Question

The enzyme that hydrolyzes acetylcholine to form acetate and choline has a Km = 9 x 10-5 M for the substrate. In an reaction flask with 5 nanomol / mL of the enzyme and 150 µM of acetylcholine, an initial reaction rate of 40 µmol / mLs was observed.

a) Calculate vmax for this amount of enzyme.

b) Calculate kcat for the enzyme.

c) Calculate the catalytic efficiency of the enzyme.

d) Does this enzyme approach “catalytic perfection?

Answer 1

This enzyme follows Michaelis-menten mechanism , Km (Michaelis-menten constant ).

we have v₀ (initial rate ) = kcat [E]₀ [S]₀ / [S]₀ + Km

[E]₀ = initial conc. of enzyme ; [S]₀ = initial conc.  of substrate

(a) by initial rate methods,

For a given [E]₀ and [S]₀ , the rate of product formation becomes
independent of [S]₀ initially, reaching a maximum value known as the maximum velocity,
v_max.

we have substrate conc. comparable with Km ;    [S]_{0 ~}   Km

we get ,  v₀ = kcat [E]₀ / 2

Since ,   v_max   = kcat [E]₀ ; so  v_max = 2* v₀

  v_max = 40 (µmol / mL-s ) *  2 = 80 µmol / mL-s

(b) kcat for enzyme,
we have, v₀ = kcat [E]₀ [S]₀ / [S]₀ + Km

putting values , 40 µmol / mL-s =  kcat* 5 nanomol / mL*150*10^-6 M / 150*10^-6 M + 9 * 10^-5 M

or 40 µmol / mL-s =  kcat* 5 nanomol / mL*150*10^-6 M / 24*10^-5 M = kcat* 5 nanomol / mL* 0.625

kcat =  (40 *10^-6 mol / mL-s ) / 3.125*10^-9 mol / mL = 12800 s-1

(c) the catalytic efficiency of the enzyme is enzyme is the ratio of kcat /Km

catalytic efficiency of the enzyme = kcat / Km = 12800 s-1 / 9 * 10^-5 mol/L = 1.42 *10⁸ L mol-1 s-1

(d)

The efficiency may reach its maximum value of the rate constant for the formation of a complex from two species that are diffusing freely in solution, the maximum efficiency is related to the maximum rate of diffusion of E and S in solution.  The limit that the rate of reaction is governed by the rate at which the reactant molecules diffuse through the solvent leads to rate
constants of about 10⁸–10⁹ L mol−1 s−1 for molecules as large as enzymes.

This enzyme with catalytic efficiency of the enzyme 1.42 *10⁸ L mol-1 s-1 and have
attained ‘catalytic perfection’, since the rate of the reaction it catalyses is
controlled only by diffusion: it catalyzes reaction as soon as a substrate make contact.