Why Enzymes Max Out

Add more sugar and an enzyme works faster, but only up to a point: every enzyme molecule ends up busy. The reaction rate approaches a ceiling it never quite reaches. That ceiling is a limit at infinity, a horizontal asymptote.

How it works

Limits at infinity

In one sentence

$\lim_{x\to\infty}f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ for all sufficiently large $x$ ; then $y = L$ is a horizontal asymptote. For a rational function, divide numerator and denominator by the highest power of $x$ .

General formula

{\htmlData{sym=0}{v(S)}} = \frac{{\htmlData{sym=1}{V_{\max}}}\,{\htmlData{sym=3}{S}}}{{\htmlData{sym=2}{K_m}} + {\htmlData{sym=3}{S}}}, \qquad \lim_{{\htmlData{sym=3}{S}}\to\infty} {\htmlData{sym=0}{v(S)}} = {\htmlData{sym=1}{V_{\max}}}

For huge $S$ the $K_m$ in the denominator is negligible, so $v \approx V_{\max}S/S = V_{\max}$ .

What the symbols mean

$v(S)$
Reaction rate
How fast product forms at substrate concentration S.
$V_{\max}$
Top speed
The rate when every enzyme is busy: the asymptote.
$K_m$
Michaelis constant
The concentration at which the rate is half of Vmax.
$S$
Substrate concentration
How much of the molecule the enzyme acts on is present.

At low substrate concentration $S$ , $v \approx \frac{V_{\max}}{K_m}S$ : the rate grows in proportion to $S$ .
For large $S$ , divide top and bottom by $S$ : $v = \frac{V_{\max}}{K_m/S + 1} \to V_{\max}$ .
So the line $v = V_{\max}$ is a horizontal asymptote: the enzyme's top speed.
$K_m$ is the concentration where the rate is half the maximum, a measure of how tightly the enzyme grabs its substrate.

Learn the math

Limits at InfinityCalc I · Lesson 10, Limits