Exercise 3.4: Modeling inducers in a toggle switch

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This problem is still a draft.

In our analysis of the Gardner and Collins synthetic toggle switch, we did not include a way to represent the addition of the inducer signals. Such signals typically function by decreasing a receptor’s affinity for its target binding site, in other words by increasing the value of the Hill repression constant \(k\). In the nondimensionalized system, we see that \(k_x\) (and correspondingly, \(k_y\)) appear both in the dimensionless concentration \(\tilde{x}\) and in the dimensionless production term \(\tilde{\beta}_x\).

Create an interactive plot representing the following scenario:

• The circuit is built with parameters \(\beta, \gamma, n\) fixed to be \(\beta = 10\), \(\gamma = 1\), \(n=10\).
• The plot has sliders for \(k_x\) and \(k_y\), which represent the presence of inducer signal to inactivate the repressors \(X\) and \(Y\), respectively. (Note that this representation assumes that the inducer’s action is linearly proportional to its concentration, which is typically not true but gives the right qualitative result).
• The plot shows how the nullclines, in dimensionless concentration space, change as you add and remove the inducer signals.

What insights did you gain from the visualization?