Long time scale simulations of Molecular Systems. Benjamin Gladwin and Thomas Huber, Department of M

1
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 Newtons 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 detailed
  trajectories Application to the folding of C
  peptide. Journal of Physical Chemistry B 103
  (1999), 899-911.
• R. Elber, A. Ghosh and A. Cardenas, Long time
  dynamics of complex systems. Accounts of Chemical
  Research 35 (2002), 396-403.

Contact gladwin_at_maths.uq.edu.au Contact
huber_at_maths.uq.edu.au Address Department of
Mathematics, The University of Queensland,
Brisbane Qld 4072, AUSTRALIA