mims-harvard Enzyme Kinetics (Michaelis-Menten)
Turn substrate concentration vs initial velocity data into Km, Vmax, kcat, and catalytic efficiency — and classify an inhibitor's mechanism.
The Michaelis-Menten model: v = Vmax·[S] / (Km + [S]).
When to use this
- You measured initial reaction rates at several substrate concentrations.
- You need Km (substrate affinity), Vmax, kcat (turnover number), or kcat/Km.
- You have ±inhibitor velocity data and want to classify the inhibition mode + Ki.
For published kinetic constants (someone else's Km/kcat), use the BRENDA tools instead — this skill is for analyzing your own measured data.
Step 1 — Prepare the data
| Issue | What to do |
|---|---|
| Initial velocities, not endpoints | v must be the initial rate (linear phase, <10% substrate consumed). Endpoint or plateaued rates give a wrong Km/Vmax. |
| Substrate range must span Km | Include [S] both well below and well above Km (ideally ~0.2×Km to ~5×Km). Points only above Km can't define Km; only below can't define Vmax. |
| Units — be consistent | One [S] unit (mM, µM) → Km comes back in that unit. One velocity unit. Keep them fixed. |
| For kcat you need [E] | kcat = Vmax / [E]total. The tool's "catalytic_efficiency" is Vmax/Km on the velocity scale; to get true kcat (per-second turnover) and kcat/Km, divide Vmax by the molar enzyme concentration yourself. |
| ≥5–7 points | Few points → unstable fit. Spread them across the range, not clustered. |
Step 2 — Fit Michaelis-Menten
tu run EnzymeKinetics_calculate '{"operation":"michaelis_menten",
"substrate_concs":[0.1,0.25,0.5,1,2,5,10],
"velocities":[8.5,18,32,52,72,90,98]}'
Returns a nonlinear_fit block (Vmax, Km, R2, SSE) — use these as the answer, a lineweaver_burk block (for reference only), catalytic_efficiency (Vmax/Km), and predicted_velocities + residuals.
Prefer the nonlinear fit, not Lineweaver-Burk. The double-reciprocal (Lineweaver-Burk) linearization distorts error (it over-weights low-
[S]points) and is only for visualization/sanity — never report its Km/Vmax as the final values. The tool gives both; citenonlinear_fit.
scripts/fit_michaelis_menten.py does the same nonlinear fit from a CSV and converts Vmax→kcat→kcat/Km when you supply the enzyme concentration.
Step 3 — Interpret
| Parameter | Meaning | Notes |
|---|---|---|
| Km | Substrate concentration at ½Vmax — apparent affinity (lower Km = tighter binding / higher affinity). | In the same units as [S]. Must lie inside your tested range to be trustworthy. |
| Vmax | Maximum velocity at saturating substrate. | Depends on [E]; not an intrinsic enzyme property. |
| kcat | Turnover number = Vmax/[E] (per second). | Requires the molar enzyme concentration; intrinsic to the enzyme. |
| kcat/Km | Catalytic efficiency / specificity constant. | The best single metric to compare enzymes or substrates; near ~10⁸–10⁹ M⁻¹s⁻¹ is diffusion-limited ("catalytically perfect"). |
| R² / SSE | Fit quality. | R²≥0.98 good; check residuals for systematic curvature (a pattern, not random scatter, means MM is the wrong model). |
Step 4 — Inhibition mechanism
Provide velocities ±inhibitor to classify the mode:
tu run EnzymeKinetics_calculate '{"operation":"inhibition",
"substrate_concs":[...],
"velocities_no_inhibitor":[...],
"velocities_with_inhibitor":[...],
"inhibitor_conc":5, "inhibition_type":"competitive"}'
| Mechanism | Effect on apparent Km | Effect on Vmax | Signature |
|---|---|---|---|
| Competitive | ↑ (increases) | unchanged | inhibitor competes at the active site; beatable by more substrate |
| Uncompetitive | ↓ (decreases) | ↓ | inhibitor binds only the ES complex |
| Non-competitive (mixed) | ~unchanged (pure) / changes (mixed) | ↓ | binds enzyme and ES; not relieved by substrate |
Ki is the inhibition constant (lower = more potent inhibitor). Decide the mechanism from how Km and Vmax shift, not from a single Lineweaver-Burk eyeball.
Step 5 — Gotchas (state these)
- Substrate inhibition (velocity rises then falls at high
[S]) breaks MM — the fit will show systematic residuals; flag it instead of forcing one Km. - Km outside the tested range → unreliable; widen
[S]. - kcat without [E] is impossible — don't report a turnover number if you only fit velocities.
- Lineweaver-Burk for final numbers is the classic error — it's for a quick plot, not the reported Km/Vmax.
Honest limitations
- MM assumes a single substrate, initial-rate, steady-state, one active site. Allosteric (sigmoidal) enzymes need the Hill equation; multi-substrate enzymes need their own formalism.
- Parameters are only as good as the substrate range and the initial-rate measurement.
Related skills
tooluniverse-dose-response— IC50/EC50 (the Hill/4PL sibling for concentration-response).tooluniverse-statistical-modeling— general nonlinear regression and model comparison.- BRENDA tools — look up published enzyme kinetic constants.