= Morphogen gradient
{title2=$c(x)$}
A <morphogen gradient> is a spatially varying <concentration> profile of a <morphogen>. The profile $c(x)$ specifies the local amount; its mathematical <gradient>, $c'(x)$ in one dimension, specifies how rapidly that amount changes with position. In the simplest positional interpretation, a <biological cell> responds to its local <concentration>, without having to measure the <gradient> across itself. Production, transport and removal together determine a <morphogen gradient>.
For a worked example, take a stationary one-dimensional tissue occupying $x\geq0$, constant <diffusion coefficient> $D>0$, and uniform <first-order reaction> removal at rate $k>0$. At <steady state>, this specialization of the <morphogen reaction-diffusion equation> gives the <ordinary differential equation> and <boundary conditions>
$$
D\frac{d^2c}{dx^2}-kc=0,\qquad c(0)=c_0,\qquad \lim_{x\to\infty}c(x)=0.
$$
Here $D$ has units of length squared per time, $k$ has units of inverse time, and $c_0$ is a prescribed source <concentration>. The <Dirichlet boundary condition> fixes the <concentration> at $x=0$. It does not prescribe a source <flux>; any required <flux> follows from the solution. These assumptions define a toy <morphogen gradient>, not a universal account of transport during <biological development>.
Define the <diffusion length> $\ell=\sqrt{D/k}$. Substitution into the <ordinary differential equation> shows that its general solution is $A e^{x/\ell}+B e^{-x/\ell}$. The vanishing <boundary condition> at infinity forces $A=0$, and the source <Dirichlet boundary condition> forces $B=c_0$. Hence
$$
c(x)=c_0e^{-x/\ell},\qquad \ell=\sqrt{D/k}.
$$
Over every additional <diffusion length>, the <concentration> falls by the same factor $e^{-1}$. Faster <diffusion> increases $\ell$; faster removal decreases $\ell$. This makes the <diffusion length> a useful bridge between microscopic transport and the spatial reach of a <morphogen gradient>.
Choose $c_0=100$ arbitrary <concentration> units and $\ell=100\,\mu\mathrm m$, where $1\,\mu\mathrm m=10^{-6}\,\mathrm m$. Then $c(x)=100e^{-x/(100\,\mu\mathrm m)}$: at $x=0$, $100\,\mu\mathrm m$ and $200\,\mu\mathrm m$, the <concentrations> are $100$, approximately $36.79$ and approximately $13.53$. The following diagram combines this <morphogen gradient> with the idealized responses calculated under <morphogen response threshold>.
\Image[/morphogen-gradient-and-thresholds.png]
{title=A toy <morphogen gradient> and three idealized developmental bands}
{description=The <steady state> profile uses $c_0=100$ arbitrary <concentration> units and <diffusion length> $100\,\mu\mathrm m$. <Morphogen response thresholds> at $50$ and $20$ place the boundaries at approximately $69.31$ and $160.94\,\mu\mathrm m$. The lower strip shows assigned fates A, B and C, not measured tissues. The plotted domain ends at $350\,\mu\mathrm m$; the mathematical domain continues to infinity.}
Back to article page
