Cezary Czaplewski

Cumulant-based expressions for the multibody terms for the correlation between local and electrostatic interactions in the united-residue force field

A general method to derive site-site or united-residue potentials is presented. The basic principle of the method is the separation of the degrees of freedom of a system into the primary and secondary ones. The primary degrees of freedom describe the basic features of the system, while the secondary ones are averaged over when calculating the potential of mean force, which is hereafter referred to as the restricted free energy (RFE) function. The RFE can be factored into one-, two-, and multibody terms, using the cluster-cumulant expansion of Kubo. These factors can be assigned the functional forms of the corresponding lowest-order nonzero generalized cumulants, which can, in most cases, be evaluated analytically, after making some simplifying assumptions. This procedure to derive coarse-grain force fields is very valuable when applied to multibody terms, whose functional forms are hard to deduce in another way (e.g., from structural databases). After the functional forms have been derived, they can be parametrized based on the RFE surfaces of model systems obtained from all-atom models or on the statistics derived from structural databases. The approach has been applied to our united-residue force field for proteins. Analytical expressions were derived for the multibody terms pertaining to the correlation between local and electrostatic interactions within the polypeptide backbone; these expressions correspond to up to sixth-order terms in the cumulant expansion of the RFE. These expressions were subsequently parametrized by fitting to the RFEs of selected peptide fragments, calculated with the empirical conformational energy program for peptides force field. The new multibody terms enable not only the heretofore predictable α-helical segments, but also regular β-sheets, to form as the lowest-energy structures, as assessed by test calculations on a model helical protein A, as well as a model 20-residue polypeptide (betanova); the latter was not possible without introducing these new terms.

Recent improvements in prediction of protein structure by global optimization of a potential energy function.

Recent improvements of a hierarchical ab initio or de novo approach for predicting both α and β structures of proteins are described. The united-residue energy function used in this procedure includes multibody interactions from a cumulant expansion of the free energy of polypeptide chains, with their relative weights determined by Z-score optimization. The critical initial stage of the hierarchical procedure involves a search of conformational space by the conformational space annealing (CSA) method, followed by optimization of an all-atom model. The procedure was assessed in a recent blind test of protein structure prediction (CASP4). The resulting lowest-energy structures of the target proteins (ranging in size from 70 to 244 residues) agreed with the experimental structures in many respects. The entire experimental structure of a cyclic α-helical protein of 70 residues was predicted to within 4.3 Å α-carbon (Cα) rms deviation (rmsd) whereas, for other α-helical proteins, fragments of roughly 60 residues were predicted to within 6.0 Å Cα rmsd. Whereas β structures can now be predicted with the new procedure, the success rate for α/β- and β-proteins is lower than that for α-proteins at present. For the β portions of α/β structures, the Cα rmsds are less than 6.0 Å for contiguous fragments of 30–40 residues; for one target, three fragments (of length 10, 23, and 28 residues, respectively) formed a compact part of the tertiary structure with a Cα rmsd less than 6.0 Å. Overall, these results constitute an important step toward the ab initio prediction of protein structure solely from the amino acid sequence.

United-residue force field for off-lattice protein-structure simulations: III. Origin of backbone hydrogen-bonding cooperativity in united-residue potentials

Based on the dipole model of peptide groups developed in our earlier work [Liwo et al., Prot. Sci., 2, 1697 (1993)], a cumulant expansion of the average free energy of the system of freely rotating peptide‐group dipoles tethered to a fixed α‐carbon trace is derived. A graphical approach is presented to find all nonvanishing terms in the cumulants. In particular, analytical expressions for three‐ and four‐body (correlation) terms in the averaged interaction potential of united peptide groups are derived. These expressions are similar to the cooperative forces in hydrogen bonding introduced by Koliński and Skolnick [J. Chem. Phys., 97, 9412 (1992)]. The cooperativity arises here naturally from the higher order terms in the power‐series expansion (in the inverse of the temperature) for the average energy. Test calculations have shown that addition of the derived four‐body term to the statistical united‐residue potential of our earlier work [Liwo et al., J. Comput. Chem., 18, 849, 874 (1997)] greatly improves its performance in folding poly‐l‐alanine into an α‐helix. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 259–276, 1998

