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A paper written by Alan Turing that describes how the reaction between chemical substances called morphogens and their different rates of diffusion through tissue can explain the phenomena of morphogenesis, the process where non-spherical shape develops in an organism from an initially spherical embryo. The paper uses mathematical modelling to show how an embryo can begin as a homogeneous structure and later develop a pattern due to triggering of random disturbances that create instability in the homogeneous equilibrium. Turing referred to his model as his “mathematical theory of embryology”. Turing’s model, also known as reaction-diffusion model, has had a large impact on developmental biology and also on the field of reaction-diffusion systems.

Turing’s model of a growing embryo describes the state of the system in two parts: mechanical and chemical. Substances, which are presumed to be produced by the action of genes, react chemically and diffuse through tissue masses. Turing referred to these substances as morphogens, to convey the idea that they are a type of substance that produces form. The paper describes a model for how morphogens diffuse though a mass of tissue and react within it following the ordinary laws of diffusion. Morphogens move from regions of highest concentration to regions of lower concentration at a rate that depends on the gradient concentration and the diffusibility of the substance. The model assumes the presence of two interesting morphogens, and no new features emerge from modelling with more than three morphogens.

Turing considered for his model a ring of cells or a continuous ring of tissue, which both produced similar results. The essential form of the model is as follows, where c(x; *t*) is vector of chemical concentrations, *D* is the diagonal matrix of diffusion coefficients, f the vector of reaction kinetics, x space, and t time. Boundary conditions could typically be periodic or zero flux.

The system begins in a stable homogeneous condition. Slight disturbances from this state occur. The reason for the disturbance is unspecified, but suggested reasons for disturbance include Brownian movement, neighboring structures, or slight irregularities in form. Turing suggests that these disturbances result in changes in reaction rates or diffusibilities of the two or three morphogens. These changes could be caused by changes in concentrations of other morphogens that could act as catalysts or fuel supply for the morphogens of interest. Growth of cells and changes in temperature are also potential causes for the system changing from stable to unstable.

Turing's model shows that six types of patterns of morphogen concentrations appear, and they are described as "waves." In one type of pattern, stationary waves appear on the ring of cells. Turing's stationary wave patterns are the type now referred to as Turing patterns. Turing noted that this type of pattern resembles the tentacle patterns on the freshwater organism, Hydra, and the pattern of whorled leaves on the Woodruff plant. In general, this type of pattern formation accounts for the emergence of branching structures and how circular symmetry is broken. When reactions and diffusions are considered on a sphere, the model can explain gastrulation, the process in which an embryo changes from a hollow sphere of cells to a multilayered and multidimensional structure called the gastrula. When considered in a two-dimensional domain, dappled patterns emerged, which could account for animal coat patterns.

The above figure is Figure 2 from Ball, Phillip, "Forging patterns and making waves from biology to geology: a commentary on Turing (1952) ‘The chemical basis of morphogenesis.’" Philisophical Transitions B R. Soc. (2015) B3702014021820140218. (a) A schematic of a 20-cell ring showing morphogen concentration differences. (b) Turing’s ‘dappled pattern’ hand-calculated using a morphogen scheme in two dimensions. (c) Qualitative resemblance of to dappled pattern to animal markings such as a cheetah.

The Turing system was one of the first to show that instability and complex phenomena can arise from the integration of fundamental units of stabilizing components. The process of diffusion is normally thought of as stabilizing, such as in the spreading out of a drop of food coloring in a cup of water until the cup of water becomes a uniform color. At the time, it was counterintuitive that adding diffusion to a system with stabilizing reaction kinetics would result in an unstable system. Turing may be considered to be one of the first systems biologists, since he showed the importance of looking at the interactions between components.

Six possible states were found that could emerge from the interaction between two morphogens, and researchers have later found real-world biological systems that generate some of these types of patterns. Type I is where the system converges to a state that is uniform and stable. Type II is a uniform state with phase oscillation of morphogen concentration. Type III is a system of stationary waves with very short wavelengths that can produce salt-and-pepper patterns. This is seen in real developing biological systems such as Drosophila embryos, when cells differentiate and inhibit differentiations of neighboring cells—a process called lateral inhibition—in which the reacting molecules are Notch and Delta. Type IV is the case where the pattern in case III oscillates, and this has not been found in a developmental system. Type V is a traveling wave, and this is seen in spiral patterns formed by aggregation of social amoeba and also when the sperm enters the egg of the African clawed (Xenopus laevis) and produces a similar wave of calcium ions that cross the egg. The type V traveling wave has also been shown to be produced by interactions between fibroblast growth factor (FGF) and bone morphogenetic protein (BMP), which result in a traveling wave of hair formation across the skin in a mutant mouse. In type VI, stationary patterns form that are now referred to as Turing patterns. A Turing pattern is a nonlinear wave that is maintained by dynamic equilibrium of the system. Interactions between molecules, and their rates of diffusion determine the wavelength. Turing pattern formation is able to generate spatial patterns seen in nature, such as in seashells, feather patterning, and skin patterns like stripes and spots.

