Title: Milestoning
Title:
Milestoning
Authors: Ron
Elber and Anthony West
Abstract
Atomically detailed simulations
provide useful information on biomolecular processes using a single
unified model. Specifically, Molecular Dynamics (MD) algorithms are
available to compute efficiently thermodynamic and equilibrium
behavior. However, MD is limited when studying non-equilibrium
processes and kinetics. Straightforward and typical trajectories of
condensed phase systems rarely exceed hundreds of nanoseconds, far
too short to investigate the kinetics of many interesting biophysical
systems. Examples are of conformational transitions, ion permeation,
protein folding and more. Extending the time scale of molecular
simulations is therefore an important research direction and has
attracted the attention of many investigators.
It is useful to classify processes of
long time dynamics into two categories: Dynamics which are (i)
activated or (ii) diffusive . Significant progress has been made in
algorithm design and theory development for activated processes
(Dellago, Bolhuis et al. 2002; Voter, Montalenti et al. 2002; Hummer
and Kevrekidis 2003; Faradjian and Elber 2004; Ren, Vanden-Eijnden et
al. 2005). In activated processes rare short
time trajectories pass over
significant free energy barriers and determine the overall kinetics.
Progress has been slower for diffusive processes (or a mixture of
activated and diffusive processes) in which the times of the
individual transitional trajectories are intrinsically long.
Diffusion on rugged energy landscapes is not necessarily associated
with a narrow transition domain between stable states. A narrow
transition domain is typical in activated processes and facilitates
the use of short time trajectories to probe reactive events. If we
probe an activated system at different time slices, in the majority
of the observations we do not observe something new. The system
remains in the reactant state until a rapid (but rare) transition is
initiated to the product state. In contrast, probing diffusive
processes show spatial progress in sequential observations.
Milestoning is a theoretical and computational approach that aims at
diffusive or mixed processes. Nevertheless, it can also handle
activated processes and therefore suggests a uniform technology for
the two types of dynamics.
A conceptual approach to long time
dynamics is that of coarse graining in space and time. Indeed a
number of groups have followed this idea, and have proposed fitting
parameters of a kinetic model (Sriraman, Kevrekidis et al. 2005;
Chodera, Swope et al. 2006) or of the diffusion equation (Yang,
Onuchic et al. 2007) based on atomically detailed simulations. For
example, it is assumed that rate constants (exponential relaxations
in time) describe transitions between the states of a Master
equation. Power law and stretched exponential kinetics were found in
biophysical kinetics (Frauenfelder, McMahon et al. 2001). Moreover,
there is no rigorous mapping from an atomically detailed description
of the system to a diffusion equation and the decision of what
exactly to fit is not unique.
In contrast to the phenomenological
modeling of the Master equation there is a rigorous approach to
spatial and temporal coarse graining by Zwanzig and Mori. It is the
Generalized Langevin Equation (Zwanzig. R. 2001) or equivalently the
Generalized Master Equation (Mori, Fujisaka et al. 1974). A memory
kernel (and not a rate constant) describes the impact of the “bath”.
Unfortunately, the numerical calculations of the memory kernel of the
Generalized Master Equation are difficult, motivating the use of the
simpler and less rigorous Master Equation. At the core of the
Milestoning approach one finds an algorithm to circumvent the
difficulty in computing the memory kernel. The function we compute is
formally equivalent but easier to estimate numerically than the rate
kernel. Therefore the Milestoning approach is equivalent to the
Generalized Master Equation and is based on a rigorous theory of
non-equilibrium processes.
Milestoning is a theory and an
algorithm to compute kinetics and thermodynamics of complex molecular
systems. It makes it possible to study general processes on rugged
energy landscapes on timescales not approachable by straightforward
Molecular Dynamics (microseconds and milliseconds). The algorithm is
based on monitoring progress along a set of discrete states
(Milestones) using short-time microscopic trajectories that capture
local dynamics. These discrete states can be (for example)
hypersurfaces perpendicular to a reaction coordinate. Transition
times between Milestones are recorded to produce
Local-First-Passage-Time-Distributions (LFPTD). The theory is based
on a non-Markovian integral equation for the probability flow between
Milestones. The integral equation is equivalent to the Generalized
Master equation. No specific model is assumed for the microscopic
dynamics. The theory uses the LFPTD to compute the overall kinetic
and thermodynamic. Complex transitions in proteins were investigated
(allosteric transition in Scapharca hemoglobin, the recovery stroke
in myosin).
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