Goodwin oscillator explained

Question

Hello I have been reading papers about the Goodwin oscillator and I found that the equations are kind of tricky. Specially the part of the hill coefficient. In his paper "An entrainment model for timed enzyme synthesis in bacteria" from 1966, in the part of the differential equation of the mRNA concentration Xi, there is the term Zi and this term also exists in other books that talk about this oscillator. However, I don't know what it means. Goodwin never mentioned what this term means. I'm guessing it is about the hill coefficient, but still I do not quite understand well. I share the link of a chapter book that explains the equations of this oscillator (page 244):

Biochemical Oscillations

Thanks for your help.

Answer 1

The ODEs for the Goodwin oscillator are:

$$\begin{align} \frac{dX}{dt}=&\ k_1\frac{K_i^n}{K_i^n+Z^n} -k_2X\\[1em] \frac{dY}{dt}=&\ k_3X-k_4Y\\[1em] \frac{dZ}{dt}=&\ k_5Y-k_6Z\end{align}$$

If you assume $X$ to be mRNA and $Y$to be protein then $Z$ can be thought of as an activated form of the protein (lets for instance assume a phosphorylated form).

The Hill coefficient, $n$ roughly tells you how many monomers of the activated protein co-operatively bind to the promoter or the number of protein monomers that form a multimeric repressor complex which inhibits the production of $X$. $K_i$ is a measure of how strongly would $X$ be inhibited by $Z$. To be precise it is equal to the concentration of $Z$ at which the production of $X$ is half its maximum rate. (Similar to Michaelis constant)

What the model says is that at higher values of $n$ (>8) there are sustained oscillations (which are called limit cycles). At lower values the oscillations damp. Note that even for low values of $n$ sustained oscillations (but not limit cycle types) can arise if there is substantial delay in the loop. However these oscillations are not as stable as limit cycles to perturbations.

Have a look at this post.

These are some advanced concepts and you may start by reading this book by Steven Strogatz called Non-linear Dynamics and Chaos*.

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*_{Strogatz, Steven H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview press, 2014. (latest edition); ISBN-13: 978-0813349107; ISBN-10: 0813349109}