How can I approximate the binding affinity #K_D# of an enzyme #E# (that follows Michaelis-Menten kinetics) to a substrate #S#?

MATHEMATICAL INTERPRETATION

#E + S stackrel(k_1)(rightleftharpoons) ES stackrel(k_2)(->) E + P#
#color(white)(aaaaaa)""^(k_(-1))#

It is #K_M#, which is an approximation to the binding affinity (but is NOT #K_D#!!), and is the equilibrium constant for the dissociation of the #ES# complex.

#K_M = (k_(-1) + k_2)/(k_1)#

You can tell because:

• #k_(-1)# is the rate constant for the dissociation of the complex into free enzyme #E# and substrate #S#
• #k_2# is the separation of the enzyme-substrate complex #ES# in such a way that product #P# is formed and enzyme #E# is freed.
• Dividing by #k_1# describes the reverse of the formation of the #ES# complex, i.e. its dissociation.

Also, #k_2#, the rate constant for #ES -> E + P#, can be divided by #K_M# to get the catalytic efficiency.

#"Catalytic efficiency" = (k_2)/(K_M)#

This describes the ratio of the rate at which the #ES# complex is consumed to SUCCESSFULLY make product #P#, compared to the rate at which the #ES# complex is consumed but NOT successfully making the product. This is useful when comparing enzymes of drastically differing reaction rates.

VISUAL INTERPRETATION

In a #v_0# vs. #[S]# plot, you have the initial rate of the reaction as it changes according to the concentration of the substrate.

Under the steady-state approximation, the equation is:

#v_0 = (k_2[E]_"tot"[S])/(K_M + [S]) = (v_"max"[S])/(K_M + [S])#

where, in the absence of an inhibitor, #[E]_"tot" = [E] + [ES]#, #[E]# is free enzyme, and #[ES]# is enzyme-substrate complex.

Standard Michaelis-Menten kinetics is hyperbolic, like myoglobin's behavior. With allosteric activity (i.e. adaptive binding-affinity changes that are due to previous binding activity), it becomes sigmoidal, like hemoglobin's behavior.

You can get #K_M# visually where #v_0 = v_"max"/2#, as shown in the red dotted lines above.