Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Stochastic stability of a compound cell-cycle model
Joanna Tyrcha, John Hertz

Last modified: 2014-03-31


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(TB).  We explored the conditions w(TB) must satisfy in order that the cell size distribution be stable in the limit of a large number of generations.

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|TB) conditional on a given TB over the distribution of TB.  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-h <exp hkTB>p/(p-kh) < 1 for some h<1, where k is the growth rate.  This condition is satisfied if the mean of TB 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 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.


cell cycle; stochastic stability; stochastic networks