An approach to the multiple-minima problem in protein folding by relaxing dimensionality. Tests on enkephalin

An algorithm for locating the region in conformational space containing the global energy minimum of a polypeptide is described. Distances are used as the primary variables in the minimization of an objective function that incorporates both energetic and distance-geometric terms. The latter are obtained from geometry and energy functions, rather than nuclear magnetic resonance experiments, although the algorithm can incorporate distances from nuclear magnetic resonance data if desired. The polypeptide is generated originally in a space of high dimensionality. This has two important consequences. First, all interatomic distances are initially at their energetically most favorable values; i.e. the polypeptide is initially at a global minimum-energy conformation, albeit a high-dimensional one. Second, the relaxation of dimensionality constraints in the early stages of the minimization removes many potential energy barriers that exist in three dimensions, thereby allowing a means of escaping from three-dimensional local minima. These features are used in an algorithm that produces short trajectories of three-dimensional minimum-energy conformations. A conformation in the trajectory is generated by allowing the previous conformation in the trajectory to evolve in a high-dimensional space before returning to three dimensions. The resulting three-dimensional structure is taken to be the next conformation in the trajectory, and the process is iterated. This sequence of conformations results in a limited but efficient sampling of conformational space. Results for test calculations on Met-enkephalin, a pentapeptide with the amino acid sequence H-Tyr-Gly-Gly-Phe-Met-OH, are presented. A tight cluster of conformations (in three-dimensional space) is found with ECEPP energies (Empirical Conformational Energy Program for Peptides) lower than any previously reported. This cluster of conformations defines a region in conformational space in which the global-minimum-energy conformation of enkephalin appears to lie.     

Interactions between nucleic acid bases. New parameters of potential functions and energy minima

The parameters of atom-atom potential functions suggested by one of the authors in 1979-1986 were slightly changed. The changes were performed to achieve a better agreement with experimental data of interaction energy values in global minima and hydrogen bond lengths. These changes resulted in better accord with experimental values of distances between the layers in DNA monomer crystals and between the base pairs in oligonucleotide duplexes. The refined potential functions were used to calculate the energy of interactions between nucleic acid bases in various mutual positions. The calculations revealed a few types of mutual base arrangements in minima of interaction energy for each pairwise base combination. A new type of minima was found, which correspond to a nearly perpendicular arrangement of base rings and the formation of the intermolecular hydrogen bond.     

Guiding conformation space search with an all-atom energy potential

The most significant impediment for protein structure prediction is the inadequacy of conformation space search. Conformation space is too large and the energy landscape too rugged for existing search methods to consistently find near-optimal minima. To alleviate this problem, we present model-based search, a novel conformation space search method. Model-based search uses highly accurate information obtained during search to build an approximate, partial model of the energy landscape. Model-based search aggregates information in the model as it progresses, and in turn uses this information to guide exploration toward regions most likely to contain a near-optimal minimum. We validate our method by predicting the structure of 32 proteins, ranging in length from 49 to 213 amino acids. Our results demonstrate that model-based search is more effective at finding low-energy conformations in high-dimensional conformation spaces than existing search methods. The reduction in energy translates into structure predictions of increased accuracy.     

Applications of simulated annealing to peptides

We report the application of a new conformation searching algorithm called simulated annealing to the location of the global minimum energy conformation of peptides. Simulated annealing is a Metropolis Monte Carlo approach to conformation generation in which both the energy and temperature dependence of the Boltzmann distribution guides the search for the global minimum. Both uphill and downhill moves are possible, which allows the molecule to escape from local minima. Applications to the 20 natural amino acid "dipeptide models" as well as to polyalanines up to Ala80 are very successful in finding the lowest energy conformation. A history file of the simulated annealing process allows reconstruction and examination of the random walk around conformation space. A separate program, Conf-Gen, reads the history file and extracts all low-energy conformations visited during the run.     

Energy-minimized conformation of gramicidin-like channels. II. Periodicity of the lowest energy conformation of an infinitely long poly-(L,D)-alanine beta 6.3-helix.

