Estimating kinetic rates from accelerated molecular dynamics simulations: alanine dipeptide in explicit solvent as a case study

Molecular dynamics (MD) simulation is the standard computational technique used to obtain information on the time evolution of the conformations of proteins and many other molecular systems. However, for most biological systems of interest, the time scale for slow conformational transitions is still inaccessible to standard MD simulations. Several sampling methods have been proposed to address this issue, including the accelerated molecular dynamics method. In this work, we study the extent of sampling of the phi/psi space of alanine dipeptide in explicit water using accelerated molecular dynamics and present a framework to recover the correct kinetic rate constant for the helix to beta-strand transition. We show that the accelerated MD can drastically enhance the sampling of the phi/psi conformational phase space when compared to normal MD. In addition, the free energy density plots of the phi/psi space show that all minima regions are accurately sampled and the canonical distribution is recovered. Moreover, the kinetic rate constant for the helix to beta-strand transition is accurately estimated from these simulations by relating the diffusion coefficient to the local energetic roughness of the energy landscape. Surprisingly, even for such a low barrier transition, it is difficult to obtain enough transitions to accurately estimate the rate constant when one uses normal MD.     

Overcoming barriers in trajectory space: mechanism and kinetics of rare events via Wang-Landau enhanced transition path sampling

Within the framework of transition path sampling (TPS), activation energies can be computed as path ensemble averages without a priori information about the reaction mechanism [C. Dellago and P. G. Bolhuis, Mol. Simul. 30, 795 (2004)]. Activation energies computed for different conditions can then be used to determine by numerical integration the rate constant for a system of interest from the rate constant known for a reference system. However, in systems with complex potential energy surfaces, multiple reaction pathways may exist making ergodic sampling of trajectory space difficult. Here, we present a combination of TPS with the Wang-Landau (WL) flat-histogram algorithm for an efficient sampling of the transition path ensemble. This method, denoted by WL-TPS, has the advantage that from one single simulation, activation energies at different temperatures can be determined even for systems with multiple reaction mechanisms. The proposed methodology for rate constant calculations does not require the knowledge of the reaction coordinate and is generally applicable to Arrhenius and non-Arrhenius processes. We illustrate the applicability of this technique by studying a two-dimensional toy system consisting of a triatomic molecule immersed in a fluid of repulsive soft disks. We also provide an expression for the calculation of activation volumes from path averages such that the pressure dependence of the rate constant can be obtained by numerical integration.     

String method in collective variables: minimum free energy paths and isocommittor surfaces

A computational technique is proposed which combines the string method with a sampling technique to determine minimum free energy paths. The technique only requires to compute the mean force and another conditional expectation locally along the string, and therefore can be applied even if the number of collective variables kept in the free energy calculation is large. This is in contrast with other free energy sampling techniques which aim at mapping the full free energy landscape and whose cost increases exponentially with the number of collective variables kept in the free energy. Provided that the number of collective variables is large enough, the new technique captures the mechanism of transition in that it allows to determine the committor function for the reaction and, in particular, the transition state region. The new technique is illustrated on the example of alanine dipeptide, in which we compute the minimum free energy path for the isomerization transition using either two or four dihedral angles as collective variables. It is shown that the mechanism of transition can be captured using the four dihedral angles, but it cannot be captured using only two of them.     

Free energy surfaces for the alpha(1 --> 4)-glycosidic linkage: implications for polysaccharide solution structure and dynamics

We present a potential of mean force surface for rotation about phi and psi dihedral angles of the alpha(1 --> 4)-glycosidic linkage in the maltose disaccharide (4-O-alpha-d-glucopyranosyl-d-glucopyranose) in aqueous solution. Comparison of the vacuum and solution free energy surfaces for maltose shows the principal effects of water to be an increase in the rotational freedom of the alpha(1 --> 4) linkage brought about by lowering the energy barrier for syn to anti conformational changes as well as expansion of the range of low-energy phi,psi conformations. This free energy analysis thus provides a thermodynamic and conformational rationale for the effects of water on alpha(1 --> 4)-linked polysaccharides and carbohydrate glasses.     

