Effect of Memory and Active Forces on Transition Path Time Distributions.

An analytical expression is derived for the transition path time distribution for a one-dimensional particle crossing of a parabolic barrier. Two cases are analyzed: (i) a non-Markovian process described by a generalized Langevin equation with a power-law memory kernel and (ii) a Markovian process with a noise violating the fluctuation-dissipation theorem, modeling the stochastic dynamics generated by active forces. In case i, we show that the anomalous dynamics strongly affect the short time behavior of the distributions, but this happens only for very rare events not influencing the overall statistics. At long times the decay is always exponential, in disagreement with a recent study suggesting a stretched exponential decay. In case ii, the active forces do not substantially modify the short time behavior of the distribution but do lead to an overall decrease of the average transition path time. These findings offer some novel insights, useful for the analysis of experiments of transition path times in (bio)molecular systems.

Transition path time distributions.

Biomolecular folding, at least in simple systems, can be described as a two state transition in a free energy landscape with two deep wells separated by a high barrier. Transition paths are the short part of the trajectories that cross the barrier. Average transition path times and, recently, their full probability distribution have been measured for several biomolecular systems, e.g., in the folding of nucleic acids or proteins. Motivated by these experiments, we have calculated the full transition path time distribution for a single stochastic particle crossing a parabolic barrier, including inertial terms which were neglected in previous studies. These terms influence the short time scale dynamics of a stochastic system and can be of experimental relevance in view of the short duration of transition paths. We derive the full transition path time distribution as well as the average transition path times and discuss the similarities and differences with the high friction limit.

Secondary structure formation of homopolymeric single-stranded nucleic acids including force and loop entropy: implications for DNA hybridization.

Loops are essential secondary structure elements in folded DNA and RNA molecules and proliferate close to the melting transition. Using a theory for nucleic acid secondary structures that accounts for the logarithmic entropy -c ln m for a loop of length m, we study homopolymeric single-stranded nucleic acid chains under external force and varying temperature. In the thermodynamic limit of a long strand, the chain displays a phase transition between a low-temperature/low-force compact (folded) structure and a high-temperature/high-force molten (unfolded) structure. The influence of c on phase diagrams, critical exponents, melting, and force extension curves is derived analytically. For vanishing pulling force, only for the limited range of loop exponents 2 < c ≲ 2.479 a melting transition is possible; for c ≤ 2 the chain is always in the folded phase and for 2.479 ≲ c always in the unfolded phase. A force-induced melting transition with singular behavior is possible for all loop exponents c < 2.479 and can be observed experimentally by single-molecule force spectroscopy. These findings have implications for the hybridization or denaturation of double-stranded nucleic acids. The Poland-Scheraga model for nucleic acid duplex melting does not allow base pairing between nucleotides on the same strand in denatured regions of the double strand. If the sequence allows these intra-strand base pairs, we show that for a realistic loop exponent c ≈ 2.1 pronounced secondary structures appear inside the single strands. This leads to a lower melting temperature of the duplex than predicted by the Poland-Scheraga model. Further, these secondary structures renormalize the effective loop exponent [Formula: see text], which characterizes the weight of a denatured region of the double strand, and thus affect universal aspects of the duplex melting transition.

Fluctuations in the ensemble of reaction pathways.

The dominant reaction pathway is a rigorous framework to microscopically compute the most probable trajectories, in nonequilibrium transitions. In the low-temperature regime, such dominant pathways encode the information about the reaction mechanism and can be used to estimate nonequilibrium averages of arbitrary observables. On the other hand, at sufficiently high temperatures, the stochastic fluctuations around the dominant paths become important and have to be taken into account. In this work, we develop a technique to systematically include the effects of such stochastic fluctuations, to order k(B)T. This method is used to compute the probability for a transition to take place through a specific reaction channel and to evaluate the reaction rate.

Dominant reaction pathways in high-dimensional systems.

