Enzyme Kinetics & Regulation
A board-focused lesson on Michaelis–Menten kinetics (Km, Vmax, Lineweaver–Burk), the three inhibitor types and how to classify them from a double-reciprocal plot, and enzyme regulation — anchored by classic Step 1 vignettes (ethylene glycol/fomepizole, hexokinase vs glucokinase).
Michaelis–Menten: Km, Vmax, and the double-reciprocal plot
Most enzymes follow Michaelis–Menten kinetics: reaction velocity rises hyperbolically with substrate concentration until the enzyme saturates at Vmax. Two constants define the enzyme. Km is the substrate concentration at ½ Vmax and is inversely related to substrate affinity — a low Km means high affinity. Vmax is reached only at saturating substrate and is directly proportional to enzyme concentration ([E]).
Because the hyperbola makes Vmax hard to read directly, the boards use the Lineweaver–Burk (double-reciprocal) plot of 1/V versus 1/[S], which linearizes the data: the y-intercept = 1/Vmax, the x-intercept = −1/Km, and the slope = Km/Vmax. Seeing where the inhibited and uninhibited lines cross is the single fastest way to classify an inhibitor.
Enzymes with cooperativity (allosteric enzymes; hemoglobin binding O₂) break this model, giving a sigmoidal curve instead of a hyperbola.
- Km = [S] at ½ Vmax; ↓Km = ↑affinity, ↑Km = ↓affinity
- Vmax ∝ enzyme concentration (Vmax = k_cat × [E])
- k_cat/Km = catalytic efficiency (higher = better enzyme)
- Lineweaver–Burk: y-intercept = 1/Vmax, x-intercept = −1/Km, slope = Km/Vmax
- Competitive inhibitor → ↑Km, Vmax unchanged; overcome by ↑[S]; lines cross on the y-axis
- Noncompetitive inhibitor → ↓Vmax, Km unchanged; NOT overcome by ↑[S]; lines cross on the x-axis
- Uncompetitive inhibitor → ↓Km and ↓Vmax together (parallel lines); binds only the enzyme–substrate complex
- Sigmoidal curve = positive cooperativity (Hill coefficient >1), e.g., hemoglobin, phosphofructokinase-1
- Irreversible inhibitors kinetically resemble noncompetitive (↓ functional Vmax, Km unchanged) and need new enzyme synthesis to recover
Enzyme Inhibition — Board Comparison
| Feature | Competitive | Noncompetitive | Uncompetitive |
|---|---|---|---|
| Resembles substrate | Yes | No | No |
| Binds | Active site | Allosteric site | Enzyme–substrate complex |
| Km | ↑ | Unchanged | ↓ |
| Vmax | Unchanged | ↓ | ↓ |
| Overcome by ↑[S]? | Yes | No | No |
| Lineweaver–Burk | Cross on y-axis | Cross on x-axis | Parallel lines |
| Classic examples | Statins (HMG-CoA reductase), methotrexate (DHFR), fomepizole (alcohol dehydrogenase) | NNRTIs — efavirenz, nevirapine (reverse transcriptase) | Lithium (inositol monophosphatase) |
![Lineweaver–Burk double-reciprocal plot of 1/V versus 1/[S] as a straight line, with y-intercept equal to 1/Vmax, x-intercept equal to −1/Km, and slope equal to Km/Vmax.](https://upload.wikimedia.org/wikipedia/commons/7/70/Lineweaver-Burke_plot.svg)
Vignette: A 34-year-old man is brought in confused after drinking antifreeze. He is tachypneic; ABG shows a high anion-gap metabolic acidosis, the osmolar gap is elevated, and urine shows envelope-shaped calcium oxalate crystals. Creatinine is rising.
Diagnosis: Ethylene glycol poisoning — metabolized by alcohol dehydrogenase to glycolic and oxalic acid → anion-gap acidosis plus oxalate nephropathy/AKI. Methanol gives a similar picture but with visual loss from formic acid.
Next best step: Give fomepizole, a competitive inhibitor of alcohol dehydrogenase, to block conversion to the toxic metabolites; add hemodialysis for severe acidosis, renal failure, or high levels. Ethanol is the backup antidote — it out-competes the toxin for the same active site.
This is competitive inhibition used therapeutically: flooding the enzyme with the preferred substrate (ethanol/fomepizole) slows metabolism of the toxic one.
Vignette: In vitro, adding Drug X to an enzyme raises the Km from 2 to 8 mM while Vmax is unchanged. On a Lineweaver–Burk plot, the treated and untreated lines intersect on the y-axis. Raising substrate concentration restores the original velocity.
Question — what type of inhibition is this?
Answer: Competitive inhibition. Same Vmax → same y-intercept (1/Vmax) → lines cross on the y-axis. The higher Km means more substrate is needed to reach half-max velocity, and because drug and substrate compete for the active site, excess substrate overcomes it.
Contrast: If instead Vmax fell and Km stayed the same (lines crossing on the x-axis, unchanged −1/Km), that is noncompetitive inhibition and cannot be reversed by adding substrate. A prototypical competitive drug is a statin acting on HMG-CoA reductase.
The intercept that doesn't move tells you what the inhibitor spares:
- Competitive → cross on the y-axis. Both lines share the y-intercept 1/Vmax, so Vmax is unchanged. (Competitive = Come-back: it can be overcome by adding substrate.)
- Noncompetitive → cross on the x-axis. Both share the x-intercept −1/Km, so Km is unchanged while Vmax falls.
- Uncompetitive → parallel lines. Km and Vmax fall together, so the slope (Km/Vmax) is unchanged.
And for affinity: low Km = high affinity — it takes only a little substrate to get the enzyme halfway to Vmax, so the enzyme must "love" its substrate.
Regulation: Allostery, Covalent Modification, Cooperativity
Cells tune enzyme activity through several mechanisms the boards love to test.
Allosteric regulation: effectors bind sites away from the active site. Feedback inhibition is the classic form — the end product inhibits an early, committed step (e.g., heme inhibits δ-ALA synthase in heme synthesis; the textbook allosteric example is CTP inhibiting aspartate transcarbamoylase in pyrimidine synthesis). Allosteric enzymes show sigmoidal kinetics.
Covalent modification: reversible phosphorylation/dephosphorylation (protein kinases vs phosphatases) switches enzymes on or off — e.g., in glycogen metabolism, phosphorylation activates glycogen phosphorylase but inactivates glycogen synthase.
Cooperativity: substrate binding at one site raises affinity at the others, producing the sigmoidal curve and a Hill coefficient >1 (hemoglobin, PFK-1).
Irreversible/covalent inhibitors (e.g., aspirin on COX, penicillin on transpeptidase, omeprazole on the H⁺/K⁺-ATPase) lower functional Vmax and require new enzyme synthesis to recover.
Hexokinase vs Glucokinase — The Km Classic
| Feature | Hexokinase | Glucokinase |
|---|---|---|
| Tissue | Most tissues | Liver, pancreatic β-cells |
| Km | Low (high affinity) | High (low affinity) |
| Vmax | Low | High |
| Inhibited by glucose-6-phosphate | Yes | No |
| Induced by insulin | No | Yes |
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