Studies of Turing pattern formation in zebrafish skin

Recording morphogen signals reveals mechanisms underlying gastruloid symmetry breaking

Aggregates of stem cells can break symmetry and self-organize into embryo-like structures with complex morphologies and gene expression patterns. Mechanisms including reaction-diffusion Turing patterns and cell sorting have been proposed to explain symmetry breaking but distinguishing between these candidate mechanisms of self-organization requires identifying which early asymmetries evolve into subsequent tissue patterns and cell fates. Here we use synthetic 'signal-recording' gene circuits to trace the evolution of signalling patterns in gastruloids, three-dimensional stem cell aggregates that form an anterior-posterior axis and structures resembling the mammalian primitive streak and tailbud. We find that cell sorting rearranges patchy domains of Wnt activity into a single pole that defines the gastruloid anterior-posterior axis. We also trace the emergence of Wnt domains to earlier heterogeneity in Nodal activity even before Wnt activity is detectable. Our study defines a mechanism through which aggregates of stem cells can form a patterning axis even in the absence of external spatial cues.

Barramundi (Lates calcarifer) rare coloration patterns: a multiomics approach to understand the "panda" phenotype

The barramundi (Lates calcarifer), a significant aquaculture species, typically displays silver to bronze coloration. However, attention is now drawn to rare variants like the "panda" phenotype, characterized by blotch-like patterns of black (PB) and golden (PG) patches. This phenotype presents an opportunity to explore the molecular mechanisms underlying color variations in teleosts. Unlike stable color patterns in many fish, the "panda" variant demonstrates phenotypic plasticity, responding dynamically to unknown cues. We propose a complex interplay of genetic factors and epigenetic modifications, focusing on DNA methylation. Through a multiomics approach, we analyze transcriptomic and methylation patterns between PB and PG patches. Our study reveals differential gene expression related to melanosome trafficking and chromatophore differentiation. Although the specific gene responsible for the PB-PG difference remains elusive, candidate genes like asip1, asip2, mlph, and mreg have been identified. Methylation emerges as a potential contributor to the "panda" phenotype, with changes in gene promoters like hand2 and dynamin possibly influencing coloration. This research lays the groundwork for further exploration into rare barramundi color patterns, enhancing our understanding of color diversity in teleosts. Additionally, it underscores the "panda" phenotype's potential as a model for studying adult skin coloration.

PMEL is involved in snake colour pattern transition from blotches to stripes

Corn snakes are emerging models for animal colouration studies. Here, we focus on the Terrazzo morph, whose skin pattern is characterized by stripes rather than blotches. Using genome mapping, we discover a disruptive mutation in the coding region of the Premelanosome protein (PMEL) gene. Our transcriptomic analyses reveal that PMEL expression is significantly downregulated in Terrazzo embryonic tissues. We produce corn snake PMEL knockouts, which present a comparable colouration phenotype to Terrazzo and the subcellular structure of their melanosomes and xanthosomes is also similarly impacted. Our single-cell expression analyses of wild-type embryonic dorsal skin demonstrate that all chromatophore progenitors express PMEL at varying levels. Finally, we show that in wild-type embryos PMEL-expressing cells are initially uniformly spread before forming aggregates and eventually blotches, as seen in the adults. In Terrazzo embryos, the aggregates fail to form. Our results provide insights into the mechanisms governing colouration patterning in reptiles.

Myron Gordon Award Lecture 2023: Painting the neural crest: How studying pigment cells illuminates neural crest cell biology

It has been 30 (!!) years since I began working on zebrafish pigment cells, as a postdoc in the laboratory of Prof. Christiane Nüsslein-Volhard. There, I participated in the first large-scale mutagenesis screen in zebrafish, focusing on pigment cell mutant phenotypes. The isolation of colourless, shady, parade and choker mutants allowed us (as a postdoc in Prof. Judith Eisen's laboratory, and then in my own laboratory at the University of Bath since 1997) to pursue my ambition to address long-standing problems in the neural crest field. Thus, we have studied how neural crest cells choose individual fates, resulting in our recent proposal of a new, and potentially unifying, model which we call Cyclical Fate Restriction, as well as addressing how pigment cell patterns are generated. A key feature of our work in the last 10 years has been the use of mathematical modelling approaches to clarify our biological models and to refine our interpretations. None of this would have been possible without a hugely talented group of laboratory members and other collaborators from around the world-it has been, and I am sure will continue to be, a pleasure and privilege to work with you all!

