Jaroslaw Pillardy

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.

Enriching the sequence substitution matrix by structural information

A fundamental step in homology modeling is the comparison of two protein sequences: a probe sequence with an unknown structure and function and a template sequence for which the structure and function are known. The detection of protein similarities relies on a substitution matrix that scores the proximity of the aligned amino acids. Sequence‐to‐sequence alignments use symmetric substitution matrices, whereas the threading protocols use asymmetric matrices, testing the fitness of the probe sequence into the structure of the template protein. We propose a linear combination of threading and sequence‐alignment scoring function, to produce a single (mixed) scoring table. By fitting a single parameter (which is the relative contribution of the BLOSUM 50 matrix and the threading energy table of THOM2) we obtain a significant increase in prediction capacity in the twilight zone of homology modeling (detecting sequences with <25% sequence identity and with very similar structures). For a difficult test of 176 homologous pairs, with no signal of sequence similarity, the mixed model makes it possible to detect between 40 and 100% more protein pairs than the number of pairs that are detected by pure threading. Surprisingly, the linear combination of the two models is performing better than threading and than sequence alignment when the percentage of sequence identity is low. We finally suggest that further enrichment of substitution matrices, combing more structural descriptors such as exposed surface area, or secondary structure is expected to enhance the signal as well. Proteins 2003.

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.

Calculation of protein conformation by global optimization of a potential energy function.

A novel hierarchical approach to protein folding has been applied to compute the unknown structures of seven target proteins provided by CASP3. The approach is based exclusively on the global optimization of a potential energy function for a united‐residue model by conformational space annealing, followed by energy refinement using an all‐atom potential. Comparison of the submitted models for five globular proteins with the experimental structures shows that the conformations of large fragments (∼60 aa) were predicted with rmsds of 4.2–6.8 Å for the Cα atoms. Our lowest‐energy models for targets T0056 and T0061 were particularly successful, producing the correct fold of approximately 52% and 80% of the structures, respectively. These results support the thermodynamic hypothesis that protein structure can be computed solely by global optimization of a potential energy function for a given amino acid sequence. Proteins Suppl 1999;3:204–208.

Packing helices in proteins by global optimization of a potential energy function

An efficient method has been developed for packing α-helices in proteins. It treats α-helices as rigid bodies and uses a simplified Lennard–Jones potential with Miyazawa–Jernigan contact-energy parameters to describe the interactions between the α-helical elements in this coarse-grained system. Global conformational searches to generate packing arrangements rapidly are carried out with a Monte Carlo-with-minimization type of approach. The results for 42 proteins show that the approach reproduces native-like folds of α-helical proteins as low-energy local minima of this highly simplified potential function.

Support vector training of protein alignment models

Sequence to structure alignment is an important step in homology modeling of protein structures. Incorporation of features like secondary structure, solvent accessibility, or evolutionary information improve sequence to structure alignment accuracy, but conventional generative estimation techniques for alignment models impose independence assumptions that make these features difficult to include in a principled way. In this paper, we overcome this problem using a Support Vector Machine (SVM) method that provides a well-founded way of estimating complex alignment models with hundred-thousands of parameters. Furthermore, we show that the method can be trained using a variety of loss functions. In a rigorous empirical evaluation, the SVM algorithm outperforms the generative alignment method SSALN, a highly accurate generative alignment model that incorporates structural information. The alignment model learned by the SVM aligns 47% of the residues correctly and aligns over 70% of the residues within a shift of 4 positions.

Hierarchical energy‐based approach to protein‐structure prediction: Blind‐test evaluation with CASP3 targets

A hierarchical approach based exclusively on finding the global minimum of an appropriate potential energy function, without the aid of secondary structure prediction, multiple-sequence alignment, or threading, is proposed. The procedure starts from an extensive search of the conformational space of a protein, using our recently developed united-residue off-lattice UNRES force field and the conformational space annealing (CSA) method. The structures obtained in the search are clustered into families and ranked according to their UNRES energy. Structures within a preassigned energy cutoff are gradually converted into an all-atom representation, followed by a limited conformational search at the all-atom level, using the electrostatically driven Monte Carlo (EDMC) method and the ECEPP/3 force field including hydration. The approach was tested (in the CASP3 experiment) in blind predictions on seven targets, five of which were globular proteins with sizes ranging from 89 to 140 amino acid residues. Comparison of the computed lowest-energy structures, with the experimental structures, made available after the predictions were submitted, shows that large fragments (∼60 residues, representing 45–80% of the proteins) of those five globular proteins were predicted with the root mean square deviations (RMSDs) ranging from 4 to 7 A for the Cα atoms, with correct secondary structure and topology. These results constitute an important step toward the prediction of protein structure based solely on global optimization of a potential energy function for a given amino acid sequence.

Conformation-family Monte Carlo: A new method for crystal structure prediction

A new global optimization method, Conformation-family Monte Carlo, has been developed recently for searching the conformational space of macromolecules. In the present paper, we adapted this method for prediction of crystal structures of organic molecules without assuming any symmetry constraints except the number of molecules in the unit cell. This method maintains a database of low energy structures that are clustered into families. The structures in this database are improved iteratively by a Metropolis-type Monte Carlo procedure together with energy minimization, in which the search is biased toward the regions of the lowest energy families. The Conformation-family Monte Carlo method is applied to a set of nine rigid and flexible organic molecules by using two popular force fields, AMBER and W99. The method performed well for the rigid molecules and reasonably well for the molecules with torsional degrees of freedom.

Surmounting the Multiple-Minima Problem in Protein Folding

Protein folding is a very difficult global optimization problem. Furthermore it is coupled with the difficult task of designing a reliable force field with which one has to search for the global minimum. A summary of a series of optimization methods developed and applied to various problems involving polypeptide chains is described in this paper. With recent developments, a computational treatment of the folding of globular proteins of up to 140 residues is shown to be tractable.