Stochastic Phenotype Transition of a Single Cell in an Intermediate Region of Gene State Switching

Multiple phenotypic states often arise in a single cell with different gene-expression states that undergo transcription regulation with positive feedback. Recent experiments show that, at least in E. coli, the gene state switching can be neither extremely slow nor exceedingly rapid as many previous theoretical treatments assumed. Rather, it is in the intermediate region which is difficult to handle mathematically. Under this condition, from a full chemical-master-equation description we derive a model in which the protein copy number, for a given gene state, follows a deterministic mean-field description while the protein-synthesis rates fluctuate due to stochastic gene state switching. The simplified kinetics yields a nonequilibrium landscape function, which, similar to the energy function for equilibrium fluctuation, provides the leading orders of fluctuations around each phenotypic state, as well as the transition rates between the two phenotypic states. This rate formula is analogous to Kramers’ theory for chemical reactions. The resulting behaviors are significantly different from the two limiting cases studied previously.

Figure 1

(a) A minimal gene network with positive feedback and two different gene states. (b) The diagram of the full chemical master equation [see Eq. (1)]. (c) Deterministic mean-field model [Eq. (2)] with bistability induced by positive feedback, in which $k_1=10{\min }^{-1}$, $k_2=0.1{\min }^{-1}$, and $\gamma =0.02{\min }^{-1}$.

Figure 2

In the intermediate region of gene state switching, the full CME can be simplified to a fluctuating-rate model (a), the steady-state distribution of which corresponds to a normalized landscape function ${\tilde{\mathrm{\Phi }}}_0(x)$ (b). In the case if the gene state switching is extremely rapid, a reduced CME (c) and a different landscape function ${\tilde{\mathrm{\Phi }}}_1(x)$ (d) can also be derived. The insets in (b) and (d) are the zoom in of the functions near $x=0$. The parameters in (b) and (d) are the same as those in Fig. 1 with $K_{eq}=1/5.5$.

Figure 3

The general saddle-crossing rate formulas $k_{AB}\approx k_{AB}^0\exp (-\mathrm{\Delta }{\mathrm{\Phi }}_{AB})$ (a) and the mean transition time from the off state to the on state in the full CME and both simplified models (with different colors) are obtained as either the gene state switching is in the intermediate region (b) or the extremely rapid region (c). All the simulated data are obtained through the mean-first-passage-time method, except that from the fluctuating-rate model which is obtained through stochastic simulation of the trajectories. (d) The relative stability of the two phenotypic states with respect to ${\mathrm{\Phi }}_0$ (blue) and ${\mathrm{\Phi }}_1$ (green) as a function of $1/K_{eq}$. The multicolored lines in (b) and (c) almost overlap with each other. The insets in (b) and (c) compare the numerically determined normalized barrier heights from the full CME (solid blue) and the two simplified models (dashed black). The parameter $(k_1/k_2)=3000$ in (b), 50 in (c), and 100 in (d). $K_{eq}=1/6$ in both (b) and (c). $\gamma =0.02{\min }^{-1}$. (e) Summarizing the conditions under which the simplified models and associated landscape functions as well as saddle-crossing rate formulas are valid.