The Goodwin oscillator

During a talk by Jae Kyoung Kim at last week’s SMB meeting the speaker showed a system of equations and called it ‘a system you all know’. This revealed to me a gap in my knowledge of mathematical biology. The system is the Goodwin oscillator. It is described in Murray’s book on mathematical biology and I am sure I have read the relevant section on some level. This just shows that there a big difference between reading something and understanding its significance and being able to situate in a wider context. Now I have done my homework on this and I will write something about it here. The system, in the form it is given in Murray’s book, is an example of systems of the form $\frac{du_i}{dt}=f_i(u_{i-1})-k_iu_i$. Here the labels on the $u_i$ are supposed to be interpreted modulo $n$. In other words there are $n$ equations and $u_0$ is interpreted as $u_n$. In the Goodwin model itself $n=3$ and the functions $f_i$ are linear for $i>1$. The function $f_1$ is equal to $\frac{a}{b+u_n^m}$ for constants $a$, $b$ and $m$. This function is positive and its derivative is negative. Thus it can be interpreted as representing a negative feedback on the production of $u_1$. In the context in which it was introduced by Goodwin the quantities $u_1$, $u_2$ and $u_3$ represent concentrations of mRNA, the enzyme it codes for and the product of a reaction it catalyzes. The substrate of the enzyme is assumed present at a constant level and is not modelled explicitly.

It is known that the system admits a periodic solution if the Hill coefficient $m$ is greater than eight and not otherwise. Since this number is considered unrealistically large for the application which inspired the model modifications of this have been considered where periodic solutions can be obtained for lower values of $m$. It is proved by Hastings, Tyson and Webster (J. Diff. Eq. 25, 39) that for the Goodwin system and a larger class of similar models the following is true. The system has a unique steady state and if the linearization of the system at that point has no repeated eigenvalues and at least one eigenvalue with positive real part there exist periodic solutions. This reduces the existence question to the analysis of the linearization. The existence proof relies on the Brouwer fixed point theorem and is similar to a proof I described in a previous post. Although the Goodwin system is three dimensional the method is not restricted to that case. The proof does not give information about the stability of the periodic solutions. In the paper of Hastings et. al. they indicate that an alternative analysis using a Hopf bifurcation can give stability in some cases. However no details of the stability argument are given in that paper.

The Goodwin model was inspired by the fundamental work of Monod and Jacob on gene regulation.  Various things have given me an appetite for learning more about gene regulatory networks and this was increased by some of the talks I heard last week.