While the original reaction-diffusion model uses idealized hypothetical molecules that control synthesis of themselves and their counterparts and diffuse quickly through cellular membranes, it is accepted that in real-world systems, a Turing pattern only requires the combination of a short-range positive feedback and a long-range negative feedback. Gierer and Meinhardt introduced the condition of self-enhancement and long-ranging inhibition in 1972, while unaware of Turing's work. Although Turing did not mention inhibition, from his unpublished notes it appears that he suspected something of this nature. A real-life example of self-enhancement and long-range inhibition is the interaction between Nodal and Lefty2, which in sea urchins is involved in oral field formation. Nodal is secreted and autocatalyzes its own production, and Lefty2 is under the same control as Nodal but it acts as an inhibitor of Nodal and diffuses much faster.Other pairs of molecules that generate short-range positive feedback and long-range negative feedback include Wnt and Dkk involved in hair follicle spacing, Hydra regeneration and lung branching, Nodal and Lefty in left-right asymmetry, and TGF-β or FGF paired with BMP regulating feather bud spacing in chick, tooth pattern, lung branching, and limb skeletal patterns. Theoretical modelling has shown that diffusion of molecules is not necessary to produce Turing-like periodic patterns, as they can also be produced by cell-to-cell signals, chemotactic cell migration, mechanochemical activity, and neuronal interactions. The pigment pattern in zebrafish comes about from long-range negative feedback and short-range positive feedback between black pigment cells and yellow pigment cells.

Another prominent model in morphogenesis is Wolpert’s positional information model, where gradients of morphogens give positional information to cells that interpret this information by some fate-choice mechanism. The discovery of the Bicoid gradient in Drosophila, the control of cell fate in Xenopus by TGF superfamily protein growth factor concentration thresholds and that Hox genes encode positional information along the anterior-posterior axis of all animals fit with the positional information concept. While Turing’s reaction-diffusion and Wolpert’s positional information were long considered opposing ideas, the two systems are now thought to work together to build biological patterns.

The above image is from Green, J.B.A., and Sharpe, "Positional information and reaction-diffusion: two big ideas in developmental biology combine," *Development* (2015) 142 (7): 1203-1211. (A) The left side of the figure illustrates the Turing reaction-diffusion model with an activator-inhibitor. (i) When some cells produce a higher level of activator, these levels auto-enhance and increase the concentration. (ii) The activator also enhances production of the inhibitor, which diffuses at a faster rate. (iii) Due to the quicker diffusion, the inhibitor does not accumulate sufficiently to repress activator at the peak. Also, a higher amount of inhibitor in neighboring cells prevents the activator from increasing in these regions (lateral inhibition). (iv) New peaks form outside the region of lateral inhibition. (v) A regular array of peaks and valleys forms across the field of cells. In two dimensions, this produces spots and stripes. (B) The right side shows Wolpert’s positional information model, illustrated with the French flag pattern. (i) The fields of cells are divided into three equal regions with different cell fates represented by red, white, and blue. The cell fates are prespecified and depend on their position across the morphogen concentration ranges and thresholds.

Turing mechanisms in developmental biology were first established in skin pigmentation, hair, and feather patterning. Later, Turing features were shown in the periodic pattern of digits with the Bmp-Sox9-Wnt network. There is evidence that the Turing patterning process with the Bmp-Sox9-Wnt network is conserved between shark fin and mouse digits, but spatially reorganized to produce different skeletal patterns.

The above figure is from Onimaru, K., Marcon, L., Musy, M. et al., "The fin-to-limb transition as the re-organization of a Turing pattern." *Nat Commun* 7, 11582 (2016) . It depicts how the Bmp-Sox9-Wnt model can account for shark fins and mouse limbs. (a) Sox9 is turned on in the shark (S. canicular) and mouse in the fin bud and embryonic mouse digits, respectively. (b) Sox9 expression is shown in black and a gradient from blue to red specifies the proximal to distal portions. (c) Graphs illustrate differences between shark and mouse buds along an Fgf gradient. The Turing pattern of spots form between the threshold values of th1 and th2, where Fgf acts as a positional cue. Turing stripes form in mouse along the whole gradient with the Fgf acting as a influencing the local wavelength. Green wavy lines represent the spatial location of the Turing pattern that is established by interactions between Bmp, Sox9, and Wnt.