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Question

# Genes produce molecules called mRNA that then produce proteins. High levels of protein can inhibit the production of mRNA, resulting in a feedback that regulates gene expression. Using m and p to denote the amounts of mRNA and protein in a cell $\left(\times 10^{2} \text { copies } / \mathrm{cell}\right),$ a simple model of gene regulation is$\begin{array}{l} {d m / d t=1-p-m} \\ {d p / d t=m-p} \end{array}$Construct the phase plane, including the nullclines.

Solution

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In this exercise we want to construct the phase plane, with the nullclines, of the system

\begin{aligned} \frac{dm}{dt} & =1-p-m \\ \frac{dp}{dt} & = m-p \end{aligned}

where $m$ is the amount of mRNA in a cell, and $p$ is the amount of protein in a cell (both measured in $10^2 \mathrm{\frac{copies}{cell}}$). Notice that $m$ and $p$ cannot be negative so we will plot the phase plane and the nullclines on a viewing window where both $m$ and $p$ are nonnegative ($m$ on the horizontal axis and $p$ on the vertical axis).

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