  Posttranslational modification of proteins is key in transmission of signals
in cells. Many signaling pathways contain several layers of modification cycles
that mediate and change the signal through the pathway. Here, we study a simple
signaling cascade consisting of n layers of modification cycles, such that the
modified protein of one layer acts as modifier in the next layer. Assuming
mass-action kinetics and taking the formation of intermediate complexes into
account, we show that the steady states are solutions to a polynomial in one
variable, and in fact that there is exactly one steady state for any given
total amounts of substrates and enzymes.
  We demonstrate that many steady state concentrations are related through
rational functions, which can be found recursively. For example,
stimulus-response curves arise as inverse functions to explicit rational
functions. We show that the stimulus-response curves of the modified substrates
are shifted to the left as we move down the cascade. Further, our approach
allows us to study enzyme competition, sequestration and how the steady state
changes in response to changes in the total amount of substrates.
  Our approach is essentially algebraic and follows recent trends in the study
of posttranslational modification systems.
