Long Timescale Path Integral Molecular Dynamics from Equations of Motion.

The University of Queensland2007

Abstract

Simulation of molecular systems is often limited by the time scale over which they can be analysed. Traditionally, analysis of these systems is done using atom based models and a process called Molecular Dynamics. In this method the initial position and momentum of each atom is specified and an equation of motion, such as Newton’s equation, is integrated forward in time. In the last several years Molecular Dynamics has become a standard laboratory tool and has helped to bridge the knowledge gap between molecular structure and biological function. Despite its widespread use, accurate simulation using the initial value formulation of traditional Molecular Dynamics is limited computationally to the investigation of small molecules or short timescales of approximately tens of nanoseconds. Many interesting biological and chemical processes take place over times which can be tens of seconds and this large disparity of timescales represents one of the most significant limitations facing simulation. A surprisingly small body of work has focussed on techniques which recast the search for dynamic trajectories as a boundary value problem. Addressing the dynamics in this sense, allows a broad overview of the whole trajectory to be obtained using a course sampling of points independent of the systems natural timescale. From an experimental viewpoint, this approach requires specification of initial and final positions of the atoms, and an estimate of the timescale. These properties of the system are more accessible using laboratory techniques such as X-ray crystallography. Any system in which the atoms initial and final positions are known and the transition path is required is well suited to this technique. This includes reaction mechanics, where the educts and products are known, or functional molecular machinery such as molecular motors which undergo well defined conformational changes. Boundary condition techniques are based on the least action approach proposed by Hamilton which has been used in classical mechanics as a means for determining particle trajectories. The majority of existing techniques use conservation of energy at each time in the trajectory as a measure to determine the suitability of a particular path. Conservation of energy represents a necessary condition for a dynamic path but is not, unfortunately, sufficient to guarantee that the path will satisfy equations of motion. In a molecular setting, the least action principle has been recast in terms of the deviation of a molecules path from a classical Newtonian trajectory. By defining an approximate long time scale trajectory for each atom in terms of some adjustable initial set of parameters, it becomes possible to calculate a classical action for the approximate path. The deviation from a classical Newtonian path can be measured and the trajectory can be iteratively improved. In this form, a resulting zero action path will conserve energy at each point in the path and satisfy the classical Newton’s equation of motion. In real biological systems, the effect of solvents play an important role in the behaviour of the molecule. Ideally, this effect would be accounted for by explicit modeling of each solvent molecule however, this is very computationally expensive. One way of cheaply accounting for the influence of the solvent is to use stochastic dynamics and Langevin’s equation of motion. In stochastic dynamics the energy of the system is only conserved as the long term average and therefore arguments which utilise moment to moment energy conservation are no longer valid. One outcome of this thesis has been the development of a new approach in the stochastic setting which is analogous to molecular dynamics using least action. This technique is developed in the spirit of the classical boundary condition based molecular dynamics and accounts for solvent effects. The overall purpose of this work has been the development of techniques to find atomic trajectories with minimum deviation from those which are governed by equations of motion. These approaches provide coarse grained approximate trajectories for systems over arbitrarily long time scales, and in conjunction with existing methods allow for detailed investigation of bio-molecular processes. Implementation of this work will aid in the understanding of mechanisms that underlie molecular transitions in biochemical systems and provide details which are less easily accessible using experimental techniques. Future applications may include specifically, the investigation of processes such as ion transfer and the motion of binding sites for proteins and ligands.

Abstract

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