## Reaction networks in Copenhagen

Last week I attended a workshop on reaction network theory organized by Elisenda Feliu and Carsten Wiuf. It took place in Copenhagen from 1st to 3rd July. I flew in late on the Tuesday evening and on arrival I had a pleasant feeling of being in the north just due to the amount and quality of the light. Looking at the weather information for Mainz I was glad I had got a reduction in temperature of several degrees by making this trip. A lot of comments and extra information on the talks at this conference can be found on the blog of John Baez and that of Matteo Polettini. Now, on my own slower time scale, I will write a bit about things I heard at the conference which I found particularly interesting. The topic of different time scales is very relevant to the theme of the meeting and the first talk, by Sebastian Walcher, was concerned with it. Often a dynamical system of interest can be thought of as containing a small parameter and letting this parameter tend to zero leads to a smaller system which may be easier to analyse. Information obtained in this way may be transported back to the original system. If the parameter is a ratio of time scales then the limit will be singular. The issue discussed in the talk is that of finding a suitable small parameter in a system when one is suspected. It is probably unreasonable to expect to find a completely general method but the talk presented algorithms which can contribute to solving this type of problem.

In the second talk Gheorghe Craciun presented his proof of the global attractor conjecture, which I have mentioned in a previous post. I was intrigued by one comment he made relating the concept of deficiency zero to systems in general position. Later he explained this to me and I will say something about the direction in which this goes. The concept of deficiency is central in chemical reaction network theory but I never found it very intuitive and I feel safe in claiming that I am in good company as far as that is concerned. Gheorghe’s idea is intended to improve this state of affairs by giving the deficiency a geometric interpretation. In this context it is worth mentioning that there are two definitions of deficiency on the market. I had heard this before but never looked at the details. I was reminded of it by the talk of Jeanne Marie Onana Eloundou-Mbebi in Copenhagen, where it played an important role. She was talking about absolute concentration robustness. The latter concept was also the subject of the talk of Dave Anderson, who was looking at the issue of whether the known results on ACR for deterministic reaction networks hold in some reasonable sense in the stochastic case. The answer seems to be that they do not. But now I return to the question of how the deficiency is defined. Here I use the notation $\delta$ for the deficiency as originally defined by Feinberg. The alternative, which can be found in Jeremy Gunawardena’s text with the title ‘Chemical reaction network theory for in silico biologists’ will be denoted by $\delta'$. Gunawardena, who seems to find the second definition more natural, proves that the two quantities are equal provided a certain condition holds (each linkage class contains precisely one terminal strong linkage class). This condition is, in particular, satisfied for weakly reversible networks and this is perhaps the reason that the difference in definitions is not often mentioned in the literature. In general $\delta\ge\delta'$, so that deficiency zero in the sense of the common definition implies deficiency zero in the sense of the other definition.

For a long time I knew very little about control theory. The desire to change this motivated me to give a course on the subject in the last winter semester, using the excellent textbook of Eduardo Sontag as my main source. Since that I had never taken the time to look back on what I learned in the course of doing this and this became clearer to me only now. In Copenhagen Nicolette Meshkat gave a talk on identifiability in reaction networks. I had heard her give a talk on a similar subject at the SIAM life science conference last summer and not understood much. I am sure that this was not her fault but mine. This time around things were suddenly clear. The reason is that this subject involves ideas coming from control theory and through giving the course I had learned to think in some new directions. The basic idea of identifiability is to extract information on the parameters of a dynamical system from the input-output relation.

There was another talk with a lot of control theory content by Mustafa Khammash. He had brought some machines with him to illustrate some of the ideas. These were made of Lego, driven by computers and communicating with each other via bluetooth devices. One of these was a physical realization of one of the favourite simple examples in control theory, stabilization of the inverted pendulum. Another was a robot programmed to come to rest 30 cm in front of a barrier facing it. Next he talked about an experiment coupling living cells to a computer to form a control system. The output from a population of cells was read by a combination of GFP labeling and a FACS machine. After processing the signal the resulting input was done by stimulating the cells using light. This got a lot of media attention unter the name ‘cyborg yeast’. After that he talked about a project in which programmes can be incorporated into the cells themselves using plasmids. In one of the last remarks in his talk he mentioned how cows use integral feedback to control the calcium concentration in their bodies. I think it would be nice to incorporate this into popular talks or calculus lectures in the form ‘cows can do integrals’ or ‘cows can solve differential equations’. The idea would be to have a striking example of what the abstract things done in calculus courses have to do with the real world.

My talk at the conference was on phosphorylation systems and interestingly there was another talk there, by Andreas Weber, which had a possibly very significant relation to this. I only became aware of the existence of the corresponding paper (Errami et. al., J. Comp. Phys. 291, 279) a few weeks ago and since it involves a lot of techniques I am not too familiar with and has a strong computer science component I have only had a limited opportunity to understand it. I hope to get deeper into it soon. It concerns a method of finding Hopf bifurcations.

This conference was a great chance to maintain and extend my contacts in the community working on reaction networks and get various types of inside information on the field