On Simplified Global Nonlinear Function for Fitness Landscape: A Case Study of Inverse Protein Folding

The construction of fitness landscape has broad implication in understanding molecular evolution, cellular epigenetic state, and protein structures. We studied the problem of constructing fitness landscape of inverse protein folding or protein design, with the aim to generate amino acid sequences that would fold into an a priori determined structural fold which would enable engineering novel or enhanced biochemistry. For this task, an effective fitness function should allow identification of correct sequences that would fold into the desired structure. In this study, we showed that nonlinear fitness function for protein design can be constructed using a rectangular kernel with a basis set of proteins and decoys chosen a priori. The full landscape for a large number of protein folds can be captured using only 480 native proteins and 3,200 non-protein decoys via a finite Newton method. A blind test of a simplified version of fitness function for sequence design was carried out to discriminate simultaneously 428 native sequences not homologous to any training proteins from 11 million challenging protein-like decoys. This simplified function correctly classified 408 native sequences (20 misclassifications, 95% correct rate), which outperforms several other statistical linear scoring function and optimized linear function. Our results further suggested that for the task of global sequence design of 428 selected proteins, the search space of protein shape and sequence can be effectively parametrized with just about 3,680 carefully chosen basis set of proteins and decoys, and we showed in addition that the overall landscape is not overly sensitive to the specific choice of this set. Our results can be generalized to construct other types of fitness landscape.     

Optimal design of protein docking potentials: efficiency and limitations

Protein-protein docking is a challenging computational problem in functional genomics, particularly when one or both proteins undergo conformational change(s) upon binding. The major challenge is to define scoring function soft enough to tolerate these changes and specific enough to distinguish between near-native and "misdocked" conformations. Using a linear programming technique, we derived protein docking potentials (PDPs) that comply with this requirement. We considered a set of 63 nonredundant complexes to this aim, and generated 400,000 putative docked complexes (decoys) based on shape complementarity criterion for each complex. The PDPs were required to yield for the native (correctly docked) structure a potential energy lower than those of all the nonnative (misdocked) structures. The energy constraints applied to all complexes led to ca. 25 million inequalities, the simultaneous solution of which yielded an optimal set of PDPs that discriminated the correctly docked (up to 4.0 A root-mean-square deviation from known complex structure) structure among the 85 top-ranking (0.02%) decoys in 59/63 examined bound-bound cases. The high performance of the potentials was further verified in jackknife tests and by ranking putative docked conformation submitted to CAPRI. In addition to their utility in identifying correctly folded complexes, the PDPs reveal biologically meaningful features that distinguish docking potentials from folding potentials.     

Using a Hydrophobic Contact Potential to Evaluate Native and Near-native Folds Generated by Molecular Dynamics Simulations

There are several knowledge-based energy functions that can distinguish the native fold from a pool of grossly misfolded decoys for a given sequence of amino acids. These decoys, which are typically generated by mounting, or “threading”, the sequence onto the backbones of unrelated protein structures, tend to be non-compact and quite different from the native structure: the root-mean-squared (RMS) deviations from the native are commonly in the range of 15 to 20 Å. Effective energy functions should also demonstrate a similar recognition capability when presented with compact decoys that depart only slightly in conformation from the correct structure (i.e. those with RMS deviations of ∼5 Å or less). Recently, we developed a simple yet powerful method for native fold recognition based on the tendency for native folds to form hydrophobic cores. Our energy measure, which we call the hydrophobic fitness score, is challenged to recognize the native fold from 2000 near-native structures generated for each of five small monomeric proteins. First, 1000 conformations for each protein were generated by molecular dynamics simulation at room temperature. The average RMS deviation of this set of 5000 was 1.5 Å. A total of 323 decoys had energies lower than native; however, none of these had RMS deviations greater than 2 Å. Another 1000 structures were generated for each at high temperature, in which a greater range of conformational space was explored (4.3 Å average RMS deviation). Out of this set, only seven decoys were misrecognized. The hydrophobic fitness energy of a conformation is strongly dependent upon the RMS deviation. On average our potential yields energy values which are lowest for the population of structures generated at room temperature, intermediate for those produced at high temperature and highest for those constructed by threading methods. In general, the lowest energy decoy conformations have backbones very close to native structure. The possible utility of our method for screening backbone candidates for the purpose of modelling by side-chain packing optimization is discussed.     

