Interesting talk by Zoran Nikoloski at LCSB, bipartite: one first very theoretical part discussing concentration robustness and a second part which remained unclear to me.

Genotypes encode metabolic networks that yield particular metabolic phenotype. Metabolic phenotype is determined by fluxes of chemical reactions and metabolite levels. What determines metabolic fluxes? The concentration of substrates, of the active enzyme, and of other regulatory effectors (activators and repressors). Nikolski considers enzyme kinetics with mass-action law and thermodynamic constraints (he only mentions invertibility of reactions). The structure of the networks is resumed in a metabolic network and / or in the stoichiometric matrix. The space of feasible flux distributions is shrunk by analysis of genome, metabolome, thermodynamics, etc.

The plasticity of fluxes and robustness of concentrations is believed to be an important characteristic of metabolic networks. A very mathematical theorem by Shinar and Feinberg,

Shinar, G., and Feinberg, M., *Structural sources of robustness in biochemical reaction networks,* Science** 327** (2010).

relates robustness to the so-called deficiency of the network. An analogous criterion has been devised by Nikolski and coworkers

Eloundou-Mbebi, J. M. O.; Küken, A.; Omranian, N.; Kleessen, S.; Neigenfind, J.; Basler, G.; Nikoloski, Z.: *A network property necessary for concentration robustness,* Nature Comm. **7** (2016).

where they write:

“Our main result is based on establishing whether or not the structural deficiency changes upon removing a single component from the network. To this end, we rely on the network obtained by eliminating a given component from each complex containing the component. Removal of a component may drastically alter the network, in terms of number of nodes, linkage classes and the rank of the stoichiometric matrix. […] The idea of removing a component from biochemical reaction network has been previously employed to make statements about the possibility of the network to exhibit multistationarity. Namely, for a given set of rate constants, it has been shown that if a reaction system obtained upon removal of a component admits multiple non-degenerated positive steady states, so does the original system. Therefore, this result may be used to identify subnetworks conferring multistationarity to the entire network. Here, we establish a connection between a structural deficiency, as a key network invariant, and ACR [Absolute Concentration Robustness] for a particular component. It is this connection that allows us to apply the results to large-scale networks, typically arising in the study of metabolism.”

Therefore it appear that this criterion would have more application to actual metabolic networks than the one devised by Feinberg. Actually, the criterion they discuss is a necessary condition for ACR and not a sufficient one:

“Consider a mass action reaction system that for given rate constants admits a positive steady state with and without removal of a component S. If the system has ACR in species S, then the systems with and without removal of S have the same structural deficiencies.”

Therefore, if removing a species modifies the deficiency of a network, then that species is certainly not robust. In their paper, they also have a very nice plot showing how many metabolites are robust across several kingdoms of life.