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
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.