Michaelis–Menten Enzyme Kinetics

1D Ordinary Differential Equations

Michaelis–Menten kinetics is a foundational model in biochemical reaction dynamics. It relates substrate concentration to reaction rate using a saturating nonlinearity.

The (simplified) substrate dynamics implemented in the reference script is:

\[ \dot s = \frac{V_{\max}s}{K_m + s} \]

where \(V_{\max}\) is the maximum rate and \(K_m\) is the Michaelis constant.

Reference Implementation

See:

sessions/s01_odes_1d/michaelis_menten.py

The helper plot_michaelis_menten(...) produces:

\(s(t)\) over time
\(v(s)\) with guides at \(s=K_m\) and \(v=V_{\max}/2\)

Render-time Figure

Figure 1: Michaelis–Menten: time series \(s(t)\) and rate curve \(v(s)\).

Exploration

Increase \(V_{\max}\) and observe how the rate curve changes.
Increase \(K_m\) (lower affinity). How does the saturation point shift?

Run Locally

python sessions/s01_odes_1d/michaelis_menten.py

---
title: "Michaelis–Menten Enzyme Kinetics"
subtitle: "1D Ordinary Differential Equations"
format: html
---

Michaelis–Menten kinetics is a foundational model in biochemical reaction dynamics. It relates substrate concentration to reaction rate using a saturating nonlinearity.

The (simplified) substrate dynamics implemented in the reference script is:

$$
\dot s = \frac{V_{\max}s}{K_m + s}
$$

where $V_{\max}$ is the maximum rate and $K_m$ is the Michaelis constant.

## Reference Implementation {#sec-reference}

See:

- `sessions/s01_odes_1d/michaelis_menten.py`

The helper `plot_michaelis_menten(...)` produces:

- $s(t)$ over time
- $v(s)$ with guides at $s=K_m$ and $v=V_{\max}/2$

## Render-time Figure {#sec-render-figure}

```{python}
#| label: fig-mm
#| fig-cap: "Michaelis–Menten: time series $s(t)$ and rate curve $v(s)$."
#| fig-width: 8
#| fig-height: 6
#| echo: false
import matplotlib.pyplot as plt

from sessions.s01_odes_1d.michaelis_menten import plot_michaelis_menten

plot_michaelis_menten(s0=1.0, vmax=1.0, km=0.5, t_show=2)
plt.show()
plt.close()
```

## Exploration {#sec-exploration}

1. Increase $V_{\max}$ and observe how the rate curve changes.
2. Increase $K_m$ (lower affinity). How does the saturation point shift?

## Run Locally {#sec-run-locally}

```bash
python sessions/s01_odes_1d/michaelis_menten.py
```