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 for the vector of concentrations. In the CFSTR the function is written in the form . Here is thought of as describing the dynamics of a closed system while is supposed to describe inflow and outflow and has the form . Here the are fixed non-negative constants. PW assume that has a unique zero and that there is a function such that everywhere with equality only at that zero. This is essentially the only assumption on . 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 of its domain of definition there exists a CFSTR defined by a function , an open neighbourhood of and a diffeomorphism of onto an open subset of the domain of definition of which transforms the restriction of the original vector field to to the restriction of to the image of . The vector field is constructed by starting with an arbitrary CFSTR of the right dimension and deforming while leaving 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 is analytic (). If the function 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 is defined by a reaction network as in CRNT then this network can be extended so as to give one which defines , 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 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.

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The immortal Henrietta Lacks, part 2

I now read the book about Henrietta Lacks mentioned in a previous post. This book has perhaps three main aspects. One is the history of a scientific development. The second is the information about the family of Henrietta Lacks and the effects on them of her unusual type of fame. The third is a discussion (and some history) of medical ethics. My own bias is that I was most interested in the first point and less in the other two. Having read some reviews of the book I was not sure if I would like it. I was afraid that it might be too political in a direction I would not like, with overemphasis on polemical criticism of racism and hostility to the medical community. Fortunately my fears were not confirmed. In my opinion the intellectual quality of the book is a lot higher than that typical of the reviews I had read. The book tells many interesting stories and I recommend it to anyone with an inquiring mind.

One of the themes I found most interesting was that of contamination of human cell cultures by HeLa cells. This problem has been much more extreme in the past than I had known or expected. I did not get a picture of how it is today and I would like to read up on that sometime. The situation at one time was that it was uncovered that many cell cultures allegedly coming from different tissues and individuals had actually been colonized and taken over by HeLa cells. Embarrassingly, many papers had been published reporting experiments on the differences between the properties of cells coming from these ‘different’ cultures. I was struck by the horror story of a doctor who injected patients with HeLa cells to see if they would produce tumours. Sometimes they did. Sometimes these tumours were eventually eliminated by the patient’s immune system but sometimes they were not.

Another thing I would like to know more about after reading the book is what it is that makes HeLa cells so special. It is known that they have been genetically modified by one of the HPV, the viruses known to cause cervical cancer. What I did not see is what is known the status of these cells. How unique are they? Are there many other comparable human cell lines these days? If not, why not? As well as teaching me many facts the book has left me with a list of questions which I will try to keep in mind whenever I encounter information about human cell culture in the future.