## Archive for July, 2017

### SMB conference in Utah

July 21, 2017

I have just attended the annual meeting of the Society for Mathematical Biology in Salt Lake City. Before reporting on my experiences there I will start with an apparently unrelated subject. I studied and did my PhD at the University of Aberdeen and I am very satisfied with the quality of the mathematical education I received there. There was one direction in mathematics in which I did not get as much training as I later needed, namely analysis and partial differential equations. This was not the fault of the lecturers, who had enough expertise and enthusiasm for teaching these things. It was the fault of the students. Since it was a small department the students chose which advanced courses were to be offered from a list of suggestions. Most of the students (all of them except me?) did not like analysis and so most of the advanced courses with a significant analysis content were not chosen. By the time I got my first postdoc position I had become convinced that in the area I was working in the research based on differential geometry was a region which was overgrazed and the thing to do was to apply PDE. Fortunately the members of the group of Jürgen Ehlers which I joined were of the same opinion. The first paper I wrote after I got there was on a parabolic PDE, the Robinson-Trautman equation. I had to educate myself for this from books and one of the sources which was most helpful was a book by Avner Friedman. Here is the connection to the conference. Avner Friedman, now 84 but very lively, gave a talk. Mathematically the subject was free boundary value problems for reaction-diffusion-advection equations, Friedman’s classic area. More importantly, these PDE problems came from the modelling of combination therapies for cancer. The type of therapies being discussed included antibodies to CTLA-4 and PD-1 and Raf inhibitors, subjects I have discussed at various places in this blog. I was impressed by how much at home Friedman seemed to be with these biological and medical themes. This is maybe not so surprising in view of the fact that he did found the Institute for Mathematical Biosciences in Ohio and was its director from 2002 to 2008. More generally I was positively impressed by the extent to which the talks I heard at this conference showed a real engagement with themes in biology and medicine and evidence of a lot of cooperations with biologists and clinicians. There were also quite a number of people there employed at hospitals and with medical training. As an example I mention Gary An from the University of Chicago who is trained as a surgeon and whose thoughtful comments about the relations between mathematics, biology and medicine I found very enlightening. There was a considerable thematic overlap with the conference on cancer immunotherapy I went to recently.

Now Subgroups are being set up within the Society to concentrate on particular subjects. One of these, the Immunobiology and Infection Subgroup had its inaugural meeting this week and of course I went. There I and a number of other people learned a basic immunological fact which we found very surprising. It is well known that the thymus decreases in size with age so that presumably our capacity to produce new T cells is constantly decreasing. The obvious assumption, which I had made, is that this is a fairly passive process related to the fact that many systems in our bodies run down with age. We learned from Johnna Barnaby that the situation may be very different. It may be that the decrease in the size of the thymus is due to active repression by sexual hormones. She is involved in work on therapy for prostate cancer and said that it has been found that in men with prostate cancer who are getting drugs to reduce their testosterone levels it is seen that their thymus increases in size.

There were some recurrent themes at the conference. One was oncolytic viruses. These are genetically engineered viruses intended to destroy cancer cells. In modelling these it is common to use extensions of the fundamental model of virus dynamics which is very familiar to me. For instance Dominik Wodarz talked about some ODE models for oncolytic viruses in vitro where the inclusion of interferon production in the model leads to bistability. (In reponse to a question from me he said that it is a theorem that without the interferon bistability is impossible.) I was pleased to see how, more generally, a lot of people were using small ODE models making real contact to applications. Another recurrent theme was that there are two broad classes of macrophages which may be favourable or unfavourable to tumour growth. I should find out more about that. Naveen Vaidya talked about the idea that macrophages in the brain may be a refuge for HIV. Actually, even after talking to him I am not sure if it should not rather be microglia than macrophages. James Moore talked about the question of how T cells are eliminated in the thymus or become Tregs. His talk was more mathematical than biological but it has underlined once again that I want to understand more about positive and negative selection in the thymus and the related production of Tregs.

On a quite different subject there were two plenary talks related to coral reefs. A theme which is common in the media is that of the damage to coral due to climate change. Of course this is dominated by politics and usually not accompanied by any scientific information on what is going on. The talk of Marissa Blaskett was an excellent antidote to this kind of thing and now I have really understood something about the subject. The other talk, by Mimi Koehl, was less about the reefs themselves but about the way in which the larvae of snails which graze on the coral colonize the reef. I found the presentation very impressive because it started with a subject which seemed impossibly complicated and showed how scientific investigation, in particular mathematical modelling, can lead to understanding. The subject was the interaction of microscopic swimming organisms with the highly turbulent flow of sea water around the reefs. Investigating this involved among other things the following. Measuring the turbulent flow around the reef using Doppler velocimetry. Reconstructing this flow in a wave tunnel containing an artificial reef in order to study the small-scale structure of the transport of chemical substances by the flow. Going out and checking the results by following dye put into the actual reef. And many other things. Last but not least there was the mathematical modelling. The speaker is a biologist and she framed her talk by slides showing how many (most?) biologists hate mathematical modelling and how she loves it.

