Limiting the parameter search space for dynamic models with rational kinetics using semi-definite programming

Estimation of kinetic parameters is a key step in modelling, as direct measurements are often expensive, time-consuming or even infeasible. The class of dynamic models in
polynomial form is particularly relevant in systems biology and biochemical engineering, as
those models naturally arise from modelling biochemical reactions using for instance mass action,
Michaelis-Menten or Hill kinetics. Often the parameters are not uniquely identifiable for a given
model structure and measurement set. Thus the question of which parameters are consistent
or inconsistent with the data arises naturally. Here we present a method capable of proving
inconsistency of entire parameter regions with the data. Based on the polynomial representation
of the system, we formulate a feasibility problem that can be solved efficiently by semi-definite
programming. The feasibility problem allows us to check consistency of entire parameter regions
by using upper and lower bounds on the parameters. This drastically limits the search space for
subsequent parameter estimation methods. In contrast to similar approaches in the literature,
the here presented approach does not require a steady state assumption. Measurements at
discrete time points are used, but neither regular sampling intervals, nor a time discretisation of
the system is required. Measurement uncertainties are dealt with using upper and lower bounds
on the measured states.