# mRNA expression

> How long does an mRNA dose make protein? Translation and decay kinetics: when protein peaks, how much is made in total, and what repeated doses do.

Page: https://sciencehub.sh/sciencehub-team/mrna-expression

- Project: [sciencehub-team/mrna-expression](https://sciencehub.sh/sciencehub-team/mrna-expression.md)
- Authors: ScienceHub
- Subjects: molecular-biology, pharmacology
- Topics: mrna, gene-expression, kinetics, vaccines
- 0 stars, 0 forks
- DOI: https://doi.org/10.83371/sh.2610.n159w
- Source: https://sciencehub.sh/sciencehub-team/mrna-expression/source
- Updated 13 Oct 2026

## Notebook

Text the notebook's authors wrote, its cells as Python, and the outputs it produced when it last ran in a browser and was saved.

The script header: the dependencies the notebook declares (PEP 723) and the app it defines.

```python
# /// script
# requires-python = ">=3.12"
# dependencies = ["marimo>=0.25.1", "numpy", "matplotlib"]
# ///

import marimo

__generated_with = "0.25.1"
app = marimo.App(width="medium")
```

### How long does an mRNA dose make protein?

An mRNA therapeutic or vaccine delivers instructions, not the protein
itself. Once the mRNA reaches the cytoplasm, ribosomes translate it
until it is degraded, and the protein it encodes is degraded in turn.
The simplest model that describes single-cell transfection data has two
first-order steps (Leonhardt et al. 2014):

$$\frac{dm}{dt} = -\delta\, m, \qquad \frac{dp}{dt} = k\, m - \beta\, p$$

with mRNA $m$, protein $p$, translation rate $k$, and degradation rates
$\delta = \ln 2 / t_{1/2}^{\text{mRNA}}$ and
$\beta = \ln 2 / t_{1/2}^{\text{protein}}$. Starting from $m_0$ at
$t = 0$ the protein level is

$$p(t) = \frac{k\, m_0}{\delta - \beta}\left(e^{-\beta t} - e^{-\delta t}\right).$$

```python
import marimo as mo
import numpy as np
import matplotlib.pyplot as plt
```

```python
mrna_half_life = mo.ui.slider(0.5, 48, step=0.5, value=10, label="mRNA half-life (h)")
protein_half_life = mo.ui.slider(1, 96, step=1, value=26, label="Protein half-life (h)")
mo.vstack([mrna_half_life, protein_half_life])
```

```python
def protein(t, mrna_half, protein_half, k=1.0, m0=1.0):
    """Protein level after one dose at t = 0, in units of k * m0 * hours."""
    delta, beta = np.log(2) / mrna_half, np.log(2) / protein_half
    t = np.asarray(t, dtype=float)
    if np.isclose(delta, beta):
        level = k * m0 * t * np.exp(-beta * t)
    else:
        level = k * m0 / (delta - beta) * (np.exp(-beta * t) - np.exp(-delta * t))
    return np.where(t >= 0, level, 0.0)

def peak_time(mrna_half, protein_half):
    delta, beta = np.log(2) / mrna_half, np.log(2) / protein_half
    if np.isclose(delta, beta):
        return 1 / beta
    return np.log(delta / beta) / (delta - beta)
```

```python
hours = np.linspace(0, 168, 800)
level = protein(hours, mrna_half_life.value, protein_half_life.value)
mrna = np.exp(-np.log(2) / mrna_half_life.value * hours)
t_peak = peak_time(mrna_half_life.value, protein_half_life.value)
exposure = mrna_half_life.value * protein_half_life.value / np.log(2) ** 2

fig, ax = plt.subplots(figsize=(7, 3.6))
ax.plot(hours / 24, level / level.max(), color="#0969da", lw=2, label="protein")
ax.plot(hours / 24, mrna, color="#9a6700", lw=1.5, ls="--", label="mRNA")
ax.axvline(t_peak / 24, color="#57606a", lw=0.8)
ax.set(xlabel="days after the dose", ylabel="fraction of peak", xlim=(0, 7), ylim=(0, 1.05))
ax.legend(frameon=False)
fig.tight_layout()

mo.vstack(
    [
        fig,
        mo.md(
            f"Protein peaks **{t_peak:.1f} hours** after the dose. Total protein made "
            f"over the whole course is $k m_0 / (\\delta\\beta)$ = **{exposure:,.0f}** "
            "in units of $k m_0$ hours squared."
        ),
    ]
)
```

Output:

```text
Protein peaks 22.4 hours after the dose. Total protein made over the whole course is ||(k m_0 / (\delta\beta)||) = 541 in units of ||(k m_0||) hours squared.
```