### Conference on reaction networks and population dynamics in Oberwolfach

July 9, 2017

This is a belated report on a conference in Oberwolfach I attended a couple of weeks ago. The title includes two elements. The first held no suprises for me but the second was rather different from what I had expected. My expectation was that it would be about the evolution of populations of organisms. In fact it was rather focussed on models related to genetics, in other words with the question of how certain genetic traits spread through a population.

A talk I found very interesting was by Sebastian Walcher. I already wrote briefly about a talk of his in Copenhagen in a previous post but this time I understood a lot more. The question he was concerned with is how to find interesting small parameters in dynamical systems which allow the application of geometric singular perturbation theory. In GSPT the system written in the slow time (with the smallness parameter included as a variable) contains a whole manifold of steady states, the critical manifold. The most straightforward theory is obtained when the eigenvalues of the linearization of the system transverse to the critical manifold lie away from the imaginary axis. This corresponds to the situation of a transversely hyperbolic manifold of steady states. The first idea of Walcher’s talk is that whenever we have a transversely hyperbolic manifold of steady states in a dynamical system this is an opportunity for identifying a small parameter. This may not sound very useful at first sight because it would seem reasonable that generic dynamical systems would never contain manifolds of steady states of dimension greater than zero. There is a reason why this observation is misleading for systems arising from reaction networks. In these systems the state space is defined by positivity conditions on the concentrations and there are also certain parameters (such as reaction constants and total amounts) which are required to be positive. To have a name let us call the region defined by these positivity conditions the conventional region of the spaces of states and parameters. In the conventional region manifolds of steady states are not to be expected. On the other hand it frequently happens that they arise when we go to the boundary of that region. A familiar example is the passage to the Michaelis-Menten limit in the system describing a reaction catalysed by an enzyme. This takes us from the extended mass action kinetics for substrate, free enzyme and substrate-enzyme complex to Michaelis-Menten kinetics for the substrate alone. Roughly speaking it is the limit where the amount of the enzyme is very small compared to the amount of the substrate. I often wondered whether there could not be a kind of ‘anti-Michaelis-Menten’ limit where the amount of enzyme is very large compared to the amount of the substrate. I asked Walcher whether he knew how to do this and how it fitted into his general scheme. He gave me a positive answer to this question and some references and I must look into this in detail when I get time. The reason for being interested in this is that if we can obtain suitable information about a limiting case on the boundary it may be possible obtain information on the part of the conventional region where a certain parameter is small but non-zero.

There was one talk which did have a connection to population biology in way closer to what I had expected. It happens all the time that ecosystems are damaged by exotic species imported, deliberately or by accident, from other parts of the world. There are also well-known stories of the type that to try to control exotic species number one exotic species number two is introduced and is itself very harmful. It is nice to hear an example where this kind of introduction of an exotic species was very successful. It is the case of the cassava plant which was introduced from South America to Africa and became a staple food there. Then an insect from South America (species number one) called the mealy bug was introduced accidentally and caused enormous damage. Finally an ecologist called Hans Herren introduced a parasitic wasp (species number two) from South America, restoring the food supply and saving numerous lives (often the number 20 million is quoted). More details of this story can be found here.

I want to mention one statement made in the talk of Gheorghe Craciun in Oberwolfach which I found intriguing. I might have heard this before but it did not stick in my mind properly. The statement is that the set of dynamical systems which possess a complex balanced steady state is a variety of codimension $\delta$, where $\delta$ is the deficiency. There seemed to be some belief in the audience that this variety is actually a smooth manifold. On one afternoon we had something similar to the breakout sessions in Banff. I suggested the topic for one of these, which was Lyapunov functions. The idea was to compare classes of Lyapunov functions which people working on different classes of dynamical systems knew. This certainly did not lead to any breakthrough but I think it did lead to a useful exchange of information. I documented the discussion for my own use and I think I could profit by following some of the leads there.

To finish I want to mention a claim made by Ankit Gupta in his talk. It did not sound very plausible to me but I expect that it at least contains a grain of truth. He said that these days more papers are published on $NF\kappa B$ than on all of mathematics.