TY - JOUR
T1 - Preclusion of switch behavior in networks with mass-action kinetics
AU - Feliu, Elisenda
AU - Wiuf, Carsten
PY - 2012/11/1
Y1 - 2012/11/1
N2 - We study networks taken with mass-action kinetics and provide a Jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a network to that of the network obtained by adding outflow, or degradation, reactions for all species.
AB - We study networks taken with mass-action kinetics and provide a Jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a network to that of the network obtained by adding outflow, or degradation, reactions for all species.
KW - Degenerate steady state
KW - Injectivity
KW - Jacobian criterion
KW - Multiple steady states
KW - Stoichiometric subspace
UR - http://www.scopus.com/inward/record.url?scp=84867575860&partnerID=8YFLogxK
U2 - 10.1016/j.amc.2012.07.048
DO - 10.1016/j.amc.2012.07.048
M3 - Journal article
AN - SCOPUS:84867575860
SN - 0096-3003
VL - 219
SP - 1449
EP - 1467
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
IS - 4
ER -