How diffusion can help a pattern grow

Begin with a drop of ink

A drop of ink spreads through still water. More molecules leave the crowded region than return from the dilute surroundings. Many random movements produce a net flow from high to low concentration. Without a source, diffusion cannot make the drop sharper.

Divide the water into small neighboring patches and keep one average concentration in each. A patch richer than both its neighbors loses material. A patch poorer than both gains it. This exchange has no plan for a final stripe or spot.

Write concentration as $u$ , the amount per unit area. The diffusion coefficient $D_u$ has units of length squared per time. The operator $\nabla^2u$ measures spatial curvature; on a plane it adds the second derivatives along two perpendicular directions.

\partial_t u=D_u\nabla^2u

The concentration changes according to its difference from its surroundings. A uniform concentration has zero spatial curvature.

A reaction can restore a difference

Suppose a substance speeds up its own production. A slightly richer patch then reacts faster and becomes richer still: positive feedback. Consumption or another reacting substance can oppose this increase and restore the equilibrium of a well-mixed sample.

Space changes the competition. Material can leave a patch before its local reaction is complete. Fast and slow components give the feedback different amounts of time to act. To decide whether a pattern grows, we must follow both reaction and transport.

Let $f(u,v)$ and $g(u,v)$ be the local reaction rates, where $v$ is the second concentration. At an equilibrium both reaction rates vanish. A stationary spatial pattern can still have local reactions balanced by transport.

\partial_t u=f(u,v)+D_u\nabla^2u,\qquad \partial_t v=g(u,v)+D_v\nabla^2v

Each equation combines local chemistry with exchange between positions. The underlying material is stationary, so diffusion supplies the spatial transport.

Reading: A. M. Turing

Which disturbances should we follow?

Add a gentle ripple to a uniform concentration and ask whether its height grows. The distance between successive peaks is its wavelength. Short waves have nearby peaks and troughs, so diffusion smooths them strongly. Very long waves may let the reactions restore equilibrium. An intermediate range can grow.

A complicated disturbance can be resolved into waves of different wavelengths. Following each component reveals which one grows fastest. The visible spacing can therefore emerge from the response of the system, with no template assigning a position to every spot.

A Turing instability has a precise starting condition: the well-mixed equilibrium is stable, but diffusion makes a spatial disturbance grow. A sample that already oscillates everywhere has another kind of instability. The next chapter turns this distinction into a calculation.

Reading the colors in the experiment

The pattern experiment colors a dimensionless concentration. Bright regions are richer in the displayed substance. A moving bright region means that the concentration peak has moved. The colors are a model field; grid squares are not individual cells.

This simplification leaves a question we can answer cleanly: how do small disturbances evolve under specified reactions, diffusion and boundaries? Applying that answer to tissue then requires identifying the substances, measuring their transport and testing whether forces also move them.

Reading: J. Schnakenberg

Worked example

Exchange between two patches

Two equal-area patches have concentrations $1.2$ and $0.8$ . With an exchange rate of $0.5$ per unit concentration difference, diffusion changes the left patch at $0.5(0.8-1.2)=-0.2$ and the right patch at $+0.2$ .

Adding the changes gives zero: total material is conserved. Subtracting them shows the concentration difference shrinking. With reactions switched on, their local rates must also be added before deciding whether the original difference will grow.

Try the reasoning

Compare disturbances that fade and grow

Predict whether the spots will keep growing when both substances diffuse at the same rate.
Compare the stable and Turing presets. Follow the initial ripples and identify the positive part of the growth spectrum.
Drag a small disturbance into the field. Describe what reaction and transport each contribute to its subsequent evolution.

Exercises and explanations

Can diffusion alone sharpen a concentration peak in a periodic box?
Read the explanation
No. Diffusion carries material from the peak toward its surroundings. Periodic boundaries return outgoing material at the opposite edge, preserving the total amount. The concentration can approach a uniform value.
Is a spatially uniform sample that oscillates by itself already a Turing instability?
Read the explanation
No. A Turing instability starts from an equilibrium stable under local reactions. Diffusion then destabilizes a spatial mode. Uniform oscillation is another instability, although the two can occur near each other in parameter space.
Should changing the color palette change the wavelength? How could you check?
Read the explanation
Measure distances between concentration peaks or compare spatial spectra of the same saved field. Both should be unchanged. A change in peak spacing means that more than the display palette has changed.

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.