FR21 · L1–L4
Morphogenesis and reaction–diffusion
A smooth field can grow spots, a small stimulus can launch a spiral, and a contracting cortex can gather material into peaks. Follow the equations behind each process, then ask which observation could tell the mechanisms apart.
Follow one question at a time
Begin with local reaction and exchange between neighboring patches. Chapters 2–3 derive and measure growing modes; Chapter 4 branches into excitation and recovery. Growth and active stress then add geometry and material flow. The last two chapters test mechanisms with a synthetic intervention and recalculate a bounded result from the original Turing paper.
Each chapter states its prerequisites. Start from the phenomenon, then return for the derivation when you need it. The shared textbook plan links the mathematical background. Textbook plan
L1 · 01
How diffusion can help a pattern grow
Diffusion smooths concentration differences, yet unequal transport can help reactions amplify them. Follow the exchange between neighboring patches.
L2 · 02
Calculate which ripples will grow
Derive the equilibrium, Jacobian and Turing conditions of a specified Schnakenberg model, then calculate an unstable range of wavelengths.
L2 · 03
Measure a growth rate from a simulation record
Project saved early concentration fields onto Fourier modes and distinguish finite-box selection, grid error and nonlinear saturation.
L2 · 04
How a local stimulus becomes a rotating spiral
Use excitation and recovery to explain propagation, annihilation at collisions and the curling of a broken wavefront into a spiral.
L3 · 05
Why concentration falls as a tissue grows
Derive reaction–diffusion on an expanding square, separating changing transport scales, geometric dilution and total amount.
L3 · 06
How contraction draws more material together
Derive density, stress and flow in a one-dimensional active viscous cortex, then examine a defined local inhibition of active stress.
L4 · 07
Test a pattern mechanism with an intervention
Compare active cortex and passive diffusion using noisy synthetic intervention records, calibration and a condition withheld from fitting.
L4 · 08
Recalculate Turing’s two-cell example
Read pages 42–43 of Turing’s 1952 paper, reconstruct the two-cell growth calculation and check a full four-variable numerical record.
Four experiments to compare
Which ripples grow?
Concentration field and a measured growth spectrum
Why does a broken front rotate?
Excitation, recovery and colliding waves
What changes when the domain grows?
Expanding geometry, concentration and total amount
Can contraction gather material?
Active stress, flow and local inhibition
For additional context, explore the existing immune-surveillance experiment, which connects molecular encounters to cell and tissue scales. Its search-and-capture model addresses a different question from the reaction–diffusion equations here. Immune surveillance across scales
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.