## The Perelson-Wallwork theorem

In this post I will explore some connections between two themes I have previously written about, dynamical systems empty of information and chemical reaction network theory. In the first of these posts I mentioned a paper of Perelson and Wallwork (PW) and I now want to discuss that paper in more detail, comparing it to CRNT. The formalisms used to describe chemical systems in these two cases are somewhat different. One class of systems considered by PW are the continuous flow stirred tank reactors (CFSTR). As in CRNT these are described by a system of ODE of the form $\dot c=f(c)$ for the vector $c$ of concentrations. In the CFSTR the function $f$ is written in the form $f_{\rm int}+f_{\rm ext}$. Here $f_{\rm int}$ is thought of as describing the dynamics of a closed system while $f_{\rm ext}$ is supposed to describe inflow and outflow and has the form $(f_{\rm ext})_s=\bar c_s-c_s$. Here the $\bar c_s$ are fixed non-negative constants. PW assume that $f_{\rm int}$ has a unique zero and that there is a function $L$ such that $\nabla L\cdot f_{\rm int}\le 0$ everywhere with equality only at that zero. This is essentially the only assumption on $f_{\rm int}$. They call an object of this kind a chemical vector field. Actually I suspect that there are a couple of other implicit assumptions. These can be found in the paper but are not mentioned in the definition of a chemical vector field. The conclusion is that given any dynamical system and a point $p$ of its domain of definition there exists a CFSTR defined by a function $f'$, an open neighbourhood $U$ of $p$ and a diffeomorphism $\phi$ of $U$ onto an open subset of the domain of definition of $f'$ which transforms the restriction of the original vector field to $U$ to the restriction of $f'$ to the image of $\phi$. The vector field $f'$ is constructed by starting with an arbitrary CFSTR of the right dimension and deforming $f_{\rm int}$ while leaving $f_{\rm ext}$ unchanged.

This result can be interpreted as saying that it is possible to embed arbitrary local dynamics into a CFSTR. One possible criticism of this interpretation is to say that it is not clear whether the definition of a ‘chemical vector field’ is really enough to capture the essential properties of vector fields defined by chemical reactions. The construction uses a cut-off function and it is important that in an intermediate step a vector field is constructed which vanishes exactly on an open set but is not zero everywhere. This means that the construction is not capable of ensuring that the function $f'$ is analytic ($C^\omega$). If the function $f_{\rm int}$ was constructed from a reaction network using mass-action or Michaelis-Menten kinetics then it would be analytic. It is difficult to change this feature of the construction since it is important that the vector field is transported exactly, not only up to a small error. This is necessary to make sure that all local dynamical features are preserved and not only those which are structurally stable.

I think that if $f_{\rm int}$ is defined by a reaction network as in CRNT then this network can be extended so as to give one which defines $f$, although not uniquely. I have not checked this in detail. Supposing this is the case we can ask the question, which qualitative features of a dynamical system can be reproduced by a function $f$ arising in this more restricted way. I would not be surprised if there are results in the literature relevant to this question but I have not done a serious search yet. The wider question is that of the degree and nature of the simplification obtained by specializing from arbitrary dynamical systems to those arising in the framework of CRNT.