Optimal potentials for predicting inter-helical packing in transmembrane proteins

A set of pairwise contact potentials between amino acid residues in transmembrane helices was determined from the known native structure of the transmembrane protein (TMP) bacteriorhodopsin by the method of perceptron learning, using Monte Carlo dynamics to generate suitable "decoy" structures. The procedure of finding these decoys is simpler than for globular proteins, since it is reasonable to assume that helices behave as independent, stable objects and, therefore, the search in the conformational space is greatly reduced. With the learnt potentials, the association of the helices in bacteriorhodopsin was successfully simulated. The folding of a second TMP (the helix-dimer glycophorin A) was then accomplished with only a refinement of the potentials from a small number of decoys.     

Free energy determinants of tertiary structure and the evaluation of protein models

We develop a protocol for estimating the free energy difference between different conformations of the same polypeptide chain. The conformational free energy evaluation combines the CHARMM force field with a continuum treatment of the solvent. In almost all cases studied, experimentally determined structures are predicted to be more stable than misfolded "decoys." This is due in part to the fact that the Coulomb energy of the native protein is consistently lower than that of the decoys. The solvation free energy generally favors the decoys, although the total electrostatic free energy (sum of Coulomb and solvation terms) favors the native structure. The behavior of the solvation free energy is somewhat counterintuitive and, surprisingly, is not correlated with differences in the burial of polar area between native structures and decoys. Rather. the effect is due to a more favorable charge distribution in the native protein, which, as is discussed, will tend to decrease its interaction with the solvent. Our results thus suggest, in keeping with a number of recent studies, that electrostatic interactions may play an important role in determining the native topology of a folded protein. On this basis, a simplified scoring function is derived that combines a Coulomb term with a hydrophobic contact term. This function performs as well as the more complete free energy evaluation in distinguishing the native structure from misfolded decoys. Its computational efficiency suggests that it can be used in protein structure prediction applications, and that it provides a physically well-defined alternative to statistically derived scoring functions.     

A global machine learning based scoring function for protein structure prediction

We present a knowledge‐based function to score protein decoys based on their similarity to native structure. A set of features is constructed to describe the structure and sequence of the entire protein chain. Furthermore, a qualitative relationship is established between the calculated features and the underlying electromagnetic interaction that dominates this scale. The features we use are associated with residue–residue distances, residue–solvent distances, pairwise knowledge‐based potentials and a four‐body potential. In addition, we introduce a new target to be predicted, the fitness score, which measures the similarity of a model to the native structure. This new approach enables us to obtain information both from decoys and from native structures. It is also devoid of previous problems associated with knowledge‐based potentials. These features were obtained for a large set of native and decoy structures and a back‐propagating neural network was trained to predict the fitness score. Overall this new scoring potential proved to be superior to the knowledge‐based scoring functions used as its inputs. In particular, in the latest CASP (CASP10) experiment our method was ranked third for all targets, and second for freely modeled hard targets among about 200 groups for top model prediction. Ours was the only method ranked in the top three for all targets and for hard targets. This shows that initial results from the novel approach are able to capture details that were missed by a broad spectrum of protein structure prediction approaches. Source codes and executable from this work are freely available at http://mathmed.org /#Software and http://mamiris.com/ . Proteins 2014; 82:752–759. © 2013 Wiley Periodicals, Inc.     

Distinguishing native conformations of proteins from decoys with an effective free energy estimator based on the OPLS all-atom force field and the Surface Generalized Born solvent model

Protein decoy data sets provide a benchmark for testing scoring functions designed for fold recognition and protein homology modeling problems. It is commonly believed that statistical potentials based on reduced atomic models are better able to discriminate native-like from misfolded decoys than scoring functions based on more detailed molecular mechanics models. Recent benchmark tests on small data sets, however, suggest otherwise. In this work, we report the results of extensive decoy detection tests using an effective free energy function based on the OPLS all-atom (OPLS-AA) force field and the Surface Generalized Born (SGB) model for the solvent electrostatic effects. The OPLS-AA/SGB effective free energy is used as a scoring function to detect native protein folds among a total of 48,832 decoys for 32 different proteins from Park and Levitt's 4-state-reduced, Levitt's local-minima, Baker's ROSETTA all-atom, and Skolnick's decoy sets. Solvent electrostatic effects are included through the Surface Generalized Born (SGB) model. All structures are locally minimized without restraints. From an analysis of the individual energy components of the OPLS-AA/SGB energy function for the native and the best-ranked decoy, it is determined that a balance of the terms of the potential is responsible for the minimized energies that most successfully distinguish the native from the misfolded conformations. Different combinations of individual energy terms provide less discrimination than the total energy. The results are consistent with observations that all-atom molecular potentials coupled with intermediate level solvent dielectric models are competitive with knowledge-based potentials for decoy detection and protein modeling problems such as fold recognition and homology modeling.     

