Multilevel solution technique for biomolecular simulations

          T. Washio,

     C\&C Research Laboratories, NEC Europe Ltd.,
     D-53757 Sankt Augustin, Germany
     (email: washio @ ccrl-nece.technopark.gmd.de)

          C. W. Oosterlee and D. Hoffmann

     GMD, Institute for Algorithms and Scientific Computing,
     D-53754 Sankt Augustin, Germany
     (email: oosterlee @ gmd.de, daniel.hoffmann @ gmd.de )

  We study the Monte Carlo simulation method for obtaining 3D structures
of biomolecules in water. A new approach adopted here consists of the
modeling of the water molecules by a continuum model.
This means that we have to solve the so-called Poisson-Boltzmann
equation (PBE) to compute energy coming from the electrical interaction
between the target molecule and the water in order to accept or reject
Monte-Carlo steps.
 An efficient and accurate 3D multigrid solver with locally refined
grids around the target molecule is presented. Solving the 3D PBE for
these applications means that we have to deal with (amongst other
difficulties) jumping coefficients and with a known boundary condition
at infinity. Here, we investigate the interpolation at the interior
boundary points and the way to make the coarse grids around the original
finest grid. We construct an interpolation based on the conservative
discretization of the diffusion term on the composite grid. The solution
on the composite grid coming from this interpolation satisfies
the conservation of the flux and it is more accurate than the solution
obtained by standard cubic interpolation which is frequently used.
  In order to perform simulations with larger molecules, several
additional steps are necessary to accelerate the computations for two
reasons. The first is the relatively costly computation of the energy in
each (relatively cheap) Monte Carlo step, the other is the necessity to
perform a large number of Monte Carlo steps to obtain reasonable mean
values of the physical variables or the global minimum energy
conformation of the molecules.
 We try to reduce the computational cost further as follows. In most of
the Monte Carlo steps, we move atoms of the molecule locally and we are
interested in the difference of the energies before and after the move.
Therefore there is a possibility to use the finest grid only around the
move not around the whole molecule. Here again the handling of the
interior boundary is very important since we have many jumps of the
diffusion coefficient $\epsilon$  at the interior boundaries.
The conservative interpolation proposed makes this technique possible,
and we can reduce the number of total grid points from this technique
further.
  The second problem of reducing the number of the Monte Carlo steps
is our future work. By overcoming this difficulty, we can handle large
molecules. In this talk, we focus on the multigrid solver for the
Poisson-Boltzmann equation. We also present some interesting results for
small molecules obtained by our Monte Carlo simulation method.