If an infinitely long polymer has a primary structure characterized by an N-residue periodicity, a minimum energy conformation of the polymer under the constraint of the conformational N-residue periodicity corresponds to an equilibrium structure (energy minimal or unstable equilibrium structure) when this constraint is absent. Molecular mechanics calculations showed that with an infinitely long poly-(L,D)-alanine single-stranded beta 6.3-helix (which has a 2-residue periodicity with respect to the primary structure), its lowest energy conformation within the framework of the conformational 2-residue periodicity is also the lowest energy form of this beta 6.3-helix even when no conformational periodicity is assumed. In the course of this study, contour maps of helix parameters and conformation energies for beta structures of poly-(L,D)-alanine were examined. It was also found that beta 6.3-, beta 4.5-, alpha L,D-, and tau L,D-helices constitute the global minima in the whole conformational space of this polypeptide. In the present calculation, an improved formulation of the conformation energy was introduced to estimate the structure and conformation energy of an infinite periodic chain from results on a chain of finite length.     

A structural and free energy analysis of Ag complexes to five small peptides

The complexes of Ag⁺ with the peptides MetGly, ProGly, GlyPro, GlyHis and GlyProAla were investigated using hybrid density functional theory at the B3LYP/DZVP level. The silver ion binding free energies at 298 K to each of these peptides was calculated to be 60.8, 52.0, 54.3, 71.2 and 63.3 kcal mol⁻¹, respectively. Structural information and relative free energies are presented for several isomers for each of the five complexes. Each of the global minima found for the five complexes is a charge-solvated ion. An important finding is that the Ag⁺-ProGly is the only complex where a salt bridge structure is energetically favored occurring at 4.0 kcal mol⁻¹ higher in free energy than the global minimum. The Ag⁺ ion in this salt bridge structure is attached to the carboxylate anion of zwitterionic ProGly in which the terminal amino nitrogen is protonated. For all the other complexes studied, the salt bridge structure occurs at much higher energies. All the dipeptide complexes with Ag⁺, but one, exhibit a di- or tri-coordinate metal where the sites of attachment are amino and carbonyl groups. However, the highest coordination numbers are not always the global minima due to steric costs. The global minimum of the Ag⁺-GlyProAla complex is the only structure found in this study where the metal is tetra-coordinated, binding to the terminal amino nitrogen and all three carbonyl oxygen atoms. Silver binding to sulphur and imidazole nitrogen atoms of MetGly and GlyHis, respectively, are present in the three most energetically favored species in each of these cases.     

New Monte Carlo algorithms for protein folding.

Over the past three decades, a number of powerful simulation algorithms have been introduced to the protein folding problem. For many years, the emphasis has been placed on how to both overcome the multiple minima problem and find the conformation with the global minimum potential energy. Since the new view of the protein folding mechanism (based on the free energy landscape of the protein system) arose in the past few years, however, it is now of interest to obtain a global knowledge of the phase space, including the intermediate and denatured states of proteins. Monte Carlo methods have proved especially valuable for these purposes. As well as new, powerful optimization techniques, novel algorithms that can sample much a wider phase space than conventional methods have been established.     

A novel approach to decoy set generation: designing a physical energy function having local minima with native structure characteristics

We suggest a new approach to the generation of candidate structures (decoys) for ab initio prediction of protein structures. Our method is based on random sampling of conformation space and subsequent local energy minimization. At the core of this approach lies the design of a novel type of energy function. This energy function has local minima with native structure characteristics and wide basins of attraction. The current work presents our motivation for deriving such an energy function and also tests the derived energy function.Our approach is novel in that it takes advantage of the inherently rough energy landscape of proteins, which is generally considered a major obstacle for protein structure prediction. When local minima have wide basins of attraction, the protein's conformation space can be greatly reduced by the convergence of large regions of the space into single points, namely the local minima corresponding to these funnels. We have implemented this concept by an iterative process. The potential is first used to generate decoy sets and then we study these sets of decoys to guide further development of the potential. A key feature of our potential is the use of cooperative multi-body interactions that mimic the role of the entropic and solvent contributions to the free energy.The validity and value of our approach is demonstrated by applying it to 14 diverse, small proteins. We show that, for these proteins, the size of conformation space is considerably reduced by the new energy function. In fact, the reduction is so substantial as to allow efficient conformational sampling. As a result we are able to find a significant number of near-native conformations in random searches performed with limited computational resources.     

