A dynamical analysis of the regulation of the cell cycle in space and time

The cell division cycle is a crucial biological process, essential for development and survival of all organisms. In eukaryotes, the cell cycle is regulated by an intricate network of genes and proteins that controls the progress of the cycle. All of these interactions give rise to dynamical features such as bistability and time delay which aid in the coordination of cell cycle events. Understanding this complex interplay has been helped by mathematical models, which describe how the concentrations and activities of different components of this network change over time.

In this thesis, we describe our results on different dynamical elements that play a role in the cell cycle. We have four different chapters with results. In the first chapter we investigate the role of time delay in the oscillations that drive the early embryonic cell cycle. This time delay acts between the important mitotic kinase Cdk1 and a protein complex called the APC/C. The delay has been measured experimentally, but its origin and exact role are not yet clear. In a simple model, we show how ultrasensitivity and delay together determine whether oscillations exist. Importantly, we illustrate that different implementations of the time delay may alter the conclusions. At the end of this chapter we describe a method to turn the ultrasensitive response into a bistable response, and briefly describe the implications of this method.

In the second results chapter we show how such a bistable response curve can change dynamically in time. We use mitotic entry as a motivating example to show that bistable response curves can become time dependent if different compartments are introduced in an existing cell cycle model. We then explore the consequences of a changing bistable switch and show that it may provide robustness to cellular transitions and oscillations.

The third results chapter starts from an experimental observation, namely that the timing of the early embryonic cell cycle has a particular dependence on temperature. We explore how this scaling can be explained by the dynamics of the cell cycle oscillator. For this, we make the reaction rates of two different cell cycle models dependent on temperature, and explore how different sensitivities of different rates can reproduce the observed scaling. This chapter is the report of work in progress.

The final results chapter puts oscillators based on time delay and bistability in a spatial context. Such systems can produce traveling waves, and these waves are often generated by pacemakers: regions which oscillate faster than their surroundings. In biological systems, such waves can function to transmit information, and an important question is which elements of the oscillator and the pacemaker determine the speed of these waves. We answer this question using numerical simulation and analytical methods. In this way, we show that timescale separation is important for the speed, but only in the oscillator based on bistability. Moreover, we explain how the size and frequency difference of the pacemaker affect the speed of the waves, and the speed by which they permeate the rest of the medium. At the end of this chapter, we briefly explain what happens in a system where multiple pacemakers compete to entrain the medium.