Why concentration falls as a tissue grows

Follow a patch of material

Draw a small square on the tissue and let it move with the material. Reactions occur inside and diffusion crosses its edges. Motion of the tissue carries the square and its contents together. Conservation keeps these processes separate.

Let $c(\mathbf x,t)$ be amount per unit area, $\mathbf w$ the material velocity and $R(c)$ the local production rate. An observer at fixed spatial coordinates sees reaction, diffusion and the flux carried by the material.

\partial_tc+\nabla\cdot(c\mathbf w)=D\nabla^2c+R(c)

Expanding the divergence separates transport of the concentration field from the concentration change caused by local area expansion.

Reading: E. J. Crampin

The factor of two comes from area

Take $L(t)=L_0e^{gt}$ , with the same relative extension rate $g$ in both directions. Material velocity is $\mathbf w=g\mathbf x$ , positive along increasing coordinate directions. Adding the two velocity derivatives gives $\nabla\cdot\mathbf w=2g$ .

A line gains length; a sheet gains length and width. Dilution therefore depends on dimension. Applying the one-dimensional factor to an area concentration creates material artificially. Thickness is fixed here; a volume concentration in a growing three-dimensional body needs its own derivation.

\partial_tc+\mathbf w\cdot\nabla c=D\nabla^2c+R(c)-2gc

A material observer sees reaction, diffusion and dilution. Even a spatially uniform concentration changes under expansion.

Let the coordinates grow with the tissue

Define material coordinates $\boldsymbol\xi=\mathbf x/L(t)$ on a fixed unit square. The same material point keeps the same $\boldsymbol\xi$ . Its time derivative already follows the flow, removing the explicit advection term.

Spatial derivatives also change. The same material-coordinate ripple spans distances larger by $L$ , so the second derivative gains a factor $L^{-2}$ . For the two Schnakenberg concentrations the resulting equations are below.

Geometry and concentration evolve together. Reaction, diffusion and dilution change the field values; expansion changes physical peak separation. Both affect peak formation and contrast.

\left.\partial_tu\right|_{\boldsymbol\xi}=\frac{D_u}{L(t)^2}\nabla_{\boldsymbol\xi}^2u+a-u+u^2v-2gu,\qquad \left.\partial_tv\right|_{\boldsymbol\xi}=\frac{D_v}{L(t)^2}\nabla_{\boldsymbol\xi}^2v+b-u^2v-2gv

Periodic boundaries stay fixed in material coordinates. Setting growth to zero recovers reaction–diffusion on the same physical square.

L_0,\quad c_0

2L_0,\quad c_0/4

A conservation schematic with reactions off: twice the side length gives four times the area and one quarter of the concentration. Markers represent the same amount; they are not simulated particle trajectories.

Observe amount as well as concentration

The first species has amount $M_u=L(t)^2\int_{[0,1]^2}u\,d^2\xi$ . Without reactions, diffusion only redistributes material in a periodic domain. Total amount stays constant and mean concentration falls inversely with area.

With reactions, amount can change. Its rate must equal the spatial integral of production. An externally supplied system need not conserve its total concentration, and dilution does not mean that molecules have disappeared.

The fixed-domain equilibrium generally ceases to be an equilibrium when dilution is present. A stability calculation under growth must first find the appropriate uniform reference trajectory and linearize along it.

\frac{dM_u}{dt}=L(t)^2\int_{[0,1]^2}f(u,v)\,d^2\xi

Periodic diffusion integrates to zero. Dilution exactly cancels the changing area factor.

Growth changes which modes can fit

As the tissue grows, a fixed mode number corresponds to a longer physical wavelength. Existing peaks separate, and additional modes can enter an unstable scale range. New-peak formation depends on the growth rate relative to reaction and transport.

Slow growth can let a system approach successive nearly stationary patterns. Faster expansion can make dilution and stretching dominate. The source paper’s one-dimensional mode-doubling sequence has specific conditions. The two-dimensional square here is a teaching extension of the conservation law.

Reading: E. J. Crampin

Worked example

Double the side length

Turn reactions off and start with uniform $c_0=1$ . Double both sides of the square. Area becomes four times larger and no material escapes, so concentration becomes $1/4$ .

