text: Zero-order ultrasensitivity takes place under saturating conditions.  For example, consider an enzymatic step with a kinase, phosphatase, and substrate. Steady state levels of the phosphorylated substrate have an ultrasensitive response when there is enough substrate to saturate all available kinases and phosphatases.  Under these conditions, small changes in the ratio of kinase to phosphatase activity can dramatically change the number of phosphorylated substrate (For a graph illustrating this behavior, see ). This enhancement in sensitivity of steady state phosphorylated substrate to Km, or the ratio of kinase to phosphatase activity, is termed zero-order to distinguish it from the first order behavior described by Michaelis-Menten dynamics, wherein the steady state concentration responds in a more gradual fashion than the switch-like behavior exhibited in ultrasensitivity.
  Using the notation from Goldbeter & Koshland, let W be a certain substrate protein and let W be a covalently modified version of W. The conversion of W to W is catalyzed by some enzyme  and the reverse conversion of W' to W is catalyzed by a second enzyme  according to following equations:
  The concentrations of all necessary components (such as ATP) are assumed to be constant and represented in the kinetic constants.
  Using the chemical equations above, the reaction rate equations for each component are:
  The total concentration of each component is given by:
  The zero order mechanism assumes that the  or . In other words, the system is in a Michaelis-Menten steady state, which means, to a good approximation,  and  are constant.
  From these kinetic expressions one can solve for  at steady state defining  and 
  where  and
  When the  is plotted against the molar ratio  and  it can be seen that the W to W conversion occurs over a much smaller change in the  ratio than it would under first order (non-saturating) conditions, which is the telltale sign of ultrasensitivity.