A reliable prediction of 3D protein structures from sequence data remains a big challenge due to both theoretical and computational difficulties. We have previously shown that our kinetostatic compliance method (KCM) implemented into the Protofold package can overcome some of the key difficulties faced by other de novo structure prediction methods, such as the very small time steps required by the molecular dynamics (MD) approaches or the very large number of samples needed by the Monte Carlo (MC) sampling techniques. In this article, we improve the free energy formulation used in Protofold by including the typically underrated entropic effects, imparted due to differences in hydrophobicity of the chemical groups, which dominate the folding of most water-soluble proteins. In addition to the model enhancement, we revisit the numerical implementation by redesigning the algorithms and introducing efficient data structures that reduce the expected complexity from quadratic to linear. Moreover, we develop and optimize parallel implementations of the algorithms on both central and graphics processing units (CPU GPU) achieving speed-ups up to two orders of magnitude on the GPU. Our simulations are consistent with the general behavior observed in the folding process in aqueous solvent, confirming the effectiveness of model improvements. We report on the folding process at multiple levels; namely, the formation of secondary structural elements and tertiary interactions between secondary elements or across larger domains. We also observe significant enhancements in running times that make the folding simulation tractable for large molecules.
Sampling methods generate numerous samples in the conformation space, followed by an evaluation of their free energies. Different search algorithms are used to find the unchallenged global minimum of the free energy, assumed to be associated with the native structure according to the thermodynamic hypothesis' @cite_82 . These search methods include simulated annealing @cite_70 @cite_18 , basin hopping @cite_67 @cite_15 @cite_9 @cite_47 , evolutionary algorithms @cite_32 @cite_71 @cite_37 and MC simulation with biased moves @cite_50 @cite_75 @cite_44 . A review of conformation sampling methods for protein folding can be found in @cite_69 . Sampling methods have two major limitations, namely 1) they do not provide any information about the biological pathway; and 2) finding the global minimum is not guaranteed because of the finite number of samples.
@cite_18 @cite_67 @cite_47 @cite_37 @cite_69 @cite_70 @cite_9 @cite_32 @cite_44 @cite_71 @cite_50 @cite_15 @cite_75 @cite_82
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The results of basin-hopping global optimization simulations are presented for four small, α-helical proteins described by a coarse-grained potential. A step-taking scheme that incorporates the local conformational preferences extracted from a large number of high-resolution protein structures is compared with an unbiased scheme. In addition, the discrete path sampling method is used to investigate the folding of one of the proteins, namely, the villin headpiece subdomain. Folding times from kinetic Monte Carlo simulations and iterative calculations based on a Markovian first-step analysis for the resulting stationary-point database are in good mutual agreement, but differ significantly from the experimental values, probably because the native state is not the global free energy minimum for the potential employed. The authors present a new method for searching low free energy paths in complex molecular systems at finite temperature. They introduce two variables that are able to describe the position of a point in configurational space relative to a preassigned path. With the help of these two variables the authors combine features of approaches such as metadynamics or umbrella sampling with those of path based methods. This allows global searches in the space of paths to be performed and a new variational principle for the determination of low free energy paths to be established. Contrary to metadynamics or umbrella sampling the path can be described by an arbitrary large number of variables, still the energy profile along the path can be calculated. The authors exemplify the method numerically by studying the conformational changes of alanine dipeptide. We report the predictive all-atom folding of the 60 amino acid four-helix bacterial ribosomal protein (BRP) L20 with a stochastic evolutionary optimization method in a free-energy force field. The energetically best, as well as six of the 10 lowest conformations, converge to a near-native structure. All of the 10 best energy conformations share the secondary structure elements of the native conformation, but differ in their tertiary alignment. The best conformation has a backbone root-mean-square deviation of 4.6 A to the native conformation and reproduces most distance constraints of the NMR experiment to 1.5 A resolution. Starting from random initial conditions, the native content of the simulated population increases more than 60-fold in the course of the simulation. These data demonstrate the feasibility of predictive unbiased all-atom protein folding with present day computational resources for the BRP L20. A comparison of two methods for surmounting the multiple-minima problem, Simulated Annealing (SA) and Monte Carlo with Minimization (MCM), is presented with applications to [Met]-enkephalin in the absence and in the presence of water. SA explores a continuous space of internal variables, while MCM explores a discrete space consisting of the local energy minima on that space. Starting from random conformations chosen from the whole conformational space in both cases, it is found that, while SA converges to low-energy structures significantly faster than MCM, the former does not converge to a unique minimum whereas the latter does. Furthermore, the behavior of the RMS deviations with respect to the apparent global minimum (for enkephalin in the absence of water) shows no correlation with the observed overall energy decrease in the case of SA, whereas such a correlation is quite evident with MCM; this implies that, even though the potential energy decreases in the annealing process, the Monte Carlo SA trajectory does not proceed towards the global minimum. Possible reasons for these differences between the two methods are discussed. It is concluded that, while SA presents attractive prospects for possibly improving or refining given structures, it must be considered inferior to MCM, at least in problems where little or no structural information is available for the molecule of interest.