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; 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:

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.