A New Approximation Approach for Transient Differential Equation Models

Abstract

Ordinary differential equation (ODE) models are frequently applied to describe the dynamics of signaling in living cells. In systems biology, ODE models are typically defined by translating relevant biochemical interactions into rate equations. The advantage of such \emph{mechanistic} models is that each dynamic variable and model parameter has its counterpart in the biological process which potentiates interpretations and enables biologically relevant conclusions. A disadvantage for such mechanistic dynamic models is, however, that they become very large in terms of the number of dynamic variables and parameters if entire cellular pathways are described. Moreover, analytical solutions of the ODEs are not available and the dynamics is nonlinear which is a challenge for numerical approaches as well as for statistically valid reasoning. Here, a complementary modeling approach based on curve fitting of a tailored \emph{retarded transient function (RTF)} is introduced which exhibits amazing capabilities in approximating ODE solutions in case of transient dynamics as it is typically observed for cellular signaling pathways. A benefit of the suggested RTF is the feasibility of self-explanatory interpretations of the parameters as response time, as amplitudes, and time constants of a transient and a sustained part of the response. Nine benchmark problems for cellular signaling were analyzed to demonstrate the suggested approach in realistic systems biology settings. The presented approach can serve as alternative modeling approach of individual time courses for large systems in the case of few observables. Moreover, it enables valuable interpretations of the response for traditional ODE models. It offers a data-driven strategy for predicting the approximate dynamic responses by an explicit function that also facilitates subsequent analytical calculations. Thus, it constitutes a promising complementary mathematical modeling strategy for situations where classical ODE modeling is cumbersome or even infeasible.