Effective sampling in the configurational space of a small peptide by the multicanonical-multioverlap algorithm

Protein structure predictions by enhanced conformational sampling methods

In this Special Festschrift Issue for the celebration of Professor Nobuhiro Gō's 80th birthday, we review enhanced conformational sampling methods for protein structure predictions. We present several generalized-ensemble algorithms such as multicanonical algorithm, replica-exchange method, etc. and parallel Monte Carlo or molecular dynamics method with genetic crossover. Examples of the results of these methods applied to the predictions of protein tertiary structures are also presented.

Transformation of a design peptide between the α-helix and β-hairpin structures using a helix-strand replica-exchange molecular dynamics simulation

We investigated the transformation between the α-helix and β-hairpin structures of an 18-residue design peptide, whose sequence is INYWLAHAKAGYIVHWTA. This peptide has both α-helix and β-hairpin structures in aqueous solution. For this purpose, we proposed the helix-strand replica-exchange method. This is one of the Hamiltonian replica-exchange methods in which we exchange parameters for umbrella potentials to enhance the α-helix or β-strand structure formation. We performed an all-atom helix-strand replica-exchange molecular dynamics (MD) simulation of this peptide in explicit water solvent with five replicas. Because the suitable umbrella potential was applied, the helix-strand replica-exchange MD simulation reproduced conformations closer to experimental conformations than a temperature replica-exchange MD simulation when the same numbers of the replicas were used, while the temperature replica-exchange MD simulation does not require bias along any specific order parameter. We calculated its free-energy landscape and revealed the transformation pathways between the α-helix and β-hairpin structures and the folding pathways from an extended structure. Although the fractions of the α-helix and β-hairpin structures are less than those obtained by the experiment, the free-energy difference between the two structures is calculated to be almost zero, which agrees with the experimental results.

On-the-fly reconstruction of free-energy profiles using logarithmic mean-force dynamics

Mean-force dynamics (MFD), which is a fictitious dynamics for a set of collective variables on a potential of mean-force, is a powerful algorithm to efficiently explore free-energy landscapes. Recently, we have introduced logarithmic MFD (LogMFD) (Morishita et al., Phys. Rev. E 2012, 85, 066702) which overcomes difficulties encounterd in free-energy calculations using standard approaches such as thermodynamic integration. Here, we present a guide to implementing LogMFD calculations paying attention to the practical issues in choosing the parameters in LogMFD. A primary focus is given to the effect of the parameters on the accuracy of the reconstructed free-energy profiles. A recipe for reducing the errors due to energy dissipation is presented. We also demonstrate that multidimensional free-energy landscapes can be reconstructed on-the-fly using LogMFD, which cannot be accomplished using any other free-energy calculation techniques.

Enhanced sampling algorithms

In biomolecular systems (especially all-atom models) with many degrees of freedom such as proteins and nucleic acids, there exist an astronomically large number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. Enhanced conformational sampling techniques are thus in great demand. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present various extensions of these three generalized-ensemble algorithms. The effectiveness of the methods is tested with short peptide and protein systems.

Replica-Permutation Method with the Suwa-Todo Algorithm beyond the Replica-Exchange Method

We propose a new method for molecular dynamics and Monte Carlo simulations, which is referred to as the replica-permutation method (RPM), to realize more efficient sampling than the replica-exchange method (REM). In RPM, not only exchanges between two replicas but also permutations among more than two replicas are performed. Furthermore, instead of the Metropolis algorithm, the Suwa-Todo algorithm is employed for replica-permutation trials to minimize its rejection ratio. We applied RPM to particles in a double-well potential energy, Met-enkephalin in a vacuum, and a C-peptide analog of ribonuclease A in explicit water. For comparison purposes, replica-exchange molecular dynamics simulations were also performed. As a result, RPM sampled not only the temperature space but also the conformational space more efficiently than REM for all systems. From our simulations of C-peptide, we obtained the α-helix structure with salt bridges between Gly2 and Arg10, which is known in experiments. Calculating its free-energy landscape, the folding pathway was revealed from an extended structure to the α-helix structure with the salt bridges. We found that the folding pathway consists of the two steps: The first step is the "salt-bridge formation step," and the second step is the "α-helix formation step."

