Biomedical subjects

Eric Vanden-Eijnden

Publications and source records attributed to Eric Vanden-Eijnden.

10 recordsLinked to original sources

PubMed2007

Noise can play an organizing role for the recurrent dynamics in excitable media.

We analyze patterns of recurrent activity in a prototypical model of an excitable medium in the presence of noise. Without noise, this model robustly predicts the existence of spiral waves as the only recurrent patterns in two dimensions. With small noise, however, we found that this model is also capable of generating coherent target patterns, another type of recurrent activity that is widely observed experimentally. These patterns remain essentially deterministic despite the presence of the noise, yet their existence is impossible without it. Their degree of coherence can also be made arbitrarily high for wide ranges of the parameters, which does not require fine-tuning. Our findings demonstrate the need to reexamine current modeling approaches to active biological media.

Animals↗

PubMed2006

Illustration of transition path theory on a collection of simple examples.

Transition path theory (TPT) has been recently introduced as a theoretical framework to describe the reaction pathways of rare events between long lived states in complex systems. TPT gives detailed statistical information about the reactive trajectories involved in these rare events, which are beyond the realm of transition state theory or transition path sampling. In this paper the TPT approach is outlined, its distinction from other approaches is discussed, and, most importantly, the main insights and objects provided by TPT are illustrated in detail via a series of low dimensional test problems.

Chemistry, Physical↗

PubMed2006

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.

Journal Article↗

PubMed2006

Non-meanfield deterministic limits in chemical reaction kinetics.

A general mechanism is proposed by which small intrinsic fluctuations in a system far from equilibrium can result in nearly deterministic dynamical behaviors which are markedly distinct from those realized in the meanfield limit. The mechanism is demonstrated for the kinetic Monte Carlo version of the Schnakenberg reaction where we identified a scaling limit in which the global deterministic bifurcation picture is fundamentally altered by fluctuations. Numerical simulations of the model are found to be in quantitative agreement with theoretical predictions.

Chemical Phenomena↗

PubMed2005

Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates.

An efficient simulation algorithm for chemical kinetic systems with disparate rates is proposed. This new algorithm is quite general, and it amounts to a simple and seamless modification of the classical stochastic simulation algorithm (SSA), also known as the Gillespie [J. Comput. Phys. 22, 403 (1976); J. Phys. Chem. 81, 2340 (1977)] algorithm. The basic idea is to use an outer SSA to simulate the slow processes with rates computed from an inner SSA which simulates the fast reactions. Averaging theorems for Markov processes can be used to identify the fast and slow variables in the system as well as the effective dynamics over the slow time scale, even though the algorithm itself does not rely on such information. This nested SSA can be easily generalized to systems with more than two separated time scales. Convergence and efficiency of the algorithm are discussed using the established error estimates and illustrated through examples.

Journal Article↗

PubMed2005

Transition state theory: variational formulation, dynamical corrections, and error estimates.

Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient kappaS of the TST dividing surface are given, and it is shown that the relative error on kappaS scales as 1/square root(kappaS) when kappaS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient kappaS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise kappaS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum kappaS, but this coefficient may still be very small).

Journal Article↗

PubMed2005

Transition pathways in complex systems: Application of the finite-temperature string method to the alanine dipeptide.

The finite-temperature string method proposed by E, et al. [W. E, W. Ren, and E. Vanden-Eijnden, Phys. Rev. B 66, 052301 (2002)] is a very effective way of identifying transition mechanisms and transition rates between metastable states in systems with complex energy landscapes. In this paper, we discuss the theoretical background and algorithmic details of the finite-temperature string method, as well as the application to the study of isomerization reaction of the alanine dipeptide, both in vacuum and in explicit solvent. We demonstrate that the method allows us to identify directly the isocommittor surfaces, which are approximated by hyperplanes, in the region of configuration space where the most probable transition trajectories are concentrated. These results are verified subsequently by computing directly the committor distribution on the hyperplanes that define the transition state region.

Alanine↗

PubMed2005

Two distinct mechanisms of coherence in randomly perturbed dynamical systems.

We carefully examine two mechanisms--coherence resonance and self-induced stochastic resonance--by which small random perturbations of excitable systems with large time scale separation may lead to the emergence of new coherent behaviors in the form of limit cycles. We analyze what controls the degree of coherence in these two mechanisms and classify their very different properties. In particular we show that coherence resonance arises only at the onset of bifurcation and is rather insensitive against variations in the noise amplitude and the time scale separation ratio. In contrast, self-induced stochastic resonance may arise away from bifurcations and the properties of the limit cycle it induces are controlled by both the noise amplitude and the time scale separation ratio.

Journal Article↗

PubMed2005

Blue moon sampling, vectorial reaction coordinates, and unbiased constrained dynamics.

We give a new formula expressing the components of the mean force in terms of a conditional expectation which can be computed by Blue Moon sampling. This generalizes to the vectorial case a formula first derived by Ruiz-Montero et al. for a scalar reaction coordinate. We also discuss how to compute this conditional average by means of constrained stochastic dynamics which, unlike the usual constrained molecular dynamics, introduces no bias. Finally, we give a new perspective on bias removal by using constrained molecular dynamics.

Journal Article↗

PubMed2005

Finite temperature string method for the study of rare events.

A method is presented for the study of rare events such as conformational changes arising in activated processes whose reaction coordinate is not known beforehand and for which the assumptions of transition state theory are invalid. The method samples the energy landscape adaptively and determines the isoprobability surfaces for the transition: by definition the trajectories initiated anywhere on one of these surfaces has equal probability to reach first one metastable set rather than the other. Upon weighting these surfaces by the equilibrium probability distribution, one obtains an effective transition pathway, i.e., a tube in configuration space inside which conformational changes occur with high probability, and the associated rate. The method is first validated on a simple two-dimensional example; then it is applied to a model of solid-solid transformation of a condensed system.

Journal Article↗