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Introduction

Much is known about the physiology and genetics of the cell division cycle in budding yeast, Saccharomyces cerevisiae. Budding yeast cells generally divide unequally by shedding a small bud from the surface of the mother cell, leaving a scar on the surface of the mother cell. The division cycle of the smaller daughter is longer than that of the mother for the next division cycle, seemingly due to a minimal size requirement (not necessarily equal for all cells) on the initiation of DNA synthesis (see e.g. Hartwell & Unger 1977, Johnston et. al. 1977, Lord & Wheals 1981).

In recent years an understanding of the cell cycle on the molecular level has rapidly increased (e.g. Andrews & Herkowitz 1989, Richardson et. al. 1989, Tyers et. al. 1991, Nasmyth & Dirick, 1991, Amon et. al. 1993 and 1994, Epstein & Cross 1994). In the consensus picture of the budding yeast cell cycle (for a review see Koch & Nasmyth 1994), two families of cyclin proteins arise and disappear in a carefully regulated temporal sequence. The first family, called G1-cyclins, is necessary for the onset of transcription of genes whose products are needed for DNA synthesis and bud formation. The onset of transcription factors by G1-cyclins is defined as START, it is a signal indicating that the cell is committed irrevocably to carry out DNA synthesis, bud formation, spindle pole body duplication, ultimately followed by cell division. The second family, called G2-cyclins, is needed for the onset of both DNA synthesis and more evidently the mitotic event. Cyclins from either family must bind to Cdc28 (whose concentration does not vary much throughout the cell cycle) to render Cdc28 active as a protein kinase essential for triggering key events of the cell division cycle. Dimers of Cdc28 and G1-cyclins are denoted SPF (START promoting factor) and dimers of G2-cyclins are called MPF (mitosis promoting factor).

The transcription of G1-cyclins is controlled in part by cell size and in part by the positive feedback of SPF on its transcription factor SBF. As a cell passes through START, a transcription factor (SBF) for G1-cyclins is activated and SPF concentration reaches a peak soon thereafter. The synthesis of G2-cyclins is also controlled through a positive feedback of MPF on its transcription factor MCM1. Completion of the cell cycle is only possible if the G2-cyclins are destroyed by a specific ubiquitin pathway, UBE. The presence of MPF increases the activity of this degradation pathway, while SPF decreases its activity. The presence of MPF also causes repression of synthesis of the G1-cyclins.

Directly after division both G1-cyclins and G2-cyclins are absent and G2-cyclin destruction is active. At START (which depends on the cell size), the G1-cyclins appear and cause the inactivation of G2-cyclin destruction. As a result, G2-cyclins start to accumulate autocatalytically, which in turn causes the disappearance of the G1-cyclins. As G1-cyclins disappear and G2-cyclins increase, the UBE pathway can be turned on leading to the destruction of G2-cyclins and exit of mitosis into the next division cycle.

For balanced growth, cell division must be coupled to cell size. Unfortunately, the molecular mechanism of this coupling is not yet understood. It is found that upon an increase in growth rate, mean cell size increases (Johnston et. al. 1979), and there is a decrease in length of both the pre-START period and (to a lesser extent) the budded period (Hartwell & Unger 1977). So size has an effect on events around START and around the mitotic event. In the following we link cell cycle kinetics and cell size by assuming that SPF accumulates and acts in the nucleus while its production is proportional to cell size. The complete picture is given in Fig. 1.

    figure17
Figure 1: Schematic view of our model of the budding yeast cell cycle. Results of interactions, given by joining two lines, are in the directions of the arrows. A '+' sign indicates a catalytic action. A '-' sign indicates an inhibitory action. Non attached arrows indicate uncontrolled constant production. See the main text for an explanation of the various components.

Until now mathematical modelling of budding yeast cells has proceeded on two different paths. One focused on results from physiological experiments, e.g. cell to cell variation, cell size distributions, and scar distributions (e.g. Gani & Saunders 1977, Hartwell & Unger 1977, Adams et. al. 1981, Gyllenberg 1986). The modelling strategy was based on the Smith and Martin (1973) model, which divides the cell cycle into a variable G1 phase and a constant S-G2-M phase. The other approach concentrated on the deterministic dynamics of the molecular control system (Thron 1991, Norel & Agur 1991, Goldbeter 1993, Novak & Tyson 1995). In this paper an attempt is made to join both modelling approaches. Taking the approach of Gyllenberg (1986), we keep track of size and scar distributions by means of structured population models, and in addition, we keep track of the chemical state of the cells. In our model, progression through the cell cycle is completely deterministic (the only stochastic process is the random removal of cells by dilution). Departure from G1 is not modelled in a probabilistic fashion, but by purely deterministic processes. Furthermore, the length of the budded period is not taken as a constant, it is fully determined by the underlying chemical model.

In the full model presented below, we follow the typical movement through state space of the smallest daughter, and then its subsequent division cycles as a mother. With two different numerical approaches, one solves PDE's numerically the other one follows the branching process of individual cells, we show that in our model, as in the model of Gyllenberg, the population can reach a stable size distribution, which is qualitatively similar to (but narrower than) the observed distribution.


next up previous
Next: The model Up: A purely deterministic model Previous: A purely deterministic model

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