Problem 5.1: Robustness in a C1-FFL

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This problem is still a draft.

Consider the design principle we discussed in a previous chapter: The C1-FFL with AND logic displays an on-delay. We saw that this property of the C1-FFL holds even when the Hill coefficient for the regulation is unity. We might also ask if the delay is robust to variations in the Hill activation constants.

As a reminder, the dimensionless dynamical equations for the concentrations of Y and Z from a stimulus X are

\begin{align} \frac{\mathrm{d}y}{\mathrm{d}t} &= \beta\,\frac{(\kappa x)^{n_{xy}}}{1 + (\kappa x)^{n_{xy}}} - y, \\[1em] \gamma^{-1}\frac{\mathrm{d}z}{\mathrm{d}t} &= \frac{x^{n_{xz}} y^{n_{yz}}}{(1 + x^{n_{xz}})\,(1+ y^{n_{yz}})} - z. \end{align}

To investigate the effect of the Hill activation constants, we need only to vary the dimensionless parameter \(\kappa\), which is the ratio of the Hill activation constant for activation of Z by X to the Hill activation constant for activation of Z by Y; \(\kappa = k_{xz}/k_{yz}\).

a) Investigate the dynamics of this circuit in response to a step in X concentration to make a robustness statement about on-delay for varying \(\kappa\).

b) Make a robustness statement about the steady state levels of Z for varying \(\kappa\).