Folding Lennard-Jones proteins by a contact potential

Weighted families of contact maps to characterize conformational ensembles of (highly-)flexible proteins

MOTIVATION

Characterizing the structure of flexible proteins, particularly within the realm of intrinsic disorder, presents a formidable challenge due to their high conformational variability. Currently, their structural representation relies on (possibly large) conformational ensembles derived from a combination of experimental and computational methods. The detailed structural analysis of these ensembles is a difficult task, for which existing tools have limited effectiveness.

RESULTS

This study proposes an innovative extension of the concept of contact maps to the ensemble framework, incorporating the intrinsic probabilistic nature of disordered proteins. Within this framework, a conformational ensemble is characterized through a weighted family of contact maps. To achieve this, conformations are first described using a refined definition of contact that appropriately accounts for the geometry of the inter-residue interactions and the sequence context. Representative structural features of the ensemble naturally emerge from the subsequent clustering of the resulting contact-based descriptors. Importantly, transiently populated structural features are readily identified within large ensembles. The performance of the method is illustrated by several use cases and compared with other existing approaches, highlighting its superiority in capturing relevant structural features of highly flexible proteins.

AVAILABILITY AND IMPLEMENTATION

An open-source implementation of the method is provided together with an easy-to-use Jupyter notebook, available at https://gitlab.laas.fr/moma/WARIO.

Toward correct protein folding potentials

Empirical protein folding potentialfunctions should have a global minimum nearthe native conformationof globular proteins that fold stably, andthey should give the correct free energy offolding. We demonstrate that otherwise verysuccessful potentials fail to have even alocal minimumanywhere near the native conformation, anda seemingly well validated method ofestimatingthe thermodynamic stability of the nativestate is extremely sensitive to smallperturbations inatomic coordinates. These are bothindicative of fitting a great deal ofirrelevant detail. Here weshow how to devise a robust potentialfunction that succeeds very well at bothtasks, at least for alimited set of proteins, and this involvesdeveloping a novel representation of thedenatured state.Predicted free energies of unfolding for 25mutants of barnase are in close agreementwith theexperimental values, while for 17 mutantsthere are substantial discrepancies.

Random field model reveals structure of the protein recombinational landscape

We are interested in how intragenic recombination contributes to the evolution of proteins and how this mechanism complements and enhances the diversity generated by random mutation. Experiments have revealed that proteins are highly tolerant to recombination with homologous sequences (mutation by recombination is conservative); more surprisingly, they have also shown that homologous sequence fragments make largely additive contributions to biophysical properties such as stability. Here, we develop a random field model to describe the statistical features of the subset of protein space accessible by recombination, which we refer to as the recombinational landscape. This model shows quantitative agreement with experimental results compiled from eight libraries of proteins that were generated by recombining gene fragments from homologous proteins. The model reveals a recombinational landscape that is highly enriched in functional sequences, with properties dominated by a large-scale additive structure. It also quantifies the relative contributions of parent sequence identity, crossover locations, and protein fold to the tolerance of proteins to recombination. Intragenic recombination explores a unique subset of sequence space that promotes rapid molecular diversification and functional adaptation.

Mutation and recombination are the primary sources of genetic variation in evolving populations. The relative benefit of these two diversification mechanisms and how they complement each other has been a long-standing question in evolutionary biology. While it is clear what types of genetic diversity these two mechanisms can create, a significant challenge is relating these sequence changes to changes in fitness. The fitness landscape, which describes this mapping from genotype to phenotype, is extraordinarily complex and defined over an incomprehensibly large space of sequences. Here, we develop a model of the landscape that relies not on the details of this mapping, but rather on the statistical relationships between sequences. By studying the expected values of landscape properties, we can gain insights into the structure of the landscape that are independent of the details of how genotype dictates phenotype. We use this random field model to understand how recombination explores a functionally enriched and diverse subset of protein sequence space.

