Enzyme kinetics has two parameters and people routinely mix up what they measure. Vmax is a rate, the ceiling an enzyme approaches when it is saturated. Km is not a rate at all. It is a concentration.
v = Vmax × [S] ÷ (Km + [S])
Set the substrate concentration equal to Km and the equation gives Vmax divided by two. That is the definition: Km is the substrate concentration at which the enzyme runs at half speed. With a Vmax of 100 µmol/min and a Km of 10 mM, the Michaelis-Menten calculator gives exactly 50 µmol/min at 10 mM of substrate.
Saturation is asymptotic and expensive
At 5 mM, half of Km, the rate is 33.3 µmol/min, or a third of Vmax. At 1 mM the rate is 9.091, about 9 per cent. And to reach 90 per cent of Vmax, 90 µmol/min, the substrate has to be at 90 mM, which is nine times Km.
So the first half of the performance costs one Km of substrate and the next forty per cent costs eight more. Vmax itself is never actually reached at any finite concentration, which is why it is estimated by fitting rather than measured directly, and why the old practice of eyeballing it off a curve produced such inconsistent values.
What Km tells you
A low Km means the enzyme reaches half speed at a low substrate concentration, which is usually described as high affinity. A high Km means it needs a lot of substrate to get going. Comparing two enzymes acting on the same substrate, the one with the lower Km is the one that works at scarce concentrations.
That matters physiologically because substrate concentrations in a cell are often near Km rather than far above it. An enzyme operating around its Km sits on the steep part of the curve, where its rate responds proportionally to substrate availability. One operating far above Km is saturated and its rate barely responds at all, which makes it a poor control point and a good constant supplier.
The model and its limits
Michaelis-Menten assumes a single substrate, a steady state, no product inhibition and no cooperativity. Enzymes with multiple subunits that influence each other produce sigmoidal curves that this equation cannot describe, and inhibitors change the apparent Km, the apparent Vmax, or both depending on the mechanism.
It remains the right starting point, in the same way the ideal gas law is, because it captures the shape of the thing: a rate that rises steeply, bends over, and approaches a ceiling it never touches.
For growth rather than catalysis, bacterial growth covers the exponential phase and its doubling time, and the Q10 coefficient handles how much faster a biological rate runs when the temperature rises by ten degrees, which is the other question people ask of an enzyme.