= Morphogen response threshold
{title2=$\theta$}
A <morphogen response threshold> is a <concentration> level used to distinguish developmental responses to a <morphogen>. In an idealized threshold model, otherwise equivalent <biological cells> exposed to $c\geq\theta$ take one response and those exposed to $c<\theta$ take another. Two <morphogen response thresholds> can assign three bands: fate A for $c\geq50$, fate B for $20\leq c<50$, and fate C for $c<20$, all in the toy arbitrary <concentration> units. The A/B/C labels specify a response rule, not named cell types or a claim that every <morphogen> produces sharp boundaries. See the positional-information model in Gilbert's reference under <morphogen>.
For the decreasing <morphogen gradient> $c(x)=c_0e^{-x/\ell}$ and $0<\theta<c_0$, solve $c(x_\theta)=\theta$ using the <natural logarithm>:
$$
x_\theta=\ell\ln\!\left(\frac{c_0}{\theta}\right).
$$
With $c_0=100$ and <diffusion length> $\ell=100\,\mu\mathrm m$, the <morphogen response thresholds> $50$ and $20$ give
$$
x_{50}=100\ln2\,\mu\mathrm m\approx69.3147\,\mu\mathrm m,\qquad x_{20}=100\ln5\,\mu\mathrm m\approx160.9438\,\mu\mathrm m.
$$
Thus fate A occupies $0\leq x\leq x_{50}$, fate B occupies $x_{50}<x\leq x_{20}$, and fate C occupies $x>x_{20}$ in this idealization. The equality convention belongs to the response rule. If $\theta=c_0$, the crossing is at the source; if $\theta>c_0$, that <morphogen response threshold> is never reached anywhere in the tissue.
Now double the prescribed source <concentration> from $c_0$ to $2c_0$, keeping $D$, $k$, <diffusion length> $\ell$ and a fixed <morphogen response threshold> $0<\theta<c_0$ unchanged. The whole <morphogen gradient> doubles, but the boundary moves by an additive distance:
$$
\Delta x_\theta=\ell\ln\!\left(\frac{2c_0}{\theta}\right)-\ell\ln\!\left(\frac{c_0}{\theta}\right)=\ell\ln2.
$$
For $\theta=20$, the crossing moves from $160.9438\,\mu\mathrm m$ to $100\ln10\,\mu\mathrm m\approx230.2585\,\mu\mathrm m$, a shift of $69.3147\,\mu\mathrm m$. Every fixed <morphogen response threshold> below the original source <concentration> shifts by this same distance in the toy model. The example compares two <Dirichlet boundary conditions>, $c(0)=100$ and $c(0)=200$; doubling an experimentally supplied source <flux> is a different boundary specification.
\Image[/morphogen-source-strength-boundary-shift.png]
{title=Doubling source <concentration> shifts a toy <morphogen response threshold> boundary}
{description=Profiles with source <concentrations> $100$ and $200$ share <diffusion length> $100\,\mu\mathrm m$. The same <morphogen response threshold> $20$ is crossed at approximately $160.94$ and $230.26\,\mu\mathrm m$. The arrow marks the predicted $69.31\,\mu\mathrm m$ shift. Source strength here means prescribed boundary <concentration>, with all numbers arbitrary and illustrative.}
These calculations are useful because they separate changes in source <concentration>, <diffusion length> and response rule. Their limits are equally concrete: growing tissues change the geometry, a <transient response> can precede <steady state>, molecular fluctuations can blur a boundary, and <signaling feedback> or altered transport can change the <morphogen gradient> or its interpretation. A local instantaneous <morphogen response threshold> need not capture exposure duration, <signaling adaptation> or the receiving <biological cell>'s state. Experiments on cultured human <biological cells> found different responses to the level and rate of change of developmental signals, illustrating why signal dynamics can matter. https://pmc.ncbi.nlm.nih.gov/articles/PMC6398983/[Heemskerk et al., Rapid changes in morphogen concentration control self-organized patterning in human embryonic stem cells, eLife 8 (2019), e40526: primary experiments]{ref} Quantitative measurements of bicoid <protein> and its downstream <gene expression> also make precision a question for experiment, rather than something guaranteed by drawing threshold lines. https://pmc.ncbi.nlm.nih.gov/articles/PMC2253670/[Gregor et al., Probing the limits to positional information, Cell 130 (2007), 153–164: primary measurements]{ref} <Morphogen gradients> provide one mechanism for spatial patterning in <biological development>, alongside other signals and interactions.
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