 We explore the combinatorics of reaction networks , with a view towards the Global Attractor Conjecture. We prove that full-rank matrices with positive off-diagonal and negative diagonal entries permit a positive linear combination of the rows so that all coordinates have the same sign. Using this, we show that a reaction network has critical siphons iff it has "drainable" or "self-replicable" siphons. Further, if the minimal siphons of a reaction network are not drainable , then the dynamics is persistent. Consequently, we obtain a new, elementary proof for the persistence of non-catalytic weakly-reversible chemical reaction networks . Our results clarify that the difficulties in proving the Global Attractor Conjecture are essentially due to competition between extinction and autocatalytic growth .
The persistence conjecture is a long-standing open problem in chemical reaction network theory. It concerns the behavior of solutions to coupled ODE systems that arise from applying mass-action kinetics to a network of chemical reactions. The idea is that if all reactions are reversible in a weak sense, then no species can go extinct. A notion that has been found useful in thinking about persistence is that of "critical siphon." We explore the combinatorics of critical siphons , with a view towards the persistence conjecture. We introduce the notions of "drainable" and "self-replicable" (or autocatalytic) siphons. We show that: every minimal critical siphon is either drainable or self-replicable; reaction networks without drainable siphons are persistent; and non-autocatalytic weakly-reversible networks are persistent . Our results clarify that the difficulties in proving the persistence conjecture are essentially due to competition between drainable and self-replicable siphons .
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The persistence conjecture is a long-standing open problem in chemical reaction network theory. It concerns the behavior of solutions to coupled ODE systems that arise from applying mass-action kinetics to a network of chemical reactions. The idea is that if all reactions are reversible in a weak sense, then no species can go extinct. A notion that has been found useful in thinking about persistence is that of "critical siphon."
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reaction networks
critical siphons
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Global Attractor Conjecture. We prove that full-rank matrices with positive off-diagonal and negative diagonal entries permit a positive linear combination of the rows so that all coordinates have the same sign. Using this, we show that a reaction network has critical siphons iff it has
persistence conjecture. We introduce the notions of
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or
and
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siphons. Further, if the minimal siphons of a reaction network are not drainable , then the dynamics is persistent. Consequently, we obtain a new, elementary proof for the persistence of non-catalytic
(or autocatalytic) siphons. We show that: every minimal critical siphon is either drainable or self-replicable; reaction networks without drainable siphons are persistent; and non-autocatalytic
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chemical reaction networks
networks are persistent
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Global Attractor Conjecture
persistence conjecture
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extinction and autocatalytic growth
drainable and self-replicable siphons