σ^{F} activation overestimates the fraction of cells that sporulate. (*A*) Stochastic simulations of a mathematical model integrating phosphorelay and σ^{F} activation modules show ultrasensitive increases of σ^{F} in single cells as a function of IPTG (mean response, solid blue line; SD, shaded area; Hill equation fit, dashed black line). However, only a twofold increase in the mean active σ^{F} level is observed between 4 μM and 10 μM IPTG because σ^{F} activation has a low threshold (∼4 μM IPTG). Bimodal distributions of active σ^{F} level in the model (*B*, *Upper*) and in single-cell experiments with *PspoIIQ-mCherry* reporter (*C*, *Upper*) are shown. (*B* and *C*) Cumulative distributions (*Lower*) corresponding to the data (*Upper*) are shown (i.e., total fraction of cells with an active σ^{F} level below the given value). Threshold values separating two peaks (gray bars) are chosen to predict the fraction of cells that activate σ^{F} in *D*. a.u., arbitrary units. (*D*) Model predictions for fraction of sporulating cells based on the threshold of active σ^{F} level (black curve) are computed using the distributions for various IPTG levels and the threshold value shown in *B*. Experimental data (red triangle and green square for 4 μM and 10 μM, respectively) are obtained using the threshold and distributions in *C*. Fractions of σ^{F}-active cells computed from experimental and simulation data are in excellent agreement with one another, but both exceed observed spore fractions (purple dots; calculated from spore counts shown in using in ). (*E*) Examples of the microscopy data of strain MF3765 used to construct distributions in *C*. A significant fraction of cells show σ^{F} activity (measured by *PspoIIQ-mCherry* false-colored magenta forespore in the image) even at low IPTG concentrations (*Left*, 4 μM). This fraction increases at high IPTG (*Right*, 10 μM), but the increase is not ultrasensitive. Spo0A activity was measured by *PspoIIG-gfp* (green). The images show a field of view of 20 × 20 μm.

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