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 MPhase 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 Mphasepromoting 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.
After the activation through cyclin, cdc2 kinase induces activation of cyclin protease. Given the fact that MPF inactivates cdc2, it also leads to a decrease in activation of the cyclin protease.
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_{i}, which is the rate at which cyclin is synthesised and

v_{d} for the degradation of cyclin due to activated cyclin protease.
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.
First cycle: Constantly synthesised cyclin triggers activation of cdc2 kinase (M^{+} is the inactive form of cdc2 kinase and 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 (C) accumulates up to a certain value before the activation of cdc2 (M) starts, which constitutes the first time delay. The second time delay results from the fact that M also has to increase up to a threshold before it activates the cyclin protease (X).
The second impact of the activated cdc2 is the degradation of cyclin which leads to a reduced activation of cdc2 and consequently a decrease in protease activation.
Afterwards, the cascade starts again and the oscillating nature of this system becomes visible.

Bibliographic References

J.J. Tyson. Modeling the cell division cycle: cdc2 and cyclin interactions. Proc Natl Acad Sci U S A 88:73287332, 1991.
[SRS@EBI]

B. Novak, J.J. Tyson. Numerical analysis of a comprehensive model of Mphase control in Xenopus oocyte extracts and intact embryos. J. Cell. Sci. 106:11531168, 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 88:91079111, 1991.
[SRS@EBI]
