NOTE-427: The Chemical Basis of Morphogenesis

Contribution

Before this paper, a spherically symmetric blastula developing by chemistry and diffusion looked as if it must stay symmetric forever (§4). After it, there is a mechanism by which it need not. Diffusion, usually a smoothing process, can destabilize a homogeneous steady state of reacting chemicals, and the instability amplifies whatever small disturbances are present into a pattern. The paper:

• classifies the linear onset of such instabilities on a ring;
• gives exact conditions for the two-morphogen cases;
• treats noise acting continuously during a slow approach to instability;
• computes a non-linear example to its final pattern on the Manchester computer;
• extends the analysis to a sphere.

Key insight

The homogeneous state is stable to uniform disturbances, but it need not be stable to disturbances of a particular wave-length. When two substances react and diffuse at different rates, a mode of finite wave-length can grow while uniform and very short modes decay. The pattern’s spacing is then set by the chemistry (the “chemical wave-length”), and only its phase or orientation by the accidental disturbance, “like an upright stick falling over” (p. 66). The sustained pattern is a dynamic steady state, maintained by free energy flowing from fuel to waste (p. 66).

Assumptions

• Model. Tissue is either a ring of N cells, each well mixed, or a continuous ring (or, in §12, a thin spherical shell). Mechanics and growth are set aside; the chemistry is the subject (§1).
• Kinetics. Mass-action kinetics, reaction rates f(X, Y), g(X, Y); “fuel” substances such as A are held at large constant concentration (§10).
• Linearity assumption. Departures from homogeneity are small enough to linearize: f ≈ ax + by, g ≈ cx + dy with “marginal reaction rates” a, b, c, d (§6). The paper calls this assumption “a serious one” (p. 66).
• Diffusion. Cell-to-cell diffusion constants μ, ν (continuous ring μ′, ν′). In §9 it is assumed that ν′ ≤ μ′ > 0. If μ′ = ν′ = 0 “there is no co-operation between the cells whatever” (p. 55).
• Disturbances. In §§6–8 the state at t = 0 is displaced by disturbances, which are then ignored. §9(2) drops this and treats white noise of constant amplitude while the marginal rates drift slowly.
• Chance relations excluded. The analysis assumes no special parameter relations making several modes equally unstable (§8).

Key results

• Separation of modes (§§6–7). Fourier transforming round the ring separates the 2N equations into pairs. Mode s grows as e^{p_s t}, with p_s a root of (p − a + 4μ sin²(πs/N))(p − d + 4ν sin²(πs/N)) = bc (6.8; in the continuum (p − a + μ′U)(p − d + ν′U) = bc, 9.1, with U the squared wave-number).
• Six types of onset (§8). After a lapse of time the mode with the largest Re p dominates:
  • (a) stationary, extreme long wave-length: each cell drifts as if isolated;
  • (b) oscillatory, extreme long wave-length: synchronized oscillation;
  • (c) stationary, extreme short wave-length: neighbouring cells drift in opposite directions;
  • (d) stationary waves of finite wave-length, “the case which is of greatest interest”;
  • (e) oscillatory, finite wave-length: travelling waves, which need three or more morphogens;
  • (f) oscillatory, extreme short wave-length: neighbouring cells about 180° out of phase, which also needs three or more. Each is shown to occur with explicit parameter values.
• Conditions for case (d) (p. 55). bc < 0 and 4√(μ′ν′)/(μ′+ν′) < (d−a)/√(−bc) < (μ′+ν′)/√(μ′ν′) (9.4a). Instability requires in addition (d/√(−bc))√(μ′/ν′) − (a/√(−bc))√(ν′/μ′) > 2 (9.4b).
  • My derivation, not stated in the paper: with μ′ = ν′, (9.4a) requires 1 < (d−a)/√(−bc) < 2, while (9.4b) requires (d−a)/√(−bc) > 2. So with two morphogens, equal diffusibilities cannot give unstable stationary waves. The paper’s §10 example uses diffusion constants of 5×10⁻⁸ and 2.5×10⁻⁸ cm² s⁻¹.
• Noise (§9(2)). When instability sets in gradually, only disturbances near the moment of zero instability matter ultimately. Earlier ones are damped; later ones have less time to grow. Turing compares this to the superregenerative receiver.
• Non-linearity (§9(4)). With quadratic terms the amplitude of the leading mode obeys dz₁/dt = p₁z₁ − kz₁³. For k > 0 growth saturates at √(p₁/k); for k < 0 it is “catastrophic”, and in two dimensions it “is almost universal”.
• Worked example (§10). 20 cells of 0.1 mm diameter; the reaction set given in special units; the computed incipient and final patterns (Table 1, Fig. 3). With γ = 0, s₀ = 3.333 predicts a three-lobed pattern, the four-lobed one being the closest competitor (p₃ − p₄ = 0.0084, about 33 h to gain a factor e). Slow “cooking” gives three lobes reliably; fast “cooking” gives irregular incipient patterns and sometimes four lobes. Growth is halted when Y reaches zero in some cells. The final equilibria are “only dynamic equilibria … in order to maintain the wave pattern a continual supply of free energy is required” (pp. 65–66).
• Sphere (§12). On a growing blastula the degree-1 harmonic is first to go unstable, giving an axially symmetric pattern about a random axis, which is offered as a mechanism for gastrulation.