Improved transition path sampling methods for simulation of rare events

The free energy surfaces of a wide variety of systems encountered in physics, chemistry, and biology are characterized by the existence of deep minima separated by numerous barriers. One of the central aims of recent research in computational chemistry and physics has been to determine how transitions occur between deep local minima on rugged free energy landscapes, and transition path sampling (TPS) Monte-Carlo methods have emerged as an effective means for numerical investigation of such transitions. Many of the shortcomings of TPS-like approaches generally stem from their high computational demands. Two new algorithms are presented in this work that improve the efficiency of TPS simulations. The first algorithm uses biased shooting moves to render the sampling of reactive trajectories more efficient. The second algorithm is shown to substantially improve the accuracy of the transition state ensemble by introducing a subset of local transition path simulations in the transition state. The system considered in this work consists of a two-dimensional rough energy surface that is representative of numerous systems encountered in applications. When taken together, these algorithms provide gains in efficiency of over two orders of magnitude when compared to traditional TPS simulations.     

Hamiltonian and distance replica exchange method studies of Met-enkephalin

The conformational states of the zwitterionic form of the pentapeptide Met-enkephalin were explored with the use of explicit solvent molecular dynamics (MD). The N and C termini are ionized, as appropriate to polar solvent conditions, and consequently, there is a competition between open forms driven by polar solvation of the ammonium and carboxylate groups and closed forms driven by their salt-bridge formation. Normal MD started from an open state does not sample closed conformations. Sampling was enhanced with a distance replica exchange method (DREM) and with a Hamiltonian replica exchange method (HREM). The potential of mean force (PMF) along an end-to-end distance reaction coordinate was obtained with the DREM. The PMF shows a stable salt-bridge state and the presence of a large region of open states, as hypothesized for conformationally promiscuous small opiate peptides. The HREM systems differ by scaling the peptide-peptide and peptide-solvent electrostatic and Lennard-Jones potentials, with the goal of improving the sampling efficiency with a limited number of systems. A small number of systems were found to be sufficient to sample closed and open states. A principal component analysis (PCA) shows that the HREM-generated fluctuations are dominated by the first two principal modes. The first corresponds to the end-to-end reaction coordinate found in the DREM, and the first mode PMF is similar to the DREM PMF. The second mode describes the presence of two conformations, both of which correspond to the salt-bridge state distance. The conformers differ in the values of neighboring psi and phi dihedral angles, since such psi/phi compensation can still produce the same end-to-end distance. The two-dimensional PMF constructed from the first two PCA modes captures most of the significant backbone conformational space of Met-enkephalin.     

Obtaining reaction coordinates by likelihood maximization

We present a new approach for calculating reaction coordinates in complex systems. The new method is based on transition path sampling and likelihood maximization. It requires fewer trajectories than a single iteration of existing procedures, and it applies to both low and high friction dynamics. The new method screens a set of candidate collective variables for a good reaction coordinate that depends on a few relevant variables. The Bayesian information criterion determines whether additional variables significantly improve the reaction coordinate. Additionally, we present an advantageous transition path sampling algorithm and an algorithm to generate the most likely transition path in the space of collective variables. The method is demonstrated on two systems: a bistable model potential energy surface and nucleation in the Ising model. For the Ising model of nucleation, we quantify for the first time the role of nuclei surface area in the nucleation reaction coordinate. Surprisingly, increased surface area increases the stability of nuclei in two dimensions but decreases nuclei stability in three dimensions.     