Stochastic modeling of noninteracting probes in the protein structure space for construction of knowledge-based potentials for atom-atom interactions

Knowledge-based potentials are extensively used to represent atomic interactions in modeling the protein structure. We consider a number of problems in constructing efficient knowledge-based potentials for biopolymer modeling. We show that some limitations can be overcome by normalizing estimated interactions through the distribution of distances between noninteracting random probes in protein structure space. We demonstrate that knowledge-based potentials thus constructed can be efficiently applied for analysis of the hydration state of proteins atoms. With this approach, one can predict the locations of structural water molecules in a protein globule. We have also succeeded in recognizing the correctly folded protein structure among many misfolded decoys in cases when the interaction with water solvent is dominant for structure formation.     

Multibody coarse-grained potentials for native structure recognition and quality assessment of protein models

Multibody potentials have been of much interest recently because they take into account three dimensional interactions related to residue packing and capture the cooperativity of these interactions in protein structures. Our goal was to combine long range multibody potentials and short range potentials to improve recognition of native structure among misfolded decoys. We optimized the weights for four-body nonsequential, four-body sequential, and short range potentials to obtain optimal model ranking results for threading and have compared these data against results obtained with other potentials (26 different coarse-grained potentials from the Potentials 'R'Us web server have been used). Our optimized multibody potentials outperform all other contact potentials in the recognition of the native structure among decoys, both for models from homology template-based modeling and from template-free modeling in CASP8 decoy sets. We have compared the results obtained for this optimized coarse-grained potentials, where each residue is represented by a single point, with results obtained by using the DFIRE potential, which takes into account atomic level information of proteins. We found that for all proteins larger than 80 amino acids our optimized coarse-grained potentials yield results comparable to those obtained with the atomic DFIRE potential.     

Free energies for coarse-grained proteins by integrating multibody statistical contact potentials with entropies from elastic network models

We propose a novel method of calculation of free energy for coarse grained models of proteins by combining our newly developed multibody potentials with entropies computed from elastic network models of proteins. Multi-body potentials have been of much interest recently because they take into account three dimensional interactions related to residue packing and capture the cooperativity of these interactions in protein structures. Combining four-body non-sequential, four-body sequential and pairwise short range potentials with optimized weights for each term, our coarse-grained potential improved recognition of native structure among misfolded decoys, outperforming all other contact potentials for CASP8 decoy sets and performance comparable to the fully atomic empirical DFIRE potentials. By combing statistical contact potentials with entropies from elastic network models of the same structures we can compute free energy changes and improve coarse-grained modeling of protein structure and dynamics. The consideration of protein flexibility and dynamics should improve protein structure prediction and refinement of computational models. This work is the first to combine coarse-grained multibody potentials with an entropic model that takes into account contributions of the entire structure, investigating native-like decoy selection.     

Use of decoys to optimize an all-atom force field including hydration

A novel method of parameter optimization is proposed. It makes use of large sets of decoys generated for six nonhomologous proteins with different architecture. Parameter optimization is achieved by creating a free energy gap between sets of nativelike and nonnative conformations. The method is applied to optimize the parameters of a physics-based scoring function consisting of the all-atom ECEPP05 force field coupled with an implicit solvent model (a solvent-accessible surface area model). The optimized force field is able to discriminate near-native from nonnative conformations of the six training proteins when used either for local energy minimization or for short Monte Carlo simulated annealing runs after local energy minimization. The resulting force field is validated with an independent set of six nonhomologous proteins, and appears to be transferable to proteins not included in the optimization; i.e., for five out of the six test proteins, decoys with 1.7- to 4.0-Å all-heavy-atom root mean-square deviations emerge as those with the lowest energy. In addition, we examined the set of misfolded structures created by Park and Levitt using a four-state reduced model. The results from these additional calculations confirm the good discriminative ability of the optimized force field obtained with our decoy sets.     

