Gene regulation
Gene regulation determines when a gene is active and how much RNA or protein it produces. Mathematically, the central idea is a balance between production and removal.
Start with the simplest expression model
Let \(m(t)\) be the concentration of an mRNA. Suppose it is produced at constant rate \(\alpha\) and degraded at rate proportional to its abundance:
\[\boxed{\frac{dm}{dt}=\alpha-\delta m.}\]| Quantity | Meaning |
|---|---|
| \(m\) | mRNA concentration |
| \(\alpha\) | transcription or production rate |
| \(\delta\) | first-order degradation rate constant |
Why is degradation proportional to m?
If each mRNA molecule has approximately the same chance per unit time of being degraded, then having twice as many mRNA molecules gives roughly twice as many degradation events. This leads to
\[\text{degradation rate}=\delta m.\]Find the steady state
At steady state, production balances degradation:
\[0=\alpha-\delta m^*.\]Therefore
\[\boxed{m^*=\frac{\alpha}{\delta}.}\]This gives a direct biological interpretation: stronger production raises the steady level, while faster degradation lowers it.
How mRNA approaches its steady state
The solution of the linear equation is
\[\boxed{m(t)=m^*+[m(0)-m^*]e^{-\delta t}.}\]The difference from steady state decays exponentially.
The timescale is controlled by degradation
The characteristic response time is
\[\tau=\frac1\delta.\]A large \(\delta\) gives a short response time: mRNA turns over quickly and adjusts rapidly. A small \(\delta\) gives a slower response.
The half-life is
\[\boxed{t_{1/2}=\frac{\ln2}{\delta}.}\]Now add regulation
In real cells, production is usually not constant. It depends on regulatory molecules such as transcription factors.
Write
\[\boxed{\frac{dm}{dt}=\alpha\,F(x)-\delta m,}\]where \(x\) is the regulator concentration and \(F(x)\) describes how the regulator changes gene activity.
Activation
An activator increases transcription. A common model is the Hill activation function
\[\boxed{F_{\mathrm{act}}(x)=\frac{x^n}{K^n+x^n}.}\]Here \(K\) is the regulator concentration giving half-maximal activation, and \(n\) controls steepness.
What does K mean?
Set \(x=K\):
\[F_{\mathrm{act}}(K)=\frac{K^n}{K^n+K^n}=\frac12.\]So \(K\) is the input concentration at half-maximal activation.
What does n mean?
The Hill coefficient \(n\) controls how sharply the response changes around \(K\).
For \(n=1\), activation is gradual. Larger \(n\) gives a steeper response and can approximate switch-like regulation.
Repression
A repressor decreases transcription. A standard Hill repression function is
\[\boxed{F_{\mathrm{rep}}(x)=\frac{K^n}{K^n+x^n}=\frac1{1+(x/K)^n}.}\]At low repressor concentration the gene is active; at high repressor concentration production is strongly suppressed.
Activation and repression are complementary in the simplest Hill form
For the same \(K\) and \(n\),
\[F_{\mathrm{act}}(x)+F_{\mathrm{rep}}(x)=1.\]This is mathematically convenient, although real regulatory mechanisms can be more complicated.
Regulation changes the steady state
If the regulator \(x\) is held fixed, then
\[0=\alpha F(x)-\delta m^*.\]Hence
\[\boxed{m^*(x)=\frac{\alpha}{\delta}F(x).}\]The input–output curve of regulation therefore becomes the steady-state expression curve.
Maximum expression level
For an activator, as \(x\to\infty\),
\[F_{\mathrm{act}}(x)\to1.\]Therefore the maximum steady mRNA level is
\[m^*_{\max}=\frac{\alpha}{\delta}.\]The same production–degradation balance that appeared in the unregulated model still sets the scale.
Basal expression
Some genes are never completely off. A simple model can include a basal fraction \(b\):
\[\frac{dm}{dt}=\alpha\left[b+(1-b)F(x)\right]-\delta m,\qquad 0\le b\le1.\]When the regulator gives no activation, transcription still occurs at the fraction \(b\) of the maximal rate.
From mRNA to protein
Gene expression often involves two stages:
A simple two-stage model is
\[\frac{dm}{dt}=\alpha F(x)-\delta_m m,\]\[\frac{dp}{dt}=k_p m-\delta_p p.\]Protein production depends on mRNA, so protein usually responds more slowly than the transcriptional input.
Steady-state protein level
At steady state,
\[m^*=\frac{\alpha F(x)}{\delta_m}.\]Then
\[p^*=\frac{k_p}{\delta_p}m^*=\boxed{\frac{k_p\alpha}{\delta_m\delta_p}F(x)}.\]This shows how transcription, translation and both degradation rates combine to determine the final protein level.
Why delays appear naturally
A change in transcription factor concentration affects mRNA first. Protein then changes because translation acts on the changing mRNA pool.
Even without an explicit time-delay term, a multi-stage gene-expression model can produce delayed protein responses.
Gene switching
At the molecular level, a promoter may switch between states such as
\[\text{OFF}\rightleftharpoons\text{ON}.\]A deterministic model may replace rapid switching by an average transcription function \(F(x)\). A stochastic model can instead represent the switching events explicitly.
This distinction becomes important when molecule numbers are small.
Deterministic versus stochastic regulation
ODE models describe continuous average concentrations. In a single cell, transcription can occur in bursts and gene activation can be random.
Two genetically identical cells can therefore have different mRNA or protein counts even under the same external conditions.
Negative feedback
A gene product can repress its own production. A simple model is
\[\frac{dm}{dt}=\alpha\frac{K^n}{K^n+m^n}-\delta m.\]As \(m\) rises, production falls. This can stabilise expression and reduce sensitivity to perturbations.
Positive feedback
A gene product can also activate its own production:
\[\frac{dm}{dt}=\alpha\frac{m^n}{K^n+m^n}-\delta m.\]Strong positive feedback can create multiple steady states and switch-like behaviour. This idea is developed more fully in gene regulatory network models.
Why nullclines and stability matter
Once gene regulation is written as differential equations, the same dynamical-systems tools used elsewhere in mathematical biology apply: equilibria, Jacobians, eigenvalues, bifurcations and stochastic fluctuations.
This is why gene regulation fits naturally into dynamical-systems biology.
A practical modelling workflow
Define the molecular output being modelled. Write production and degradation separately. Decide which regulator controls production. Choose an activation or repression function with a biological interpretation. Estimate \(K\), \(n\), production and degradation parameters. Analyse steady states and response times. Add mRNA–protein stages, feedback or stochastic switching only when the biological question requires them.