Source
Turing (1952), LIT-535, “The Chemical Basis of Morphogenesis”, Phil. Trans. R. Soc. Lond. B 237:37–72, read in full in NOTE-427. Cross & Hohenberg (1993), LIT-527, skimmed in NOTE-418, for the later chemical realisations and the status of morphogens.
What was actually shown
The modes (§§6–7). On a ring of cells, Fourier transforming separates the linearised equations into pairs, and mode s grows as e^{p_s t} with (p − a + μ′U)(p − d + ν′U) = bc in the continuum (9.1), a, b, c, d the marginal reaction rates and U the squared wave-number. After a while the mode with the largest Re p dominates.
Six types of onset (§8). Stationary or oscillatory, at extreme long, finite or extreme short wave-length. Case (d), stationary waves of finite wave-length, is “the case which is of greatest interest”. Travelling waves (e) and short-wave oscillation (f) need three or more substances. Each case is shown with explicit parameters.
Conditions for case (d) (p. 55). bc < 0, with 4√(μ′ν′)/(μ′+ν′) < (d−a)/√(−bc) < (μ′+ν′)/√(μ′ν′) (9.4a) and, for instability, (d/√(−bc))√(μ′/ν′) − (a/√(−bc))√(ν′/μ′) > 2 (9.4b). NOTE-427 substitutes μ′ = ν′: (9.4a) then needs (d−a)/√(−bc) < 2 and (9.4b) needs it > 2, so two substances with equal diffusibility cannot give the pattern. The §10 example uses diffusion constants of 5×10⁻⁸ and 2.5×10⁻⁸ cm² s⁻¹.
The spacing. The pattern’s wave-length is the “chemical wave-length”, set by the constants, not by tissue size. Only phase or orientation comes from the disturbance, “like an upright stick falling over” (p. 66).
A worked case (§10). Twenty cells, computed on the Manchester computer: the linear theory predicts three lobes, with four the closest competitor, and slow onset gives three reliably. The final states are “only dynamic equilibria … in order to maintain the wave pattern a continual supply of free energy is required” (pp. 65–66).
Later realisation (LIT-527, §X.B.5). Stationary chemical Turing structures were found only in 1989–1991, in gels and in the chlorite–iodide–malonic-acid reaction without macroscopic gradients.
What could have come out otherwise. Diffusion, a smoothing process, could only have stabilised a homogeneous state. It does not always.
What this does not say
- That development works this way. No real reaction is identified; the §10 chemistry is chosen to give the required rates. Turing concedes the biological examples are “very limited” (p. 72). Cross and Hohenberg report that “the chemical identification of the morphogens has proved elusive” (p. 1050).
- What happens far from onset. Everything outside §10 is linear about homogeneity, an assumption Turing calls “a serious one”. In two dimensions he asserts, without derivation, that the non-linear instability is almost always “catastrophic”.
- That activator–inhibitor structure is necessary in general. It is necessary within the linear two-species model; Cross and Hohenberg call it “by no means a general necessary condition” for this type of instability (§XI.A).
- That the pattern is an equilibrium. Turing’s “equilibrium” means a steady state. The patterns are fed, which places them among dissipative structures before the term existed.
Connections
- Dissipative structures (THEORY-692). Prigogine names this instability of the Brusselator the “Turing bifurcation” and credits this paper with first seeing it. Turing’s criterion is purely kinetic, which fits the lecture’s conclusion that far from equilibrium the kinetics decides.
- Spiral defects in cortex (THEORY-088) treats slow cortical activity as a pattern-forming medium, of the kind this paper founded.
- Non-reciprocal metabolic cycles (THEORY-634) uses the same linearise-and-read-the-growing-mode method, with conserved catalysts in place of reactants.
- Organisms against flames (THEORY-072). The mechanism is the same in an embryo and an unstirred reactor, so it cannot be what marks organisms off; NOTE-427 reads it as a pattern-forming step inside organisms.