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<TITLE> Distance Computation in MD-Simulations</TITLE>
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<P>
<H1><A NAME=SECTION00070000000000000000> Distance Computation in MD-Simulations</A></H1>
<P>
<A NAME=secdistanceestimate> </A>
Once a density function <IMG ALIGN=MIDDLE ALT="" SRC="img59.gif"> is computed by <tt> SOLVATE</tt> and the
parameters defining <b>f</b> are stored in the file <tt> boundary.lis</tt>,
any MD-program can access that file and use <b>f</b> to compute
boundary forces acting on water molecules near the surface of the
solute/solvent system. This force is always a function of the
distance to that surface, which therefore must be computed efficiently.
In this section we describe how this can be done to an accuracy
sufficient for most purposes.
<P>
For the computation of the distance <b>r</b> between the location of a water
molecule, <IMG ALIGN=MIDDLE ALT="" SRC="img60.gif">, and the surface point closest to
<IMG ALIGN=MIDDLE ALT="" SRC="img61.gif"> we first have to find
that surface point. We assume that, from the construction of <b>f</b>
as a sum of few gaussians, this surface point is sufficiently
close to the intersection
of a line through <IMG ALIGN=MIDDLE ALT="" SRC="img62.gif">, the direction of which is given by the
gradient <IMG ALIGN=MIDDLE ALT="" SRC="img63.gif">, with the surface
<IMG ALIGN=MIDDLE ALT="" SRC="img64.gif">. So we have to compute that intersection point.
<P>
<A href="f_l.gif"><IMG ALIGN=RIGHT ALT=" " SRC="f_s.gif"></A><BR>
Since this cannot be done analytically, and to avoid a possible (but slow)
iterative method, we approximate <b>f</b> for points on the line defined above
beween <IMG ALIGN=MIDDLE ALT="" SRC="img65.gif"> and the intersection point. In that region,
<b>f</b> (solid white line) is
by construction (a sum of gaussians, red, dashed lines)
dominated by the gaussian next to <IMG ALIGN=MIDDLE ALT="" SRC="img66.gif"> (thin, vertical line) and thus
can be approximated by a simple (one-dimensional) gaussian function <b>g</b>
(white dashed line),
<P><IMG ALIGN=BOTTOM ALT="" SRC="img67.gif"><P>
The parameters <b>k</b>,
<b>a</b>, and
<IMG ALIGN=BOTTOM ALT="" SRC="img68.gif">, which define
<b>g</b>, are obtained
from the value <IMG ALIGN=MIDDLE ALT="" SRC="img69.gif">,
the gradient <IMG ALIGN=MIDDLE ALT="" SRC="img70.gif">,
as well as the second derivative along the gradient,
<IMG ALIGN=MIDDLE ALT="" SRC="img71.gif">
(where <b>J</b>
is the Jacobian matrix of
<b>f</b> at
<IMG ALIGN=MIDDLE ALT="" SRC="img72.gif">),
<P><IMG ALIGN=BOTTOM ALT="" SRC="img73.gif"><P>
The intersection point <IMG ALIGN=MIDDLE ALT="" SRC="img74.gif">, approximated by
requiring <IMG ALIGN=MIDDLE ALT="" SRC="img75.gif">,
can now be calculated analytically, and the distance <b>r</b> we are interested in
reads
<P><IMG ALIGN=BOTTOM ALT="" SRC="img76.gif"><P>
<P>
As a rule of thumb, all distances required in an MD-simulation are
computed with an error smaller than 0.2 Å, which becomes even smaller
(0.02 Å) for water molecules close to the surface.
<P>
Through a similar approximation, also the average curvature
of the solvent at the intersection point, which may enter the
boundary forces as a second parameter, can be obtained.
<P>
<BR> <HR>
<P><ADDRESS>
<I>Helmut Grubmueller <BR>
Wed Jun 19 19:00:00 MET DST 1996</I>
</ADDRESS>
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