Monte Carlo minimization with thermalization for global optimization of polypeptide conformations in cartesian coordinate space.

A new minimization procedure for the global optimization in cartesian coordinate space of the conformational energy of a polypeptide chain is presented. The Metropolis Monte Carlo minimization is thereby supplemented by a thermalization process, which is initiated whenever a structure becomes trapped in an area containing closely located local minima in the conformational space. The method has been applied to the endogenous opioid pentapeptide methionine enkephalin. Five among 13 different starting conformations led to the same apparent global minimum of an in-house developed energy function, a type II' reverse turn, the central residues of which are Gly-3-Phe-4. A comparison between the ECEPP/2 global minimum conformation of methionine enkephalin and the apparent one achieved by the present method shows that minimum-energy conformations having a certain similarity can be generated by relatively different force fields.     

Influence of local interactions on protein structure. I. Conformational energy studies of N-acetyl-N′-methylamides of pro-X and X-pro dipeptides

Conformational energy calculations using an Empirical Conformational Energy Program for Peptides (ECEPP) were carried out on the N-acetyl-N′-methylamides of Pro-X, where X = Ala, Asn, Asp, Gly, Leu, Phe, Ser, and Val, and of X-Pro, where X = Ala, Asn, Gly, and Pro. The conformational energy was minimized from starting conformations which included all combinations of low-energy single-residue minima and several standard bend structures. It was found that almost all resulting minima are combinations of low-energy single-residue minima, suggesting that intra residue interactions predominate in determining conformation. The calculations also indicate, however, that inter residue interactions can be important. In addition, librational entropy was found to influence the relative stabilities of some minima. Because of the existence of 10–100 low-energy minima for each dipeptide, the normalized statistical weight of an individual minimum rarely exceeds 0.3, suggesting that these dipeptides have considerable conformational flexibility and exist as statistical ensembles of low-energy structures. The propensity of each dipeptide to form bend conformations was calculated, and the results were compared with available experimental data. It was found that bends are favored in Pro-X dipeptides because ?_Pro is fixed by the pyrrolidine ring in a conformation which is frequently found in bends, but that bends are not favored in X-Pro dipeptides because interactions between the X residue and the pyrrolidine ring restrict the X residue to conformations which are not usually found in bends.     

Structural basis of hierarchical multiple substates of a protein. II: Monte Carlo simulation of native thermal fluctuations and energy minimization

Conformational fluctuations in a globular protein, bovine pancreatic trypsin inhibitor, in the time range between picoseconds and nanoseconds are studied by a Monte Carlo simulation method. Multiple energy minima are derived from sampled conformations by minimizing their energy. They are distributed in clusters in the conformational space. A hierarchical structure is observed in the simulated dynamics. In the time range between 10(-14) and 10(-10) seconds dynamics is well represented by a superposition of vibrational motions within an energy well with transitions among minima within each cluster. Transitions among clusters take place in the time range of nanoseconds or longer.     

Evolutionary-inspired probabilistic search for enhancing sampling of local minima in the protein energy surface

Background

Despite computational challenges, elucidating conformations that a protein system assumes under physiologic conditions for the purpose of biological activity is a central problem in computational structural biology. While these conformations are associated with low energies in the energy surface that underlies the protein conformational space, few existing conformational search algorithms focus on explicitly sampling low-energy local minima in the protein energy surface.

Methods

This work proposes a novel probabilistic search framework, PLOW, that explicitly samples low-energy local minima in the protein energy surface. The framework combines algorithmic ingredients from evolutionary computation and computational structural biology to effectively explore the subspace of local minima. A greedy local search maps a conformation sampled in conformational space to a nearby local minimum. A perturbation move jumps out of a local minimum to obtain a new starting conformation for the greedy local search. The process repeats in an iterative fashion, resulting in a trajectory-based exploration of the subspace of local minima.

