Last modified: 2014-03-31

#### Abstract

We update a stochastic cell-cycle model (Tyson and Hannsgen 1986) to incorporate features of the START transition observed more recently in budding yeast. The original model divided the cycle into two parts: a one of random duration, denoted *A, *and one of fixed duration, denoted *B*. It assumed exponential growth and a rate of transition from *A* to *B* that was a constant, *p*, provided the size exceeded a fixed threshold. Stability of the cell size distribution could be proved, provided the size-doubling time was greater than the total mean time between divisions (Tyrcha 1988). However, Di Talia *et al* (2007) found that START in budding yeast proceeds in two distinct steps, both of which are stochastic. In the first, which is size-controlled through Cln3 concentration, Whi5 leaves the nucleus. In the second, which is size-independent, a feedback loop is established in which Cln1 and Cln2 promote their own expression.

Earlier theoretical work had showed that stability was not possible in a model with exponential growth lacking size control, raising the question of the stability of a system with both size-dependent and size-independent stochastic transitions. Therefore, in the work reported here, we extended the original model to include a second, size-independent, random step. Part *A* of the cycle* *now represents G1 phase up to and including the size-dependent part of START, and part *B* is* *the rest of the cell cycle up to division, including the size-independent part of START, the duration of which is assumed to have a distribution *w*(*T _{B}*). We explored the conditions

*w*(

*T*) must satisfy in order that the cell size distribution be stable in the limit of a large number of generations.

_{B}A key object in the analysis is the kernel *K*(*x*,*y*) that relates the probability density of the cell size *y* in one generation to the corresponding probability density of the size *x* in the next. In our extended model, it can be written as an integral of a kernel *K*(*x*,*y*|*T _{B}*) conditional on a given

*T*over the distribution of

_{B}*T*. This fact allowed us to follow the method used by Tyrcha (1988) for the Tyson-Hannsgen model, employing a theorem of Lasota and Mackey (1984) to show that the stationary distribution is stable if the quantity 2

_{B}^{-h }<exp

*hkT*>

_{B}*p*/(

*p*-

*kh*) < 1 for some

*h*<1, where

*k*is the growth rate. This condition is satisfied if the mean of

*T*is finite and the mean time between divisions is smaller than the size-doubling time. This result can be generalized to any model in which the transition rate function

_{B}*r*(

*x*) has a nonzero positive lower bound as the size

*x*goes to infinity. The treatment described so far assumes equal mother and daughter sizes, as in the original Tyson-Hannsgen model, but it can be extended to allow unequal sizes.