01 / From noise to order
No pattern
was drawn.
We wanted to understand how spatial order can emerge without drawing a pattern into the starting conditions. We began with nearly uniform concentrations, added a small disturbance and let local reactions and diffusion evolve together.
We compared two mathematical models along a line, extended them to a surface, and then explored a growing domain. Each step asked a different question: how do the fields relate, when do patterns emerge, and what changes when their available space expands?
Perturbation
t = 0 s
Spots emerge
t = 40 s
Later pattern
t = 1000 s
Grenoble INP – Phelma · Numerical methods · March 2024 · Team project
02 / The mechanism
Two fields.
Two feedback loops.
The fields u and v represent concentrations at each position. Reactions change them locally; diffusion spreads them between neighbours. In Gierer–Meinhardt, u promotes both fields and v limits further production of u. In Schnakenberg, production of u consumes v. Comparing these feedbacks lets us test how the mechanism shapes the pattern.
Here, d = Du / Dv = 0.025: v diffuses 40 times faster than u. No-flux boundaries keep both fields inside the domain.
Local amplification, coupled inhibition.
u → u and v
More u increases its own production and produces v.
v suppresses u production
The denominator in u²/v provides inhibition.
a = 0.5
a controls the linear removal of u.
Local amplification consumes a substrate.
u²v produces u
The autocatalytic reaction depends on both concentrations.
The same reaction consumes v
High u can coincide with local depletion of v.
a = 1 in this experiment
Here a supplies v; its role differs from the other model.
At the boundary: ∂u/∂n = ∂v/∂n = 0. No diffusive flux crosses an edge.
03 / One dimension
Do the peaks meet,
or alternate?
We started along a line so the relationship between the fields could be read as two concentration profiles. Gierer–Meinhardt produced peaks at similar positions. Schnakenberg produced alternating peaks: high u coincided with low v.
That difference is consistent with the reaction rules: one mechanism produces both fields together, while the other consumes v as u is produced. Peak alignment alone is not a test of activation or inhibition; the equations define those relationships.
u above · v below
Gierer–Meinhardt · select an image to inspect its axes and concentration scale.
MATLAB profiles: position along the horizontal axis, concentration vertically; u above and v below. The fields have separate concentration scales. Playback steps through saved snapshots; read the displayed time because time gaps and model timestamps differ.
Why begin with finite differences?
We represented the line with a regular grid and estimated diffusion from neighbouring values. Explicit time stepping advances all concentrations from the previous state. It is straightforward to implement, but the time step must stay small enough for numerical stability. We limited this experiment to 40 s; the displayed snapshots end earlier. A convergence study—repeating the calculation with finer grids and smaller time steps—would be needed to establish numerical precision.
04 / A surface
The same rules
organise a surface.
Would the relationships seen along a line also organise a surface? We used COMSOL, a finite-element simulation tool, to evolve the same coupled equations on a 10 × 10 square. No-flux edges prevent either field from diffusing out of the domain.
Step through the recorded times and compare the two fields. Gierer–Meinhardt shows sharper u peaks within broader v peaks; Schnakenberg shows high u where v is depleted.
u · concentration field
v · concentration field
Gierer–Meinhardt · select an image to inspect its axes and concentration scale.
Colour represents concentration across the square; u is shown alongside v. Playback steps through five saved times at equal viewing intervals, although simulation time gaps are unequal. Read each colour scale separately: its range changes between frames and fields.
05 / A window for instability
Diffusion can
destabilise uniformity.
A pattern at one setting does not tell us when it can form. We examined a more specific question: can a uniform state settle down under local reactions, yet become unstable once diffusion is added? This is diffusion-driven instability: small spatial differences can grow instead of disappearing.
Change the reaction parameter and diffusion ratio to explore the conditions for that instability. The map calculates linear stability from the equations. It explains how small disturbances initially behave; it does not rerun the saved simulations or predict their final pattern.
Reading the selected point
Diffusion-driven instability is possible.
The uniform state is stable without diffusion, but some spatial wavelengths can grow when diffusion is added.
- Uniform concentrations (u*, v*)
- 2.00, 4.00
- Local stability indicator (trace)
- −0.500
- Diffusion ratio
- v diffuses 40.0× faster than u
This calculation considers small disturbances around uniform, positive concentrations. A growing disturbance must also have a wavelength that fits the domain. Outside the green region, this particular route to instability is absent; other changing or nonlinear patterns may still occur.
The stability calculation and its boundary case
The reaction Jacobian J describes how each local reaction changes when u or v changes slightly. Its trace (tr J) and determinant (det J) test stability around the uniform equilibrium. With f and g denoting the reactions for u and v, and subscripts denoting partial derivatives, strict Turing conditions require:
tr J < 0, det J > 0
B = fu + d gv > 0
B² − 4d det J > 0
For Gierer–Meinhardt this gives 0 < a < 1 and 0 < d < (3 − 2√2)a. For Schnakenberg it gives a > 1 and 0 < d < (3 − 2√2)/a². The map uses these boundaries.
The Schnakenberg experiment uses a = 1, where tr J = 0. This is a boundary case: the uniform state is not strictly stable without diffusion. We can inspect its simulated patterns, but they do not establish the strict instability conditions shown in green.
Compare patterns across parameter settings
06 / A growing domain
What if the canvas
keeps growing?
Finally, we asked what happens when the space available for a pattern changes. We replaced the fixed square with an expanding disk and a moving computational mesh. This let the geometry grow while the concentration fields continued to evolve.
The setup doubles the radius over 100 s. Follow the spatial axes to see the disk expand and compare how the concentration pattern changes within it.
u · concentration field
v · concentration field
Growing domain · select an image to inspect its axes and concentration scale.
Paired u and v fields at 4, 16, 29, 45 and 100 s. The axes show spatial extent and colour shows concentration. Playback steps through these saved times; the physical time gaps are unequal and each image has its own colour scale.
What the moving mesh establishes
An arbitrary Lagrangian–Eulerian (ALE) formulation lets the computational mesh move with the geometry. For a quantitative growth model, we would also need to establish how expansion transports and dilutes the concentrations. The snapshots show an exploration of changing geometry; they do not validate tissue growth.
What we learned
A mechanism to explore.
An explanation to test.
The comparison made the role of the mechanism visible: different reaction rules produced different relationships between the two concentration fields. Extending the model showed why parameters and geometry belong in the investigation alongside those local rules.
The next step would be to check numerical convergence and test a specified model against biological measurements. Similar-looking spots alone cannot establish that living tissue uses the same mechanism.
Team
Mélina Roge-Fakhar · Eloïse Soule · Héloïse Ployon · Julien Le Boulch
Supervision
David Jauffres · Grenoble INP – Phelma
Project
Numerical methods · March 2024
The academic foundations