A general Michaelis-Menten Uptake Model is a pair of functions S(t), P(t), with parameters h, k, and v, defined as follows:

S(t) is the "substrate" function, P(t) is the "product" function, and we will use this type of model in several kinetics situations including both drug kinetics and enzyme kinetics. Each of these models has the following rate equations, the first of which we derived in LW Exercise 1.6.

Example 1.

 

In kinetics applications, this type of model represents some sort of saturated process, which usually occurs when some initially first or second order process becomes saturated. Below we have superimposed our model in Example 1 on a plot containing a 1st order process (see RK Exercise 4), in order to provide a visual comparison between these two types of models.

The parametric (navy) plot below is the standard Michaelis-Menten Uptake plot, which displays the reaction rate P'(t) as a function of the substrate concentration S(t). Another way to describe this is as a plot of the velocity of the production of the product P vs the concentration of the substrate S. Note that t is not explicity displayed, since this is a parametric 2D plot, however the time flow does starts at the right of the diagram and advances to the left. In addition, again we have superimposed a plot containing a 1st order process (see RK Exercise 4), in order to provide a visual comparison between these two types of models.

Further, it is interesting to notice that the relationship between P' and S for an unsaturated first order process is apparently parabolic, while for a saturated Michaelis-Menten process the relationship appears to be hyperbolic. We will explore this further in LW Exercise 2.