Exponential growth gives the same check: doubling means $gt=\log2$ and $c(t)=c_0e^{-2gt}=1/4$ . An incorrect one-dimensional dilution term gives $1/2$ , creating twice the original amount when multiplied by the fourfold area.

For a nonuniform field diffusion changes shape, but area-weighted total amount must still remain constant without reactions or boundary flux. This tests dilution more directly than a comparison of colors.

Try the reasoning

Weigh a growing pattern

In the no-reaction preset, predict the changes in mean concentration and total amount as the side length grows.
Compare the expanding domain with the concentration and amount traces. Explain their difference through area growth.
Restore reaction and compare fixed, slowly growing and rapidly growing domains. Record peak spacing and the time of new-peak formation.

Exercises and explanations

A sheet grows to one and a half times its initial side length. Without reaction, what is the final uniform concentration?
Read the explanation
Area increases by $1.5^2=2.25$ , so concentration becomes $1/2.25\approx0.444$ of its initial value. Total amount is unchanged.
Why does eliminating explicit advection in material coordinates not eliminate dilution?
Read the explanation
Moving coordinates account for transport of the concentration field. Each material patch still changes area, so the amount per unit area changes.
Does increasing total amount in a supplied reaction system prove that the solver violates conservation?
Read the explanation
No. Compare the amount change with integrated reaction production. Alternatively, turn off all reactions and test a source-free limit.

Course sources and calculation scope

The derivations, diagrams and numerical examples are authored for this course. Source locations below distinguish adopted equations, teaching extensions and the one exact published example recalculated here. Interactive fields and noisy observations are model-generated.

A. M. Turing, The Chemical Basis of Morphogenesis (1952), Philosophical Transactions B 237, 37–72
Read §4, pp. 42–43, for the two-cell numerical example and §6 for mode analysis. The reproduced result is early linear growth in §4. No source figure is copied; the Schnakenberg experiments use a different model.
J. Schnakenberg, Simple Chemical Reaction Systems with Limit Cycle Behaviour (1979), Journal of Theoretical Biology 81, 389–400
Original reaction-model reference. The dimensionless irreversible teaching form is specified here; its equilibrium, Jacobian and numerical examples are derived in the course. Parameters are not biological measurements.
D. Barkley, A Model for Fast Computer Simulation of Waves in Excitable Media (1991), Physica D 49, 61–70
Read the two-variable equations in §2 and the discussion of applicability. The experiment solves those continuous equations with its own discretization and teaching initial conditions; source images and rotation frequencies are not reproduced.
E. J. Crampin, E. A. Gaffney and P. K. Maini, Reaction and Diffusion on Growing Domains (1999), Bulletin of Mathematical Biology 61, 1093–1120
Read §3, equations 9–20, on conservation, transport and growth. The paper mainly studies one-dimensional growth. The two-dimensional uniformly expanding square is a separately derived teaching extension, not a reproduction of its mode-doubling figures.
M. Nishikawa, S. R. Naganathan, F. Jülicher and S. W. Grill, Controlling Contractile Instabilities in the Actomyosin Cortex (2017), eLife 6, e19595
Read Appendix equations 4–9 on density flux, active viscous stress, saturation and force balance. This teaching reduction uses one density, linear turnover and local stress inhibition. It does not fit embryo data or reproduce Figure 2. The source is CC BY.
A. L. Krause, M. A. Ellis and R. A. Van Gorder, Influence of Curvature, Growth, and Anisotropy on the Evolution of Turing Patterns on Growing Manifolds (online 2018; issue 2019)
Read equations 4–6 for moving-domain conservation, 14–15 for material coordinates and 44–45 for Schnakenberg reactions and Jacobian. The periodic square and parameters are teaching choices; ellipse geometries and Figures 1–4 are not reproduced.
L. Giomi, Geometry and Topology of Turbulence in Active Nematics, arXiv:1409.1555v2 (2015); Physical Review X 5, 031003
Frontier branch reading on two-dimensional incompressible active nematics, vortex statistics and defects. Those orientation and fluid equations belong to FR22; their results are outside the one-dimensional scalar cortex model.