A modular approach for modeling the cell cycle based on functional response curves

Abstract

Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way.

Cell cycle oscillations driven by two interlinked bistable switches

“…In what follows, we will introduce and elaborately study the dynamics of different oscillator models built on each of the bistable switches in Figure 1, as well as both switches combined. Note that we introduced a similar approach in De Boeck et al (2021), which allowed for greater control over the response curve. However, here we limit ourselves to a cubic nonlinear function as it is likely one of the simplest expressions that can be used to reproduce a bistable switch.…”

“…This triggers the question of what the advantage might be of interlinking bistable switches to generate cell cycle oscillations when combined with negative feedback? We recently found that having multiple bistable switches can increase the parameter range of oscillations, while also allowing to switch between regions of qualitatively different oscillations (in terms of period and amplitude) De Boeck et al (2021). Having multiple positive feedback loops (which can lead to bistability) has been shown to allow for improving speed and resistance to noise Brandman et al (2005), which could also indicate a potential advantage of interlinked bistable switches.…”

“…We only consider deterministic systems and do not explicitly implement the various molecular interactions that lead to the different bistable switches (those have been studied elsewhere, see e.g. Novak and Tyson (1993); Pomerening et al (2003); Sha et al (2003); Mochida et al (2016); Rata et al (2018); Kamenz et al (2021); De Boeck et al (2021)). Instead, we approximate each experimentally measured bistable response curve by using a single cubic function [see Figure 1(e)-(f)].…”

Spatially coordinated collective phosphorylation filters spatiotemporal noises for precise circadian timekeeping

“…20 ( ! , which was used to describe the ultrasensitive response previously 44,47 . We set 𝐾 : = 400 and 𝑛 = 70…”

“…After the arrival at the perinucleus, 𝑅 " undergoes phosphorylation with one of the three phosphorylation mechanisms (Figure 2A) and enters the nucleus. To describe these different types of phosphorylation, we adopted the modular approach, which describes complex multistep reactions with a single phenomenological module that qualitatively describes the dynamical behavior 47,52 . This allowed us to simulate the complex phosphorylation process, which is a necessary step for nuclear entry (see STAR Methods).…”