Simplified method for the computation of parameters of power-law rate equations from time-series

Modeling biological processes from time-series data is a resourceful procedure which has received much attention in the literature. For models established in the context of non-linear differential equations, parameter-dependent phenomenological tentative response functions are tested by comparing would-be solutions of those models to the experimental time-series. Those values of the parameters for which a tested solution is a best fit are then retained. It is done with the help of some appropriate optimization algorithm which simplifies the searching procedure within the range of variability of the parameters that are to be estimated. The procedure works well in problems with a small number of adjustable parameters or/and with narrow searching ranges. However, it may start to be problematic for models with a large number of problem parameters inasmuch as convergence to the best fit is not necessarily ensured. In this case, a reduction in size of the parameter estimation problem must be undertaken. We presently address this issue by proposing a systematic procedure that does so in problems in which the system's response to a sufficiently small pulse perturbation of steady-state can be obtained. The response is then assumed to be a solution of the linearized equations, the Jacobian of which can be retrieved by a simple multilinear regression. The calculated n(2) Jacobian entries provide as many relationships among problem parameters, thus cutting substantially the size of the starting problem. After this preliminary treatment is applied, only (kappa-n(2)) of the initial kappa adjustable parameters are left for evaluation by means of a non-linear optimization procedure. The benefits of the present variant are both in economy of computation and in accuracy in determining the parameter values. The performance of the method is established under different circumstances. It is illustrated in the context of power-law rates, although this does not preclude its applicability to more general functional responses.