tooluniverse-enzyme-kinetics

# 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.