Fig. 3.

Response times of molecular switches governed by activation, derepression, and concerted mechanisms. (A) The response (resp.) time of the switch increases as the response time of the receptor increases. Differences in the signaling mechanisms are more prominent when the receptor response is fast (i.e., the relative [rel.] response time of the receptor goes to zero). In this limit, activation decrease the response time, derepression increases the response time, and the concerted mechanism leaves it unchanged in comparison to the basal response time. For each signaling mechanism, the response time is computed by using the analytical result in Eq. 9 (solid lines) and numerically validated by using Eq. 6 (dashed lines). To ensure the same basal response and basal response time of the switches across signaling mechanisms, we chose the parameters as $k_3=1/9$, $k_4=1$, $k_6=0$, and $k_5{X}_T=10$ for activation; $k_3=1/9$, $k_4=0$, $k_5=0$, and $k_6{X}_T=1$ for derepression; and $k_3=1/9$, $k_4=0$, and $k_5{X}_T=k_6{X}_T=1$ for concerted. The receptor response time was $k_{1}S+k_2$ varied through $k_2$, while maintaining $k_{1}S/k_2=1$. (B) With increase in the stimulus level, response time decreases for activation, increases for derepression, and does not change for the concerted mechanism. The comparison is controlled by setting same response time at half-maximal (half-max.) stimulus ${\mathrm{\Theta }}_{Y^{*}}$. The following parameters were chosen to have same basal response, but different basal response times: $k_3=1$, $k_4=9$, $k_5{X}_T=90$, and $k_6=0$ for activation; $k_3=10,k_4=0,k_5=0$, and $k_6{X}_T=90$ for derepression; and $k_3=10$, $k_4=0$, and $k_5{X}_T=k_6{X}_T=90$ for concerted. The receptor occupancy was varied by changing $k_{1}S/k_2$ while maintaining the receptor response time $1/\left(k_{1}S+k_2\right)$, which was chosen to be 100 times faster than the response time of the switches at their respective half-maximal stimulus levels. (C) Comparing the system’s response to a pulse of stimulus provides a method for distinguishing mechanisms. The response time for activation (derepression) following exposure to stimulus is shorter (longer) than following removal of stimulus. The time series for each mechanism is normalized to its steady state in the presence of stimulus. We used parameters such that each mechanism has the same response time of the switch in the absence of the stimulus and the stimulus strength produced a half-maximal response. Specific values of parameters are: $k_1=1$, $k_2=1$, $X_T=100$, and $Y_T=100$ for both mechanisms; $k_3=1/9$, $k_4=1$, $k_5=0.1$, and $S=0.1$ for activation; and $k_3=1/9$, $k_4=0$, $k_6=0.01$, and $S=10$ for derepression.