This method is based on the fact that the length of the G_{1} phase of budding yeast is more sensitive to starvation. Many key steps in the budding yeast cell cycle have been reported in the literature. In the G_{1} phase cells need to produce some regulatory protein such as Cln3, whose abundance is governed by the availability of nutrient. By phosphorylating Whi5 and then activating SBF/MBF, Cln3 promotes the transition to the S phase. So the key step in the G_{1} phase may be the accumulation of regulatory proteins. Starvation will slow down these courses. The key steps in the non-G_{1} phase should be the cell division process such as DNA replication, spindle pole body duplication, spindle assembly, chromatin segregation *etc.*, which have little connection to extracellular nutrition.

To account for the observed synchronization of the cell cycle phases under the poor-rich medium modulation, we first measured the distribution of the length (time duration) of G

_{1} and non-G

_{1} phases for populations in rich and poor media, and for mother and daughter cells, respectively (, vertical bars, and Table S1, ESI

†). From these data, we observe that (1) in all cases there is a significant variability in the length of the cell cycle phase;

^{26} (2) both the G

_{1} and non-G

_{1} lengths are prolonged in poor medium compared with those in rich medium, but the G

_{1} phase lengths are prolonged more significantly; (3) mother and daughter cells have distinct length distributions in G

_{1}, and both vary under the nutrient conditions (); (4) mother and daughter cells have very similar length distribution for non-G

_{1} phase (ESI

†), which only changes under the nutrient conditions (Table S1, ESI

†). These observations led us to construct the following phenomenological stochastic model.

In the model the cell cycle process is considered as a series of ordered events or steps. The progression of the cell cycle is then successive transitions from one step to the next. The parameters of the model are the number of steps and the transition probability. Specifically, a phase (G

_{1}/non-G

_{1}) of a cell (mother/daughter) cultivated in a growth medium (rich/poor) was divided into

*k* hypothetical steps, and cell cycle progression was interpreted as a “leap forward” event along these steps irreversibly. At every time step of evolution, a cell had a chance

*p* to jump from the current step to the next on the events chain. In this model, the duration

*j* (in terms of the time steps) of a cell cycle phase satisfies the negative binomial distribution (the probability that the cell traveled through a cell cycle phase in

*j* simulation steps):

To derive the parameters

*k* and

*p* for different cases (mother/daughter cell, rich/poor medium, G

_{1}/non-G

_{1} phase), we fitted the negative binomial distribution to the measured phase length distribution data. Given the similarity of phase length distributions for daughter and mother cells in the non-G

_{1} phase, six sets of parameters were obtained (Table S2, ESI

†). In this model,

*k/p* is the expected simulation steps of a phase,

*k*(1 −

*p*)/

*p*^{2} is the variance,

*t*_{0}k/p is the expected duration of the phase (here one simulation step represents

*t*_{0} of time). With the same expected duration, the smaller the

*k* is, the larger the fluctuation of a phase is. Both the expected value and the variance of duration are important for the phenomena, the parameters (

*k, p*) determine these two most important features of negative binomial distribution. The fitted distribution is compared with the experimental data in . One observes that the phase length distribution for all cases agrees well with the negative binomial distribution.

By performing simulations using the obtained parameters (see ESI

† for details), we qualitatively reproduced the modulation effect of different schemes: the strongest synchronization is achieved by modulation schemes with a period of around 150 min; elongating or shortening the modulation period will decrease the degree of synchronization. The simulation results corresponding to three typical modulation schemes are plotted in .