Coarse-grain Protein Models

Coarse-graining is a powerful approach for modeling biomolecules that, over the last few decades, has been extensively applied to proteins. Coarse-grain models offer access to large systems and to slow processes without becoming computationally unmanageable. In addition, they are very versatile, enabling both the protein representation and the energy function to be adapted to the biological problem in hand. This review concentrates on modeling soluble proteins and their assemblies. It presents an overview of the coarse-grain representations, of the associated interaction potentials, and of the optimization procedures used to define them. It then shows how coarse-grain models have been used to understand processes involving proteins, from their initial folding to their functional properties, their binary interactions, and the assembly of large complexes.

Universal distribution of protein evolution rates as a consequence of protein folding physics

The hypothesis that folding robustness is the primary determinant of the evolution rate of proteins is explored using a coarse-grained off-lattice model. The simplicity of the model allows rapid computation of the folding probability of a sequence to any folded conformation. For each robust folder, the network of sequences that share its native structure is identified. The fitness of a sequence is postulated to be a simple function of the number of misfolded molecules that have to be produced to reach a characteristic protein abundance. After fixation probabilities of mutants are computed under a simple population dynamics model, a Markov chain on the fold network is constructed, and the fold-averaged evolution rate is computed. The distribution of the logarithm of the evolution rates across distinct networks exhibits a peak with a long tail on the low rate side and resembles the universal empirical distribution of the evolutionary rates more closely than either distribution resembles the log-normal distribution. The results suggest that the universal distribution of the evolutionary rates of protein-coding genes is a direct consequence of the basic physics of protein folding.

Selection of optimal variants of Gō-like models of proteins through studies of stretching

The Gō-like models of proteins are constructed based on the knowledge of the native conformation. However, there are many possible choices of a Hamiltonian for which the ground state coincides with the native state. Here, we propose to use experimental data on protein stretching to determine what choices are most adequate physically. This criterion is motivated by the fact that stretching processes usually start with the native structure, in the vicinity of which the Gō-like models should work the best. Our selection procedure is applied to 62 different versions of the Gō model and is based on 28 proteins. We consider different potentials, contact maps, local stiffness energies, and energy scales--uniform and nonuniform. In the latter case, the strength of the nonuniformity was governed either by specificity or by properties related to positioning of the side groups. Among them is the simplest variant: uniform couplings with no i, i + 2 contacts. This choice also leads to good folding properties in most cases. We elucidate relationship between the local stiffness described by a potential which involves local chirality and the one which involves dihedral and bond angles. The latter stiffness improves folding but there is little difference between them when it comes to stretching.

Mapping folding energy landscapes with theory and experiment

The detailed characterization of the overall free energy landscape associated with the folding process of a protein is the ultimate goal in protein folding studies. Modern experimental techniques and all-atom simulations provide a way to obtain accurate thermodynamic and kinetic measurements, but they are oftentimes restricted to probe limited regions of a protein landscape. Although simplified protein models can access larger regions of the landscape, they are built on assumptions and approximations that can affect the accuracy of the results. We review here recent promising approaches that allow to combine the complementary strengths of theory and experiment for a more complete characterization of a protein folding landscape at multiple resolutions. Recent results and possible applications are discussed.

Atomically detailed potentials to recognize native and approximate protein structures

Atomically detailed potentials for recognition of protein folds are presented. The potentials consist of pair interactions between atoms. One or three distance steps are used to describe the range of interactions between a pair. Training is carried out with the mathematical programming approach on the decoy sets of Baker, Levitt, and some of our own design. Recognition is required not only for decoy-native structural pairs but also for pairs of decoy and homologous structures. Performance is tested on the targets of CASP5 using templates from the Protein Data Bank, on two test ab initio decoy sets from Skolnick's laboratory, and on decoy sets from Moult's laboratory. We conclude that the newly derived potentials have significant recognition capacity, comparable to the best models derived from other techniques. The new potentials require a significantly smaller number of parameters. The enhanced recognition capacity extends primarily to the identification of structures generated by ab initio simulation and less to the recognition of approximate shapes created by homology.

