---
title: "Recalculate Turing’s two-cell example"
description: "Read pages 42–43 of Turing’s 1952 paper, reconstruct the two-cell growth calculation and check a full four-variable numerical record."
canonical_url: "https://sandboxphysics.com/stories/morphogenesis/turing-paper-guide/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/morphogenesis/turing-paper-guide/"
last_modified: "2026-10-28"
---

[Morphogenesis and reaction–diffusion](https://sandboxphysics.com/stories/morphogenesis/)

L4 · Chapter 8

# Recalculate Turing’s two-cell example

Sandbox Physics · 2026-10-28

Open §4, pages 42–43, of Turing’s 1952 paper The Chemical Basis of Morphogenesis. Its two-cell example is small enough to calculate reaction, exchange and growth term by term. We reproduce that early positive-concentration interval and provide the complete numerical record.

## Before you begin

Matrix growth rates from Chapter 2 and record comparisons from Chapter 3. These reactions differ from the Schnakenberg experiments, so variables and exchange rates are defined afresh.

- Recover the full four-variable system from the source reactions and exchange rates.
- Check growth through symmetric and antisymmetric disturbances and report the reproduced interval.

## Start with the definitions on page 42

Equal-area cells have concentrations $X_1,Y_1$ and $X_2,Y_2$. Local reactions are $5X-6Y+1$ and $6X-7Y+1$, with exchange rates $0.5$ and $4.5$ for the two species. This abstract example uses its own concentration and time scales.

All four concentrations equal to one give zero reaction and zero exchange. Write exchange as the other cell minus the local cell to obtain the full equations.

$$
\begin{aligned}\dot X_1&=5X_1-6Y_1+1+0.5(X_2-X_1),\\\dot Y_1&=6X_1-7Y_1+1+4.5(Y_2-Y_1),\\\dot X_2&=5X_2-6Y_2+1+0.5(X_1-X_2),\\\dot Y_2&=6X_2-7Y_2+1+4.5(Y_1-Y_2).\end{aligned}
$$

Exchange conserves the total of each species across the cells. Local reactions can change those totals.

Reading: A. M. Turing

## Page 43 follows the difference between cells

Identical disturbances in both cells produce no exchange and follow the local reaction matrix. Equal and opposite disturbances make exchange equal to minus twice the exchange coefficient times the local disturbance.

The two subspaces have matrices $J_0$ and $J_-$. Local eigenvalues are both $-1$, giving asymptotic uniform stability. Antisymmetric eigenvalues are $2$ and $-14$. The growing eigenvector has component ratio $(3,1)$.

Set cell disturbances to $(3C,C)$ and $(-3C,-C)$. Then $\dot C=2C$. The source initial value $C=0.02$ lies along this pure growing direction.

$$
J_0=\begin{pmatrix}5&-6\\6&-7\end{pmatrix},\qquad J_-=\begin{pmatrix}4&-6\\6&-16\end{pmatrix}
$$

Antisymmetric exchange subtracts one and nine from the diagonal entries. Four variables separate into two two-variable systems.

$$
C(t)=0.02e^{2t},\qquad (X_1,Y_1,X_2,Y_2)=(1+3C,1+C,1-3C,1-C)
$$

This expression satisfies the full system while all concentrations remain positive.

Reading: A. M. Turing

## Read a newly computed record column by column

The downloadable JSON and CSV contain 101 simulated times. Time comes first, followed by the four concentrations. Computation directly integrates the full four-variable system. A separate full-matrix exponential checks it before comparison with the reduced analytic mode.

The table selects three times and rounds to six decimals. Error checks use unrounded values. There are no observational error bars: these are deterministic numerical solutions with integration, matrix and floating-point errors.

Newly computed early trajectory in dimensionless concentrations. This table and these sampling times are not experimental data from the source paper.

| $t$ | $X_1$ | $Y_1$ | $X_2$ | $Y_2$ |
| --- | --- | --- | --- | --- |
| $0$ | $1.060000$ | $1.020000$ | $0.940000$ | $0.980000$ |
| $0.25$ | $1.098923$ | $1.032974$ | $0.901077$ | $0.967026$ |
| $0.5$ | $1.163097$ | $1.054366$ | $0.836903$ | $0.945634$ |

## Why the saved interval ends at one half

The paper modifies behavior when a concentration reaches zero. Our reproduced linear interval ends at $t=0.5$, with every concentration positive. Extrapolating the exponential makes $X_2$ reach zero at $t=\tfrac12\log(1/0.06)\approx1.41$; beyond that point the same linear expression cannot describe nonnegative concentrations.

