  Simulations of biophysical systems inevitably include steps that correspond
to time integrations of ordinary differential equations. These equations are
often related to enzyme action in the synthesis and destruction of molecular
species, and in the regulation of transport of molecules into and out of the
cell or cellular compartments. Enzyme action is almost invariably modeled with
the quasi-steady-state Michaelis-Menten formula or its close relative, the Hill
formula: this description leads to systems of equations that may be stiff and
hard to integrate, and poses unusual computational challenges in simulations
where a smooth evolution is interrupted by the discrete events that mark the
cells' lives. This is the case of a numerical model (Virtual Biophysics Lab -
VBL) that we are developing to simulate the growth of three-dimensional tumor
cell aggregates (spheroids). The program must be robust and stable, and must be
able to accept frequent changes in the underlying theoretical model: here we
study the applicability of known integration methods to this unusual context
and we describe the results of numerical tests in situations similar to those
found in actual simulations.