Claims

id	claim	strength	support
C1	Reaction plus diffusion can destabilize a stable homogeneous state into a spatial pattern	strong	linear analysis §§6–9; worked example §10
C2	On a ring there are exactly six types of onset, (e) and (f) needing three or more morphogens	strong for the linear, generic case	§8 examples; §9 conditions for (a)–(d)
C3	Stationary finite-wave-length patterns (case d) occur under (9.4a, b); with two morphogens this needs unequal diffusibilities	strong (the conditions); the corollary is mine, by substitution	p. 55
C4	The pattern’s spacing is a “chemical wave-length” set by reaction and diffusion constants, not by tissue size	strong in the linear regime	§8, (9.3)
C5	Only disturbances near the moment instability begins have lasting effect	moderate	approximation (9.17) under stated conditions
C6	In two dimensions the non-linear instability is almost always catastrophic	asserted	§9(4), no derivation given
C7	The resulting patterns are dynamic equilibria needing a continual supply of free energy	strong as a physical remark	pp. 65–66
C8	The mechanism may account for Hydra tentacles, whorled leaves, dappling, phyllotaxis and gastrulation	suggestion	§§11–12; “It must be admitted that the biological examples … are very limited” (p. 72)

Concepts

• morphogen: a “form producer”, any substance whose reaction and diffusion figures in the theory. Genes are a special, non-diffusing case; hormones and evocators are typical ones (p. 38).
• marginal reaction rates: the entries of the linearized reaction matrix at the homogeneous equilibrium; dimension of reciprocal time (p. 47).
• chemical wave-length: the limiting wave-length as the ring is enlarged, or the one giving the largest instability (p. 51).
• instability (I): the real part of the dominant p (p. 51).
• equilibrium: Turing’s word for a steady state, including the non-uniform “dynamic equilibria” of §10. It is not thermodynamic equilibrium.

