We formulate a structured model in which we keep track of size (mass), chemical state (spf,mpf,ube) and family pedigree (number of scars) of the cells. We split the population into two distinct subclasses (pre-START and post-START) for each scar class. The number of scars can be infinite. The pre-START (or G1) subclass will be represented by a density distribution on the 4 dimensional state x=(mass,spf,mpf,ube) with state space . For the post-START (or mitotic) cells, we add the time spent in the post-START (budded) period as an extra i-state for computational convenience. This subclass will be represented by a density distribution on the 5 dimensional state on the state space . The densities are defined as:
The state of an individual yeast cell changes in time according to a rate equation which is a simplification of a higher dimensional model presented by Val et al. (in prep). It is only valid for parameter values which guarantee that both spf,mpf are sufficiently smaller than 1.
in which denotes the rate of background synthesis of spf denotes the rate of synthesis of spf of relative activity of the transcription machinery for spf, and denotes natural decay of species s. Similar notation and meanings for parameter values is used for mpf (m) and ube (u).
The dynamics of the transcription machinery of spf and mpf was modelled as a Goldbeter-Koshland switch (see the equation for ube) in pseudo-equilibrium, i.e. the positive root of the solution of (e.g. spf)
where A is the activator and I is the inhibitor of the transcription machinery. For spf , . The total concentration of the transcription factor is assumed to be constant throughout the cell cycle. and are rate constants, and are Michaelis constants, and are additional species (which are present at constant concentrations during the cell cycle) involved in the control of transcription.
Cells pass START upon the rapid onset of the transcription machinery SBF. In our model this moment is taken when
i.e. half of the machinery is turned on and still being activated further while spf is increasing. Note that nuclear storage and action of SPF is resembled by the product , so we implicitly assume that nuclear size is constant.
After passing START the cells are shifted to subclass of post-START cells where the time spent in the budded period is added as a state variable. This time is 0 when cells just pass START and forms together with a boundary subspace.
The post-START cells follow the chemical cascade leading to division as presented in the Introduction and given by (1). In the model, division is marked by the loss of the possibility to keep the transcription machinery for MPF on, i.e.
Note that all mothers have the same MPF concentration when they exit from post-START, so is a boundary in . Both mothers and daughters retain that concentration upon entrance in the pre-START subclass, and so cells starting a new cycle all enter the pre-START subclass on the boundary .
We are now ready to present the model on the population level. As noted before, cells only enter classes at boundaries in , and when they are in the interior marked by these boundaries we need to keep track of movement and losses only. Following Metz & Diekmann (1986), we can write, as a linear population model, the temporal changes of the pre-START and post-START populations are,
and
where d(t,x) is the death rate of cells being in state x and . Here we set , representing balanced growth.
In order to complete the model we have to provide the proper boundary conditions. The boundary condition for post-START cells is
i.e. the cells flow from pre-START to post-START compartment without any transformations of state space.
The boundary condition for the pre-START cells is not as simple. We assume that all the mass accumulated in the post-START phase goes to the daughter cell (cf. Hartwell & Unger, 1977). So if cells pass START at size and need time to reach the point where division takes place, then the masses of mother and daughter (for the next cycle) just prior to cell separation are
which are equal to the fractions and of the mass of the cell at division.
At division mother and daughter inherit the MPF and UBE concentration of mother cell prior to division. The MPF concentration is equal for all cells right at the moment of division, but the UBE concentration may differ from other mother-daughter pairs (3 ). For SPF the story is different. It is assumed that SPF is localized in the nucleus and, in order to be consistent, both mother and daughter will inherit half of the content of SPF in the cell. So the concentrations for SPF in mother and daughter cells right after division are
(Note, if we assume, on the other hand, that the concentration of SPF for mother and daughter is the same after divison, i.e., SPF is divided between mother and daughter on the basis of cell size, the general outcome of the model is not affected. In this case the G1 period of the daughter cell is longer while initially it has less SPF, and visa versa for the mother. )
We now have all the ingredients to form the boundary condition for births into the population of pre-START cells which occur at the hypersurface
where
are functions transferring the state of the cell at the beginning of a new cell cycle to the state of the mother prior to division for mothers and daughters .
For the dynamics of pre-START and post-START cells (sum over all scars in each subclass), , and , we can write