text: Enzymes are proteins that act as biological catalysts by accelerating chemical reactions. Enzymes act on small molecules called substrates, which an enzyme converts into products. Almost all metabolic processes in the cell need enzyme catalysis in order to occur at rates fast enough to sustain life. The study of how fast an enzyme can transform a substrate into a product is called enzyme kinetics. 
  The rate of reaction of many chemical reactions shows a linear response as function of the concentration of substrate molecules. Enzymes however display a saturation effect where,, as the substrate concentration is increased the reaction rate reaches a maximum value. Standard approaches to describing this behavior are based on models developed by Michaelis and Menten as well and Briggs and Haldane. Most elementary formulations of these models assume that the enzyme reaction is irreversible, that is product is not converted back to substrate. However, this is unrealistic when describing the kinetics of enzymes in an intact cell because there is product available. Reversible Michaelis–Menten kinetics, using the reversible form of the Michaelis–Menten equation, is therefore important when developing computer models of cellular processes involving enzymes.
  In enzyme kinetics, the Michaelis–Menten kinetics kinetic rate law that describes the conversion of one substrate to one product, is often commonly depicted in its irreversible form as:
  where  is the reaction rate,  is the maximum rate when saturating levels of the substrate are present,  is the Michaelis constant and  the substrate concentration.
  In practice, this equation is used to predict the rate of reaction when little or no product is present. Such situations arise in enzyme assays. When used to model enzyme rates , for example, to model a metabolic pathway, this representation is inadequate because under these conditions product is present.  As a result, when building computer models of metabolism or other enzymatic processes, it is better to use the reversible form of the Michaelis–Menten equation. 
  To model the reversible form of the Michaelis–Menten equation, the following reversible mechanism is considered:
  To derive the rate equation, it is assumed that the concentration of enzyme-substrate complex is at steady-state, that is . 
  Following current literature convention, we will be using lowercase Roman lettering to indicate concentrations (this avoids cluttering the equations with square brackets). Thus  indicates the concentration of enzyme-substrate complex, ES. 
  The net rate of change of product (which is equal to ) is given by the difference in forward and reverse rates:
  The total level of enzyme moiety is the sum total of free enzyme and enzyme-complex, that is . Hence the level of free  is given by the difference in the total enzyme concentration,  and the concentration of complex, that is:
  Using mass conservation we can compute the rate of change of  using the balance equation:
  where  has been replaced using . This leaves  as the only unknown. Solving for  gives:
  Inserting  into the rate equation  and rearranging gives:
  The following substitutions are now made:
  and
  after rearrangement, we obtain the reversible Michaelis–Menten equation in terms of four constants: