  We discuss chemical reaction networks and metabolic pathways based on
stoichiometric network analysis, and introduce deformed toric ideal constraints
by the algebraic geometrical approach. This paper concerns steady state flux of
chemical reaction networks and metabolic pathways. With the deformed toric
ideal constraints, the linear combination parameters of extreme pathways are
automatically constrained without introducing ad hoc constraints. To illustrate
the effectiveness of such constraints, we discuss two examples of chemical
reaction network and metabolic pathway; in the former the flux and the
concentrations are constrained completely by deformed toric ideal constraints,
and in the latter, it is shown the deformed toric ideal constrains the linear
combination parameters of flux at least partially. Even in the latter case, the
flux and the concentrations are constrained completely with the additional
constraint that the total amount of enzyme is constant.
