pseudocontact shifts as constraints for energy minimization and molecular dynamics calculations on solution structures of paramagnetic metalloproteins

The pseudocontact shifts of NMR signals, which arise from the magnetic susceptibility anisotropy of paramagnetic molecules, have been used as structural constraints under the form of a pseudopotential in the SANDER module of the AMBER 4.1 molecular dynamics software package. With this procedure, restrained energy minimization (REM) and restrained molecular dynamics (RMD) calculations can be performed on structural models by using pseudocontact shifts. The structure of the cyanide adduct of the Met80Ala mutant of the yeast iso-1-cytochrome c has been used for successfully testing the calculations. For this protein, a family of structures is available, which was obtained by using NOE and pseudocontact shifts as constraints in a distance geometry program. The structures obtained by REM and RMD calculations with the inclusion of pseudocontact shifts are analyzed. Proteins 29:68–76, 1997. © 1997 Wiley-Liss, Inc.     

Bundles of amphipathic transmembrane alpha-helices as a structural motif for ion-conducting channel proteins: studies on sodium channels and acetylcholine receptors

Channel proteins are transmembrane symmetric (or pseudosymmetric) oligomers organized around a central ionic pore. We present here a molecular model of the pore forming structures of two channel proteins with different primary structures and oligomeric size: the voltage-sensitive sodium channel and the nicotinic cholinergic receptor. We report low-energy arrangements of alpha-helical bundles calculated by semiempiricial potential energy functions and optimization routines and further refined using molecular dynamics. The ion-conducting pore is considered to be a symmetric or pseudosymmetric homooligomer of 3-5 amphipathic alpha-helices arranged such that the polar residues line a central hydrophilic pathway and the apolar residues face the hydrophobic bilayer interior. The channel lining exposes either charged (Asp, Glu, Arg, Lys) or polar-neutral (Ser, Thr) residues. A bundle of four parallel helices constrained to C4 symmetry, the helix axis aligned with the symmetry axis, and the helices constrained to idealized dihedral angles, produces a structure with a pore of the size inferred for the sodium channel protein (area approximately 16 A2). Similarly, a pentameric array optimized with constraints to maintain C5 symmetry and backbone torsions characteristic of alpha-helices adopts a structure that appears well suited to form the lining of the nicotinic cholinergic receptor (pore area approximately 46 A2). Thus, bundles of amphipathic alpha-helices satisfy the structural, energetic, and dynamic requirements to be the molecular structural motif underlying the function of ionic channels.     

3Drefine: Consistent protein structure refinement by optimizing hydrogen bonding network and atomic‐level energy minimization

One of the major limitations of computational protein structure prediction is the deviation of predicted models from their experimentally derived true, native structures. The limitations often hinder the possibility of applying computational protein structure prediction methods in biochemical assignment and drug design that are very sensitive to structural details. Refinement of these low‐resolution predicted models to high‐resolution structures close to the native state, however, has proven to be extremely challenging. Thus, protein structure refinement remains a largely unsolved problem. Critical assessment of techniques for protein structure prediction (CASP) specifically indicated that most predictors participating in the refinement category still did not consistently improve model quality. Here, we propose a two‐step refinement protocol, called 3Drefine, to consistently bring the initial model closer to the native structure. The first step is based on optimization of hydrogen bonding (HB) network and the second step applies atomic‐level energy minimization on the optimized model using a composite physics and knowledge‐based force fields. The approach has been evaluated on the CASP benchmark data and it exhibits consistent improvement over the initial structure in both global and local structural quality measures. 3Drefine method is also computationally inexpensive, consuming only few minutes of CPU time to refine a protein of typical length (300 residues). 3Drefine web server is freely available at http://sysbio.rnet.missouri.edu/3Drefine/ . Proteins 2013. © 2012 Wiley Periodicals, Inc.     

