The present and future of Turing models in developmental biology

The Turing model or reaction-diffusion model is a mathematical model that can explain spatial pattern formation during development (Turing, 1952). Briefly, Turing showed mathematically that robust positional information can arise spontaneously in a system in which multiple diffusing chemical factors control the synthesis and degradation of one another, and introduced the idea that this may be the basis for the morphogenesis of living organisms.

First published in 1952, the model was purely mathematical without any experimental evidence. At the time, the proposal was controversial and has remained so over the decades (Murray, 1989; Meinhardt, 1982). However, over the years, experimental evidence has begun to accumulate, and researchers are increasingly recognizing the Turing model as a basic principle of pattern formation (Meinhardt, 1982; Maini et al., 2006; Kondo and Miura, 2010; Green, 2021). Typical examples of developmental phenomena for which the Turing model has been applied and experimental evidence obtained include vertebrate skin patterns (Watanabe and Kondo, 2015), feather follicle patterns of chick (Jung et al., 1998), and palatal (Economou et al., 2012), hair (Sick et al., 2006) and digit (Raspopovic et al., 2014) patterns of mice. Overall, the Turing model seems to be moving from the stage where its existence is being debated to the stage where it is being used as a working hypothesis for various experimental systems.

When the Turing model began to be used as a working hypothesis to guide real experiments, it was recognized that the original model was inconvenient to use as is. The real-life mechanisms revealed by experiments are often not simple activator-inhibitor systems, but complex systems involving three or more factors, and mechanisms other than diffusion often contribute to long-range interactions (Watanabe and Kondo, 2015; Marcon et al., 2016; Veerman et al.; 2021). In order to make the simulation more in line with experimental data, the model needs to be modified. Moreover, these modified Turing models are sometimes incorporated as an element of more complex model systems when reproducing the pattern formation by cell interaction and 3D morphogenesis.

I will summarize some of the recent research trends related to Turing models. Given space constraints, I will only be able to touch on some examples, but this Spotlight is intended to provide useful guidelines for researchers who want to introduce Turing models into their research.

The components of the original Turing model are two diffusible factors that control (react with) the synthesis and degradation of one another (Turing, 1952). However, experimental investigations over the years have provided evidence that diffusible molecules cannot account for pattern formation in many cases. For example, in the case of fish stripes (Watanabe and Kondo, 2015; Eom and Parichy, 2017), stimuli (interactions) are transmitted by protrusions extending from two types of pigment cells. The length of the protrusions corresponds mathematically to the diffusion rate (Vasilopoulos and Painter, 2016; Hadjivasiliou et al., 2016). There are also several examples in which mechanical stress (Veerman et al., 2021; Gan and Motegi, 2021) or electrical potential (Blackiston et al., 2009; Harris, 2021) have been proposed to act as long-range stimuli and generate spatial patterns.

However, in the simulations reproducing the specific pattern, the Turing model can still be used as a simplified model (Meinhardt and Gierer, 1974; Oster, 1988). The conditions necessary for stable pattern formation to occur in the classical reaction-diffusion model are that the diffusion coefficient of the activator must be small and the diffusion rate of the inhibitor must be large. As long as the LALI (local activation and lateral inhibition) condition is satisfied, the nature of the pattern formed is similar regardless of the propagation method of the interaction.

It should be noted for the researcher who is not familiar with the mathematical modelling that, even if a spatial patterning is successfully reproduced by a reaction-diffusion model, it may not be clear whether or not a diffusion factor is responsible. In cases where the propagation method is unknown, it may be advisable to use a more abstract model (Kondo, 2017) that does not depend on the signalling method. The combination of computational approaches and experiment can help to distinguish between possible underlying mechanisms.

Another important principle for understanding pattern formation in development is the positional information model, first articulated by Lewis Wolpert as the French Flag problem (Wolpert, 1969). Although this model was published some 20 years after the Turing model, it very quickly gained experimental support and wide acceptance (Wolpert, 1989). In the original version of the positional information model, signalling molecules (morphogens) are locally produced and create a concentration gradient across a tissue by diffusion; the concentration of morphogen(s) at each position provides positional information. The major difference from the Turing model is that the positional information model does not provide a self-organising mechanism, but rather defines a system that would depend on earlier heterogeneities or polarities across the tissue (Green and Sharpe, 2015). As a result of the highlighted difference in whether the origin of the pattern is self-generating or pre-existing, the two models have often been considered as opposing principles (Akam, 1989, Green and Sharpe, 2015).