Computational techniques for efficient conformational sampling of proteins

In this review, we summarize the computational methods for sampling the conformational space of biomacromolecules. We discuss the methods applicable to find only lowest energy conformations (global minimization of the potential-energy function) and to generate canonical ensembles (canonical Monte Carlo method and canonical molecular dynamics method and their extensions). Special attention is devoted to the use of coarse-grained models that enable simulations to be enhanced by several orders of magnitude.

A method for optimizing potential-energy functions by a hierarchical design of the potential-energy landscape: Application to the UNRES force field

A method for optimizing potential-energy functions of proteins is proposed. The method assumes a hierarchical structure of the energy landscape, which means that the energy decreases as the number of native-like elements in a structure increases, being lowest for structures from the native family and highest for structures with no native-like element. A level of the hierarchy is defined as a family of structures with the same number of native-like elements (or degree of native likeness). Optimization of a potential-energy function is aimed at achieving such a hierarchical structure of the energy landscape by forcing appropriate free-energy gaps between hierarchy levels to place their energies in ascending order. This procedure is different from methods developed thus far, in which the energy gap and/or the Z score between the native structure and all non-native structures are maximized, regardless of the degree of native likeness of the non-native structures. The advantage of this approach lies in reducing the number of structures with decreasing energy, which should ensure the searchability of the potential. The method was tested on two proteins, PDB ID codes 1FSD and 1IGD, with an off-lattice united-residue force field. For 1FSD, the search of the conformational space with the use of the conformational space annealing method and the newly optimized potential-energy function found the native structure very quickly, as opposed to the potential-energy functions obtained by former optimization methods. After even incomplete optimization, the force field obtained by using 1IGD located the native-like structures of two peptides, 1FSD and betanova (a designed three-stranded β-sheet peptide), as the lowest-energy conformations, whereas for the 46-residue N-terminal fragment of staphylococcal protein A, the native-like conformation was the second-lowest-energy conformation and had an energy 2 kcal/mol above that of the lowest-energy structure.

Investigation of Protein Folding by Coarse-Grained Molecular Dynamics with the UNRES Force Field

Coarse-grained molecular dynamics simulations offer a dramatic extension of the time-scale of simulations compared to all-atom approaches. In this article, we describe the use of the physics-based united-residue (UNRES) force field, developed in our laboratory, in protein-structure simulations. We demonstrate that this force field offers about a 4000-times extension of the simulation time scale; this feature arises both from averaging out the fast-moving degrees of freedom and reduction of the cost of energy and force calculations compared to all-atom approaches with explicit solvent. With massively parallel computers, microsecond folding simulation times of proteins containing about 1000 residues can be obtained in days. A straightforward application of canonical UNRES/MD simulations, demonstrated with the example of the N-terminal part of the B-domain of staphylococcal protein A (PDB code: 1BDD, a three-alpha-helix bundle), discerns the folding mechanism and determines kinetic parameters by parallel simulations of several hundred or more trajectories. Use of generalized-ensemble techniques, of which the multiplexed replica exchange method proved to be the most effective, enables us to compute thermodynamics of folding and carry out fully physics-based prediction of protein structure, in which the predicted structure is determined as a mean over the most populated ensemble below the folding-transition temperature. By using principal component analysis of the UNRES folding trajectories of the formin-binding protein WW domain (PDB code: 1E0L; a three-stranded antiparallel beta-sheet) and 1BDD, we identified representative structures along the folding pathways and demonstrated that only a few (low-indexed) principal components can capture the main structural features of a protein-folding trajectory; the potentials of mean force calculated along these essential modes exhibit multiple minima, as opposed to those along the remaining modes that are unimodal. In addition, a comparison between the structures that are representative of the minima in the free-energy profile along the essential collective coordinates of protein folding (computed by principal component analysis) and the free-energy profile projected along the virtual-bond dihedral angles gamma of the backbone revealed the key residues involved in the transitions between the different basins of the folding free-energy profile, in agreement with existing experimental data for 1E0L .