Protein Structure Evaluation using an All-Atom Energy Based Empirical Scoring Function

Arriving at the native conformation of a polypeptide chain characterized by minimum most free energy is a problem of long standing interest in protein structure prediction endeavors. Owing to the computational requirements in developing free energy estimates, scoring functions--energy based or statistical--have received considerable renewed attention in recent years for distinguishing native structures of proteins from non-native like structures. Several cleverly designed decoy sets, CASP (Critical Assessment of Techniques for Protein Structure Prediction) structures and homology based internet accessible three dimensional model builders are now available for validating the scoring functions. We describe here an all-atom energy based empirical scoring function and examine its performance on a wide series of publicly available decoys. Barring two protein sequences where native structure is ranked second and seventh, native is identified as the lowest energy structure in 67 protein sequences from among 61,659 decoys belonging to 12 different decoy sets. We further illustrate a potential application of the scoring function in bracketing native-like structures of two small mixed alpha/beta globular proteins starting from sequence and secondary structural information. The scoring function has been web enabled at www.scfbio-iitd.res.in/utility/proteomics/energy.jsp.

Recognizing protein folds by cluster distance geometry

Cluster distance geometry is a recent generalization of distance geometry whereby protein structures can be described at even lower levels of detail than one point per residue. With improvements in the clustering technique, protein conformations can be summarized in terms of alternative contact patterns between clusters, where each cluster contains four sequentially adjacent amino acid residues. A very simple potential function involving 210 adjustable parameters can be determined that favors the native contacts of 31 small, monomeric proteins over their respective sets of nonnative contacts. This potential then favors the native contacts for 174 small, monomeric proteins that have low sequence identity with any of the training set. A broader search finds 698 small protein chains from the Protein Data Bank where the native contacts are preferred over all alternatives, even though they have low sequence identity with the training set. This amounts to a highly predictive method for ab initio protein folding at low spatial resolution.

Thermal folding and mechanical unfolding pathways of protein secondary structures

Mechanical stretching of secondary structures is studied through molecular dynamics simulations of a Go-like model. Force versus displacement curves are studied as a function of the stiffness and velocity of the pulling device. The succession of stretching events, as measured by the order in which contacts are ruptured, is compared to the sequencing of events during thermal folding and unfolding. Opposite cross-correlations are found for an alpha-helix and a beta-hairpin structure. In a tandem of two alpha-helices, the two constituent helices unravel nearly simultaneously. A simple condition for simultaneous versus sequential unraveling of repeat units is presented.

A protein folding potential that places the native states of a large number of proteins near a local minimum

BACKGROUND

We present a simple method to train a potential function for the protein folding problem which, even though trained using a small number of proteins, is able to place a significantly large number of native conformations near a local minimum. The training relies on generating decoys by energy minimization of the native conformations using the current potential and using a physically meaningful objective function (derivative of energy with respect to torsion angles at the native conformation) during the quadratic programming to place the native conformation near a local minimum.

RESULTS

We also compare the performance of three different types of energy functions and find that while the pairwise energy function is trainable, a solvation energy function by itself is untrainable if decoys are generated by minimizing the current potential starting at the native conformation. The best results are obtained when a pairwise interaction energy function is used with solvation energy function.

CONCLUSIONS

We are able to train a potential function using six proteins which places a total of 42 native conformations within approximately 4 A rmsd and 71 native conformations within approximately 6 A rmsd of a local minimum out of a total of 91 proteins. Furthermore, the threading test using the same 91 proteins ranks 89 native conformations to be first and the other two as second.

Classification of amino acids based on statistical results of known structures and cooperativity of protein folding

It has been found that the 20 kinds of amino acids have different frequencies of occurrence in alpha,beta, and coil structures [P. Y. Chou and G. D. Fasman, Biochemistry 13, 211 (1974)]. Based on more known structures of proteins, frequencies for each amino acid in alpha and beta secondary structures are recalculated. Next step, under the approximation ignoring the chain connectivity of proteins, energy parameters to form alpha and beta secondary structures for each amino acid are obtained. According to the hydrophobicity and energies in alpha and beta secondary structures, 20 kinds of amino acids are classified. The results suggest that dividing amino acids to five or nine groups is desirable. At last, a protein model considering both two-body hydrophobic interaction and one-body energy to form secondary structures, hydrophobic-polar alphabeta model, is introduced. It is shown that the consistency among various energy terms makes the cooperativity of protein folding closer to the experiments.

