15 Longer negative feedback networks
A simplified 3-variable model
We saw in chapter 13 that a two-component negative feedback circuit (the auto-repressive gene system) possesses a unique steady state, which is always stable. This underlies homeostasis; the tendency of a negative feedback system to return to its equilibrium when perturbed (for small perturbations, at least).
However, negative feedback circuits can also generate sustained oscillations. One way this can occur is if we extend our two-variable model to a model including at least three variables. This was first proposed in the context of regulated gene expression in the 1960s, and the prototype three-component model is often referred to as the Goodwin oscillator (see chapter references). There are a number of ways of thinking about what might underlie a model of auto-regulatory gene expression that would contain more than two variables. One example would be that in fact the protein product of the gene is translated/produced in one place – the cytoplasm – but regulates transcription in another – the nucleus. We could therefore consider three variables: mRNA, cytoplasmic protein and nuclear protein.
Here, we will look at a slightly simplified model to Goodwin’s original, but the principles remain the same. Rather than focusing on a specific example of gene expression, we will consider a simple model of three interacting components, [latex]X[/latex], [latex]Y[/latex], and [latex]Z[/latex], which form a negative feedback loop by regulating each other. In particular:
- Increasing [latex]X[/latex] leads to increased production of [latex]Y[/latex],
- Increasing [latex]Y[/latex] leads to increased production of [latex]Z[/latex],
- Increasing [latex]Z[/latex] leads to decreased production of [latex]X[/latex].
Put together there is therefore a negative feedback in the system (since increasing [latex]X[/latex] ultimately leads to decreased production of [latex]X[/latex]). For simplicity we will assume that the degradation rates of all three components are the same, so our model can be written as,
[latex]\begin{align} \frac{dX}{dt} &= f_1(Z) - \mu X\\ \frac{dY}{dt} &= f_2(X) - \mu Y\\ \frac{dZ}{dt} &= f_3(Y) - \mu Z. \end{align}[/latex]
Each of the functions [latex]f_i[/latex] are non-negative and bounded as we assumed before. Given our assumptions in the bullet points we also have that [latex]df_1/dZ\lt0[/latex] and [latex]df_2/dx,df_3/dY\gt0[/latex].
Equilibria of this system occur where the following three conditions are simultaneously satisfied:
[latex]\begin{align} & \mu X = f_1(Z)\\ & \mu Y = f_2(X)\\ & \mu Z = f_3(Y). \end{align}[/latex]
How many equilibria can we expect? If we let [latex]F_i=f_i/\mu[/latex], then we have [latex]X=F_1(F_3(F_2(X)))=G(X)[/latex]. We know that [latex]G(X)[/latex] must be a decreasing function of [latex]X[/latex] (intuitively it must be because we know we have a negative feedback loop, but we can also show it by the chain rule). By the same argument as in the