This paper is devoted to the development of a theoretical and computational framework denominated dominant reaction pathways (DRPs) to efficiently sample the statistically significant thermally activated reaction pathways, in multidimensional systems. The DRP approach is consistently derived from the Langevin equation through a systematic expansion in the thermal energy, k(B)T. Its main advantage with respect to existing simulation techniques is that it provides a natural and rigorous framework to perform the path sampling using constant displacement steps, rather than constant time steps. In our previous work, we have shown how to obtain the set of most probable reaction pathways, i.e., the lowest order in the k(B)T expansion. In this work, we show how to compute the corrections to the leading order due to stochastic fluctuations around the most probable trajectories. We also discuss how to obtain predictions for the evolution of arbitrary observables and how to generate conformations, which are representative of the transition state ensemble. We illustrate how our method works in practice by studying the diffusion of a point particle in a two-dimensional funneled external potential.

Simulating stochastic dynamics using large time steps.

We present an approach to investigate the long-time stochastic dynamics of multidimensional classical systems, in contact with a heat bath. When the potential energy landscape is rugged, the kinetics displays a decoupling of short- and long-time scales and both molecular dynamics or Monte Carlo (MC) simulations are generally inefficient. Using a field theoretic approach, we perform analytically the average over the short-time stochastic fluctuations. This way, we obtain an effective theory, which generates the same long-time dynamics of the original theory, but has a lower time-resolution power. Such an approach is used to develop an improved version of the MC algorithm, which is particularly suitable to investigate the dynamics of rare conformational transitions. In the specific case of molecular systems at room temperature, we show that elementary integration time steps used to simulate the effective theory can be chosen a factor approximately 100 larger than those used in the original theory. Our results are illustrated and tested on a simple system, characterized by a rugged energy landscape.

Quantitative protein dynamics from dominant folding pathways.

We develop a theoretical approach to the protein-folding problem based on out-of-equilibrium stochastic dynamics. Within this framework, the computational difficulties related to the existence of large time scale gaps are removed, and simulating the entire reaction in atomistic details using existing computers becomes feasible. We discuss how to determine the most probable folding pathway, identify configurations representative of the transition state, and compute the most probable transition time. We perform an illustrative application of these ideas, studying the conformational evolution of alanine dipeptide, within an all-atom model based on the empiric GROMOS96 force field.

Anharmonicity and self-similarity of the free energy landscape of protein G.

The near-native free-energy landscape of protein G is investigated through 0.4-micros-long atomistic molecular dynamics simulations in an explicit solvent. A theoretical and computational framework is used to assess the time dependence of salient thermodynamical features. While the quasiharmonic character of the free energy is found to degrade in a few ns, the slow modes display a very mild dependence on the trajectory duration. This property originates from a striking self-similarity of the free-energy landscape embodied by the consistency of the principal directions of the local minima, where the system dwells for several ns, and of the virtual jumps connecting them.

Dominant pathways in protein folding.

We present a method to investigate the kinetics of protein folding and the dynamics underlying the formation of secondary and tertiary structures during the entire reaction. By writing the solution of the Fokker-Planck equation in terms of a path integral, we derive a Hamilton-Jacobi variational principle from which we are able to compute the most probable pathway of folding. The method is applied to the folding of the Villin headpiece subdomain simulated using a Go model. An initial collapsing phase driven by the initial configuration is followed by a rearrangement phase, in which secondary structures are formed and all computed paths display strong similarities. This completely general method does not require the prior knowledge of any reaction coordinate and is an efficient tool to perform simulations of the entire folding process with available computers.

Variational charge renormalization in charged systems.

We apply general variational techniques to the problem of the counterion distribution around highly charged objects where strong condensation of counterions takes place. Within a field-theoretic formulation using a fluctuating electrostatic potential, the concept of surface-charge renormalization is recovered within a simple one-parameter variational procedure. As a test, we reproduce the Poisson-Boltzmann surface potential for a single-charged planar surface both in the weak-charge and strong-charge regime. We then apply our techniques to non-planar geometries where closed-form solutions of the non-linear Poisson-Boltzmann equation are not available. In the cylindrical case, the Manning charge renormalization result is obtained in the limit of vanishing salt concentration. However, for intermediate salt concentrations a slow crossover to the non-charged-renormalized regime (at high salt) is found with a quasi-power-law behavior which helps to understand conflicting experimental and theoretical results for the electrostatic persistence length of polyelectrolytes. In the spherical geometry charge renormalization is only found at intermediate salt concentrations, in agreement with previous numerical results.