Coulomb replica-exchange method: handling electrostatic attractive and repulsive forces for biomolecules

We propose a new type of the Hamiltonian replica-exchange method (REM) for molecular dynamics (MD) and Monte Carlo simulations, which we refer to as the Coulomb REM (CREM). In this method, electrostatic charge parameters in the Coulomb interactions are exchanged among replicas while temperatures are exchanged in the usual REM. By varying the atom charges, the CREM overcomes free-energy barriers and realizes more efficient sampling in the conformational space than the REM. Furthermore, this method requires only a smaller number of replicas because only the atom charges of solute molecules are used as exchanged parameters. We performed Coulomb replica-exchange MD simulations of an alanine dipeptide in explicit water solvent and compared the results with those of the conventional canonical, replica exchange, and van der Waals REMs. Two force fields of AMBER parm99 and AMBER parm99SB were used. As a result, the CREM sampled all local-minimum free-energy states more frequently than the other methods for both force fields. Moreover, the Coulomb, van der Waals, and usual REMs were applied to a fragment of an amyloid-β peptide (Aβ) in explicit water solvent to compare the sampling efficiency of these methods for a larger system. The CREM sampled structures of the Aβ fragment more efficiently than the other methods. We obtained β-helix, α-helix, 3(10)-helix, β-hairpin, and β-sheet structures as stable structures and deduced pathways of conformational transitions among these structures from a free-energy landscape.

pH replica-exchange method based on discrete protonation states

We propose a new algorithm for obtaining proton titration curves of ionizable residues. The algorithm is a pH replica-exchange method (PHREM), which is based on the constant pH algorithm of Mongan et al. (J Comput Chem 2004;25:2038-2048). In the original replica-exchange method, simulations of different replicas are performed at different temperatures, and the temperatures are exchanged between the replicas. In our PHREM, simulations of different replicas are performed at different pH values, and the pHs are exchanged between the replicas. The PHREM was applied to a blocked amino acid and to two protein systems (snake cardiotoxin and turkey ovomucoid third domain), in conjunction with a generalized Born implicit solvent. The performance and accuracy of this algorithm and the original constant pH method (PHMD) were compared. For a single set of simulations at different pHs, the use of PHREM yields more accurate Hill coefficients of titratable residues. By performing multiple sets of constant pH simulations started with different initial states, the accuracy of predicted pK(a) values and Hill coefficients obtained with PHREM and PHMD methods becomes comparable. However, the PHREM algorithm exhibits better samplings of the protonation states of titratable residues and less scatter of the titration points and thus better precision of measured pK(a) values and Hill coefficients. In addition, PHREM exhibits faster convergence of individual simulations than the original constant pH algorithm.

Effect of using suboptimal alignments in template-based protein structure prediction

Computational protein structure prediction remains a challenging task in protein bioinformatics. In the recent years, the importance of template-based structure prediction is increasing because of the growing number of protein structures solved by the structural genomics projects. To capitalize the significant efforts and investments paid on the structural genomics projects, it is urgent to establish effective ways to use the solved structures as templates by developing methods for exploiting remotely related proteins that cannot be simply identified by homology. In this work, we examine the effect of using suboptimal alignments in template-based protein structure prediction. We showed that suboptimal alignments are often more accurate than the optimal one, and such accurate suboptimal alignments can occur even at a very low rank of the alignment score. Suboptimal alignments contain a significant number of correct amino acid residue contacts. Moreover, suboptimal alignments can improve template-based models when used as input to Modeller. Finally, we use suboptimal alignments for handling a contact potential in a probabilistic way in a threading program, SUPRB. The probabilistic contacts strategy outperforms the partly thawed approach, which only uses the optimal alignment in defining residue contacts, and also the re-ranking strategy, which uses the contact potential in re-ranking alignments. The comparison with existing methods in the template-recognition test shows that SUPRB is very competitive and outperforms existing methods.

