The Michaelis-Menten model of enzyme catalysis is
E + S stackrelcolor(blue)(k_f)(⇌) ES stackrelcolor(blue)(k_"cat"color(white)(m))(→) E + P
color(white)(mmmm)stackrelcolor(blue)(k_r)(color(white)(l))
If the enzyme concentration is much less than the substrate concentration, the rate of product formation is
(d[P])/dt = V_"max" ([S])/(K_"M" + [S]) = k_"cat" [E]_0 ([S])/(K_"M" + [S])
The maximum rate V_"max" is attained when all of the enzyme molecules are bound to the substrate.
It follows from the Michaelis-Menten equation that
color(blue)(|bar(ul(color(white)(a/a) V_"max" = k_"cat"[E]_0color(white)(a/a)|)))" "
[E]_0 is the initial enzyme concentration and k_"cat" is the turnover number — the maximum number of substrate molecules converted to product per enzyme molecule per second.
k_"cat" = V_"max"/[E]_0 = (30 × 10^"-6" color(red)(cancel(color(black)("mol·L"^"-1")))"min"^"-1")/(2 × 10^"-9" color(red)(cancel(color(black)("mol·L"^"-1")))) = "15 000 min"^"-1" = "250 s"^"-1"
