Homework 7.1: Robustness to copy number (45 pts)

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Segall-Shapiro, Sontag, and Voigt (Nat. Biotech., 2018) proposed a simple circuit that they claim can reduce the impact of changes in the copy number of a gene to the concentration of the gene product. To be clear, by copy number of a gene, we mean the number of pieces of DNA in the cell that code for the gene. You may get higher copy number with more plasmids, for example, or for genomic DNA, you may get variation in copy number depending on how many replication forks are active.

The circuit consists of two genes, X and Y, with X being unregulated and Y being repressed by X. The production rate or X is proportional to its gene copy number, \(c\). Segall-Shapiro and coworkers put Y very close to X in the genome, such that the unregulated production rate of Y is also proportional to \(c\).

a) Write down a system of ODEs for the circuit and solve for the steady state concentration of Y.

b) Over what operating regime is the steady state concentration of Y robust to changes in copy number \(c\)?

c) Consider another simple circuit wherein Y is autorepressive (with the production rate still being proportional to \(c\) in the \(y\to 0\) limit). There is no X in this circuit. Over what operating regime is the steady state concentration of Y robust to changes in copy number \(c\) in this circuit? How does this compare to the circuit proposed by Segall-Shapiro and coworkers?