The reproduced object is a published mathematical example with a fixed location, equations, initial state and interval. Source figures, cell images and experimental data are not copied. The later spiral, growing-square and scalar-cortex models are separate teaching models.

The paper is linked for reading and is not redistributed as an attachment. Tables and records are generated by the course code; cite the original example and the new calculation separately.

Reading: A. M. Turing

## Carry the method to the next paper

Find variables, units, geometry and boundaries first. Establish whether the result concerns uniform stability, early growth or a saturated pattern. Record the source version and parameters, and identify whether each vertical axis shows concentration, amplitude, power or growth rate.

Choose a limit or numerical value that can be checked independently, and preserve raw records with processing code. Separate published conclusions, new numerical evidence and open questions. Orientation defects and active turbulence require additional orientation and fluid fields; the Giomi reference offers a different frontier branch.

Reading: L. Giomi · M. Nishikawa

Worked example

## The initial derivative already reveals growth

At first-cell state $(1.06,1.02)$, the first reaction contributes $0.18$ and exchange $-0.06$, giving $0.12$. The second reaction contributes $0.22$ and exchange $-0.18$, giving $0.04$.

The second cell has opposite derivatives. The initially richer cell grows richer and the poorer one grows poorer. Analytically $\dot C(0)=0.04$, which multiplied by $(3,1,-3,-1)$ gives the same derivatives. This algebraic check catches an exchange-sign error or a missing factor of two.

## Download the authored records

[JSON](https://sandboxphysics.com/data/morphogenesis/turing-two-cell.json) · [CSV](https://sandboxphysics.com/data/morphogenesis/turing-two-cell.csv)

## Try the reasoning

### [Transfer the two-cell reasoning to a spatial mode](https://sandboxphysics.com/labs/turing-patterns/)

1. Recalculate the source initial derivative using all four equations.
2. In the pattern experiment choose one discrete spatial mode and compare reaction-only and transport-modified matrices.
3. Identify the shared stability argument and distinguish the reactions, geometry and saturation mechanisms.

## Exercises and explanations

1. What happens to exchange for identical disturbances in both cells?

   Read the explanation

   It stays zero. The local reaction matrix controls evolution and its eigenvalues have negative real parts, so uniform disturbances decay asymptotically. Spatial growth belongs to the antisymmetric subspace.

2. Does halving the initial $C$ also halve the growth rate?

   Read the explanation

   No. The rate remains $2$. Reaching the same small amplitude takes an additional $\log2/2$ time units within this linear positive-concentration interval.

3. Does agreement of one trajectory with the formula reproduce the paper’s entire morphogenesis theory?

   Read the explanation

   No. This checks the §4 two-cell early-growth example. Rings, other instability types and more complex reactions require their own parameters, records and validation.

[Previous: Test a pattern mechanism with an intervention](https://sandboxphysics.com/stories/morphogenesis/testing-mechanisms/)

## 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.

1. [A. M. Turing, The Chemical Basis of Morphogenesis (1952), Philosophical Transactions B 237, 37–72](https://www.damtp.cam.ac.uk/user/gold/pdfs/teaching/turing1952.pdf)

   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.

2. [J. Schnakenberg, Simple Chemical Reaction Systems with Limit Cycle Behaviour (1979), Journal of Theoretical Biology 81, 389–400](https://doi.org/10.1016/0022-5193(79%2990042-0)

   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.

3. [D. Barkley, A Model for Fast Computer Simulation of Waves in Excitable Media (1991), Physica D 49, 61–70](https://warwick.ac.uk/fac/sci/maths/people/staff/dwight_barkley/home_page/papers/barkley_-_1991_-_phys_d_nonlinear_phenom.pdf)

   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.

4. [E. J. Crampin, E. A. Gaffney and P. K. Maini, Reaction and Diffusion on Growing Domains (1999), Bulletin of Mathematical Biology 61, 1093–1120](https://people.maths.ox.ac.uk/maini/PKM%20publications/109.pdf)

   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.

5. [M. Nishikawa, S. R. Naganathan, F. Jülicher and S. W. Grill, Controlling Contractile Instabilities in the Actomyosin Cortex (2017), eLife 6, e19595](https://pmc.ncbi.nlm.nih.gov/articles/PMC5354522/)

   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.

6. [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)](https://pmc.ncbi.nlm.nih.gov/articles/PMC6373535/)

   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.

7. [L. Giomi, Geometry and Topology of Turbulence in Active Nematics, arXiv:1409.1555v2 (2015); Physical Review X 5, 031003](https://arxiv.org/abs/1409.1555v2)

   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.