A probabilistic fragment-based protein structure prediction algorithm

Conformational sampling is one of the bottlenecks in fragment-based protein structure prediction approaches. They generally start with a coarse-grained optimization where mainchain atoms and centroids of side chains are considered, followed by a fine-grained optimization with an all-atom representation of proteins. It is during this coarse-grained phase that fragment-based methods sample intensely the conformational space. If the native-like region is sampled more, the accuracy of the final all-atom predictions may be improved accordingly. In this work we present EdaFold, a new method for fragment-based protein structure prediction based on an Estimation of Distribution Algorithm. Fragment-based approaches build protein models by assembling short fragments from known protein structures. Whereas the probability mass functions over the fragment libraries are uniform in the usual case, we propose an algorithm that learns from previously generated decoys and steers the search toward native-like regions. A comparison with Rosetta AbInitio protocol shows that EdaFold is able to generate models with lower energies and to enhance the percentage of near-native coarse-grained decoys on a benchmark of [Formula: see text] proteins. The best coarse-grained models produced by both methods were refined into all-atom models and used in molecular replacement. All atom decoys produced out of EdaFold's decoy set reach high enough accuracy to solve the crystallographic phase problem by molecular replacement for some test proteins. EdaFold showed a higher success rate in molecular replacement when compared to Rosetta. Our study suggests that improving low resolution coarse-grained decoys allows computational methods to avoid subsequent sampling issues during all-atom refinement and to produce better all-atom models. EdaFold can be downloaded from http://www.riken.jp/zhangiru/software/.     

Comparison of two optimization methods to derive energy parameters for protein folding: perceptron and Z score

Two methods were proposed recently to derive energy parameters from known native protein conformations and corresponding sets of decoys. One is based on finding, by means of a perceptron learning scheme, energy parameters such that the native conformations have lower energies than the decoys. The second method maximizes the difference between the native energy and the average energy of the decoys, measured in terms of the width of the decoys' energy distribution (Z-score). Whereas the perceptron method is sensitive mainly to "outlier" (i.e., extremal) decoys, the Z-score optimization is governed by the high density regions in decoy-space. We compare the two methods by deriving contact energies for two very different sets of decoys: the first obtained for model lattice proteins and the second by threading. We find that the potentials derived by the two methods are of similar quality and fairly closely related. This finding indicates that standard, naturally occurring sets of decoys are distributed in a way that yields robust energy parameters (that are quite insensitive to the particular method used to derive them). The main practical implication of this finding is that it is not necessary to fine-tune the potential search method to the particular set of decoys used.     

Statistical mechanics-based method to extract atomic distance-dependent potentials from protein structures

In this study, we have developed a statistical mechanics-based iterative method to extract statistical atomic interaction potentials from known, nonredundant protein structures. Our method circumvents the long-standing reference state problem in deriving traditional knowledge-based scoring functions, by using rapid iterations through a physical, global convergence function. The rapid convergence of this physics-based method, unlike other parameter optimization methods, warrants the feasibility of deriving distance-dependent, all-atom statistical potentials to keep the scoring accuracy. The derived potentials, referred to as ITScore/Pro, have been validated using three diverse benchmarks: the high-resolution decoy set, the AMBER benchmark decoy set, and the CASP8 decoy set. Significant improvement in performance has been achieved. Finally, comparisons between the potentials of our model and potentials of a knowledge-based scoring function with a randomized reference state have revealed the reason for the better performance of our scoring function, which could provide useful insight into the development of other physical scoring functions. The potentials developed in this study are generally applicable for structural selection in protein structure prediction.     

Capturing native/native like structures with a physico-chemical metric (pcSM) in protein folding

Specification of the three dimensional structure of a protein from its amino acid sequence, also called a “Grand Challenge” problem, has eluded a solution for over six decades. A modestly successful strategy has evolved over the last couple of decades based on development of scoring functions (e.g. mimicking free energy) that can capture native or native-like structures from an ensemble of decoys generated as plausible candidates for the native structure. A scoring function must be fast enough in discriminating the native from unfolded/misfolded structures, and requires validation on a large data set(s) to generate sufficient confidence in the score. Here we develop a scoring function called pcSM that detects true native structure in the top 5 with 93% accuracy from an ensemble of candidate structures. If we eliminate the native from ensemble of decoys then pcSM is able to capture near native structure (RMSD < = 5 ?) in top 10 with 86% accuracy. The parameters considered in pcSM are a C-alpha Euclidean metric, secondary structural propensity, surface areas and an intramolecular energy function. pcSM has been tested on 415 systems consisting 142,698 decoys (public and CASP—largest reported hitherto in literature). The average rank for the native is 2.38, a significant improvement over that existing in literature. In-silico protein structure prediction requires robust scoring technique(s). Therefore, pcSM is easily amenable to integration into a successful protein structure prediction strategy. The tool is freely available at http://www.scfbio-iitd.res.in/software/pcsm.jsp.     