Conformational transition pathway of polymerase beta/DNA upon binding correct incoming substrate

The closing conformational transition of wild-type polymerase beta bound to DNA template/primer before the chemical step (nucleotidyl transfer reaction) is simulated using the stochastic difference equation (in length version, "SDEL") algorithm that approximates long-time dynamics. The order of the events and the intermediate states during pol beta's closing pathway are identified and compared to a separate study of pol beta using transition path sampling (TPS) (Radhakrishnan, R.; Schlick, T. Proc. Natl. Acad. Sci. USA 2004, 101, 5970-5975). Results highlight the cooperative and subtle conformational changes in the pol beta active site upon binding the correct substrate that may help explain DNA replication and repair fidelity. These changes involve key residues that differentiate the open from the closed conformation (Asp192, Arg258, Phe272), as well as residues contacting the DNA template/primer strand near the active site (Tyr271, Arg283, Thr292, Tyr296) and residues contacting the beta and gamma phosphates of the incoming nucleotide (Ser180, Arg183, Gly189). This study compliments experimental observations by providing detailed atomistic views of the intermediates along the polymerase closing pathway and by suggesting additional key residues that regulate events prior to or during the chemical reaction. We also show general agreement between two sampling methods (the stochastic difference equation and transition path sampling) and identify methodological challenges involved in the former method relevant to large-scale biomolecular applications. Specifically, SDEL is very quick relative to TPS for obtaining an approximate path of medium resolution and providing qualitative information on the sequence of events; however, associated free energies are likely very costly to obtain because this will require both successful further refinement of the path segments close to the bottlenecks and large computational time.     

Generation of initial trajectories for transition path sampling of chemical reactions with ab initio molecular dynamics

Transition path sampling is an innovative method for focusing a molecular dynamics simulation on a reactive event. Although transition path sampling methods can generate an ensemble of reactive trajectories, an initial reactive trajectory must be generated by some other means. In this paper, the authors have evaluated three methods for generating initial reactive trajectories for transition path sampling with ab initio molecular dynamics. The authors have tested each of these methods on a set of chemical reactions involving the breaking and making of covalent bonds: the 1,2-hydrogen elimination in the borane-ammonia adduct, a tautomerization, and the Claisen rearrangement. The first method is to initiate trajectories from the potential energy transition state, which was effective for all reactions in the test set. Assigning atomic velocities found using normal mode analysis greatly improved the success of this method. The second method uses a high temperature molecular dynamics simulation and then iteratively reduces the total energy of the simulation until a low temperature reactive trajectory is found. This was effective in generating a low temperature trajectory from an initial trajectory run at 3000 K of the tautomerization reaction, although it failed for the other two. The third uses an orbital based bias potential to find a reactive trajectory and uses this trajectory to initiate an unbiased trajectory. The authors found that a highest occupied molecular orbital-lowest unoccupied molecular orbital bias could be used to find a reactive trajectory for the Claisen rearrangement, although it failed for the other two reactions. These techniques will help make it practical to use transition path sampling to study chemical reaction mechanisms that involve bond breaking and forming.     

Transition path sampling for discrete master equations with absorbing states

Transition path sampling (TPS) algorithms have been implemented with deterministic dynamics, with thermostatted dynamics, with Brownian dynamics, and with simple spin flip dynamics. Missing from the TPS repertoire is an implementation with kinetic Monte Carlo (kMC), i.e., with the underlying dynamics coming from a discrete master equation. We present a new hybrid kMC-TPS algorithm and prove that it satisfies detailed balance in the transition path ensemble. The new algorithm is illustrated for a simplified Markov State Model of trp-cage folding. The transition path ensemble from kMC-TPS is consistent with that obtained from brute force kMC simulations. The committor probabilities and local fluxes for the simple model are consistent with those obtained from exact methods for simple master equations. The new kMC-TPS method should be useful for analysis of rare transitions in complex master equations where the individual states cannot be enumerated and therefore where exact solutions cannot be obtained.     

