Bert W. Rust, ACMD
Robert W. Ashton, Chemical Science and Technology Laboratory
An important inverse problem in time series modeling is to fit a system of
ordinary differential equations (ODEs) to a set of observed time series
data which are corrupted by stochastic measuring errors.
Let the given data be one or more measured time series
, where the subscripts
correspond to times 
and the superscript
is the time series index.
Suppose that these data are to
be modeled by a system of 
first order ODEs
where q of the 
are to be fit to the
measured data, and
is an n-vector of parameter values to be determined by
fitting. If the initial measurements 
are corrupted by
measurement error, it
is necessary to also include the p-vector 
of initial
conditions
as fitting parameters in order not to bias the fit. In most cases the system of ODEs cannot be solved in closed form so it is necessary to combine a numerical integrator with parameter fitting program. This is accomplished by using David Kahaner's integrator SDRIV1 together with the Stanford nonlinear least squares code VARPRO to minimize
Figure 13: Simultaneous fits to the two thrombin
concentration time series
The least squares code also requires the partial derivatives of 
with respect to each of the fitting parameters. These can be computed
exactly by numerically integrating the system of variational equations
and
Since each of the 
are implicit functions of all of the 
and all of the 
, the partial derivatives of the right hand sides
can become complicated and numerous.
A recent biochemical application of this program at NIST arose in connection
with a study of the ability of anhydrothrombin (
), a derivative of the
enzyme thrombin (
), to compete with thrombin for the binding of a potent
thrombin inhibitor hirudin (
). The system of ODEs describing the
kinetics of the reactions can be written
where 
and 
are the concentrations of the molecules
thrombin-hirudin complex and anhydrothrombin complex, respectively, and
, 
and 
are parameters to be determined
by the fit. The initial values 
were known exactly so there were
only 15 variational equations for the partial derivatives
.
The measured data were two time series of thrombin concentrations,
corresponding to two separate experiments with different values of the initial
conditions 
. If the concentrations are expressed in units of
%Activity, then one experiment started with
and the other with
The data for the two experiments were combined and the two time series were fit simultaneously to determine
The fits accounted for 
of the combined total variance in the two
measured records.
The measurements and the fits are shown in Figure 13 where thrombin concentration 
is plotted against t measured in hours.