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A computational approach is essential whenever the complexity of the process under study is such that direct theoretical or experimental approaches are not viable. This is the case for protein folding, for which a significant amount of data are being collected. This paper reports on the essential role of in silico methods and the unprecedented interplay of computational and theoretical approaches, which is a defining point of the interdisciplinary investigations of the protein folding process. Besides giving an overview of the available computational methods and tools, we argue that computation plays not merely an ancillary role but has a more constructive function in that computational work may precede theory and experiments. More precisely, computation can provide the primary conceptual clues to inspire subsequent theoretical and experimental work even in a case where no preexisting evidence or theoretical frameworks are available. This is cogently manifested in the application of machine learning methods to come to grips with the folding dynamics. These close relationships suggested complementing the review of computational methods within the appropriate theoretical context to provide a self-contained outlook of the basic concepts that have converged into a unified description of folding and have grown in a synergic relationship with their computational counterpart. Finally, the advantages and limitations of current computational methodologies are discussed to show how the smart analysis of large amounts of data and the development of more effective algorithms can improve our understanding of protein folding.

In this article we use mutation studies as a benchmark for a minimal model of the folding process of helical proteins. The model ascribes a pivotal role to the collisional dynamics of a few crucial residues (foldons) and predicts the folding rates by exploiting information drawn from the protein sequence. We show that our model rationalizes the effects of point mutations on the kinetics of folding. The folding times of two proteins and their mutants are predicted. Stability and location of foldons have a critical role as the determinants of protein folding. This allows us to elucidate two main mechanisms for the kinetic effects of mutations. First, it turns out that the mutations eliciting the most notable effects alter protein stability through stabilization or destabilization of the foldons. Secondly, the folding rate is affected via a modification of the foldon topology by those mutations that lead to the birth or death of foldons. The few mispredicted folding rates of some mutants hint at the limits of the current version of the folding model proposed in the present article. The performance of our folding model declines in case the mutated residues are subject to strong long-range forces. That foldons are the critical targets of mutation studies has notable implications for design strategies and is of particular interest to address the issue of the kinetic regulation of single proteins in the general context of the overall dynamics of the interactome., ((c) 2006 Wiley-Liss, Inc.)

A diffusion-collision-like model is proposed for helical proteins with three-state folding dynamics. The model generalizes a previous scheme based on the dynamics of putatively essential parts of the protein (foldons) that was successfully tested on proteins with two-state folding. We show that the extended model, unlike the original one, allows satisfactory calculation of the folding rate and reconstruction of the salient steps of the folding pathway of two proteins with three-state folding (Im7 and p16). The dramatic reduction of variables achieved by focusing on the foldons makes our model a good candidate for a minimal description of the folding process also for three-state folders. Finally, the applicability of the foldon diffusion-collision model to two-state and three-state folders suggests that different folding mechanisms are amenable to conceptually homogeneous descriptions. The implications for a unification of the variety of folding theories so far proposed for helical proteins are discussed in the final discussion.

Computational tools can bridge the gap between sequence and protein 3D structure based on the notion that information is to be retrieved from the databases and that knowledge-based methods can help in approaching a solution of the protein-folding problem. To this aim our group has implemented neural network-based predictors capable of performing with some success in different tasks, including predictions of the secondary structure of globular and membrane proteins, the topology of membrane proteins and porins and stable alpha-helical segments suited for protein design. Moreover we have developed methods for predicting contact maps in proteins and the probability of finding a cysteine in a disulfide bridge, tools which can contribute to the goal of predicting the 3D structure starting from the sequence (the so called ab initio prediction). All our predictors take advantage of evolution information derived from the structural alignments of homologous (evolutionary related) proteins and taken from the sequence and structure databases. When it is necessary to build models for proteins of unknown spatial structure, which have very little homology with other proteins of known structure, non-standard techniques need to be developed and the tools for protein structure predictions may help in protein modeling. The results of a recent simulation performed in our lab highlights the role of high performing computing technology and of tools of computational biology in protein modeling and peptidomimetic design.

The most stringent test for predictive methods of protein secondary structure is whether identical short sequences that are known to be present with different conformations in different proteins known at atomic resolution can be correctly discriminated. In this study, we show that the prediction efficiency of this type of segments in unrelated proteins reaches an average accuracy per residue ranging from about 72 to 75% (depending on the alignment method used to generate the input sequence profile) only when methods of the third generation are used. A comparison of different methods based on segment statistics (2nd generation methods) and/or including also evolutionary information (3rd generation methods) indicate that the discrimination of the different conformations of identical segments is dependent on the method used for the prediction. Accuracy is similar when methods similarly performing on the secondary structure prediction are tested. When evolutionary information is taken into account as compared to single sequence input, the number of correctly discriminated pairs is increased twofold. The results also highlight the predictive capability of neural networks for identical segments whose conformation differs in different proteins.

A data base of minimally frustrated alpha helical segments is defined by filtering a set comprising 822 non redundant proteins, which contain 4783 alpha helical structures. The data base definition is performed using a neural network-based alpha helix predictor, whose outputs are rated according to an entropy criterion. A comparison with the presently available experimental results indicates that a subset of the data base contains the initiation sites of protein folding experimentally detected and also protein fragments which fold into stable isolated alpha helices. This suggests the usage of the data base (and/or of the predictor) to highlight patterns which govern the stability of alpha helices in proteins and the helical behavior of isolated protein fragments.