A simple formalism on dynamics of proteins on potential energy landscapes

We present a simple formalism for the dynamics of proteins on a potential energy landscape, using connectedness of configurational domains as an order parameter. This formalism clearly shows that the energy bias required to form a unit correct contact toward the native configuration of a two-state folder, to overcome Levinthal's paradox, is E(bias) congruent with RT ln 2. This result agrees well with earlier studies and indicates that the bias is mainly due to hydrophobic interaction. Further investigations have shown that the landscape funnel could be experimentally mapped onto a two-dimensional space formed by denaturant concentration and the connectedness of configurational domains. The theoretical value of the depth-of-folding funnel in terms of denaturant concentration has been calculated for a model protein (P450cam), which agrees well with the experimental value. Using our model, it is also possible to explain the turnover nature of heat-capacity change upon unfolding of proteins and the existence of enthalpy and entropy convergence temperatures during unfolding without any strict assumptions as proposed in earlier studies.     

Incoherent neutron scattering of copper azurin: a comparison with molecular dynamics simulation results

The low-frequency dynamics of copper azurin has been studied at different temperatures for a dry and deuterium hydrated sample by incoherent neutron scattering and the experimental results have been compared with molecular dynamics (MD) simulations carried out in the same temperature range. Experimental Debye-Waller factors are consistent with a dynamical transition at approximately 200 K which appears partially suppressed in the dry sample. Inelastic and quasielastic scattering indicate that hydration water modulates both vibrational and diffusive motions. The low-temperature experimental dynamical structure factor of the hydrated protein shows an excess of inelastic scattering peaking at about 3 meV and whose position is slightly shifted downwards in the dry sample. Such an excess is reminiscent of the “boson peak” observed in glass-like materials. This vibrational peak is quite well reproduced by MD simulations, although at a lower energy. The experimental quasielastic scattering of the two samples at 300 K shows a two-step relaxation behaviour with similar characteristic times, while the corresponding intensities differ only by a scale factor. Also, MD simulations confirm the two-step diffusive trend, but the slow process seems to be characterized by a decay faster than the experimental one. Comparison with incoherent neutron scattering studies carried out on proteins having different structure indicates that globular proteins display common elastic, quasielastic and inelastic features, with an almost similar hydration dependence, irrespective of their secondary and tertiary structure. Received: 12 October 1998 / Revised version: 19 February 1999 / Accepted: 1 March 1999     

Relationship between energy distribution and fold stability: Insights from molecular dynamics simulations of native and mutant proteins

Most proteins must fold to a well-defined structure with a minimal stability to perform their function. Here we use a simple, molecular dynamics-based, energy decomposition approach to map the principal energetic interactions in a set of proteins representative of different folds. This work involves the all-atom simulation and analysis of the native structures and mutants of five different proteins representative of an all-alpha (yACPB, Protein A), all-beta (SH3), and a mixed alpha/beta fold (Proteins G and L). Given a certain structure, a native sequence and a set of mutants, we show that our model discriminates the ability of a mutation to yield a more or less stable protein, in agreement with experimental data, catching the principal energetic determinants of protein stabilization. Our approach identifies the interaction determinants responsible to define a fold and shows that mutations can either modulate the strength of pair-wise coupling between residues important for folding, or modify the profile of the principal interactions. Furthermore, we address the question of how to evaluate the fitness of a sequence to a given structure by comparing the information contained in the energy map, which recapitulates the chemistry of the sequence, to that contained in the contact map, which recapitulates the fold topology. The results show that the better fit between the energetic properties of the sequence and the fold topology corresponds to a higher stabilization of the protein. We discuss the relevance of these observations to the analysis of protein designability and to the rational evolution of new sequences.     

Hydrophilicity of cavities in proteins

Water molecules inside cavities in proteins constitute integral parts of the structure. We have sought a quantitative measure of the hydrophilicity of the cavities by calculating energies and free energies of introducing a water molecule into these cavities. A threshold value of the water-protein interaction energy at −12 kcal/mol was found to be able to distinguish hydrated from empty cavities. It follows that buried waters have entropy comparable to that of liquid water or ice. A simple consistent picture of the energetics of the buried waters provided by this study enabled us to address the reliability of buried waters assigned in experiments.     