In fact, however, the two are not entirely contradictory (Kondo and Miura, 2010; Green and Sharpe, 2015). As the Turing model is expressed in terms of partial differential equations, initial and boundary conditions must be entered in order to calculate it, and these have a significant impact on the final pattern that is formed. For example, if a stripe pattern is formed, it is not the intrinsic properties of the system that determine its direction, but the initial configuration of the factors, which is equivalent to the role of initial configuration of the morphogens in the positional information model (Kondo and Miura, 2010). Roughly speaking, the positional information model can be said to be the one in which the interaction between the factors in the reaction-diffusion equation is omitted. Conversely, adding the interaction between morphogens to the usual positional information model makes it homologous to the Turing model. Interestingly, this possibility was suggested in an early paper by Wolpert (1969).

The results of a simulation as a working hypothesis for experiments is the gap between the simplicity of the classical Turing model and the complexity of reality. In the classical Turing model (Turing, 1952), only two factors are involved in a differential equation. However, it is difficult to use the classical model when there are many factors (genes) involved in a phenomenon or when behaviour that cannot be expressed by changes in the concentration of factors, such as cell movement or division, is affected. In such cases, agent-based modelling (ABM) is often used.

ABM was originally created to evaluate the impact of the actions of autonomous agents (individuals or groups) and their resulting interactions on the system (society) as a whole (Bonabeau, 2002). One application of ABM in biology is to model the collective behaviour of cells in response to stimuli. Given that most mathematical models describing pattern formation are expressed in terms of differential equations, the space subject to computation is a continuum. Therefore, it is difficult to represent phenomena (active migration, division, differentiation, etc.) that occur in individual elements (cells). By assuming the cells are the agent in the ABM, it is possible to reproduce experimental phenomena almost as they are in simulation (Glen et al., 2019).

On the other hand, ABM also has inherent disadvantages. The calculation becomes more complex as the number of variables increases. This causes two serious problems. One is that even if one reproduces the pattern with a complex model, one may still not understand the underlying patterning mechanism. This is because people do not feel they understand a complex phenomenon unless it can be explained by a simple cause-and-effect relationship. Another problem is that a model that includes many variables contains so many assumptions. Even if a phenomenon can be reproduced, it is difficult to evaluate its significance.

Therefore, when using ABM, rather than inputting all experimental data, it is more realistic to use only the focal items and use a simplified model for the rest. For example, recent applications of this model successfully avoid this problem by using ABM to focus on the significance of field expansion and cell migration in skin pattern formation, and substituting diffusion for signalling methods (Bullara and De Decker, 2015; Caicedo-Carvajal and Shinbrot, 2008; Volkening and Sandstede, 2015, 2018).

The attempt to create new functional systems by assembling cells and genes like machine parts began in the 2000s and has now been established as a new research field under the name of synthetic biology (Flores Bueso and Tangeney, 2017). In the early days, the focus was on relatively simple gene expression and regulation (Elowitz and Leibler, 2000; Gardner et al., 2000), but gradually the field began to deal with more-complex problems of spatial patterning of cell populations. In studying such artificial systems, it would be useful to have indicators that would allow anyone to see at a glance whether the artificial system works as the researcher expects or not. In the artificial gene network, oscillations and toggle switches act as such indicators. Periodic stripe patterns are universally present in living organisms and yet their quality can be judged at a glance. Therefore, they are very suitable as indicators of the perfection of an artificial patterning system.

By incorporating artificial genetic circuits into E. coli and controlling cell density and migration capacity, Liu et al. (2011) have succeeded in creating a beautiful stripe pattern autonomously on a gel medium. This spatial structure results from repeated aggregation processes in front of a continuously expanding cell population. They have also successfully varied the width of the stripe pattern by regulating the expression of a single gene. This system is not a Turing pattern, as Liu et al. have not shown the ability to modify it in response to disturbances. However, it is an excellent example showing the future of artificial biosystems in controlling spatial structure.

