A remark on the number of steady states in a multiple futile cycle

This note studies the number of positive steady states in biomolecular reactions consisting of activation/deactivation futile cycles, such as those arising from phosphorylations and dephosphorylations at each level of a MAPK cascade. It is shown that (1) for some parameter ranges, there are at least n + 1 (if n is even) or n (if n is odd) steady states; (2) There never are more than 2n steady states; (3) for parameters near the standard Michaelis-Menten quasi-steady state conditions, there are at most n + 1 steady states; and (4) for parameters far from the standard Michaelis-Menten quasi-steady state conditions, there is at most one steady state.