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

bash
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; cite nonlinear_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:

bash
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.