Alternative designs for a genetic switch: analysis of switching times using the piecewise power-law representation

Some genes are thought to be switched discontinuously ON or OFF in response to environmental or developmental stimuli, whereas other genes are thought to be switched in a continuously variable fashion. We have previously identified criteria that distinguish between discontinuous and continuous genetic switches for an inducible catabolic pathway. These two types of switches exhibit several additional characteristics, beyond their qualitatively distinct behaviors, that influence their natural selection. These characteristics include threshold value, magnitude of the input signal required for switching ('switching effort'), magnitude of the corresponding output signal, duty cycle, switching time, and robustness. In order to characterize the biological design principles governing such switches, we have developed mathematical models of generic gene circuits and analyzed their behavior. Here we report the results of a comparative study designed to identify essential differences in switching time. This study has been greatly facilitated by use of the piecewise power-law representation, which was first developed by systems engineers in the 1940s and adapted for biochemical systems in the early 1970s. With this approach, we have been able to derive analytical expressions for switching time. When the alternative designs are made as nearly equivalent as possible, by the method of mathematically controlled comparison, we find that the switching times for the continuous case are less than that for the corresponding discontinuous case. We also find that ON times are faster than OFF times in all cases. These results are discussed in the specific context of the inducible lactose operon of Escherichia coli.