Results and conclusions

The analysis of PLOW's performance shows that, by navigating only the subspace of local minima, PLOW is able to sample conformations near a protein's native structure, either more effectively or as well as state-of-the-art methods that focus on reproducing the native structure for a protein system. Analysis of the actual subspace of local minima shows that PLOW samples this subspace more effectively that a naive sampling approach. Additional theoretical analysis reveals that the perturbation function employed by PLOW is key to its ability to sample a diverse set of low-energy conformations. This analysis also suggests directions for further research and novel applications for the proposed framework.

     

Empirical solvation models in the context of conformational energy searches: application to bovine pancreatic trypsin inhibitor.

Continuum solvation models that estimate free energies of solvation as a function of solvent accessible surface area are computationally simple enough to be useful for predicting protein conformation. The behavior of three such solvation models has been examined by applying them to the minimization of the conformational energy of bovine pancreatic trypsin inhibitor. The models differ only with regard to how the constants of proportionality between free energy and surface area were derived. Each model was derived by fitting to experimentally measured equilibrium solution properties. For two models, the solution property was free energy of hydration. For the third, the property was NMR coupling constants. The purpose of this study is to determine the effect of applying these solvation models to the nonequilibrium conformations of a protein arising in the course of global searches for conformational energy minima. Two approaches were used: (1) local energy minimization of an ensemble of conformations similar to the equilibrium conformation and (2) global search trajectories using Monte Carlo plus minimization starting from a single conformation similar to the equilibrium conformation. For the two models derived from free energy measurements, it was found that both the global searches and local minimizations yielded conformations more similar to the X-ray crystallographic structures than did searches or local minimizations carried out in the absence of a solvation component of the conformational energy. The model derived from NMR coupling constants behaved similarly to the other models in the context of a global search trajectory. For one of the models derived from measured free energies of hydration, it was found that minimization of an ensemble of near-equilibrium conformations yielded a new ensemble in which the conformation most similar to the X-ray determined structure PTI4 had the lowest total free energy. Despite the simplicity of the continuum solvation models, the final conformation generated in the trajectories for each of the models exhibited some of the characteristics that have been reported for conformations obtained from molecular dynamics simulations in the presence of a bath of explicit water molecules. They have smaller root mean square (rms) deviations from the experimentally determined conformation, fewer incorrect hydrogen bonds, and slightly larger radii of gyration than do conformations derived from search trajectories carried out in the absence of solvent.     

Free energy distributions in proteins.

Protein molecules in solution have a broad distribution of enthalpy states. A good approximation to the distribution function for enthalpy states can be calculated, using the maximum-entropy method, from the moments of the distribution that, in turn, are obtained from the experimental temperature dependence of the heat capacity. In the present paper, we show that the enthalpy probability distribution can then be formulated in terms of a free energy function that gives the free energy of the protein corresponding to a particular value of the enthalpy. By the location of the minima in this function, the free energy distribution graphically indicates the most probable values of the enthalpy for the protein. We find that the behavior of the free energy functions for proteins falls somewhere between two different cases: a two-state like function with two minima, the relative levels of the two states changing with temperature; and, a single-minimum function where the position of the minimum shifts to higher enthalpy values as the temperature is increased. We show that the temperature dependence of the free energy function can be expressed in terms of a central free energy distribution for a given, fixed temperature (which is most conveniently chosen as the temperature of the maximum in the heat capacity). The nature of this central free energy function for a given protein thus yields all of the thermodynamic behavior of that protein over the temperature range of the denaturation process.     