Potential of mean force of association of large hydrophobic particles: toward the nanoscale limit.

The potentials of mean force (PMFs) were determined, in both water with the TIP3P water model and in vacuo, for systems involving formation of nonpolar dimers composed of bicyclooctane, adamantane (both an all-atom model and a sphere with the radius of 3.4 A representing adamantane), and fullerene, respectively. A series of umbrella-sampling molecular dynamics simulations with the AMBER force field were carried out for each pair under both environmental conditions. The PMFs were calculated by using the weighted histogram analysis method. The results were compared with our previously determined PMF for neopentane. The shape of the PMFs for dimers of all four nonpolar molecules is characteristic of hydrophobic interactions with contact and solvent-separated minima and desolvation maxima. The positions of all these minima and maxima change with the size of the nonpolar molecule; for larger molecules they shift toward larger distances. Comparison of the PMFs of the bicyclooctane, adamantane, and fullerene dimers in water and in vacuo shows that hydrophobic interactions in each dimer are different from that for the dimer of neopentane. Interactions in the bicyclooctane, adamantane, and fullerene dimers are stronger in vacuo than in water. These dimers cannot be treated as classical, spherical, hydrophobic objects. The solvent contribution to the PMF was also computed by subtracting the PMF determined in vacuo from that in explicit solvent. The solvent contribution to the PMFs of bicyclooctane, adamantane, and fullerene is positive, as opposed to that of neopentane. The water molecules in the first solvation sphere of both adamantane and neopentane dimers are more ordered as compared to bulk water, with their dipole moments pointing away from the surface of the dimers. The average number of hydrogen bonds per water molecule in the first hydration shell of adamantane is smaller compared to that in bulk water, but this shell is thicker for all-atom adamantane than for neopentane or a spherical model of adamantane. In the second hydration shell, the average number of hydrogen bonds is greater compared to that in bulk water only for neopentane and a spherical model of adamantane but not for the all-atom model. The strength of the hydrophobic interactions shows a linear dependence on the number of carbon atoms both in water and in vacuo. Smaller nonpolar particles interact more strongly in water than in vacuo. For larger molecules, such as bicyclooctane, adamantane and fullerene, the reversed tendency is observed.

Efficient parallel algorithms in global optimization of potential energy functions for peptides, proteins, and crystals

Global optimization is playing an increasing role in physics, chemistry, and biophysical chemistry. One of the most important applications of global optimization is to find the global minima of the potential energy of molecules or molecular assemblies, such as crystals. The solution of this problem typically requires huge computational effort. Even the fastest processor available is not fast enough to carry out this kind of computation in real time for the problems of real interest, e.g., protein and crystal structure prediction. One way to circumvent this problem is to take advantage of massively parallel computing. In this paper, we provide several examples of parallel implementations of global optimization algorithms developed in our laboratory. All of these examples follow the master/worker approach. Most of the methods are parallelized on the algorithmic (coarse-grain) level and one example of fine-grain parallelism is given, in which the function evaluation itself is computationally expensive. All parallel algorithms were initially implemented on an IBM/SP2 (distributed-memory) machine. In all cases, however, message passing is handled through the standard Message Passing Interface (MPI); consequently the algorithms can also be implemented on any distributed- or shared-memory system that runs MPI. The efficiency of these implementations is discussed.

