Turing models, 70 years later

Alan Turing, often considered one of the fathers of modern computing, has been widely recognized for his contributions to the field of computing. Notably, Turing developed the Turing machine, a hypothetical machine that manipulates symbols on an infinite, one-dimensional tape according to an array of rules. A Turing machine can simulate any computer algorithm and provide the mathematical formulation for today’s digital computers. At a lecture in 1947, Turing alluded to the concept of computer intelligence—probably one of the earliest mentions of this concept—stating that “what we want is a machine that can learn from experiment” and stating that “the ability to let the machine modify its own instructions provides the mechanism for this”¹. Turing eventually came up with what is known as the Turing test, a method for determining whether a machine can demonstrate human intelligence, which has been very influential and has been the subject of much discourse in the field of intelligence artificial.

Perhaps less well known, though equally remarkable, are Turing’s contributions to the field of mathematical biology. On August 14, 1952, Turing published his only research paper in the field of biology titled “The Chemical Basis of Morphogenesis”.², in which he proposed a mechanism to explain the patterns observed in nature. Specifically, Turing noted that a unique property of biological systems is asymmetry, which cannot arise solely due to physical laws. Taking the development of an embryo as an example, he said that an embryo begins as a symmetrical mass of cells and that the laws of physics could not explain, in most cases, the development of organs such as the limbs and eyes in specific places. Instead, he proposed that the development of asymmetry in biological systems could result from signaling molecules that emerge from some source in the tissue and move away from their source, leading to a concentration gradient and to patterns within a system. Turing referred to these signaling molecules as morphogens. However, any pattern appearing in a system of molecules moving according to the fundamental laws of physics would eventually disappear, leaving behind no observable patterns: it would require a process that would generate and amplify patterns in the system in a stable way. To explain the development of the models, Turing proposed the addition of instability to a linear system of molecular motion by introducing the diffusion of morphogens at specific times. Based on theoretical calculations, the addition of scattering led to the destabilization of the system and the development of patterns, now called Turing patterns. While model development due to the introduction of diffusion seemed counter-intuitive at the time, it is now a widely accepted system known as the reaction-diffusion system.

Turing’s study is landmark for several reasons. First, the study introduced the concept of morphogens and reaction-diffusion systems, which had a tremendous impact not only on the field of developmental biology, but also on various other fields, including chemistry, physics and ecology. Additionally, the study influenced the field of mathematical biology to a great extent given its unique use of numerical analysis to study biologically observed phenomena.

Interestingly, Turing’s work on morphogenesis remained largely unknown until over 25 years later when researchers pointed to the existence of morphogenic gradients.³. Nüsslein–Volhard and Wieschaus showed that mutations at 15 loci in a Drosophila melanogaster disrupted the segmental pattern of the larva and that these mutations had three distinct levels of spatial organization. Ultimately, the results indicated that a morphological gradient was responsible for forming the segmental pattern in the system. Experimental confirmation of Turing’s patterns came even later, in 2014, when researchers were able to reproduce them in chemical cells.⁴.

But as noted above, applications of Turing models are not just seen in developmental biology – or biology in general. To name a few, Turing models have explained the shell structure and patterns seen in aquatic molluscs⁵and they have also been used to better understand human settlements⁶ and design water filters^seven. Experimentally, Turing models were able to explain the spaced transverse ridges of the palate in mammals⁸. In 2021, researchers showed that an atomic bismuth monolayer strained on the surface of niobium diselenide displays Turing patterns⁹, an observation that can play a crucial role in the development of microdevices. Fascinatingly, artists have also used Turing models to create generative art.

Today, August 14, 2022, the manuscript “The chemical bases of morphogenesis” celebrates its 70th anniversary. Unfortunately, Turing died two years after it was published and was unable to see the vast implications of his study. And at first glance, we’ve barely begun to scratch the surface of the potential applications of Turing’s models.