Complexity of free energy landscapes of peptides revealed by nonlinear principal component analysis

Employing the recently developed hierarchical nonlinear principal component analysis (NLPCA) method of Saegusa et al. (Neurocomputing 2004;61:57-70 and IEICE Trans Inf Syst 2005;E88-D:2242-2248), the complexities of the free energy landscapes of several peptides, including triglycine, hexaalanine, and the C-terminal beta-hairpin of protein G, were studied. First, the performance of this NLPCA method was compared with the standard linear principal component analysis (PCA). In particular, we compared two methods according to (1) the ability of the dimensionality reduction and (2) the efficient representation of peptide conformations in low-dimensional spaces spanned by the first few principal components. The study revealed that NLPCA reduces the dimensionality of the considered systems much better, than did PCA. For example, in order to get the similar error, which is due to representation of the original data of beta-hairpin in low dimensional space, one needs 4 and 21 principal components of NLPCA and PCA, respectively. Second, by representing the free energy landscapes of the considered systems as a function of the first two principal components obtained from PCA, we obtained the relatively well-structured free energy landscapes. In contrast, the free energy landscapes of NLPCA are much more complicated, exhibiting many states which are hidden in the PCA maps, especially in the unfolded regions. Furthermore, the study also showed that many states in the PCA maps are mixed up by several peptide conformations, while those of the NLPCA maps are more pure. This finding suggests that the NLPCA should be used to capture the essential features of the systems.     

Stretching the immunoglobulin 27 domain of the titin protein: the dynamic energy landscape

Molecular simulations are carried out on the Immunoglobulin 27 domain of the titin protein. The energy landscape is mapped out using an implicit solvent model, and molecular dynamics simulations are run with the solvent explicitly modeled. Stretching a protein is shown to produce a dynamic energy landscape in which the energy minima move in configuration space, change in depth, and are created and destroyed. The connections of these landscape changes to the mechanical unfolding of the Immunoglobulin 27 domain are addressed. Hydrogen bonds break upon stretching by either intrabasin processes associated with the movement of energy minima, or interbasin processes associated with transitions between energy minima. Intrabasin changes are reversible and dominate for flexible interactions, whereas interbasin changes are irreversible and dominate for stiff interactions. The most flexible interactions are Glu-Lys salt bridges, which can act like tethers to bind strands even after all backbone interactions between the strands have been broken. As the protein is stretched, different types of structures become the lowest energy structures, including structures that incorporate nonnative hydrogen bonds. Structures that have flat energy versus elongation profiles become the lowest energy structures at elongations of several Angstroms, and are associated with the unfolding intermediate state observed experimentally.     

Determining minimum energy conformations of polypeptides by dynamic programming

A combinatorial optimization approach is used for solving the multiple-minima problem when determining the low-energy conformations of short polypeptides. Each residue is represented by a finite number of discrete states corresponding to single residue local minima of the energy function. These precomputed values constitute a search table and define the conformational space for discrete minimization by a generalized dynamic programming algorithm that significantly limits the number of intermediate conformations to be generated during the search. Since dynamic programming involves stagewise decisions, it results in buildup-type procedures implemented in two different forms. The first procedure predicts a number of conformations by a completely discrete search and these are subsequently refined by local minimization. The second involves limited continuous local minimization within the combinatorial algorithm, generally restricted to two dihedral angles in a buildup step. Both procedures are tested on 17 short peptides previously studied by other global minimization methods but involving the same potential energy function. The discrete method is extremely fast, but proves to be successful only in 14 of the 17 test problems. The version with limited local minimization finds, however, conformations in all the 17 examples that are close to the ones previously presented in the literature or have lower energies. In addition, results are almost independent of the cutoff energy, the most important parameter governing the search. Although the limited local minimization increases the number of energy evaluations, the method still offers substantial advantages in speed.     

The energy computation paradox and ab initio protein folding

The routine prediction of three-dimensional protein structure from sequence remains a challenge in computational biochemistry. It has been intuited that calculated energies from physics-based scoring functions are able to distinguish native from nonnative folds based on previous performance with small proteins and that conformational sampling is the fundamental bottleneck to successful folding. We demonstrate that as protein size increases, errors in the computed energies become a significant problem. We show, by using error probability density functions, that physics-based scores contain significant systematic and random errors relative to accurate reference energies. These errors propagate throughout an entire protein and distort its energy landscape to such an extent that modern scoring functions should have little chance of success in finding the free energy minima of large proteins. Nonetheless, by understanding errors in physics-based score functions, they can be reduced in a post-hoc manner, improving accuracy in energy computation and fold discrimination.