Lessons from application of the UNRES force field to predictions of structures of CASP10 targets

Significance With the example of the coarse-grained United Residue model of polypeptide chains, this paper demonstrates that the physics-based approach for protein-structure prediction can lead to exceptionally good results when correct domain packing is an issue, even for a highly homologous target. The reason for this is probably that emphasis is placed on energetically favorable residue–residue interactions, including those with residues in relatively flexible linker regions; these regions are usually very different in the target compared with those of proteins in the databases used for template-based modeling. The results suggest that a combination of bioinformatics and a physics-based approach could result in a major increase in the prediction capacity of existing approaches. The performance of the physics-based protocol, whose main component is the United Residue (UNRES) physics-based coarse-grained force field, developed in our laboratory for the prediction of protein structure from amino acid sequence, is illustrated. Candidate models are selected, based on probabilities of the conformational families determined by multiplexed replica-exchange simulations, from the 10th Community Wide Experiment on the Critical Assessment of Techniques for Protein Structure Prediction (CASP10). For target T0663, classified as a new fold, which consists of two domains homologous to those of known proteins, UNRES predicted the correct symmetry of packing, in which the domains are rotated with respect to each other by 180° in the experimental structure. By contrast, models obtained by knowledge-based methods, in which each domain is modeled very accurately but not rotated, resulted in incorrect packing. Two UNRES models of this target were featured by the assessors. Correct domain packing was also predicted by UNRES for the homologous target T0644, which has a similar structure to that of T0663, except that the two domains are not rotated. Predictions for two other targets, T0668 and T0684_D2, are among the best ones by global distance test score. These results suggest that our physics-based method has substantial predictive power. In particular, it has the ability to predict domain–domain orientations, which is a significant advance in the state of the art.

Molecular modeling of the human vasopressin V2 receptor/agonist complex.

The V2 vasopressin renal receptor (V2R), which controls antidiuresis in mammals, is a member of the large family of heptahelical transmembrane (7TM) G protein-coupled receptors (GPCRs). Using the automated GPCR modeling facility available via Internet (http://expasy.hcuge.ch/swissmod/SWISS-MODEL.html) for construction of the 7TM domain in accord with the bovine rhodopsin (RD) footprint, and the SYBYL software for addition of the intra- and extracellular domains, the human V2R was modeled. The structure was further refined and its conformational variability tested by the use of a version of the Constrained Simulated Annealing (CSA) protocol developed in this laboratory. An inspection of the resulting structure reveals that the V2R (likewise any GPCR modeled this way) is much thicker and accordingly forms a more spacious TM cavity than most of the hitherto modeled GPCR constructs do, typically based on the structure of bacteriorhodopsin (BRD). Moreover, in this model the 7TM helices are arranged differently than they are in any BRD-based model. Thus, the topology and geometry of the TM cavity, potentially capable of receiving ligands, is in this model quite different than it is in the earlier models. In the subsequent step, two ligands, the native [arginine8]vasopressin (AVP) and the selective agonist [d-arginine8]vasopressin (DAVP) were inserted, each in two topologically non-equivalent ways, into the TM cavity and the resulting structures were equilibrated and their conformational variabilities tested using CSA as above. The best docking was selected and justified upon consideration of ligand-receptor interactions and structure-activity data. Finally, the amino acid residues were indicated, mainly in TM helices 3-7, as potentially important in both AVP and DAVP docking. Among those Cys112, Val115-Lys116, Gln119, Met123 in helix 3; Glu174 in helix 4; Val206, Ala210, Val213-Phe214 in helix 5; Trp284, Phe287-Phe288, Gln291 in helix 6; and Phe307, Leu310, Ala314 and Asn317 in helix 7 appeared to be the most important ones. Many of these residues are invariant for either the GPCR superfamily or the neurophyseal (vasopressin V2R, V1aR and V1bR and oxytocin OR) subfamily of receptors. Moreover, some of the equivalent residues in V1aR have already been found critical for the ligand affinity [Mouillac et al., J. Biol. Chem, 270 (1995) 25771].