Differential Equations and Nonlinear Regression

Stewart N. Abramson sna at prophet.pharm.pitt.edu
Wed Apr 13 11:42:06 EST 1994

Howdy out there in Bionet land:

We would like to fit experimental data to a system of differential
equations.   The process involves numerical integration of the system of
differential equations, and nonlinear least squares fitting of experimental
data to the numerically integrated equations.  We are currently using the
NIH-sponsored PROPHET software to do this, but I wonder if there is more
"user friendly" software out there that will perform the same functions.  A
*simple* example of what we would like to do is as follows (we actually
want to use this process for more complicated situations):

If a receptor (R) can interact reversibly with an agonist (A) and an
inhibitor (I), then these reactions are described below where k1 and k2 are
association and dissociation constants for the agonist, while k3 and k4 are
the association and dissociation constants for the inhibitor.

k1                                     k3
A  +  R   <=>   AR     and      I  + R   <=>   IR
k2                                     k4

The differential equations that describe this system are

d(AR)/dt    =   (k1*A*R) - (k2*AR)
d((IR)/dt   =   (k3*I*R) - (k4*IR)

A, I, and R in the above equations can be subtituted with the conservation
of mass equations

Ao  =   A  +  AR
Io   =   I  +  IR
Ro  =  R  +  AR  +  IR

where Ao, Io, and Ro are the total concentrations of A, I, and R.

In our system, we know Ao, Io, Ro, k2, and k4, and we experimentally
measure IR over time.  Therefore, we have 2 differential equations and 2
unknowns (k1 and k2).

My question again is:  Does anyone know of any software that makes it
"easy" to numerically integrate a system of differential equations and fit
those equations to experimental data using nonlinear least squares curve
fitting procedures??

Any feedback would be appreciated.

Stewart N. Abramson
Assistant Professor
Department of Pharmacology
University of Pittsburgh

sna at prophet.pharm.pitt.edu