Optimization of partial multicanonical molecular dynamics simulations applied to an alanine dipeptide in explicit water solvent

The partial multicanonical algorithm for molecular dynamics and Monte Carlo simulations samples a wide range of an important part of the potential energy. Although it is a strong technique for structure prediction of biomolecules, the choice of the partial potential energy has not been optimized. In order to find the best choice, partial multicanonical molecular dynamics simulations of an alanine dipeptide in explicit water solvent were performed with 15 trial choices for the partial potential energy. The best choice was found to be the sum of the electrostatic, Lennard-Jones, and torsion-angle potential energies between solute atoms. In this case, the partial multicanonical simulation sampled all of the local-minimum free-energy states of the P(II), C(5), α(R), α(P), α(L), and C states and visited these states most frequently. Furthermore, backbone dihedral angles ϕ and ψ rotated very well. It is also found that the most important term among these three terms is the electrostatic potential energy and that the Lennard-Jones term also helps the simulation to overcome the steric restrictions. On the other hand, multicanonical simulation sampled all of the six states, but visited these states fewer times. Conventional canonical simulation sampled only four of the six states: The P(II), C(5), α(R), and α(P) states.

The SAAP force field: development of the single amino acid potentials for 20 proteinogenic amino acids and Monte Carlo molecular simulation for short peptides

Molecular simulation by using force field parameters has been widely applied in the fields of peptide and protein research for various purposes. We recently proposed a new all-atom protein force field, called the SAAP force field, which utilizes single amino acid potentials (SAAPs) as the fundamental elements. In this article, whole sets of the SAAP force field parameters in vacuo, in ether, and in water have been developed by ab initio calculation for all 20 proteinogenic amino acids and applied to Monte Carlo molecular simulation for two short peptides. The side-chain separation approximation method was employed to obtain the SAAP parameters for the amino acids with a long side chain. Monte Carlo simulation for Met-enkephalin (CHO-Tyr-Gly-Gly-Phe-Met-NH2) by using the SAAP force field revealed that the conformation in vacuo is mainly controlled by strong electrostatic interactions between the amino acid residues, while the SAAPs and the interamino acid Lennard-Jones potentials are predominant in water. In ether, the conformation would be determined by the combination of the three components. On the other hand, the SAAP simulation for chignolin (H-Gly-Tyr-Asp-Pro-Glu-Thr-Gly-Thr-Trp-Gly-OH) reasonably reproduced a native-like beta-hairpin structure in water although the C-terminal and side-chain conformations were different from the native ones. It was suggested that the SAAP force field is a useful tool for analyzing conformations of polypeptides in terms of intrinsic conformational propensities of the single amino acid units.

Partial multicanonical algorithm for molecular dynamics and Monte Carlo simulations

Partial multicanonical algorithm is proposed for molecular dynamics and Monte Carlo simulations. The partial multicanonical simulation samples a wide range of a part of the potential-energy terms, which is necessary to sample the conformational space widely, whereas a wide range of total potential energy is sampled in the multicanonical algorithm. Thus, one can concentrate the effort to determine the weight factor only on the important energy terms in the partial multicanonical simulation. The partial multicanonical, multicanonical, and canonical molecular dynamics algorithms were applied to an alanine dipeptide in explicit water solvent. The canonical simulation sampled the states of P(II), C(5), alpha(R), and alpha(P). The multicanonical simulation covered the alpha(L) state as well as these states. The partial multicanonical simulation also sampled the C(7) (ax) state in addition to the states that were sampled by the multicanonical simulation. In the partial multicanonical simulation, furthermore, backbone dihedral angles phi and psi rotated more frequently than those in the multicanonical and canonical simulations. These results mean that the partial multicanonical algorithm has a higher sampling efficiency than the multicanonical and canonical algorithms.

Amyloid-beta(29-42) dimer formations studied by a multicanonical-multioverlap molecular dynamics simulation

Amyloid-beta peptides are known to form amyloid fibrils and are considered to play an important role in Alzheimer's disease. Amyloid-beta(29-42) is a fragment of the amyloid-beta peptide and also has a tendency to form amyloid fibrils. In order to study the mechanism of amyloidogenesis of this fragment, we applied one of the generalized-ensemble algorithms, the multicanonical-multioverlap algorithm, to amyloid-beta(29-42) dimer in aqueous solution. We obtained a detailed free-energy landscape of the dimer system. From the detailed free-energy landscape, we examined monomer and dimer formations of amyloid-beta(29-42) and deduced dimerization processes, which correspond to seeding processes in the amyloidogenesis of amyloid-beta(29-42).