Persistence via reachability vs. via Petri nets

I have received and responded to an email question regarding my paper Reachability, Persistence, and Constructive Chemical Reaction Networks. The question and the response could be beneficial to others. I am posting them here with the kind permission of the person who asked the question.

QUESTION (4 December 2009):
Can you give a couple of interesting examples that can be proved to be persistent (or not to be) using your theorem, but not using the Petri net methods developed by David Angeli, Patrick De Leenheer and Eduardo Sontag?  That will help appreciate the impact of the work.

RESPONSE (14 December 2009):

I do not have examples in response to this question, but I have some relevant comments.

I actually prove “vacuous persistence”. (I would welcome suggestions for better terminology.) This is important because boundary initial states are a real concern. One can prove vacuous persistence by proving persistence for all “closed” subnetworks. But it’s a lot of extra work. I actually had to do essentially that in the paper [1] on the monotonicity of IL-1 receptor-ligand binding equilibrium. This was in response to a request from a referee. I felt at the time that this made the paper unnecessarily longer but I did not have solid arguments for that hunch.

I wanted to be able to tell whether a reaction network is vacuously persistent just by “looking at it” and not having to compute siphons. This is what I achieved with the result that the absence of isomerism among the elementary species in an explicitly-reversibly constructive network implies persistence, and the result that futile cascaded binary enzymatic networks are persistent.

I suspect that the “each siphon contains the support of a P-semiflow” condition is in fact equivalent to vacuous persistence for conservative networks. Other than just intuition, I observe that the networks that were shown in [2] to be persistent through this condition are in fact vacuously persistent. Furthermore, the examples (I know of) for which proving persistence required more effort are not vacuously persistent. I am referring to the following networks from [3], one of which is also mentioned in [2]:
(a) 2A+B -> A+2B , B -> A ;
(b) 2A+B -> C -> A+2B -> D -> 2A+B ;
(c) A -> B , B+C -> 2A , 2B -> 2C .
I also observe in these three examples that elementary species are isomeric to each other and that explicit or implicit isomerizations occur without catalytic (e.g. enzymatic) help.



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