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The model

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 tex2html_wrap_inline713 . 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 tex2html_wrap_inline715 on the state space tex2html_wrap_inline717 . The densities are defined as:

tex2html_wrap_inline719 density of pre-START cells with j scars, in state x and at time t.

tex2html_wrap_inline727 density of post-START cells with j scars, in state x, which at time t have spent time tex2html_wrap_inline735 in the post-START phase.

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.

   eqnarray32

in which tex2html_wrap_inline749 denotes the rate of background synthesis of spf tex2html_wrap_inline751 denotes the rate of synthesis of spf of relative activity of the transcription machinery tex2html_wrap_inline753 for spf, and tex2html_wrap_inline755 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)

displaymath705

where A is the activator and I is the inhibitor of the transcription machinery. For spf tex2html_wrap_inline765 , tex2html_wrap_inline767 . The total concentration of the transcription factor tex2html_wrap_inline769 is assumed to be constant throughout the cell cycle. tex2html_wrap_inline771 and tex2html_wrap_inline773 are rate constants, tex2html_wrap_inline775 and tex2html_wrap_inline777 are Michaelis constants, tex2html_wrap_inline779 and tex2html_wrap_inline781 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

   eqnarray80

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 tex2html_wrap_inline785 , 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 tex2html_wrap_inline735 is 0 when cells just pass START and forms together with tex2html_wrap_inline789 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.

   eqnarray95

Note that all mothers have the same MPF concentration when they exit from post-START, so tex2html_wrap_inline695 is a boundary in tex2html_wrap_inline793 . 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 tex2html_wrap_inline695 .

We are now ready to present the model on the population level. As noted before, cells only enter classes at boundaries in tex2html_wrap_inline797 , 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,

  equation111

and

  equation116

where d(t,x) is the death rate of cells being in state x and tex2html_wrap_inline803 . Here we set tex2html_wrap_inline805 , 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

  equation123

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 tex2html_wrap_inline807 and need tex2html_wrap_inline809 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

   eqnarray126

which are equal to the fractions tex2html_wrap_inline811 and tex2html_wrap_inline813 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

   eqnarray134

(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 tex2html_wrap_inline815

   eqnarray148

where

   eqnarray156

are functions transferring the state of the cell at the beginning of a new cell cycle tex2html_wrap_inline817 to the state of the mother prior to division for mothers tex2html_wrap_inline819 and daughters tex2html_wrap_inline821 .

For the dynamics of pre-START and post-START cells (sum over all scars in each subclass), tex2html_wrap_inline823 , and tex2html_wrap_inline825 , we can write

   eqnarray173


next up previous
Next: Results Up: A purely deterministic model Previous: Introduction

John Val
Fri Apr 26 16:33:55 EDT 1996