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A morphogen gradient is a spatially varying concentration profile of a morphogen. The profile $c(x)$ specifies the local amount; its mathematical gradient, $c^{\prime }(x)$ 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 $x\ge 0$, 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 conditionsHere $D$ 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 $B=c_0$. HenceOver 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.

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 $c\ge \theta$ take one response and those exposed to $c<\theta$ take another. Two morphogen response thresholds can assign three bands: fate A for $c\ge 50$, fate B for $20\le c<50$, 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 $c(x)=c_0{e}^{-x/\ell }$ and $0<\theta <c_0$, solve $c(x_{\theta })=\theta$ using the natural logarithm:With $c_0=100$ and diffusion length $\ell =100\mu \mathrm{m}$, the morphogen response thresholds $50$ and $20$ giveThus fate A occupies $0\le x\le x_{50}$, fate B occupies $x_{50}<x\le x_{20}$, 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:For $\theta =20$, the crossing moves from $160.9438\mu \mathrm{m}$ to $100\ln 10\mu \mathrm{m}\approx 230.2585\mu \mathrm{m}$, 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.

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. ^[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. ^[ref] Morphogen gradients provide one mechanism for spatial patterning in biological development, alongside other signals and interactions.

At $227^{\circ }\mathrm{C}\simeq 500\mathrm{K}$, $1000/T\simeq 2$. Reading the carbon line gives $D_C\sim 2\times 10^{-15}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ to graphical accuracy. With $t=60\mathrm{s}$, the characteristic diffusion length isIf “distance” means a one-coordinate root-mean-square displacement, use $\sqrt{2Dt}\sim 0.49\mu \mathrm{m}$; the three-dimensional radial root-mean-square is $\sqrt{6Dt}\sim 0.85\mu \mathrm{m}$. The graph does not justify precision between these conventions. Equal diffusion distance in equal time requires equal diffusion coefficients. The copper line reaches this diffusivity around $1000/T\simeq 0.96$, giving $T_{\mathrm{Cu}}\sim 1.0\times 10^3\mathrm{K}$, about $770^{\circ }\mathrm{C}$. Carbon moves interstitially through ferrite, whereas copper self-diffusion is substitutional and vacancy-mediated, with a much higher effective activation barrier.

Use a planar saline Stefan problem with upward coordinate $z$, the initial contact at $z=0$, and phase boundaries $z=a(t)$ and $z=b(t)$ enclosing pure ice, where $a<b$. Fresh liquid occupies $z>b$ and brine occupies $z<a$. We assume negligible bulk flow, equal constant mass density $\rho$, specific heat capacity $c_p$ and thermal diffusivity $\kappa$ in all regions, and zero salt content and salt transport in the ice. Equal densities remove phase-change volume flow; suppressing convection isolates the molecular-transport mechanism. Different material properties would change the numerical coefficients. Set $K=\rho c_p\kappa$ for the common thermal conductivity and $\Delta T=T_m-T_{\infty }>0$. Denote the brine salt diffusion coefficient by $D$.

Use a linear ice liquidus, $T_L(C)=T_m-mC$, with $m>0$ and $C$ the salt mass fraction. Assume the initial brine is a stable liquid, $T_{\infty }>T_L(C_0)$, and all concentrations considered are below the eutectic composition. Thus $R=m{C}_0/\Delta T>1$. These assumptions exclude independent nucleation in a supercooled brine or an eutectic system event, neither of which is specified by the initial data alone. Start with a negligible seed of ice and use local phase equilibrium without interfacial kinetics or curvature. Temperature differences may be in Celsius, since only differences enter this calculation.

WriteThe paper calls $S$ a Stefan number; it is the latent-to-sensible heat ratio, reciprocal to another common Stefan number convention. At the lower phase boundary, define $T_i=T_L(C_i)$ and letThe liquidus relation is $\vartheta =Rc$.

A similarity solution gives an explicit calculation of the two positions. Putand use $u=z/(2\sqrt{\kappa t})$, $v=z/(2\sqrt{Dt})$. The temperatures and salinities areThese error function and complementary error function profiles satisfy the heat equation and the salt diffusion equation, their far-field conditions, and the interfacial temperature conditions. The fresh-water temperature is constant because the interface and the far field are both at $T_m$.

