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Determination of rate law for enzyme kinetics

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Determination of Rate Law for Enzyme Kinetics

by Robert Geil, Vanderbilt University, USA,

This worksheet demonstrates the use of Maple for determining kinetic rate laws in Biochemical Engineering.

Reaction Mechanism:

Consider the following enzyme reaction mechanism [1]:

E + S ------ k1 ------> ES

ES ------ k2 ------> E + S

ES + W ------ k3 ------> P + E

Where: E : Enzyme; S : Substrate; W : Water; ES : Enzyme-substrate complex; P : Reaction product

Maple will be used to determine the rate law for the given enzyme-substrate reaction mechanism

> restart;

Rate Law Evaluation:

Rate of disappeatrance of substrate,

-r[s] = k1*C[E]*C[S]-k2*C[ES]

Net rate of formation of enzyme-substrate system:

r[ES] = k1*C[E]*C[S]-k2*C[ES]-k3*C[W]*C[ES]

Total concentration of enzyme in the system:

C[Et] = C[E]+C[ES]

> eq1:=-r[s]=k1*C[E]*C[S]-k2*C[ES]:eq1;

> eq2:=r[ES]=k1*C[E]*C[S]-k2*C[ES]-k3*C[W]*C[ES]:eq2;

-r[s] = k1*C[E]*C[S]-k2*C[ES]

r[ES] = k1*C[E]*C[S]-k2*C[ES]-k3*C[W]*C[ES]

Total Enzyme Concentration:

> eq3:=C[Et]=C[E]+C[ES]:eq3;

C[Et] = C[E]+C[ES]

> eq4:=isolate(%,C[E]):eq4;

C[E] = C[Et]-C[ES]

Substituting eq4 into eq2:

> eq5:=subs(%,eq2):eq5;

r[ES] = k1*(C[Et]-C[ES])*C[S]-k2*C[ES]-k3*C[W]*C[ES...

Applying Psuedo-steady State Hypotheis , r[ES] = 0:

> assign(r[ES]=0);

> eq5;

0 = k1*(C[Et]-C[ES])*C[S]-k2*C[ES]-k3*C[W]*C[ES]

Solving for C[ES] :

> eq6:=isolate(eq5,C[ES]):eq6;

C[ES] = k1*C[S]*C[Et]/(k1*C[S]+k2+k3*C[W])

Substituting enzyme concentration into substrate rate law:

> eq7:=subs(eq4,eq1):eq7;

-r[s] = k1*(C[Et]-C[ES])*C[S]-k2*C[ES]

Subtracting the following equations gives a simplified substrate rate law:

> eq8:=eq7-eq5:eq8;

-r[s] = k3*C[W]*C[ES]

Substituting enzyme complex rate law into substrate rate law:

> eq9:=subs(eq6,eq8):eq9;

-r[s] = k3*C[W]*k1*C[S]*C[Et]/(k1*C[S]+k2+k3*C[W])

Applying the following simplifications, %?

( V[max] is the maximum rate of the reaction for a given total enzyme concentration)

After simplifications, -r[s] has form of Michaelis-Menton equation where K[m] , the Michaelis constant, can be substituted for (k5+k2)/k1

> eq10:=simplify(eq9, {k3*C[W]=k4,k4*C[Et]=V[max],k2+k4=k5}):eq10;

-r[s] = k1*C[S]*V[max]/(k1*C[S]+k5)

Assigning values to parameters:

> k1:=0.1; k5:=0.05; V[max]:=2;

k1 := .1

k5 := .5e-1

V[max] := 2

> eq11:=k1*C[S]*V[max]/(k1*C[S]+k5):eq11;


Reaction rate of the substrate over a range of substrate concentrations:

> plot(eq11, C[S]=0..25, labels=["C[S]", "-r[S]"]);

[Maple Plot]

Notice, as C[S] approaches infinity, -r[s] approaches 2, the value of V[max]

Conclusion : Maple eliminates the tedious process of algebraically manipulating equations to determine rate laws.

Disclaimer: While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material.


[1] Folger, Scott H. Elements of Chemical Reaction Engineering . New Jersey: Prentice Hall, 1999.