13 Autoregulation 1: auto-repression
regulatory feedback loops
In our first example, transcription of mRNA was due to some external signal or resource. However, many genes encode transcription factors that directly regulate the rate of transcription of the coding gene. This produces a small feedback circuit. Representing the concentrations of mRNA and protein by [latex]M(t)[/latex] and [latex]P(t)[/latex] again, we can now write,
[latex]\begin{align} \frac{dM}{dt} &= f(P) - \mu M\\ \frac{dP}{dt} &= kM - \nu P. \end{align}[/latex]
Now the concentration of protein directly influences the transcription of mRNA, producing this autoregulation feedback loop. The function [latex]f(P)[/latex] must be bounded above, since transcription must have some maximal rate. Furthermore [latex]f(P) \geq 0[/latex] (the production rate cannot be negative). Therefore [latex]f(P)[/latex] must be non-linear, which makes the system defined by the equations above difficult, if not impossible, to solve explicitly.
To gain insight into the possible dynamics of the system, we will use qualitative analysis based around the construction of the phase portraits and linear stability analysis.
Constructing the phase portrait
Nullclines
The nullclines of the system are given by,
[latex]\begin{align} &\frac{dM}{dt}=0 \implies M=f(P)/\mu\\ & \frac{dP}{dt}=0\implies M=\nu P/k. \end{align}[/latex]
These are both single lines/curves in the phase plane.
Equilibria
Putting the two nullcline equations equal to one another, we find a single equation for the possible equilibrium values of [latex]P[/latex]:
[latex]\begin{equation} \frac{\mu\nu}{k}P^*=f(P^*). \end{equation}[/latex]
In that case we can then find [latex]M^*=\nu P^*/k[/latex] to be the corresponding equilibrium for mRNA. We can therefore focus our attention on determining the nature of any solutions for [latex]P^*[/latex] for different forms of the function [latex]f(P)[/latex].
Linearisation
As in previous examples, we assess stability around an equilibrium by linearisation, through looking at the Jacobian. For this model our Jacobian is,
[latex]\begin{align*} J=&\left( \begin{array}{cc} -\mu & f'(P^*)\\ k & -\nu \end{array} \right). \end{align*}[/latex]
We then check the trace and determinant to assess stability:
- [latex]tr(J)=-\mu-\nu[/latex]
- [latex]\det(J)=\mu\nu-kf'(P^*)[/latex].
The trace is always negative, limiting stability to a stable spiral or node, or an unstable saddle. Which outcome we get depends on the determinant, which in turn depends on the gradient of our transcription function at the equilibrium, [latex]f'(P^*)[/latex]. We will now consider a few examples of what this function might look like.
Auto-repression: a negative feedback
If the protein product of a gene acts to reduce the rate of transcription, we describe it as a transcriptional repressor. The feedback circuit is described as auto-repression. In this case, [latex]f'(P)\leq 0[/latex], i.e. [latex]f(P)[/latex] is a decreasing function. For the most p