Stochastic threshold in cell size control

Classic models of cell size control consider that cells divide when reaching a threshold, e.g., size, age, or size extension. The molecular basis of the threshold involves multiple layers of regulation as well as gene noise. In this paper, we study the cell cycle as a first-passage problem with a stochastic threshold and discover that such stochasticity affects the intergeneration statistics, which bewilders the criteria to distinguish the types of size control models. The analytic results show that the autocorrelation in the threshold can drive a sizer model to the adderlike and even timerlike intergeneration statistics, which is supported by simulations. Following the picture that the autocorrelation in the threshold can propagate to the intergeneration statistics, we further show that the adder model can be driven to the timerlike one by a positively autocorrelated threshold, and even to the sizerlike one when the threshold is negatively autocorrelated. This work highlights the importance of examining gene noise in size control.

Figure 1

The intergeneration correlation observed in experiments reflects both the regulatory mechanism and the correlation in the stochastic threshold. The three mechanisms with cell size, size extension, and cell age as the cell cycle indicator [see (a)] led to the native intergeneration correlation as sizer, adder, and timer [see (b)]. The accumulation model describes the cell cycles as the first-passage process, as illustrated in (c) with sizer mechanism as an example. The birth size $x_b$, the division size $x_d$, the added size $\mathrm{\Delta }$, and the intergeneration time (or generation time) $\tau$ are depicted. (d) The autocorrelation of the stochastic threshold $s\left(t\right)$ could drive the intergeneration correlation either from sizer to adder to timer [with positive autocorrelation, i.e., $\langle s\left(t\right)s(t+\tau )\rangle -{\langle s\rangle }^2>0$] or the reverse [with negative autocorrelation, i.e., $\langle s\left(t\right)s(t+\tau )\rangle -{\langle s\rangle }^2<0$].

Figure 2

The distribution of interdivision time $P\left(\tilde{\tau }\right|{\tilde{x}}_b)$ for the Ornstein-Uhlenbeck case with $\epsilon =1, \sigma =0.1$, and various birth sizes ${\tilde{x}}_b$. The solid lines are given by Eq. (27), and the symbols are from simulations.

Figure 3

The intergeneration correlation in the Ornstein-Uhlenbeck case. (a) The mean division size $\langle {\tilde{x}}_d\rangle$ conditioned by the given birth size ${\tilde{x}}_b$ for $\sigma =0.1$ and various $\epsilon$. The dashed lines show the typical value ${\widehat{x}}_d$ solved from Eq. (33), which almost collapse to the mean values. (b) Same as (a), but for the mean added size $\langle \tilde{\mathrm{\Delta }}\rangle$. The dash-dotted line shows the perfect adder correlation for guidance. (c) Same as (a), but for the mean intergeneration time $\langle \tilde{\tau }\rangle$.

Figure 4

The statistics of simulation of ${10}^4$ cell cycles of the sizer mechanism with the threshold $s\left(t\right)$ following the OU process with $\epsilon =1, s_0=1$, and $\sigma =0.1$. (a) A scatterplot of the birth sizes vs the added sizes is shown by the gray circles. The mean added sizes for the given birth size are shown by the red circles. The black dash-dotted line indicates the added size for the perfect adder. (b) The distribution of the added size conditioned by various birth sizes, $P\left(\tilde{\mathrm{\Delta }}\right|{\tilde{x}}_b)$ for $\epsilon =1$ and $\sigma =0.1$. The solid lines are the analytic results. The symbols are from the simulation.

Figure 5

The intrinsic correlation modifies the intergeneration correlation of the accumulator mechanics, shown by the $x_b\text{−}\mathrm{\Delta }$ scatterplot of simulation data (gray circles). The red circles denote the mean added size $\langle \mathrm{\Delta }\rangle$ for the given $x_b$. The dash-dotted lines show the perfect sizer, adder, and timer correlations for guidance. (a) The oscillating threshold $s\left(t\right)$ drives the native adder correlation to the sizerlike one when the oscillation period $T\simeq 2{\tau }_c$. (b) The native adder correlation of the accumulator with stochastic threshold $s\left(t\right)$ in the uncorrelated limit with $\epsilon =t_{c}ln2/{\tau }_c=0.1$. (c) The positively correlated threshold $s\left(t\right)$ drives the native adder correlation to the timerlike one in the strong-correlation limit with $\epsilon =10$.

Figure 6

The autocorrelation in the threshold modifies the intergeneration correlation of the sizer mechanism. The slope $k=d\mathrm{\Delta }/d{x}_b$ as a function of $\epsilon =\lambda /\gamma =t_{c}ln2/\tau$. The symbols are from the experimental data of Tanouchi et al. [7], Taheri-Araghi et al. [15], and Wallden et al. [34], assuming $t_c=1.2\mathrm{h}$. The black dashed lines show the perfect sizer, adder, and timer correlations for guidance.