General formalism for phase combination and phase refinement: a statistical thermodynamics approach in reciprocal space.

The mean-field optimization methodology has been used to recast in a single formalism the problem of phase optimization using an arbitrary energy function in the presence of an experimentally determined phase probability distribution function. It results naturally in the generalization of the notions of figure of merit and centroid phase where the weight of the energy refinement is controlled by an effective temperature in a self-consistent manner. In the limit of high temperature, the formalism reduces of course to the Blow & Crick [Acta Cryst. (1959), 12, 794-802] classical treatment. If a model is available, Sim's [Acta Cryst. (1960), 13, 511-512] weighting scheme for a combined map appears as the first step of a refinement to be conducted until self-consistency is achieved. Assuming that MIR phases exist and that they agree reasonably well with the phases of the model, a first-order expansion gives an estimate of the changes of weights and phases to be performed for the Fourier synthesis. This provides for a new way of doing phase combination that might prove useful in challenging cases of model refinement, e.g. in large macromolecular complexes. Thermodynamic considerations have been used to discuss the best determination of weights in phase refinement; they also suggest that a variational expression of maximum likelihood is best suited as a target for refinement because it is the free energy of the system. The formalism readily allows use of solvent flattening, density averaging and the atomicity criterion to refine phases, and automatically assigns a figure of merit to each reflection. Numerical tests of the method are presented in an attempt to resolve the phase-ambiguity problem of protein crystallography in the centrosymmetric P¿1¿ space group using an energy derived from the Sayre equation.

Partially folded states of proteins: characterization by X-ray scattering.

Partially folded states of proteins are found to occur with a wide variety of degrees of unfolding, ranging from the compact molten globule to the fully unfolded forms, depending on solvent conditions and the specific protein involved. Small to intermediate angle X-ray scattering from partially folded states of proteins yields low resolution scattering profiles that may be used to explore the degree of folding of a protein under given solution conditions. By Monte Carlo simulation of a highly simplified homopolymer model, we show that such partially folded states will yield a characteristic scattering profile that may be written as a linear superposition of scattering from a compact core and of scattering from random coil loops that emerge from this core. We also find a term resulting from interference of X-rays scattering from the core with those scattering from the loops. This interference term oscillates in sign and tends to enhance the core portion of the scattering profile. We compare the model calculations of the scattering profile with measurements of the scattering profile as a function of salt concentration for cytochrome c at pH 2. Because of our characterization of the scattering profiles, we suggest that these results may be re-interpreted in terms of the presence of a range of partially folded states as a function of pH and salt concentration, and that the observed scattering profiles are consistent with the characterization of the partially folded states in terms of random coil loops emerging from a compact core with the loop fraction increasing as the salt concentration is decreased. This characterization is consistent with data on amide protection against H-2H exchange of compact regions within partially folded states observed for a number of proteins, including cytochrome c.

Biasing a Monte Carlo chain growth method with Ramachandran's plot: application to twenty-L-alanine.

In this paper, we explore the possibility of using experimental observations in the Monte Carlo chain growth method that we have previously developed. In this method, the macromolecule (peptide, protein, nuclei acid, etc.) is grown atom-by-atom (or residue-by-residue, etc.) and partial chains are replicated according to their Boltzmann weights. Once the molecule completed, we are left with a Boltzmann-distributed ensemble of configurations. For long molecules, an efficient sampling of the (extremely large) phase space is difficult for obvious reasons (existence of many local minima, limited computer memory, etc.). In the case in which one is mainly interested in the low energy conformations, we have incorporated in the growth scheme experimental observations taken from the Protein Data Banks. More precisely, we have considered the case of twenty-L-alanine and we have used the (experimental) Ramachandran's plot for this residue. The biased growth procedure goes as follows: (a) each time one adds along the main backbone chain, either a carbon atom belonging to a carbonyl group, or a nitrogen atom, its dihedral angle (theta) or (psi) is drawn with a probability law that reflects the experimental Ramachandran (theta, psi) plot; (b) the bias introduced in this way is canceled through an extra term in the energy (replication energy = true energy + bias energy); (c) the configurations, generated at T = 1000 K, are then energy minimized.(ABSTRACT TRUNCATED AT 250 WORDS)