where is supplied by an external source at the rate which is supposed to remain constant during the reaction. From the equations it is clear that is irreversibly converted into the product molecules . The product is then removed by an irreversible sink at the rate . The free enzyme E is inactive unless when it has product molecules bounded, forming the complex . Assuming that the sink of the product behaves as a first order reaction, and including in the description the transport of the substrate and product molecules by diffusion (the transport of the enzyme and its complexes can be neglected) the reaction-diffusion system can be written as:
where
with , , , , , and . Adding Eqs. (61), (62) and (63), the constancy of the total amount of the enzyme becomes evident: . That allows us to reduce the dimensionality of the system by one.
According to Selkov's data, the concentration of E and its complexes is 3 to 4 orders of magnitude below those of ATP and ADP. So, we can rescale the concentrations by a small parameter (of the order of ), as done in . In order to discuss the validity of the approximation in different limits we will introduce a new parameter such that and and use the expansion to simplify the notation.
We proceed as described in Sec. V of the paper in order to get rid of , and in favor of and . The set of algebraic relations can be solved. Its solutions are given by
where we have defined the rescaled concentrations, and , as
In this case, as we are dealing with more than one fast reaction, the equations for the first correction terms, , can be written together in matrix form. Their expression reads,
from which we get:
where .
Introducing the expansions , with and given by Eqs. (71) and (74), into Eqs. (59)-(60), and using the rescaled variables (72) we finally find:
where we have defined