Flexibility of "polyunsaturated fatty acid chains" and peptide backbones: A comparative ab initio study

The conformational properties of omega-3 type of polyunsaturated fatty acid (PUFA) chains and their fragments were studied using Hartree-Fock (RHF/3-21G) and DFT (B3LYP/6-31G(d)) methods. Comparisons between a unit (U) fragment of the PUFA chain and a mono N-Ac-glycine-NHMe residue show that both structures have the same sequence of sp2-sp3-sp2 atoms. The flexibility of PUFA originates in the internal rotation about the above pairs of sigma bonds. Therefore, potential energy surfaces (PESs) were generated by a scan around the terminal dihedral angles (phi t1 and phi t2) as well as the phi 1 and psi 1 dihedrals of both 1U congeners (Me-CHCH-CH2-CHCHMe and MeCONH-CH2-CONHMe) at the RHF/3-21G level of theory. An interesting similarity was found in the flexibility between the cis allylic structure and the trans peptide models. A flat landscape can be seen in the cis 1U (hepta-2,5-diene) surface, implying that several conformations are expected to be found in this (PES). An exhaustive search carried out on the 1U and 2U models revealed that straight chain structures such as trans and cis beta (phi 1 approximately psi 1 approximately 120 degrees; phi 2 approximately psi 2 approximately -120 degrees) or trans and cis extended (phi 1 approximately psi 1 approximately phi 2 approximately psi 2 approximately 120 degrees) can be formed at the lowest energy of both isomers. However, forming helical structures, such as trans helix (phi 1 approximately -120 degrees, psi 1 approximately 12 degrees; phi 2 approximately -120 degrees, psi 2 approximately 12 degrees) or cis helix (phi 1 approximately -130 degrees, psi 1 approximately 90 degrees; phi 2 approximately -145 degrees, psi 2 approximately 90 degrees) will require more energy. These six conformations, found in 2U, were selected to construct longer chains such as 3U, 4U, 5U, and 6U to obtain the thermochemistry of secondary structures. The variation in the extension or compression of the chain length turned out to be a factor of 2 between the helical and nonhelical structures. The inside diameter of the "tube" of cis helix turned out to be 3.5 A after discounting the internal H atoms. Thermodynamic functions were computed at the B3LYP/6-311+G(2d,p)//B3LYP/6-31G(d). The cis-trans isomerization energy of 1.7 +/- 0.2 kcal mol(-1) unit(-1) for all structure pairs indicates that the conformer selection was consistent. A folding energy of 0.5 +/- 0.1 kcal mol(-1) unit(-1) has been extracted from the energy comparison of the helices and most extended nonhelical structures. The entropy change associated with the folding (Delta S(folding)) is decreases faster with the degree of polymerization (n) for the cis than for the trans isomer. As a consequence, the linear relationships between (Delta G(folding)) and n for the cis and trans isomer crossed at about n = 3. This suggested that the naturally occurring cis isomer less ready to fold than the trans isomer since a greater degree of organization is exhibited by the cis isomer during the folding process. The result of this work leads to the question within the group additivity rule: could the method applied in our study of the folding of polyallylic hydrocarbons be useful in investigating the thermochemistry of protein folding?     

Extending the treatment of backbone energetics in protein force fields: limitations of gas-phase quantum mechanics in reproducing protein conformational distributions in molecular dynamics simulations

Computational studies of proteins based on empirical force fields represent a powerful tool to obtain structure-function relationships at an atomic level, and are central in current efforts to solve the protein folding problem. The results from studies applying these tools are, however, dependent on the quality of the force fields used. In particular, accurate treatment of the peptide backbone is crucial to achieve representative conformational distributions in simulation studies. To improve the treatment of the peptide backbone, quantum mechanical (QM) and molecular mechanical (MM) calculations were undertaken on the alanine, glycine, and proline dipeptides, and the results from these calculations were combined with molecular dynamics (MD) simulations of proteins in crystal and aqueous environments. QM potential energy maps of the alanine and glycine dipeptides at the LMP2/cc-pVxZ//MP2/6-31G* levels, where x = D, T, and Q, were determined, and are compared to available QM studies on these molecules. The LMP2/cc-pVQZ//MP2/6-31G* energy surfaces for all three dipeptides were then used to improve the MM treatment of the dipeptides. These improvements included additional parameter optimization via Monte Carlo simulated annealing and extension of the potential energy function to contain peptide backbone phi, psi dihedral crossterms or a phi, psi grid-based energy correction term. Simultaneously, MD simulations of up to seven proteins in their crystalline environments were used to validate the force field enhancements. Comparison with QM and crystallographic data showed that an additional optimization of the phi, psi dihedral parameters along with the grid-based energy correction were required to yield significant improvements over the CHARMM22 force field. However, systematic deviations in the treatment of phi and psi in the helical and sheet regions were evident. Accordingly, empirical adjustments were made to the grid-based energy correction for alanine and glycine to account for these systematic differences. These adjustments lead to greater deviations from QM data for the two dipeptides but also yielded improved agreement with experimental crystallographic data. These improvements enhance the quality of the CHARMM force field in treating proteins. This extension of the potential energy function is anticipated to facilitate improved treatment of biological macromolecules via MM approaches in general.     

