Optimal Paths Between Parameter Estimates in Non-linear ODE Systems Using the Nudged Elastic Band Method - Tönsing, Christian; Timmer, Jens; Kreutz, Clemens (Frontiers in Physics, 2019)
Ordinary differential equation (ODE) models are frequently used to mathematically represent the dynamic behavior of cellular components, e.g. for describing biochemical reaction networks. Solutions of these ODE models depend non-linearly on parameters, which can be estimated using experimental data by minimizing the discrepancy between data and the model trajectories. In realistic applications, only relative, sparse and noisy data is available which makes model fitting a challenging optimization problem. In order to take account for the non-convexity of the objective function and to reveal the existence of local optima within the parameter search space, optimization is performed with multiple initial guesses. For statistically valid conclusions it is of general interest, whether distinct optimization outcomes are correctly identified as local optima originating from the non-convexity of the objective function as typically presumed, or if they are only a result of incompletely converged optimization runs and could be merged by a connecting path to a single optimum by e.g. fine-tuning of the numeric algorithms. To clarify this question in application settings, we present an approach for finding optimal paths between parameter estimates in complex objective function landscapes. By analyzing the profiles of these paths, conclusions about non-trivial connections of parameter estimates can be made and by this, it can be reliably determined if two different optimization results belong to the same or to distinct local optima. For optimal path finding, we adapt the nudged elastic band (NEB) method and apply our approach to a benchmark model with a suboptimal optimization result, which yields a preferable grouping of fits afterwards.