Deterministic and stochastic mathematical modeling and analysis of cellular systems with time delay

Abstract

The intracellular biochemical processes are generally inbuilt with time delay postponing signal transduction. This time delay plays a critical role in the dynamics of intracellular systems, in particular, involving negative feedback. For instance, negative feedback with time delay can lead to oscillations in gene expression level with ~24hr period. This controls daily physiological rhythms, such as wake-sleep cycle, called circadian rhythms, which are essential for the survival of humans. Given such importance, many experimental studies have been conducted and proteins that constitute the systems have been identified. However, despite the progress, understanding of the time-delay systems remains far from complete. How long are delays generated in the systems? How can delays be generated? How is dynamics of time-delay systems altered when the length of delays changes?To address the questions, we developed theoretical frameworks and mathematical models. Specifically, we proposed a Bayesian inference method for estimation of time delay, which is based on queueing theory and mixed-effects modeling. By applying our method to time-lapse live-cell imaging data, we estimated time delay for cellular stress signal responses including SOS response. This revealed that the number of rate-limiting steps determining time delay is positively correlated with the magnitude of the cell-to-cell variability in signal activation rate amplified by the rate-limiting reactions. We also developed spatiotemporal stochastic model of the circadian clock by using agent-based modeling approach. By simulating the model, we identified a potential mechanism of the circadian clock compensating for the spatiotemporal noise in cells and leading to the robust time delay, which was confirmed by follow-up experiments. Lastly, we developed a systems pharmacology model based on ordinary differential equations to explore the effect of change of delay length by a chemical compound on the circadian rhythms. The model simulation revealed that its effect strongly depends on the abundance of a clock protein, PERIOD2, which was validated by experiments. These frameworks and mathematical models would provide a clear mechanistic insight into time-delay systems.