The analysis of the information flow in a feed-forward neural network suggests that the output of the network can be used to compute a structural entropy for the sequence-to-secondary structure mapping. On this basis, we formulate a minimum entropy criterion for the identification of minimally frustrated traits with helical conformation that correspond to initiation sites of protein folding. The entropy of protein segments can be viewed as a nucleation propensity that is useful to characterize putative regions where folding is likely to be initiated with the formation of stretches of alpha-helices under the predominant influence of local interactions. Our procedure is successfully tested in the search for initiation sites of protein folding for which independent experimental and computational evidence exists. Our results lend support to the view that folding is a hierarchical event in which, in harmony with the minimal frustration principle, the final conformation preserves structural modules formed in the early stages of the process.

Back-propagation, feed-forward neural networks are used to predict alpha-helical transmembrane segments of proteins. The networks are trained on the few membrane proteins whose transmembrane alpha-helix domains are known to atomic or nearly atomic resolution. When testing is performed with a jackknife procedure on the proteins of the training set, the fraction of total correct assignments is as high as 0.87, with an average length for the transmembrane segments of 20 residues. The method correctly fails to predict any transmembrane domain for porin, whose transmembrane segments are beta-sheets. When tested on globular proteins, lower and upper limits of 1.6 and 3.5% for a total of 26826 residues are determined for the mispredicted cases, indicating that the predictor is highly specific for alpha-helical domains of membrane proteins. The predictor is also tested on 37 membrane proteins whose transmembrane topology is partially known. The overall accuracy is 0.90, two percentage points higher than that obtained with statistical methods. The reliability of the prediction is 100% for 60% of the total 18242 predicted residues of membrane proteins. Our results show that the local directional information automatically extracted by the neural networks during the training phase plays a key role in determining the accuracy of the prediction.

In this work we describe a parallel system consisting of feed-forward neural networks supervised by a local genetic algorithm. The system is implemented in a transputer architecture and is used to predict the secondary structures of globular proteins. This method allows a wide search in the parameter space of the neural networks and the determination of their optimal topology for the predictive task. Different neural network topologies are selected by the genetic algorithm on the basis of minimal values of mean square errors on the testing set. When the alpha-helix, beta-strand and random coil motifs of secondary structures are discriminated, the maximal efficiency obtained is 0.62, with correlation coefficients of 0.35, 0.31 and 0.37 respectively. This level of accuracy is similar to that previously attained by means of neural networks without hidden layers and using single protein sequences as input. The results validate the neural network topologies used for the prediction of protein secondary structures and highlight the relevance of the input information in determining the limit of their performance.

Radial basis function neural networks are trained on a data base comprising 38 globular proteins of well resolved crystallographic structure and the corresponding free energy contributions to the overall protein stability (as computed partially from chrystallographic analysis and partially with multiple regression from experimental thermodynamic data by Ponnuswamy and Gromiha (1994)). Starting from the residue sequence and using as input code the percentage of each residue and the total residue number of the protein, it is found with a cross-validation method that neural networks can optimally predict the free energy contributions due to hydrogen bonds, hydrophobic interactions and the unfolded state. Terms due to electrostatic and disulfide bonding free energies are poorly predicted. This is so also when other input codes, including the percentage of secondary structure type of the protein and/or residue-pair information are used. Furthermore, trained on the computed and/or experimental delta G values of the data base, neural networks predict a conformational stability ranging from about 10 to 20 kcal mol-1 rather independently of the residue sequence, with an average error per protein of about 9 kcal mol-1.

Back-propagation, feed-forward neural networks are used to predict the secondary structures of membrane proteins whose structures are known to atomic resolution. These networks are trained on globular proteins and can predict globular protein structures having no homology to those of the training set with correlation coefficients (Ci) of 0.45, 0.32 and 0.43 for alpha-helix, beta-strand and random coil structures, respectively. When tested on membrane proteins, neural networks trained on globular proteins do, on average, correctly predict (Qi) 62%, 38% and 69% of the residues in the alpha-helix, beta-strand and random coil structures. These scores rank higher than those obtained with the currently used statistical methods and are comparable to those obtained with the joint approaches tested so far on membrane proteins. The lower success score for beta-strand as compared to the other structures suggests that the sample of beta-strand patterns contained in the training set is less representative than those of alpha-helix and random coil. Our analysis, which includes the effects of the network parameters and of the structural composition of the training set on the prediction, shows that regular patterns of secondary structures can be successfully extrapolated from globular to membrane proteins.

Experimental investigations showed linear relations between flows and forces in some biological energy converters operating far from equilibrium. This observation cannot be understood on the basis of conventional nonequilibrium thermodynamics. Therefore, the efficiencies of a linear and a nonlinear mode of operation of an energy converter (a hypothetical redox-driven H+ pump) were compared. This comparison revealed that at physiological values of the forces and degrees of coupling (1) the force ratio permitting optimal efficiency was much higher in the linear than in the nonlinear mode and (2) the linear mode of operation was at least 10(6)-times more efficient that the nonlinear one. These observations suggest that the experimentally observed linear relations between flows and forces, particularly in the case of oxidative phosphorylation, may be due to a feedback regulation maintaining linear thermodynamic relations far from equilibrium. This regulation may have come about as the consequence of an evolutionary drive towards higher efficiency.