E + S → ES → E + P
++
I I
I I
I I
EI ESI
K_I = [E][I]/[EI]

K_i = [ES]/[ESI]

ET = E + ES + ESI + EI
= [E] + [ES] + [ES][I]/K_I + [E][I]/K_I
= [E] (1 + I/K_i) + [ES](1 + I/K_i)
= K_m [ES]/[S] + [ES](1 + I/K_I)
= [ES](K_m/[S] + 1) (1 + I/K_I)

For enzyme-catalysed reaction:

Substituting (8) in (5), we get

V_max /V = (K_m/[S] +)(1 + I/K_I)
V_max /V = (K_m + [S]/[S])(1 + I/K_I)
V/V_max = [S]/K_m + [S]) × 1/1 + I/K_I)
V = V_max[S]/(K_m + [S]) × 1/1 + I/K_I)

Dividing by (1 + I/K_I) in numerator and denominator,

Where

V_max^app = V_max/1 + I/K_i

This is the M–M equation for non-competitive inhibitor refer Table 6.9.

From equation (10), it is clear that when an enzyme is inhibited non-competitive, V_max decreases and K_m value does not change; that is, the combination with either [S] or [I] does not affect the affinity for the other as shown in Figure 6.22.

 

Table 6.9 M–M equation (Slope, X-intercept, Y-intercept)

Slope(M)	Km(1 + I/KI)/Vmax
Y intercept	1 + I/KI/Vmax
X intercept	−1/Km

Figure 6.22 Effect of Non-competitive Inhibitor