The Michaelis-Menten equations describe a biochemical reaction in which an enzyme E and substrate S bind to form a complex C. This complex can then either dissociate back into its original components or undergo a reaction in which a product P is produced along with the free enzyme:

E 1 SK C l E 1 P This can be expressed by the differential equations

where M is the total number of enzymes (both free and bound), x and z are the numbers of substrate and product molecules, y is the fraction of the enzyme pool that is free, and the ki’s are positive constants.

(a) Explain all the terms in this system of differential equations.

(b) Although this is a system of three differential equations, its dynamics can be understood by constructing a phase plane for the variables x and y alone. Explain why.

(c) Construct the phase plane mentioned in part (b), including all nullclines and equilibria, and indicate the direction of movement in the plane.

Transcribed Image Text:

dx -kxyM+k,(1- y)M dt dy -k;xyM + k,(1 - y)M+kea(1 - y)M dt dz kcat(1-y)M dt