Temperature weighted histogram analysis method, replica exchange, and transition paths

We analyzed the data from a replica exchange molecular dynamics simulation using the weighted histogram analysis method to combine data from all of the temperature replicas (T-WHAM) to obtain the room-temperature potential of mean force of the G-peptide (the C-terminal beta-hairpin of the B1 domain of protein G) in regions of conformational space not sampled at room temperature. We were able to determine the potential of mean force in the transition region between a minor alpha-helical population and the major beta-hairpin population and identify a possible transition path between them along which the peptide retains a significant amount of secondary structure. This observation provides new insights into a possible mechanism of formation of beta-sheet secondary structures in proteins. We developed a novel Bayesian statistical uncertainty estimation method for any quantity derived from WHAM and used it to validate the calculated potential of mean force. The feasibility of estimating regions of the potential of mean force with unfavorable free energy at room temperature by T-WHAM analysis of replica exchange simulations was further tested on a system that can be solved analytically and presented some of the same challenges found in more complex chemical systems.     

Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes

A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.     

Reaction path potential for complex systems derived from combined ab initio quantum mechanical and molecular mechanical calculations

Combined ab initio quantum mechanical and molecular mechanical calculations have been widely used for modeling chemical reactions in complex systems such as enzymes, with most applications being based on the determination of a minimum energy path connecting the reactant through the transition state to the product in the enzyme environment. However, statistical mechanics sampling and reaction dynamics calculations with a combined ab initio quantum mechanical (QM) and molecular mechanical (MM) potential are still not feasible because of the computational costs associated mainly with the ab initio quantum mechanical calculations for the QM subsystem. To address this issue, a reaction path potential energy surface is developed here for statistical mechanics and dynamics simulation of chemical reactions in enzymes and other complex systems. The reaction path potential follows the ideas from the reaction path Hamiltonian of Miller, Handy and Adams for gas phase chemical reactions but is designed specifically for large systems that are described with combined ab initio quantum mechanical and molecular mechanical methods. The reaction path potential is an analytical energy expression of the combined quantum mechanical and molecular mechanical potential energy along the minimum energy path. An expansion around the minimum energy path is made in both the nuclear and the electronic degrees of freedom for the QM subsystem internal energy, while the energy of the subsystem described with MM remains unchanged from that in the combined quantum mechanical and molecular mechanical expression and the electrostatic interaction between the QM and MM subsystems is described as the interaction of the MM charges with the QM charges. The QM charges are polarizable in response to the changes in both the MM and the QM degrees of freedom through a new response kernel developed in the present work. The input data for constructing the reaction path potential are energies, vibrational frequencies, and electron density response properties of the QM subsystem along the minimum energy path, all of which can be obtained from the combined quantum mechanical and molecular mechanical calculations. Once constructed, it costs much less for its evaluation. Thus, the reaction path potential provides a potential energy surface for rigorous statistical mechanics and reaction dynamics calculations of complex systems. As an example, the method is applied to the statistical mechanical calculations for the potential of mean force of the chemical reaction in triosephosphate isomerase.     

The reweighted path ensemble

We introduce a reweighting scheme for the path ensembles in the transition interface sampling framework. The reweighting allows for the analysis of free energy landscapes and committor projections in any collective variable space. We illustrate the reweighting scheme on a two dimensional potential with a nonlinear reaction coordinate and on a more realistic simulation of the Trp-cage folding process. We suggest that the reweighted path ensemble can be used to optimize possible nonlinear reaction coordinates.     