Protein structure refinement by optimization

Knowledge‐based protein potentials are simplified potentials designed to improve the quality of protein models, which is important as more accurate models are more useful for biological and pharmaceutical studies. Consequently, knowledge‐based potentials often are designed to be efficient in ordering a given set of deformed structures denoted decoys according to how close they are to the relevant native protein structure. This, however, does not necessarily imply that energy minimization of this potential will bring the decoys closer to the native structure. In this study, we introduce an iterative strategy to improve the convergence of decoy structures. It works by adding energy optimized decoys to the pool of decoys used to construct the next and improved knowledge‐based potential. We demonstrate that this strategy results in significantly improved decoy convergence on Titan high resolution decoys and refinement targets from Critical Assessment of protein Structure Prediction competitions. Our potential is formulated in Cartesian coordinates and has a fixed backbone potential to restricts motions to be close to those of a dihedral model, a fixed hydrogen‐bonding potential and a variable coarse grained carbon alpha potential consisting of a pair potential and a novel solvent potential that are b‐spline based as we use explicit gradient and Hessian for efficient energy optimization. Proteins 2015; 83:1616–1624. © 2015 Wiley Periodicals, Inc.     

Analysis of anisotropic side-chain packing in proteins and application to high-resolution structure prediction

pi-pi, Cation-pi, and hydrophobic packing interactions contribute specificity to protein folding and stability to the native state. As a step towards developing improved models of these interactions in proteins, we compare the side-chain packing arrangements in native proteins to those found in compact decoys produced by the Rosetta de novo structure prediction method. We find enrichments in the native distributions for T-shaped and parallel offset arrangements of aromatic residue pairs, in parallel stacked arrangements of cation-aromatic pairs, in parallel stacked pairs involving proline residues, and in parallel offset arrangements for aliphatic residue pairs. We then investigate the extent to which the distinctive features of native packing can be explained using Lennard-Jones and electrostatics models. Finally, we derive orientation-dependent pi-pi, cation-pi and hydrophobic interaction potentials based on the differences between the native and compact decoy distributions and investigate their efficacy for high-resolution protein structure prediction. Surprisingly, the orientation-dependent potential derived from the packing arrangements of aliphatic side-chain pairs distinguishes the native structure from compact decoys better than the orientation-dependent potentials describing pi-pi and cation-pi interactions.     

Automated use of mutagenesis data in structure prediction

In the absence of experimental structural determination, numerous methods are available to indirectly predict or probe the structure of a target molecule. Genetic modification of a protein sequence is a powerful tool for identifying key residues involved in binding reactions or protein stability. Mutagenesis data is usually incorporated into the modeling process either through manual inspection of model compatibility with empirical data, or through the generation of geometric constraints linking sensitive residues to a binding interface. We present an approach derived from statistical studies of lattice models for introducing mutation information directly into the fitness score. The approach takes into account the phenotype of mutation (neutral or disruptive) and calculates the energy for a given structure over an ensemble of sequences. The structure prediction procedure searches for the optimal conformation where neutral sequences either have no impact or improve stability and disruptive sequences reduce stability relative to wild type. We examine three types of sequence ensembles: information from saturation mutagenesis, scanning mutagenesis, and homologous proteins. Incorporating multiple sequences into a statistical ensemble serves to energetically separate the native state and misfolded structures. As a result, the prediction of structure with a poor force field is sufficiently enhanced by mutational information to improve accuracy. Furthermore, by separating misfolded conformations from the target score, the ensemble energy serves to speed up conformational search algorithms such as Monte Carlo-based methods.     

The victor/FRST function for model quality estimation.

Scoring functions are widely used in the final step of model selection in protein structure prediction. This is of interest both for comparative modeling targets, where it is important to select the best model among a set of many good, "correct" ones, as well as for other (fold recognition or novel fold) targets, where the set may contain many incorrect models. A novel combination of four knowledge-based potentials recognizing different features of native protein structures is introduced and tested. The pairwise, solvation, hydrogen bond, and torsion angle potentials contain largely orthogonal information. Of these, the torsion angle potential is found to show the strongest correlation with model quality. Combining these features with a linear weighting function, it was possible to construct a robust energy function capable of discriminating native-like structures on several benchmarking sets. In a recent blind test (CAFASP-4 MQAP), the scoring function ranked consistently well and was able to reliably distinguish the correct template from an ensemble of high quality decoys in 52 of 70 cases (33 of 34 for comparative modeling). An executable version of the Victor/FRST function for Linux PCs is available for download from the URL http://protein.cribi.unipd.it/frst/.