Protein structure refinement based on paramagnetic NMR shifts: applications to wild-type and mutant forms of cytochrome c.

A new approach to NMR solution structure refinement is introduced that uses paramagnetic effects on nuclear chemical shifts as constraints in energy minimization or molecular dynamics calculations. Chemical shift differences between oxidized and reduced forms of horse cytochrome c for more than 300 protons were used as constraints to refine the structure of the wild-type protein in solution and to define the structural changes induced by a Leu 94 to Val mutation. A single round of constrained minimization, using the crystal structure as the starting point, converged to a low-energy structure with an RMS deviation between calculated and observed pseudo-contact shifts of 0.045 ppm, 7.5-fold lower than the starting structure. At the same time, the procedure provided stereospecific assignments for more than 45 pairs of methylene protons and methyl groups. Structural changes caused by the mutation were determined to a precision of better than 0.3 A. Structure determination based on dipolar paramagnetic (pseudocontact) shifts is applicable to molecules containing anisotropic paramagnetic centers with short electronic relaxation times, including numerous naturally occurring metalloproteins, as well as proteins or nucleic acids to which a paramagnetic metal ion or ligand may be attached. The long range of paramagnetic shift effects (up to 20 A from the iron in the case of cytochrome c) provides global structural constraints, which, in conjunction with conventional NMR distance and dihedral angle constraints, will enhance the precision of NMR solution structure determination.     

Empirical free energy calculation: comparison to calorimetric data.

An effective free energy potential, developed originally for binding free energy calculation, is compared to calorimetric data on protein unfolding, described by a linear combination of changes in polar and nonpolar surface areas. The potential consists of a molecular mechanics energy term calculated for a reference medium (vapor or nonpolar liquid), and empirical terms representing solvation and entropic effects. It is shown that, under suitable conditions, the free energy function agrees well with the calorimetric expression. An additional result of the comparison is an independent estimate of the side-chain entropy loss, which is shown to agree with a structure-based entropy scale. These findings confirm that simple functions can be used to estimate the free energy change in complex systems, and that a binding free energy evaluation model can describe the thermodynamics of protein unfolding correctly. Furthermore, it is shown that folding and binding leave the sum of solute-solute and solute-solvent van der Waals interactions nearly invariant and, due to this invariance, it may be advantageous to use a nonpolar liquid rather than vacuum as the reference medium.     

Progressive structure-based alignment of homologous proteins: Adopting sequence comparison strategies

Comparison of multiple protein structures has a broad range of applications in the analysis of protein structure, function and evolution. Multiple structure alignment tools (MSTAs) are necessary to obtain a simultaneous comparison of a family of related folds. In this study, we have developed a method for multiple structure comparison largely based on sequence alignment techniques. A widely used Structural Alphabet named Protein Blocks (PBs) was used to transform the information on 3D protein backbone conformation as a 1D sequence string. A progressive alignment strategy similar to CLUSTALW was adopted for multiple PB sequence alignment (mulPBA). Highly similar stretches identified by the pairwise alignments are given higher weights during the alignment. The residue equivalences from PB based alignments are used to obtain a three dimensional fit of the structures followed by an iterative refinement of the structural superposition. Systematic comparisons using benchmark datasets of MSTAs underlines that the alignment quality is better than MULTIPROT, MUSTANG and the alignments in HOMSTRAD, in more than 85% of the cases. Comparison with other rigid-body and flexible MSTAs also indicate that mulPBA alignments are superior to most of the rigid-body MSTAs and highly comparable to the flexible alignment methods.     