#### Why mRNA stability matters

The area under the protein curve is $k m_0 / (\delta \beta)$: it is
proportional to the mRNA half-life times the protein half-life. Doubling
how long the mRNA survives doubles the total protein made from the same
dose, without changing the dose. Much of the engineering of therapeutic
mRNA (modified nucleosides such as N1-methylpseudouridine, optimised
untranslated regions, the cap structure) works on $k$ and $\delta$.
The curves below keep the protein half-life you chose and vary only the
mRNA half-life.

```python
days = np.linspace(0, 10, 800)
fig2, ax2 = plt.subplots(figsize=(7, 3.4))
for _half, _shade in zip((2, 6, 12, 24), ("#d0d7de", "#8c959f", "#57606a", "#0969da")):
    ax2.plot(
        days,
        protein(days * 24, _half, protein_half_life.value),
        color=_shade,
        lw=2,
        label=f"mRNA half-life {_half} h",
    )
ax2.set(xlabel="days after the dose", ylabel="protein (k m₀ h)")
ax2.legend(frameon=False)
fig2.tight_layout()
fig2
```

![Figure (image/png)](https://pppqewd2askr4srw3c4biffsvi.notebooks.sciencehubusercontent.com/_outputs/eyJuYW1lc3BhY2VJZCI6Im5zLXdicjZya2tkOWZqMG1neGUiLCJwcm9qZWN0SWQiOiJwcm9qLXJ6dGEwZnlkZ3FxOGtkNTEiLCJub3RlYm9va0lkIjoibmItYWQ5bmMyODdzMXQ0MGFheiIsImhhc2giOiI1MTQ1YmE4MWFhOTQ1MWI0MzFlMzBlNWY3NTU4M2JhMiJ9.0.YO0sQEtMT41-bA67UwwGsixZlmeE6ghvuYI84O_BHIs)

#### Repeated doses

The model is linear, so repeated doses add up. The protein level after
a schedule is the sum of the single-dose curves, each shifted to its
dose time. Choose an interval and a number of doses to see whether
expression stays above a threshold between doses.

```python
interval = mo.ui.slider(6, 96, step=6, value=24, label="Dose every (h)")
doses = mo.ui.slider(1, 10, step=1, value=5, label="Number of doses")
threshold = mo.ui.slider(0.05, 1.0, step=0.05, value=0.5, label="Threshold (fraction of one-dose peak)")
mo.vstack([mo.hstack([interval, doses], justify="start"), threshold])
```

```python
span = interval.value * doses.value + 96
t = np.linspace(0, span, 1600)
single_peak = protein(np.linspace(0, 400, 4000), mrna_half_life.value, protein_half_life.value).max()
total = sum(
    protein(t - n * interval.value, mrna_half_life.value, protein_half_life.value)
    for n in range(doses.value)
) / single_peak
covered = t[(t > 0) & (t < interval.value * doses.value)]
above = np.mean(np.interp(covered, t, total) >= threshold.value) if covered.size else 0.0

fig3, ax3 = plt.subplots(figsize=(7, 3.4))
ax3.plot(t / 24, total, color="#0969da", lw=2)
ax3.axhline(threshold.value, color="#cf222e", lw=0.8, ls="--")
for _n in range(doses.value):
    ax3.axvline(_n * interval.value / 24, color="#d0d7de", lw=0.8)
ax3.set(xlabel="days", ylabel="protein / one-dose peak")
fig3.tight_layout()
mo.vstack(
    [
        fig3,
        mo.md(
            f"During dosing, expression is above the threshold **{above:.0%}** of the time. "
            f"The highest level reached is **{total.max():.2f}** times a single dose's peak."
        ),
    ]
)
```

Output:

```text
During dosing, expression is above the threshold 96% of the time. The highest level reached is 2.77 times a single dose's peak.
```

---
**Model and limits.** One compartment, first-order kinetics, all mRNA
released at $t = 0$. Real delivery is spread over hours, cells differ
widely in all three rates, and the immune response to a vaccine is not
modelled. Levels are relative; set $k$ and $m_0$ from your own data to
get absolute amounts. The default protein half-life, 26 hours, is the
value reported for EGFP in mammalian cells (Corish and Tyler-Smith 1999).
**References.** A. Leonhardt et al., *Nanomedicine: NBM* 10 (2014) 679;
P. Corish and C. Tyler-Smith, *Protein Eng.* 12 (1999) 1035;
K. Karikó et al., *Mol. Ther.* 16 (2008) 1833.
Fork it to fit the model to your own time-course data.

## Versions

- [ver_01M4SH2J4KJZJE7G8JZQK7HWBH](https://sciencehub.sh/sciencehub-team/mrna-expression/history/ver_01M4SH2J4KJZJE7G8JZQK7HWBH), 13 Oct 2026: Edited (current)
- [ver_01M4SGES848X1P2SR8DNY3E037](https://sciencehub.sh/sciencehub-team/mrna-expression/history/ver_01M4SGES848X1P2SR8DNY3E037), 13 Oct 2026: Created

## Cite

ScienceHub (2026). mRNA expression \[Computer software\]. ScienceHub. https://doi.org/10.83371/sh.2610.n159w

```bibtex
@software{mrna_expression_2026,
  author = {ScienceHub},
  title = {mRNA expression},
  year = {2026},
  publisher = {ScienceHub},
  doi = {10.83371/sh.2610.n159w}
}
```