A Variational Approach to Parameter Estimation in Ordinary Differential Equations

TL;DR

This paper introduces a variational calculus-based method for estimating entire courses of network components in systems modeled by ordinary differential equations, enabling more comprehensive and accurate parameter estimation in biological and chemical systems.

Contribution

It presents a novel variational approach that estimates both courses and parameters simultaneously, accounting for input uncertainties and enabling independent analysis of network motifs.

Findings

Accurate course and parameter estimation with uncertainty quantification.

Enables independent analysis of network motifs.

Integrates control theory and statistics methods.

Abstract

Ordinary differential equations are widely-used in the field of systems biology and chemical engineering to model chemical reaction networks. Numerous techniques have been developed to estimate parameters like rate constants, initial conditions or steady state concentrations from time-resolved data. In contrast to this countable set of parameters, the estimation of entire courses of network components corresponds to an innumerable set of parameters. The approach presented in this work is able to deal with course estimation for extrinsic system inputs or intrinsic reactants, both not being constrained by the reaction network itself. Our method is based on variational calculus which is carried out analytically to derive an augmented system of differential equations including the unconstrained components as ordinary state variables. Finally, conventional parameter estimation is applied to the augmented system resulting in a combined estimation of courses and parameters. The combined estimation approach takes the uncertainty in input courses correctly into account. This leads to precise parameter estimates and correct confidence intervals. In particular this implies that small motifs of large reaction networks can be analysed independently of the rest. By the use of variational methods, elements from control theory and statistics are combined allowing for future transfer of methods between the two fields.