Long time scale simulations of Molecular System s. Benjamin Gladwin ¥ and Thomas Huber ‡ , Department of Mathematics §

1. 1. 2. Lennard-Jones Potential Experiments: Seven Particle Rotation • The two examples shown here are used to demonstrate the effectiveness of the method described in the theory section. Each of the examples has the following properties. • Particles with equal masses. • Forced Rotation. • Non-bonded Lennard-Jones potential energy function. Three Particle Rotation Discussion of Results: The two graphs in the figure above show the final path for a seven particle system. Due to the larger number of interactions the computation time for large numbers of expansions was significantly longer. The path shown above uses 10 expansions and was converged to rapidly. Due to the larger constraints placed on the paths by the increased number of interactions, higher order frequencies contribute less to the final path. The Least Action Approach. The Principle of Least Action states that any real path will occur in such a way as to make the action an extremum. Hamilton defined action mathematically as: 2 exp 27 exp 9 exp The first of the two plots in the figure above shows the time independent position of the three particles during their rotation. The second plot shows how the potential changes as the number of expansions is increased. A straight line initial path approximation had a maximum potential of 90000 kJ at the center of the path. The least action path has a maximum potential of 3 kJ. The figure to the left shows a close-up of one section of the path. The three paths shown were calculated using 2, 9 and 27 expansions. From this figure it is seen that in this simple example, with increasing numbers of expansions the path quickly converges to the final result. In the discrete case, this is the sum across all of the time points for a particular path. Here the action is reformulated as the deviation of the path from a Newtonian path where the force term is derived from the potential field (this reformulation follows Olender and Elber (1996) [1]). Minimisation of the Action: In this formulation, a conjugate gradient minimisation technique has been used to find the optimum set of coefficients. For this, the gradient of the action with respect to the coefficients needs to be known. Starting with the Action function specified, The Path and the Potential. The potential energy (V) experienced by the system is determined by the interactions of all particles. The interactions considered in the simulation need to be defined for each particle. The interaction types specified in this analysis are: Two body terms: bond interaction, Lennard-Jones, Coloumb potential energy. Three Bodied term: bond angle interactions. Four Body terms: proper and improper dihedral angle interactions. For M particles and N expansions, there are 3MN derivatives which need to be evaluated at each point in time and at each optimisation step. Computational difficulty arises from the derivatives of the force in this expression. To calculate the Action function above it is necessary to specify the coordinates of each particle at every point in time. The present formulation does this by defining the path as a Fourier Sine series. In addition, a linear term defines the trend between the boundary conditions. which is a second derivative of the potential function. Though mathematically involved, the reduction in time to locate an optimum path using gradient information justifies the additional complexity. Using this, the action can now be calculated for some given set of coefficients. A Newtonian path will have a set of coefficients (bn) which minimises the action. Long time scale simulations of Molecular Systems.Benjamin Gladwin¥ and Thomas Huber‡, Department of Mathematics§, Introduction and Motivation. Modelling of molecular systems is traditionally achieved through molecular dynamics. The position and momentum of each particle are determined experimentally and Newton’s equations of motion are integrated forward in time. This process relies on the approximation that the path is linear over small enough time increments. If the system is complex, it is necessary to make a large number of calculations to maintain sufficient accuracy. Large bio-molecules are subject to high numbers of atomic interactions. In simulating these systems, a high degree of computationally complexity is experienced. At present these limitations mean that the timeframes involved in simulating many biologically interesting molecular processes are much larger than is feasible using present computer hardware. Our approach computes molecular trajectories from experimentally determined start and end points.We find physically realisable paths without placing a constraint on the size of the time step employed. This approach provides information throughout the path and can be used to highlight significant events in the process and analyse them in greater detail. With Aij and Bij chosen arbitrarily as Conclusions. The boundary value approach allows us to specify an initial path across the whole time frame. One advantage of this is that we gain a broad overview of the whole trajectory using a coarse sampling of points. This initial set of coefficients can establish a general behavior for the system and any sudden transitions will be highlighted. By increasing the resolution across any of these areas we will get a better idea of the behavior of the molecule in these transitions. The practical advantage of this approach is that it only requires the specification of initial and final positions of the atoms, which are experimentally more accessible than the momenta of the particles. As a consequence any system in which the atoms initial and final positions are know is well suited to this technique. This includes reaction mechanics, where the educts and products are known, or molecular machinery, such as molecular motors, where the molecules making up the system go through some cycle such that their initial and final positions are the same.  Long term we look to apply this approach to the design and analysis of molecular motors in nano-technology. One example of a molecular motor is the F1-ATPase molecule. This is one domain of a transmembrane protein called ATP synthase, which acts to pump hydrogen ions across the membrane against their concentration gradients. Its structure is experimentally well understood and it is found in animals, plants and bacteria. Understanding the mechanism behind this process will not only increase our understanding of cellular energetics but also strengthen our understanding of molecular interactions. • Advantages. • Conceptual: • Smaller step sizes increases time resolution. • More expansions increases path accuracy. • No step size limitation. • Always have a stable solution (trajectories). • Computational: • Allows hierarchical optimization (unlike Molecular Dynamics). • Well suited to parallel processing. • Minimises search space by directing transition. • Disadvantages. • Conceptual: • Boundary Constraints may impose artificial forces. • Computational: • Large step sizes may inadequately sample search path. • References: • R. Olender and R. Elber, Calculation of classical trajectories with a very large time step: Formalism and numerical examples. Journal of Chemical Physics 105 (1996), 9299-9315. • M. Parrinello, Action-derived molecular dynamics in the study of rare events. Abstracts of Papers of the American Chemical Society 221 (2001), 140-PHYS. • D. Passerone and M. Parrinello, Action-derived molecular dynamics in the study of rare events. Physical Review Letters 8710 (2001), art. no.-108302. • R. Elber, J. Meller and R. Olender, Stochastic path approach to compute atomically detailedtrajectories: Application to the folding of C peptide. Journal of Physical Chemistry B103 (1999), 899-911. • R. Elber, A. Ghosh and A. Cardenas, Long time dynamics of complex systems. Accounts ofChemical Research 35 (2002), 396-403. ¥ Contact: gladwin@maths.uq.edu.au ‡ Contact: huber@maths.uq.edu.au § Address: Department of Mathematics, The University of Queensland, Brisbane Qld 4072, AUSTRALIA