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 equation. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which 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 account for events such as DNA damage and coupling to the circadian clock in a phenomenological way.

Spatially coordinated collective phosphorylation filters spatiotemporal noises for precise circadian timekeeping

“…To investigate the advantage of having such bistable phosphorylation of R p in reducing the heterogeneity in the distribution of t n , we compare it with the other typical phosphorylation mechanisms: ultrasensitive phosphorylation and linear phosphorylation. Ultrasensitive phosphorylation also has a switch-on threshold for synchronous phosphorylation, similar to bistable phosphorylation, but it does not have a distinct switch-off threshold (Figure 2A, blue) 44,47 . Thus, the fraction of the monotonically increases as the total amount of PER in the perinucleus increases, with the sigmoidal pattern having a steep increase near the switch-on threshold.…”

“…After the arrival at the perinucleus, R p 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).…”

“…Furthermore, it is challenging to model ultrasensitive and bistable phosphorylations since they are generated from complex combinations of multiple reactions. To resolve this, we utilized a modular approach that describes those complex multistep reactions with a phenomenological module, which was developed in work by De Boeck et al 47 . This allowed us to effectively capture the key dynamics of the phosphorylation mechanisms without comprehensively modeling the underlying complex reactions.…”

“…Then, is either dephosphorylated to R p with the rate of or enters the nucleus with the rate of λ n . To simply describe the phosphorylation based on multiple reactions, we adopted the modular approach 47 , which treated the phosphorylation reaction as a single phenomenological module. That is, the module takes the total amount of PER complex in the perinucleus as the input and gives the fraction of the in the perinucleus as the output.…”

“…To describe ultrasensitive phosphorylation, the propensity function was used. Thus, the fraction of quickly equilibrates to , which was used to describe the ultrasensitive response previously 44,47 . We set K u = 400 and n = 70 to obtain a steep increase in the fraction of when T reaches 400 (Figure 2A, blue).…”