Turing patterns, reaction-diffusion and autocatalysis

Bayliss, in Principles of General Physiology (1915), described the plotted course of an autocatalytic reaction as S-shaped. The rate is slow at first, becomes quicker, then slows again.¹ He attributed this to too little catalyst at the start and too little substrate at the end. He also warned that the rate must not be confused with the velocity constant, which rises steadily throughout.¹ Thompson, in On Growth and Form, described the case in which the catalyst is itself formed as a product or by-product of the main reaction. The reaction velocity then tends to be steadily accelerated instead of dwindling away.⁴

Loeb, in The Organism as a Whole (1916), drew a biological inference in a footnote. If enzymes in the cell also make molecules of their own kind, the synthetic processes of the cell are of the nature of autocatalysis.² Lotka, in Elements of Physical Biology (1925), reported a terminology proposed by Ostwald. Any growth in which a substance or structure acts as nucleus for more of itself would be called autocatakinetic. The narrower terms autocatalysis and autocatalytic growth would be kept for chemical cases.³ Lotka found this useful because it left the mechanism of growth unspecified.³

Ball's 2015 commentary states the central point. Autocatalysis is positive feedback, and the more that is made, the faster it appears.⁵ Meinhardt and Gierer devised a theory of pattern formation by diffusing reagents in 1972 that paralleled Turing's. Ball reports that they learned of Turing's work only when a referee pointed it out.⁶ Their model has an autocatalytic activator and an inhibitor that suppresses it.⁶ The inhibitor must diffuse faster. The activator's self-amplification is then confined to local patches, and the inhibitor stops another patch forming too close.⁶

Pietak and Levin (2016) modelled a positive feedback loop in a cell collective. They concluded that such feedback can generate strong membrane voltage gradients, and may be one way bioelectric pre-patterns arise and can be manipulated.⁸ Pietak and colleagues (2019) noted a limit of reaction-diffusion models. Their patterns often depend on tissue size, so they are not sufficient for regenerating planaria. The authors argued that adding vector transport on nerve polarity maps improves on this.⁷ Nunn, Guy and Bell (2022) cited evidence of reflexively autocatalytic networks in microbial metabolism, in a discussion of the origin of life.⁹