DeBoeck2021 - Modular approach to modelling the cell cycle

January 2024, Model of the Month by Dr Jan Rombouts Lendert Gelens’ Lab, University of Leuven, Belgium

Original models - MODEL2212060001, MODEL2212060002.

Introduction

To multiply, a cell goes through a sequence of phases in the cell division cycle. The transitions between these phases are quick and robust due to a so-called bistable switch (Novák and Tyson 2021). For example, entry into mitosis is characterized by the fast activation of the kinase Cdk1 when cyclin B levels cross a threshold concentration. Moreover, lowering cyclin B values does not immediately lead to mitotic exit – the threshold for inactivation of Cdk1 is lower than its activation threshold, leading to hysteresis. When plotting the response of Cdk1 activity to cyclin B concentration, an S-shaped curve results with a region where two outputs are possible for a single input value - hence 'bistable' switch (Pomerening, Sontag, and Ferrell 2003).

 

These biochemical bistable switches with S-shaped responses are ultimately generated by feedback loops between genes and proteins (Ferrell and Ha 2014). Other response curves, such as ultrasensitive responses, are also generated by interaction networks. However, the reactions leading to an ultrasensitive response are not often modelled explicitly. Rather, the whole interaction is summarized by a Hill function. This functional form is also often used when the shape of the response is known, but the underlying reaction network is not. In contrast to ultrasensitive responses, bistable response curves don’t have a straightforward implementation in terms of simple mathematical expressions like the Hill function.

 

In our paper (De Boeck, Rombouts, and Gelens 2021) we propose a solution to this problem. We modify a Hill-type ultrasensitive response such that it produces a bistable response (Fig 1A, B). These bistable responses can, just like Hill functions, be included directly in larger models.

 

Model

The essence of our approach is summarized by the equation shown in Fig. 1A. We start from a Hill-type response, but make the threshold dependent on the output variable (represented by the function ξ(Y)). If ξ(Y) is constant, the steady state of this ODE would be a Hill-type response of Y as a function of X. However, by changing the function ξ(Y), a variety of bistable response curves can be obtained (Fig. 1C).

Figure 1: The modeling approach. A) The main equation: plotting the steady state of Y as a function of X reveals a bistable response curve, whose shape can be manipulated through the function ξ. B) The main idea is to modify a Hill-type response into a bistable response. C) Some examples of correspondence between functions ξ and resulting response functions.

 

 

The BioModels database contains two concrete models to illustrate the approach.

The first one (MODEL2212060001) is inspired by the early embryonic cell cycle of Xenopus laevis, which is driven by the accumulation and degradation of cyclin B. In the model, there is one variable representing the cyclin B-Cdk1 complex, which is proportional to cyclin B concentrations and governed by production and degradation. The degradation is mediated by the protein complex APC/C, which has a bistable response as a function of cyclin B-Cdk1 (Fig 2A).

 

The second model (MODEL2212060002) is a model of a somatic cell cycle with different phases. It uses three bistable modules to describe the transitions between cell cycle phases (Fig 2C).

 

Results

The simple cell cycle model produces relaxation-type oscillations, with slow dynamics for cyclin B-Cdk1 and fast dynamics for APC/C (Fig 2B). These oscillations are similar to those seen in Fitzhugh-Nagumo type systems.

 

Figure 2: Two examples of models with a bistable module. A) A model for the early embryonic cell cycle of Xenopus laevis with two variables. B) Time series of the simple cell cycle model. C) Interaction diagram for a model of the mammalian cell cycle with three bistable switches. D) Time series of two cycles of the model described in panel C. All panels are directly taken from the paper by De Boeck et al. (2021).

 

The model of the somatic cell cycle produces the cell cycle as a complicated limit cycle, with clear transitions between different phases (Fig 2D). Using this model, we studied the effect of DNA damage and coupling to the circadian clock. The benefit of modelling this system with the bistable modules is that we do not need to know exactly how, for example, the circadian clock affects the cell cycle on a molecular level. In our model, we made the assumption (based on the literature) that the activation threshold of one of the bistable switches is influenced by the circadian clock. This leads to the entrainment of the cell cycle by the circadian clock.

 

Discussion

 

We propose an approach to modelling bistable responses in biochemical systems based on a high-level description. This makes it unnecessary to know the details of the reactions that generate the bistability. The bistable response can be included directly in a model, very similar to how Hill functions are commonly used to introduce a steep response. In fact, our bistable module has as limiting case the Hill function response if one takes the threshold to be constant.

 

We think this approach can be useful if the focus is not on the mechanism generating the bistability, but on how this bistability interacts with other parts of the system. Another use would be in situations where data is available from a hysteresis experiment, such that thresholds and the overall shape of the response are known, but the underlying chemistry is unknown. In this case, one can fit the S-shaped response curve from the data, and use it directly in a model.

 

The bistable module introduced here could also have applications outside of biology, since it is an abstract description of input-output systems with a region of bistability. For example, the bistable units do not need to be linked up in a chain, like in the somatic cell cycle example, but could form a network with arbitrary connections. Studying such networks, and how they respond collectively as some input parameters change, could lead to interesting insights into robustness, domino-like switching, etc.. This may have applications in other fields such as climate dynamics (Wunderling et al. 2021).


References

De Boeck, Jolan, Jan Rombouts, and Lendert Gelens. 2021. “A Modular Approach for Modeling the Cell Cycle Based on Functional Response Curves.” PLOS Computational Biology 17 (8): e1009008. https://doi.org/10.1371/journal.pcbi.1009008.

Ferrell, James E. Jr., and Sang Hoon Ha. 2014. “Ultrasensitivity Part III: Cascades, Bistable Switches, and Oscillators.” Trends in Biochemical Sciences 39 (12): 612–18. https://doi.org/10.1016/j.tibs.2014.10.002.

Novák, Béla, and John J. Tyson. 2021. “Mechanisms of Signalling-Memory Governing Progression through the Eukaryotic Cell Cycle.” Current Opinion in Cell Biology 69 (April): 7–16. https://doi.org/10.1016/j.ceb.2020.12.003.

Pomerening, Joseph R., Eduardo D. Sontag, and James E. Ferrell. 2003. “Building a Cell Cycle Oscillator: Hysteresis and Bistability in the Activation of Cdc2.” Nature Cell Biology 5 (4): 346–51. https://doi.org/10.1038/ncb954.

Wunderling, Nico, Jonathan F. Donges, Jürgen Kurths, and Ricarda Winkelmann. 2021. “Interacting Tipping Elements Increase Risk of Climate Domino Effects under Global Warming.” Earth System Dynamics 12 (2): 601–19. https://doi.org/10.5194/esd-12-601-2021.