# Goldbeter (1991), Minimal mitotic oscillator

April 2008, model of the month by *Anika Oellrich*

Original models: BIOMD0000000003 and BIOMD0000000004

BioModels database contains several models dealing with the cell cycle and related mitotic oscillation. Besides BIOMD0000000005 [1] (see also here) or BIOMD0000000107 [2] (see also here), which are representing M-Phase in fission yeast, one of these models is BIOMD0000000003 [3] which focuses on mitosis in early amphibian embryos. As shown in figure 1, this simple model of the mitotic oscillator contains only three variables: -
**C**- cyclin -
**M**- cdc2 kinase (inactive or active) and -
**X**- cyclin protease (inactive or active).
Cdc2 kinase is activated by cyclin when it accumulates up to a certain threshold. Both cyclin and cdc2 kinase are forming a complex which is named the M-phase-promoting factor (MPF). The degradation of cyclin, the associated inactivation of cdc2 and, subsequently, mitosis are caused by MPF. The latter leads to a new cell cycle. In addition to the three variables and to the activation and inactivation rates of cdc2 kinase and cyclin protease, respectively, there are two more rate parameters: -
**v**, which is the rate at which cyclin is synthesised and_{i} -
**v**for the degradation of cyclin due to activated cyclin protease._{d}
The behaviour of the system can be described with the following three differential equations: -
dC/dt = v _{i}- v_{d}X(C/(K_{d}+ C)) - k_{d}C -
dM/dt = V _{1}((1 - M)/(K_{1}+ (1 - M))) - V_{2}(M/(K_{2}+ M)) -
dX/dt = V _{3}((1 - X)/(K_{3}+ (1 - X))) - V_{4}(X/(K_{4}+ X))
| Figure 1: Bicyclic cascade model for oscillating behaviour in Mitosis. M is the active form) by reversible dephosphorylation. Second cycle: activated cdc2 kinase leads to activation of cyclin protease ( X is the inactive form of cyclin protease and ^{+}X is the active form). The active cyclin protease degrades cyclin. |

Figure 2: Resulting graph of the minimal mitotic oscillator (parameter values see [1]). | Figure 2 shows the resulting output graph of the minimal mitotic oscillator. The graph shows two time delays: - time delay between
**C**and**M**, and - time delay between
**M**and**X**.
Cyclin ( |

## Bibliographic References

- J.J. Tyson. Modeling the cell division cycle: cdc2 and cyclin interactions.
*Proc Natl Acad Sci U S A*88:7328-7332, 1991. [SRS@EBI] - B. Novak, J.J. Tyson. Numerical analysis of a comprehensive model of M-phase control in Xenopus oocyte extracts and intact embryos.
*J. Cell. Sci.*106:1153-1168, 1993. [SRS@EBI] - A. Goldbeter. A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase.
*Proc Natl Acad Sci U S A*88:9107-9111, 1991. [SRS@EBI]