# Michaelis-Menten Enzyme Kinetics Calculator

> Computes the initial reaction velocity of an enzyme with Michaelis–Menten kinetics, v = Vmax × [S] / (Km + [S]), the fraction of Vmax reached, the substrate concentrations for 50 % and 90 % of Vmax, and optionally a rate table for a list of substrate concentrations.

- Calculator id: `michaelis-menten` · Category: Biology, Earth & Space (`science`) · Tool name: `calculate_michaelis_menten_rate`
- Canonical page: https://tttkmbb.com/science/michaelis-menten · This document: https://tttkmbb.com/science/michaelis-menten.md · JSON definition: https://tttkmbb.com/science/michaelis-menten.json

## Purpose

Computes the initial reaction velocity of an enzyme with Michaelis–Menten kinetics, v = Vmax × [S] / (Km + [S]), the fraction of Vmax reached, the substrate concentrations for 50 % and 90 % of Vmax, and optionally a rate table for a list of substrate concentrations.

**Use when:** You know an enzyme's Vmax and Km and need the rate at a given substrate concentration, or want to tabulate the saturation curve.

**Do not use when:** The enzyme shows cooperativity (Hill equation), inhibition, or substrate depletion over time (integrated rate law), or you need to fit Vmax and Km from data (use linear-regression on a Lineweaver–Burk transform).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `vmax` | number |  | required | Maximum reaction velocity at saturating substrate, in any rate unit (e.g. µmol/min or µM/s); the rate output uses the same unit. (> 0, max 1000000000000) |
| `km` | number |  | required | Michaelis constant: substrate concentration at half Vmax, in the same concentration unit as substrate_concentration. (> 0, max 1000000000000) |
| `substrate_concentration` | number |  | required | Substrate concentration in the same unit as Km. (min 0, max 1000000000000) |
| `substrate_list` | number_list |  | optional | Optional list of up to 20 substrate concentrations (same unit as Km) to tabulate rates. |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `reaction_rate` | number |  | Vmax × [S] / (Km + [S]), in the unit of vmax. |
| `fraction_of_vmax_percent` | number | % | 100 × [S] / (Km + [S]). |
| `substrate_for_half_vmax` | number |  | Equals Km. |
| `substrate_for_90_percent_vmax` | number |  | 9 × Km. |
| `reciprocal_substrate` | number |  | 1 / substrate_concentration (when [S] > 0). |
| `reciprocal_rate` | number |  | 1 / reaction_rate (when [S] > 0). |
| `table` | list |  | Rows of substrate, rate and percent_of_vmax for each value of substrate_list. |

## Formula

`v = Vmax × [S] / (Km + [S]); fraction = [S] / (Km + [S]); [S] at 50 % Vmax = Km; [S] at 90 % Vmax = 9 Km; Lineweaver-Burk: 1/v = (Km / Vmax) × 1/[S] + 1/Vmax`

Michaelis–Menten (1913) steady-state model for a single-substrate enzyme without cooperativity or inhibition; v is the initial velocity before significant substrate depletion or product inhibition. Reaching 99 % of Vmax needs [S] = 99 Km.

## Data Sources

- Wikipedia – Michaelis–Menten kinetics — https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics (reference, retrieved 2026-09-24)
- Johnson KA & Goody RS (2011) The original Michaelis constant: translation of the 1913 Michaelis-Menten paper, Biochemistry 50(39):8264-8269 — https://pubmed.ncbi.nlm.nih.gov/21888353/ (peer_reviewed, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/michaelis-menten?vmax=…&km=…&substrate_concentration=…`
- `POST https://tttkmbb.com/api/v1/calculate/michaelis-menten` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/michaelis-menten · OpenAPI operationId `calculate_michaelis_menten_rate` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "michaelis-menten", "inputs": {…}}`

## Example

- Vmax 100, Km 5, [S] 5 with table: inputs `{"vmax":100,"km":5,"substrate_concentration":5,"substrate_list":[1,2,5,10,20,50]}` → `{"reaction_rate":50,"fraction_of_vmax_percent":50,"substrate_for_half_vmax":5,"substrate_for_90_percent_vmax":45,"table":[{"substrate":1,"rate":16.6667},{"substrate":2,"rate":28.5714},{"substrate":5,"rate":50},{"substrate":10,"rate":66.6667},{"substrate":20,"rate":80},{"substrate":50,"rate":90.9091}]}`
- Vmax 100, Km 5, [S] 20: inputs `{"vmax":100,"km":5,"substrate_concentration":20}` → `{"reaction_rate":80,"fraction_of_vmax_percent":80,"reciprocal_substrate":0.05,"reciprocal_rate":0.0125}`

```
GET https://tttkmbb.com/api/v1/calculate/michaelis-menten?vmax=100&km=5&substrate_concentration=5&substrate_list=1%2C2%2C5%2C10%2C20%2C50
```

## Limitations

The enzyme shows cooperativity (Hill equation), inhibition, or substrate depletion over time (integrated rate law), or you need to fit Vmax and Km from data (use linear-regression on a Lineweaver–Burk transform). Michaelis–Menten (1913) steady-state model for a single-substrate enzyme without cooperativity or inhibition; v is the initial velocity before significant substrate depletion or product inhibition. Reaching 99 % of Vmax needs [S] = 99 Km. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**What does Km mean?**

The substrate concentration at which the rate is half of Vmax; a lower Km means the enzyme reaches half-saturation at lower substrate concentration (often read as higher apparent affinity).

**Which units should I use?**

Any, as long as Km and [S] share a concentration unit; the rate comes out in the unit of Vmax (µmol/min, µM/s, absorbance/min).

**How do I get Vmax and Km from experimental data?**

Fit v against [S] by non-linear regression, or use the Lineweaver–Burk transform (1/v vs 1/[S]: slope Km/Vmax, intercept 1/Vmax) with the linear-regression calculator, keeping in mind that the double-reciprocal plot amplifies errors at low [S].

## Related

- [Arrhenius Equation Calculator](https://tttkmbb.com/chemistry/arrhenius-equation.md) — Temperature dependence of the rate constant.
- [Molarity Calculator](https://tttkmbb.com/chemistry/molarity.md) — Prepare substrate solutions at the required concentration.
- [Linear Regression Calculator](https://tttkmbb.com/statistics/linear-regression.md) — Fit Vmax and Km from a Lineweaver-Burk plot.