Nonlinear differential equations are widely used for characterising functional forms of processes that govern complex biological pathway systems. Over the past decade, validation and further development of these models became possible due to data collected via high-throughput experiments using methods from molecular biology. While these data are very beneficial, they are typically incomplete and noisy, so that inferring parameter values for differential equation models is associated with serious computational challenges. Fortunately, many biological systems have embedded linear mathematical features, which may be exploited, thereby improving fits and leading to better convergence of optimization algorithms.
In this work, we explored inference for dynamic models using a venerable method of separable nonlinear least-squares optimization (SLS), applied however in a new setting and a novel fashion, with some ideas from non-parametric smoothing blended in. SLS was specifically designed to exploit partial linearity inherent in some optimisation problems. The numerical results from our extensive simulations suggest that the proposed approach to inference in ODE models is at least as accurate as the traditional nonlinear least-squares, but usually superior, while also enjoying a substantial reduction in computational time.
All the simulations were performed in the simode package in R, which I highly recommend to anybody interested in inference for differential equations.