Salt conservation at the moving lower phase boundary requiresFor upward motion this is dilution by melting salt-free ice, rather than salt rejection by freezing brine. Substitution gives the dilution function for a melting saline Stefan front:For $A>0$, $0<c<1$, so the interfacial brine is fresher than the remote liquid.

The Stefan condition at the upper boundary is $\rho L\dot{b}=K{T}_{s,z}(b)$, since the fresh liquid has no temperature gradient. At the lower boundary it isThese signs follow from the jump in enthalpy: the upper front freezes liquid while a positive $\dot{a}$ melts solid. With $E=\mathrm{erf}B-\mathrm{erf}\delta$, they reduce toTogether with $\delta =ϵA$, these equations determine $A,B,\vartheta$, and hence both positions, without discarding the salt-diffusion correction.

For the stated large latent-to-sensible heat ratio, the layer is thin relative to the thermal diffusion length. Expanding the thermal equations for $B,\delta \ll 1$ givesConsequently a useful leading calculation iswhere the positive $A$ is obtained fromThe left side decreases with $A\ge 0$ while the right side increases, and their values at zero and infinity guarantee a unique positive root for large $S$. Thus this leading calculation includes the translation of the lower boundary as well as the increasing thickness.

If the scale separation also obeys $ϵSA\ll 1$, the familiar simpler result iswithThis last simplification needs the logarithmic refinement $ϵS\sqrt{\log (RS)}\ll 1$. The algebraic ordering $ϵ\ll S^{-1}$ alone should not be used to discard an arbitrarily large logarithmic correction; the preceding coupled equations remain the appropriate calculation when that refinement is unavailable. This distinction is captured by large latent heat in a freezing and melting ice layer.

The illustration uses $S=40$, $ϵ=2.5\times 10^{-4}$ and $R=2$. Solving the full equations gives $A=1.73707$, $B=0.0143053$ and $\vartheta =0.0158766$. It shows the ice layer between freshwater and cold brine, with both fronts advancing upward and a strongly diluted lower liquid boundary.

The phase diagram shows the spatial path from remote brine to the lower interface: temperature first changes over $\sqrt{\kappa t}$ at almost unchanged salinity, then salinity changes over its much smaller diffusion scale at nearly constant temperature. The liquid path ends on the liquidus at $(C_i,T_i)$; the solid has $C=0$ and its temperature rises from $T_i$ to $T_m$. The eutectic position is schematic, with $C_E=2C_0$ in the illustration, and is not used in the calculation. A point initially in the fresh layer freezes when $b$ reaches it, cools in the ice, and later melts when $a$ reaches it. The inset indicates this temporal path in the opposite direction through the solid branch and into the liquid.

The mechanism is simultaneous upper freezing and lower melting, with net growth of the ice thickness. Cold brine accepts most of the latent heat released by upper freezing. Salt reaching the lower contact lowers its equilibrium melting temperature, and melting freshens that liquid until its liquidus is close to $T_m$. The heat delivered through the ice supplies the much smaller melting demand there. The remote brine remains liquid even though it is below the pure-water melting temperature, because it lies above its own saline liquidus. Thus “cold” does not by itself decide which phase is stable. These conclusions apply while the layers remain effectively deep and the stated no-convection, pure-ice and phase-equilibrium assumptions hold.

Each larva undergoes diffusion over its exponentially distributed lifetime of mean $1/\mu$ and deposits eggs at rate $\lambda$ wherever it is found. The characteristic diffusion length before death is $\sqrt{D/\mu }$, which explains the exponential spatial decay. Integrating the remnant over space givesThis is also immediate physically: each of the $N$ larvae lives for mean time $1/\mu$ and therefore lays an expected $\lambda /\mu$ eggs. The eggs remain as a stationary record because they neither die nor diffuse.

Absorption of photons with $h\nu \gtrsim E_g$ creates electron-hole pairs, the upward band-diagram arrow. In the depletion region the built-in field drives electrons towards n material and holes towards p material; diffusion also delivers carriers generated within a diffusion length outside that region. Selective contacts collect the two signs. Illumination maintains a nonequilibrium quasi-Fermi-level splitting, allowing a voltage and useful current through an external load; it is the light, rather than the equilibrium barrier, that supplies the energy.

Taking forward diode current as positive, an ideal illuminated junction haspower is delivered where $VI<0$ in this convention. Reflection, below-gap transmission, carrier thermalization, recombination and resistive losses limit real conversion; even an ideal single junction has radiative detailed-balance limits.^[ref]