Analysis of multiple folding routes of proteins by a coarse-grained dynamics model

Langevin dynamics of a protein molecule with Go-type potentials is developed and used to analyze long time-scale events in the folding of cytochrome c. Several trajectories are generated, starting from random coil configurations and going to the native state, that are a few angstroms root mean square deviation (RMSD) from the native structure. The dynamics is controlled, to a large scale, by the two terminal helices that are in contact in the native state. These two helices form very early during folding, and depending on the trajectory, they either stabilize rapidly or break and re-form in going over steric barriers. The extended initial chain exhibits a rapid folding transition into a relatively compact shape, after which the helices are reorganized in a highly correlated manner. The time of formation of residue pair contacts strongly points to the hierarchical nature of folding; i.e., secondary structure forms first, followed by rearrangements of larger length scales at longer times. The kinetics of formation of native contacts is also analyzed, and the onset of a stable globular configuration, referred to as the molten globule in the literature, is identified. Predictions of the model are compared with extensive experimental data on cytochrome c.

Potential energy function for continuous state models of globular proteins

One of the approaches to protein structure prediction is to obtain energy functions which can recognize the native conformation of a given sequence among a zoo of conformations. The discriminations can be done by assigning the lowest energy to the native conformation, with the guarantee that the native is in the zoo. Well-adjusted functions, then, can be used in the search for other (near-) natives. Here the aim is the discrimination at relatively high resolution (RMSD difference between the native and the closest nonnative is around 1 A) by pairwise energy potentials. The potential is trained using the experimentally determined native conformation of only one protein, instead of the usual large survey over many proteins. The novel feature is that the native structure is compared to a vastly wider and more challenging array of nonnative structures found not only by the usual threading procedure, but by wide-ranging local minimization of the potential. Because of this extremely demanding search, the native is very close to the apparent global minimum of the potential function. The global minimum property holds up for one other protein having 60% sequence identity, but its performance on completely dissimilar proteins is of course much weaker.

On the transferability of atomic solvation parameters: Ab initio structural prediction of cyclic heptapeptides in DMSO

A statistical mechanics methodology for predicting the solution structures and populations of peptides developed recently is based on a novel method for optimizing implicit solvation models, which was applied initially to a cyclic hexapeptide in DMSO (C. Baysal and H. Meirovitch, Journal of American Chemical Society, 1998, vol. 120, pp. 800-812). Thus, the molecule has been described by the simplified energy function E(tot) = E(GRO) + summation operator(k) sigma(k)A(k), where E(GRO) is the GROMOS force-field energy, sigma(k) and A(k) are the atomic solvation parameter (ASP) and the solvent accessible surface area of atom k, respectively. In a more recent study, these ASPs have been found to be transferable to the cyclic pentapeptide cyclo(D-Pro(1)-Ala(2)-Ala(3)-Ala(4)-Ala(5)) in DMSO (C. Baysal and H. Meirovitch, Biopolymers, 2000, vol. 53, pp. 423-433). In the present paper, our methodology is applied to the cyclic heptapeptides axinastatin 2 [cyclo(Asn(1)-Pro(2)-Phe(3)-Val(4)-Leu(5)-Pro(6)-Val(7))] and axinastatin 3 [cyclo(Asn(1)-Pro(2)-Phe(3)-Ile(4)-Leu(5)-Pro(6)-Val(7))], in DMSO, which were studied by nmr by Mechnich et al. (Helvetica Chimica Acta, 1997, vol. 80, pp. 1338-1354). The calculations for axinastatin 2 show that special ASPs should be optimized for the partially charged side-chain atoms of Asn while the rest of the atoms take their values derived in our previous work; this suggests that similar optimization might be needed for other side chains as well. The solution structures of these peptides are obtained ab initio (i.e., without using experimental restraints) by an extensive conformational search based on E(GRO) alone and E(*)(tot), which consists of the new set of ASPs. For E(*)(tot), the theoretical values of proton-proton distances, (3)J coupling constants, and other properties are found to agree very well with the nmr results, and they are always better than those based on E(GRO).