In the first part of this exercise, we used three rules equipped with mass action kinetics. The overall effect of these rules can itself be written as a rule
E(s~1), S(s~0) -> E(s~1), S(s~1)
but how would we express the desired rate of this rule?
In general, this cannot be done exactly; however, we can use something called the quasi-steady state approximation (or QSSA) to obtain an approximate rate law that is remarkably accurate under certain conditions. An excellent explanation of the QSSA and its range of validity can be found in a 1988 paper by Lee Segel. The rate law obtained from the QSSA is called Michaelis-Menten kinetics.
If we define variables
%var: 'KM' ('BRK' + 'MOD') / 'BND'%var: 'S0' S(s~0)
then we can write our rule, with the Michaelis-Menten rate law, as follows:
E(s~1), S(s~0) -> E(s~1), S(s~1) @ 'MOD' / ('KM' + 'S0')
This overall rate law is a good approximation to standard mass action kinetics provided that E1 / (KM + S0) << 1 where E1 is the total enzyme
E(s~1) and KM and S0 are as defined above.
Try running the attached system; compare the results with those obtained using mass action kinetics. Can you explain the shape of the curve tracking the production of active substrate? Vary the parameters of the model to see how the validity of the approximation varies.
%agent: E(s~1~0) %agent: S(s~0~1) E(s~1), S(s~0) -> E(s~1), S(s~1) @ 0.1 / (200 + 'S0') %init: 100 E(s~1) %init: 5000 S(s~0) %obs: 'S0' S(s~0) %obs: 'S1' S(s~1)