## Archive for July, 2018

### ECMTB 2018 in Lisbon, part 2

July 27, 2018

In the mini-workshops at the conference related to chemical reaction network theory the most striking new result to be announced was by Balazs Boros. His preprint on this is arXiv:1710.04732. In fact it is necessary to say in what sense this is new but I will postpone that point and first discuss the mathematics. This result is very easy to formulate and I will try to make the discussion here as self-contained as possible. We start with a chemical reaction network consisting of reactions and complexes (the expressions on the left and right hand sides of the reactions like $X+Y$). This network can be represented by a directed graph where the nodes are the complexes and the edges are the reactions, oriented from the complex on the LHS to that on the RHS. The network is called weakly reversible if whenever we can get from node $X$ to node $Y$ by following directed edges we can get from $Y$ to $X$. If we assume mass action kinetics and choose a positive reaction rate for each reaction we get a system of ODE describing the evolution of the concentrations of the substances belonging to the network in a standard way. Because of the interpretation we are interested in positive solutions, i.e. solutions for which all concentrations are positive. The theorem proved by Boros says: if the network is weakly reversible then the corresponding ODE system with mass action kinetics has at least one positive steady state. Actually he proves that the stronger (and more natural) statement holds that there is a solution in each positive stoichiometric compatibility class. Evidently the hypotheses only involve the graph of the network and require no details of the form of the complexes or the values of the reaction constants. Thus it is a remarkably strong result. In contrast to the statement of the theorem the proof is not at all easy. It involves reducing the desired statement to an application of the Brouwer fixed point theorem. Returning to the question of the novelty of the result, it was announced in a preprint of Deng et al. in 2011 (arXiv:1111.2386). It has never been published and it seems that the proof proposed by the authors is far from complete. Furthermore, the authors do not seem to be willing and able to repair the argument. Thus this result has been blocked for seven years. For anyone else it is an ungrateful task to provide a proof since a positive reaction from the authors of the original paper is doubtful. Furthermore other people not familiar with the background may also fail to give due credit to the author of the new paper. I think that with this work Balazs has done a great service to the reaction network community and we who belong to this community should take every opportunity to express our gratitude for this.

There was a nice talk by Ilona Kosiuk on her work with Peter Szmolyan on NF$\kappa$B. She expressed doubts about the derivation of the three-dimensional system mentioned in a previous post from the four-dimensional system. The work she explained in some detail concerned the four-dimensional system and uses GSPT to investigate the existence of periodic solutions of that system.

I feel that I got a lot more out of this conference than that I did out of that in Nottingham two years ago. I found more points of contact with my own research. This fact perhaps has less to do with the conference itself than it does with me. It is simply that I have penetrated a lot more deeply into mathematical biology during the last two years.