Activation processes with memory

We propose a mathematical treatment of the activated processes governed by stochastic Langevin dynamics with a colored random force, corresponding to a noise generated by an Ornstein-Uhlenbeck process. Such non-Markovian dynamics take place in a variety of chemical and biological systems. Using the path integral approach, we constructed the conditional probability for passing between two stationary states in configurational space. Our relations can be used for Monte Carlo sampling of evolution trajectories for systems with many degrees of freedom as well as for determining the reaction coordinate used in transition state theory. On the basis of our relation for a conditional probability, we generalize the method of determining the most probable path to the case of colored random force. Using the simple three-hole potential, we examine numerically the effect of nonzero correlation time (memory) on the evolution of the most probable path for a finite temperature.     

How effective for fold recognition is a potential of mean force that includes relative orientations between contacting residues in proteins?

We estimate the statistical distribution of relative orientations between contacting residues from a database of protein structures and evaluate the potential of mean force for relative orientations between contacting residues. Polar angles and Euler angles are used to specify two degrees of directional freedom and three degrees of rotational freedom for the orientation of one residue relative to another in contacting residues, respectively. A local coordinate system affixed to each residue based only on main chain atoms is defined for fold recognition. The number of contacting residue pairs in the database will severely limit the resolution of the statistical distribution of relative orientations, if it is estimated by dividing space into cells and counting samples observed in each cell. To overcome such problems and to evaluate the fully anisotropic distributions of relative orientations as a function of polar and Euler angles, we choose a method in which the observed distribution is represented as a sum of delta functions each of which represents the observed orientation of a contacting residue, and is evaluated as a series expansion of spherical harmonics functions. The sample size limits the frequencies of modes whose expansion coefficients can be reliably estimated. High frequency modes are statistically less reliable than low frequency modes. Each expansion coefficient is separately corrected for the sample size according to suggestions from a Bayesian statistical analysis. As a result, many expansion terms can be utilized to evaluate orientational distributions. Also, unlike other orientational potentials, the uniform distribution is used for a reference distribution in evaluating a potential of mean force for each type of contacting residue pair from its orientational distribution, so that residue-residue orientations can be fully evaluated. It is shown by using decoy sets that the discrimination power of the orientational potential in fold recognition increases by taking account of the Euler angle dependencies and becomes comparable to that of a simple contact potential, and that the total energy potential taken as a simple sum of contact, orientation, and (phi,psi) potentials performs well to identify the native folds.     

Harmonic Fourier beads method for studying rare events on rugged energy surfaces

We present a robust, distributable method for computing minimum free energy paths of large molecular systems with rugged energy landscapes. The method, which we call harmonic Fourier beads (HFB), exploits the Fourier representation of a path in an appropriate coordinate space and proceeds iteratively by evolving a discrete set of harmonically restrained path points-beads-to generate positions for the next path. The HFB method does not require explicit knowledge of the free energy to locate the path. To compute the free energy profile along the final path we employ an umbrella sampling method in two generalized dimensions. The proposed HFB method is anticipated to aid the study of rare events in biomolecular systems. Its utility is demonstrated with an application to conformational isomerization of the alanine dipeptide in gas phase.     

The dynamics of water exchange in gadolinium DOTA complexes studied by transition path sampling and potential of mean force methods

The mechanism of water exchange at the Gd centre of the two isomers of [Gd(iii)DOTA](-) (gadolinate(1-), [1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetato(4-)-N1,N4,N7,N10,O1,O4,O7,O10]) has been explored using transition path sampling and potential of mean force methods to sample those regions of phase space inaccessible to standard molecular dynamics simulation. We find that there are definite differences in the details of the solvent rearrangement accompanying the exchange of the capping water molecule for the two isomers. We conclude that these solvent effects, rather than any differences in the binding energy of the capping water, are central in determining the exchange rate. We find that the potential of mean force studies yield absolute and relative rates of water exchange for the two isomers that are in good agreement with experiment.