A New Approximation Approach for Transient Differential Equation Models - Kreutz, Clemens (Frontiers in Physics, 2020)
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 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 with respect to 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 proves to be challenging for numerical approaches as well as for statistically valid reasoning. Here, a complementary modeling approach based on curve fitting of a tailored 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. In order to demonstrate the performance of this approach in realistic systems biology settings, nine benchmark problems for cellular signaling have been analyzed. The presented approach can serve as an alternative modeling approach of individual time courses for large systems in the case of few observables. Moreover, it not only facilitates the interpretation of the model response of traditional ODE models, but also offers a data-driven strategy for predicting the approximate dynamic responses by an explicit function that, in addition, facilitates subsequent analytical calculations. Thus, it constitutes a promising complementary mathematical modeling strategy for situations where classical ODE modeling is cumbersome or even infeasible.