  The multiple futile cycle is an important building block in networks of
chemical reactions arising in molecular biology. A typical process which it
describes is the addition of n phosphate groups to a protein. It can be
modelled by a system of ordinary differential equations depending on
parameters. The special case n = 2 is called the dual futile cycle. The main
result of this paper is a proof that there are parameter values for which the
system of ODE describing the dual futile cycle has two distinct stable
stationary solutions. The proof is based on bifurcation theory and geometric
singular perturbation theory. An important entity built of three coupled
multiple futile cycles is the MAPK cascade. It is explained how the ideas used
to prove bistability for the dual futile cycle might help to prove the
existence of periodic solutions for the MAPK cascade.