Connections

• Prigogine’s Nobel lecture (LIT-516), read. The lecture names the stationary diffusive instability of the Brusselator the “Turing bifurcation” and credits this paper as the first to see it. The two frameworks meet at one point. Turing’s case (d) is a bifurcation from the “thermodynamic branch”, and his remark that the pattern needs a continual supply of free energy is the defining property of a dissipative structure. Turing gives no entropy-production analysis; his criterion is purely kinetic, which is in line with the lecture’s own conclusion that far from equilibrium “the form of chemical kinetics plays an essential role”.
• Nicolis and Prigogine (LIT-524), Deferred. The lecture cites its ch. VII for the Brusselator, the Brussels school’s standard example of Turing’s case (d).
• Cross and Hohenberg (LIT-527), skimmed. Their type-I_s instability (finite wave-vector, zero frequency) is Turing’s case (d), and they credit this paper as first to connect macroscopic pattern formation with linear instabilities (fn. 1.1). They report the first stationary chemical Turing patterns, 1989–1991, and record that the identification of actual morphogens “has proved elusive” (§XI.A).
• The Belousov–Zhabotinsky reaction (LIT-531), read. Turing guessed at travelling waves (case e) but could cite only a spermatozoon’s tail. BZ waves are the later chemical realization of travelling concentration waves, though of the excitable, strongly non-linear kind his linear theory does not describe. Zhabotinsky’s review records that BZ in microemulsion produces genuine Turing structures.
• Primitive metabolic cycles (LIT-036), read. As NOTE-079 records, its authors contrast their non-reciprocal catalytic mechanism with Turing patterning. The method, linearizing about a homogeneous state and reading the growing mode off the eigenvalues, is the one this paper introduced.
• Bioelectric selves (LIT-439), read. Levin’s account of how cells coordinate into larger “Selves” by bioelectric signalling is an alternative to diffusing chemical morphogens as the carrier of positional information. The two are not in dispute: Turing explicitly sets aside “electrical properties” (p. 38).

Bearing on the record

• LIT-192 (demarcation). The paper’s mechanism is the same in an embryo and in an unstirred reactor. That is a direct case of what Nahas and Sachs report: dissipative pattern formation cannot be what demarcates organisms from candle flames, because it is shared. It is also a case of the “scientific orientation” they describe, an operational model that is explicitly “a simplification and an idealization, and consequently a falsification” (p. 37) and makes no metaphysical claim about organisms.
• LIT-211 (“Dissipative Self-Organizing Systems”). The paper supplies one of that cluster’s paradigm mechanisms (spontaneous order in a free-energy-fed chemical system). It also shows the mechanism’s limits as a mark of life: Turing’s own examples are pattern-forming steps inside organisms, not definitions of them.
• No THEORY is indicated by this reading alone. No ML instruction.

Limitations

• Linear and near onset. Everything except §10 is a linearization about homogeneity. The paper says the linearity assumption “is a serious one”, justified only by the hope that early patterns resemble later ones (p. 66), and that most of development is “from one pattern into another, rather than from homogeneity into a pattern” (pp. 71–72).
• Imaginary chemistry. No real reactions are identified; the §10 reactions are chosen to give the required rates (p. 60).
• Biology by suggestion. Turing concedes the biological examples are “very limited” (p. 72). No experiment is reported.
• One and two dimensions only in part. The two-dimensional theory is “not expounded here” (p. 69), and phyllotaxis is deferred to a later paper.

Open questions

• Which real morphogens, if any, implement case (d) in development? The paper leaves this open, and Cross and Hohenberg report it as still open in 1993.
• What governs the transition from one pattern to another far from homogeneity? Turing proposes digital computation of particular cases (p. 72).
• Does the catastrophic non-linear instability (§9(4)) generically select a different pattern from the linear one?

Corrections

• none to a seeded skim (there was no seed)
• Citation details. The brief’s citation is right: Phil. Trans. R. Soc. Lond. B 237(641):37–72, published 14 August 1952; the DOI is 10.1098/rstb.1952.0012. The full series title is “Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences”.
• “Turing pattern” requires unequal diffusion. This is often stated as Turing’s result. The paper states (9.4a, b) and leaves the corollary implicit; the derivation above is mine.
• A subscript slip on p. 58. The reduced amplitude equation is first printed “dz₁/dt = p₀z₁ − kz₁³” and, a few lines later, “dz₁/dt = p₁z₁ − kz₁³”. The second is the one consistent with the equations just above it and with the saturation value √(p₁/k).