Membrane-bound ARF1 peptide: interpretation of neutron diffraction data by molecular dynamics simulation methods

Adenosine diphosphate ribosylation factor-1 (ARF1) is activated by cell membrane binding of a self-folding N-terminal domain. We have previously presented four possible conformations of the membrane bound, human ARF1 N-terminal peptide in planar lipid bilayers of DOPC and DOPG (7:3 molar ratio), determined from lamellar neutron diffraction and circular dichroism data. In this paper we analyse the four possible conformations by molecular dynamics simulations. The aim of these simulations was to use MD to distinguish which of the four possible membrane bound structures was the most likely. The most likely conformation was determined according to the following criteria: (a) location of label positions on the peptide in relation to the bilayer, (b) lowest mean square displacement from the initial structure, (c) lowest system energy, (d) most peptide-lipid headgroup hydrogen bonding, (e) analysis of phi/psi angles of the peptide. These findings demonstrate the application of molecular dynamics simulations to explore neutron diffraction data.     

A novel parameterization scheme for energy equations and its use to calculate the structure of protein molecules

A novel scheme for the parameterization of a type of “potential energy” function for protein molecules is introduced. The function is parameterized based on the known conformations of previously determined protein structures and their sequence similarity to a molecule whose conformation is to be calculated. Once parameterized, minima of the potential energy function can be located using a version of simulated annealing which has been previously shown to locate global and near-global minima with the given functional form. As a test problem, the potential was parameterized based on the known structures of the rubredoxins from Desulfovibrio vulgaris, Desulfovibrio desulfuricans, and Clostridium pasteurianum, which vary from 45 to 54 amino acids in length, and the sequence alignments of these molecules with the rubredoxin sequence from Desulfovibrio gigas. Since the Desulfovibrio gigas rubredeoxin conformation has also been determined, it is possible to check the accuracy of the results. Ten simulated-annealing runs from random starting conformations were performed. Seven of the 10 resultant conformations have an all-C_α rms deviation from the crystallographically determined conformation of less than 1.7 Å. For five of the structures, the rms deviation is less than 0.8 Å. Four of the structures have conformations which are virtually identical to each other except for the position of the carboxy-terminal residue. This is also the conformation which is achieved if the determined crystal structure is minimized with the same potential. The all-C_α rms difference between the crystal and minimized crystal structures is 0.6 Å. It is further observed that the “energies” of the structures according to the potential function exhibit a strong correlation with rms deviation from the native structure. The conformations of the individual model structures and the computational aspects of the modeling procedure are discussed. © 1993 Wiley-Liss, Inc.     

Thermitase, a thermostable subtilisin: comparison of predicted and experimental structures and the molecular cause of thermostability

The Subtilisin family of proteases has four members of known sequence and structure: subtilisin Carlsberg, Subtilisin novo, proteinase K, and thermitase. Using thermitase as a test case, we ask two questions. How good are methods for model building a three-dimensional structure of a protein based on sequence homology to a known structure? And what are the molecular causes of thermostability? First, we compare predicted models of thermitase, refined by energy minimization and varied by molecular dynamics, with the preliminary crystal structure. The predictions work best in the conserve structural core and less well in seven loop regions involving insertions and deletions relative to Subtilisin. Here, variation of loop regions by molecular dynamics simulation in vacuo followed by energy minimization does not improve the prediction since we find no correlation between in vacuo energy and correctness of structure when comparing local energy minima. Second, in order to identify the molecular case of thermostability we confront hypotheses erived by calculation of the details of interatomic interactions with inactivation experiments. As a result, we can exclude salt bridges and hydrophobic interactions as main cause of thermostability. Based on a combination of theoretical and experimental evidence, the unusually tight binding of calcium by thermitase emerges as the most likely single influence responsible for its increased thermostability.     