Artificial pattern formation has also been attempted in mammalian cells. Sekine et al. sought to determine whether Nodal-Lefty, which is believed to function as an activator-inhibitor system in left-right axis determination in mice, can also form spatial patterns in vitro (Sekine et al., 2018; Müller et al., 2012). They showed that HEK293 cells, in which the artificial network and luminescence detection system were incorporated, create a periodic spatial pattern of about 200 μm. It is difficult at present to determine whether the pattern produced is a Turing pattern because it is very irregular. However, the attempt to reproduce the phenomenon in vitro is interesting and may be useful for confirming what is actually happening in vivo.

Turing's original paper is entitled ‘The chemical basis of morphogenesis’ (Turing, 1952). This might suggest that his goal was not merely uncovering the principle of two-dimensional pattern formation, but a principle that explains morphogenesis in general; indeed, this is a direction in which recent research has moved. However, it must be noted that there are technical difficulties in dealing with 3D morphogenesis in an expanding or deforming field compared with 2D pattern formation in a motionless field.

In order to calculate the deformation of a cell sheet, it is first necessary to represent the cell sheet in an appropriate physical model. Specifically, the cell population is represented as a fine mesh or a collection of small cubes, and how they deform according to known physical laws is calculated. The patterning mechanism can then be applied on the deforming cell sheet. Overall, such models are much more complex than the 2D models, but there are many examples of studies that have successfully reproduced real phenomena by determining appropriate parameter values (Hannezo et al., 2011; Heisenberg and Bellaiche, 2013; Fletcher et al., 2014).

As an example of a theoretical study applying Turing models to deforming 3D structures, Brinkmann et al. successfully transformed a spherical structure resembling a spore embryo into a variety of 3D shapes by combining mechanical calculations using a tensor model and pattern formation using the Turing model (Brinkmann et al., 2018). Okuda et al. showed that the combination of mechanical calculations using 3D vertex and reaction-diffusion models can reproduce the formation of protrusions and branching, in addition to the deformation of spherical structures (Okuda et al., 2018).

Many simulations have been performed that mimic organogenesis. For example, Iber and colleagues have succeeded in reproducing the branching of kidney and lung by controlling the field expansion to the pattern created by reaction-diffusion (Iber, 2021; Menshykau et al., 2014, 2019), while Salazar Ciudad and Jernvall have computationally reproduced the pattern of mouse teeth (Salazar-Ciudad and Jernvall, 2010). These examples demonstrate that, although challenging, Turing models can be applied in 3D developmental contexts.

Finally, I would like to discuss how the theoretical study of morphogenesis will develop in the future. One of the hottest topics of the past decade in the scientific community has been the dramatic development of artificial intelligence (AI) technology. The success of the Alphafold system (Senior et al., 2020) has shown that AI will undoubtedly enter the field of life sciences in a big way in the future, and theoretical research on morphogenesis is no exception. To date, theoretical pattern formation research has involved modellers logically deriving mathematical formulae, matching parameters and reproducing the phenomenon. As it is difficult to find a good parameter set by hand, in some studies parameter search has been carried out by computer (Diego et al., 2018; Scholes et al., 2019). This method is useful for studying the effects of different parameter values and for increasing the objectivity of the model. However, if AI is used in earnest, it should, in principle, be feasible to explore not only the parameter values, but also the mathematical formulae of the model itself. As long as there is an algorithm that automatically evaluates the quality of the forms (patterns) produced as a result of the calculations, it should be possible to automatically generate a mathematical model.

Of course, the results produced from such a ‘model’ are only so-called computer graphics, as they lack a logical basis. However, if the computer graphics reproduce a form that more closely resembles the phenomenon than any human-made model, it would be difficult to ignore. The Turing model also went through a long period without experimental support. However, one of the reasons why it has continued to gain supporters in this situation is that the similarity between the 2D patterns produced by the model and the real animal patterns is extremely striking. Models that create shapes that are extremely similar to reality are likely to capture, in some way, the essence of the phenomenon. Perhaps the next breakthrough mathematical model will not come from the brain of a single genius, but from a computer chip.