Quantitative Analysis of Dynamical Systems Models of Biological Systems

Mathematical models are powerful tools for understanding physical systems. If the dynamics of the system are of interest, these models are commonly parameterized systems of ordinary differential equations (ODEs). For the model to be useful, it should reproduce known properties of the experimental system and ideally make meaningful predictions. In general, an arbitrary choice of parameters will not result in a model that is consistent with the physical system. As a first step, a suitable set of parameters must be determined such that the model can reproduce properties of the physical system, and this can represent a significant undertaking in itself.

The process of determining a set of parameters that can reproduce the known properties of the system is an example of an inverse problem. In general, inverse problems do not admit unique solutions; there can be no solutions or multiple solutions that are equally valid. Due to this indeterminacy, Monte Carlo Markov Chain (MCMC) methods are commonly used to estimate a probability distribution over the solution space to provide both a set of optimal solutions and a measure of uncertainty. The probability distributions that arise from inverse problems involving parameterized ODE models tend to be poorly conditioned in ways that degrade the performance of general MCMC methods. We first present a specialized method for sampling the distributions that arise from inverse problems involving parameterized ODEs and demonstrate its performance gains over conventional MCMC methods.

Our specialized MCMC method enables the treatment of ODE models that are difficult for conventional MCMC methods to handle without extensive tuning of hyperparameters. We apply our method to a model of the Kai oscillator. The sampling results make an unconventional prediction that the T432 phosphorylation site of KaiC is not necessary for oscillatory dynamics, but instead keeps the period of the oscillator robust over a wider range of KaiA concentrations. We confirm this prediction experimentally. To understand the mechanism by which robustness arises in the Kai oscillator, we reduce the optimized model to a simple minimal model that we analyze using the method of matched asymptotic expansions. The asymptotic analysis reveals that the robustness in the Kai oscillator is governed by the balance between the rate of autocatalytic KaiA release and the rate of KaiA sequestration, and that the presence of T432 allows the balance to persist across a wider range of KaiA concentrations.

Lastly, we consider how to analyze noisy systems modeled with stochastic differential equations (SDEs). The presence of noise requires us to compute averages of quantities of interest instead of the deterministic quantities in the case of ODEs. We extend the framework of transition path theory, which has traditionally been used to analyze molecular simulations, to compute statistics regarding the trajectories of the system.