A computational comparison of the atomic models of the actomyosin interface

Several atomic models of the actomyosin interface have been proposed based on the docking together of their component structures using electron microscopy and resonance energy-transfer measurements. Although these models are in approximate agreement in the location of the binding interfaces when myosin is tightly bound to actin, their relationships to molecular docking simulations based on computational free-energy calculations are investigated here. Both rigid-docking and flexible-docking conformational search strategies were used to identify free-energy minima at the interfaces between atomic models of myosin and actin. These results suggest that the docking model produced by resonance energy-transfer data is closer to a free-energy minimum at the interface than are the available atomic models based on electron microscopy. The conformational searches were performed using both scallop and chicken skeletal muscle myosins and identified similarly oriented actin-binding interfaces that serve to validate that these models are at the global minimum. These results indicate that the existing docking models are close to but not precisely at the lowest-energy initial contact site for strong binding between myosin and actin that should represent an initial contact between the two proteins; therefore, conformational changes are likely to be important during the transition to a strongly bound complex.     

Protein dynamics in solution and powder measured by incoherent elastic neutron scattering: the influence of Q-range and energy resolution

Incoherent elastic neutron scattering (IENS) has been widely used to measure intramolecular atomic mean square displacements (MSDs) of proteins in powder and in solution. The instrumental energy resolution and the wave vector transfer (Q-range) determine, respectively, the time and length scales of observable motions. In order to investigate contributions of diffusive motions to MSDs measured by this method, we calculated the elastic intensity for several simple scattering functions. We showed that continuous translational diffusion contributes to MSDs in a Q-range where the energy width of the scattering function is of the order of the instrumental energy resolution. We discuss the choice of instruments adapted to focus on intramolecular motions in the presence of solvent or global macromolecular diffusion. The concepts developed are applied to interpret experimental data from H₂O- and D₂O-hydrated proteins. Finally, analogies between the Gaussian approximation in IENS and the Guinier approximation in small-angle scattering are discussed.     

A tale of two paralogs: human Transformer2 proteins with differential RNA-binding affinities

The Transformer2 (Tra2) proteins in humans are homologues of the Drosophila Tra2 protein. One of the two RNA-binding paralogs, Tra2β, has been very well-studied over the past decade, but not much is known about Tra2α. It was very recently shown that the two proteins demonstrate the phenomenon of paralog compensation. Here, we provide a structural basis for this genetic backup circuit, using molecular modelling and dynamics studies. We show that the two proteins display similar binding specificities, but differential affinities to a short GAA-rich RNA stretch. Starting from the 6-nucleotide RNA in the solution structure, close to 4000 virtual mutations were modelled on RNA and the domain–RNA interactions were studied after energy minimisation to convergence. Separately, another known 13-nucleotide stretch was docked and the domain–RNA interactions were observed through a 100-ns dynamics trajectory. We have also demonstrated the ‘compensatory’ mechanism at the level of domains in one of the domain repeat-containing RNA-binding proteins.     

Differences in the adsorption and diffusion behaviour of water and non-polar gases in nanoporous carbon: role of cooperative effects of pore confinement and hydrogen bonding

We investigate the effect of pore confinement and molecular geometry on the adsorption and self-diffusion of H₂O, CO₂, Ar, CH₄, C₃H₆, SF₆ and C₅H₁₂, in a realistic model of nanoporous silicon carbide derived carbon (SiC-DC), constructed using hybrid reverse Monte Carlo simulation. Adsorption isotherms, adsorbate–adsorbate and adsorbate–adsorbent contributions to the isosteric heat of adsorption are determined to study the effect of pore confinement, microporosity and molecular geometry on adsorption of these gases. We describe the cooperative effect of pore confinement and hydrogen bonding on the formation of water clusters and anomalous adsorption behaviour of water compared with non-polar gases. We find that, in contrast to literature results based on the slit-pore model, pore-filling does not occur below the saturation pressure in hydrophobic amorphous carbon materials such as SiC-DC and activated carbon fibre. We also compare self-diffusivities and activation energy barriers of water and non-polar gases in the microporous structure of SiC-DC to identify underlying correlations with molecular properties. We demonstrate that the self-diffusivity of water deviates considerably from the correlation between diffusivity and molecular kinetic diameter observed for non-polar gases. This is attributed to the reduced diffusivity of water, and its relatively large energy barrier at high loadings despite its small kinetic diameter, which is due to the blocking effect of water clusters at pore entries.