Dynamics, Chaos and Molecular Modeling
One important feature of any biological or chemical system
is that it is in constant motion. While we study this with
molecular modelling and molecular dynamics calculations,
it is often worth looking at simpler systems to understand the
expected behavior.
The Lorenz attractor is
a classic example of a strange attractor. The time evolution of
this hops between two minima. For any given moment, the time which
the point stays in one minimum or the other is not predictable.
However, the distribution of total time spent in either minima
is, at least in principle, predictable. Molecular dynamics shows
similar behavior. Thermodynamic properties are approximated by
a time average over a simulation of a local or kinetic representation
of a kinetic system. Interpretation of the results depends on
whether the calculation is an average over time or a point (local)
estimate. (see Weber & Harrison 1996, 1997 for more details).
The Rossler system shown here is simpler, but similar to the Lorenz
system.
The Henon-Heiles system is defined by the Hamiltonian:
H = 1/2( p1^2 +p2^2 +q1^2 + q2^2) + q1^2q2 + 1/3(q2^3)
where q1,q2 are the positions and p1,p2 are the momenta (unit
mass). Above a critical energy many paths are chaotic. This is
one such path. While almost periodic in the beginning, and occasionally
during the run, the long term behavior of this system is unpredictable.
(by the way the integrator does track periodic solutions for long
times without significant error).
These two particles start at "almost" the same point
(0.5,0.1 and 0.501,0.099) but rapidly diverge over time. This
shows the sensitive dependence of Chaos on initial conditions.
Ergodicity, the connection between a time average and a space average in
statistical mechanics, requires that the trajectory of a system eventually
visits all of the available space. Chaos ensures that a volume in space
"expands" to visit all of the space and therefor ensures that the system
is ergodic.
References:
Weber, I.T. and Harrison, R.W. "Molecular Mechanics Calculations
on HIV-1 Protease with Peptide Substrates Correlate with Experimental
Data." (1996) Protein Engineering, 9: 679-690.
Weber, I.T. and Harrison, R.W. (1997) "Molecular Mechanics
Calculations on Rous sarcoma virus Protease with Peptide Substrates"
Prot. Sci. accepted.
Weber, I.T. and Harrison, R.W. (1997) "Molecular Mechanics
Calculations on Protein-Ligand Complexes." in 3D QSAR in
Drug Design. Recent Advances. eds. Kubinyi, H., Folkers, G. and
Martin, Y. ESCOM Science Publishers, Leiden, the Netherlands,
in press.
Author: Robert Harrison
Sources are available from www.cs.gsu.edu