  Mass-action chemical reaction systems are frequently used in Computational
Biology. The corresponding polynomial dynamical systems are often large
(consisting of tens or even hundreds of ordinary differential equations) and
poorly parametrized (due to noisy measurement data and a small number of data
points and repetitions). Therefore, it is often difficult to establish the
existence of (positive) steady states or to determine whether more complicated
phenomena such as multistationarity exist. If, however, the steady state ideal
of the system is a binomial ideal, then we show that these questions can be
answered easily. The focus of this work is on systems with this property, and
we say that such systems have toric steady states. Our main result gives
sufficient conditions for a chemical reaction system to have toric steady
states. Furthermore, we analyze the capacity of such a system to exhibit
positive steady states and multistationarity. Examples of systems with toric
steady states include weakly-reversible zero-deficiency chemical reaction
systems. An important application of our work concerns the networks that
describe the multisite phosphorylation of a protein by a kinase/phosphatase
pair in a sequential and distributive mechanism.
