## Absolute concentration robustness

In the past years I have been on the committees for many PhD examinations. A few days ago, for the first time, I was was on the committee for a thesis on a subject belonging to the area of mathematical biology. This was the thesis of Jost Neigenfind and it was concerned with a concept called absolute concentration robustness (ACR).

The concentration of a given substance in cells of a given type varies widely between the individual cells. (Cf. also this previous post). It is of interest to identify mechanisms which can ensure that the steady state concentration of a particular substance is independent of initial data. (This is a way in which the output of a system can be independent of background variation.) In saying this I am assuming implicitly that more general solutions converge to steady states. A more satisfactory formulation can be obtained as follows. In a chemical reaction network there are usually a number of conserved quanitities, say $C_\alpha$. These define affine subspaces of the state space, the stoichiometric compatibility classes. For many systems there is exactly one stationary solution in each stoichiometric compatibility class. The condition of interest here is that the value of one of the concentrations, call it $x_1$, in the steady state solution is independent of the parameters $C_\alpha$. (The other concentrations $x_i,i>1$ will in general depend on the $C_\alpha$.) This property is ACR. I first heard of this in a talk by Uri Alon at the SMB conference in Krakow in the summer of 2011. The basic idea is explained clearly in a paper of Shinar and Feinberg (Science 327, 1389). They present a general theoretical approach but also describe some biological systems where ACR (in a suitable approximate sense) has been observed experimentally. In the terminology of Chemical Reaction Network Theory (CRNT) the examples they discuss have deficiency one. They mention that ACR is impossible in systems of deficiency zero. There is no reason why it should not occur in systems of deficiency greater than one but in those more complicated dynamics make it more difficult to decide whether the property holds or not.

The result of Shinar and Feinberg only covers a class of reaction networks which is probably very restricted. What Neigenfind does in his thesis is to develop more general criteria for ACR and computer algorithms which can check these criteria for given systems. The phenomenon of ACR is interesting since it is a feature which may be more common in reaction systems coming from biology than in generic systems. At least there is a good potential reason why this might be the case.