Instabilities and self-organization in spatiotemporal epidemic dynamics driven by nonlinearity and noise

Theoretical analysis of epidemic dynamics has attracted significant attention in the aftermath of the COVID-19 pandemic. In this article, we study dynamic instabilities in a spatiotemporal compartmental epidemic model represented by a stochastic system of coupled partial differential equations (SPDE). Saturation effects in infection spread-anchored in physical considerations-lead to strong nonlinearities in the SPDE. Our goal is to study the onset of dynamic, Turing-type instabilities, and the concomitant emergence of steady-state patterns under the interplay between three critical model parameters-the saturation parameter, the noise intensity, and the transmission rate. Employing a second-order perturbation analysis to investigate stability, we uncover both diffusion-driven and noise-induced instabilities and corresponding self-organized distinct patterns of infection spread in the steady state. We also analyze the effects of the saturation parameter and the transmission rate on the instabilities and the pattern formation. In summary, our results indicate that the nuanced interplay between the three parameters considered has a profound effect on the emergence of dynamical instabilities and therefore on pattern formation in the steady state. Moreover, due to the central role played by the Turing phenomenon in pattern formation in a variety of biological dynamic systems, the results are expected to have broader significance beyond epidemic dynamics.

Diffusion-driven instabilities in the BT-GN oscillatory carbonylation reaction network

This study explores the role of diffusion in creating instabilities in the Bruk Temkin-Gorodsky Novakovic (BT-GN) oscillatory carbonylation reaction network. Stoichiometric network analysis and numerical methods revealed the presence of two destabilizing feedback cycles responsible for these instabilities. Analysis of a spatially uniform system showed that the saddle-node bifurcation can be simulated within the reaction network. The introduction of diffusion results in two types of instabilities: one occurs when a spatially uniform system is already unstable, leading to a reaction-diffusion front; and another involves diffusion-driven instabilities where introducing diffusion destabilizes a stable spatially uniform system. Slower PdI2 diffusion plays a key role in inducing these instabilities. Equations describing conditions for the emergence of the instabilities in both cases were derived.

Linking discrete and continuous models of cell birth and migration

Self-organization of individuals within large collectives occurs throughout biology. Mathematical models can help elucidate the individual-level mechanisms behind these dynamics, but analytical tractability often comes at the cost of biological intuition. Discrete models provide straightforward interpretations by tracking each individual yet can be computationally expensive. Alternatively, continuous models supply a large-scale perspective by representing the 'effective' dynamics of infinite agents, but their results are often difficult to translate into experimentally relevant insights. We address this challenge by quantitatively linking spatio-temporal dynamics of continuous models and individual-based data in settings with biologically realistic, time-varying cell numbers. Specifically, we introduce and fit scaling parameters in continuous models to account for discrepancies that can arise from low cell numbers and localized interactions. We illustrate our approach on an example motivated by zebrafish-skin pattern formation, in which we create a continuous framework describing the movement and proliferation of a single cell population by upscaling rules from a discrete model. Our resulting continuous models accurately depict ensemble average agent-based solutions when migration or proliferation act alone. Interestingly, the same parameters are not optimal when both processes act simultaneously, highlighting a rich difference in how combining migration and proliferation affects discrete and continuous dynamics.

Phase textures of metal-oxide nanocomposites self-orchestrated by atomic diffusions through precursor alloys

Metal-oxide nanocomposites (MONs) are of pivotal importance as electrode materials, yet lack a guiding principle to tune their phase texture. Here we report that the phase texture of MONs can be tuned at the nanoscale by controlling the nanophase separation of precursor alloys. In situ transmission electron microscopy (in situ TEM) has demonstrated that a MON material of platinum (Pt) and cerium oxide (CeO2) is obtained through promoted nanophase separation of a Pt5Ce precursor alloy in an atmosphere containing oxygen (O2) and carbon monoxide (CO). The Pt-CeO2 MON material comprised an alternating stack of nanometre-thick layers of Pt and CeO2 in different phase textures ranging from lamellae to mazes, depending on the O2 fraction in the atmosphere. Mathematical simulations have demonstrated that the phase texture of MONs originates from a balance in the atomic diffusions across the alloy precursor, which is controllable by the O2 fraction, temperature, and composition of the precursor alloys.

Minimal reaction schemes for pattern formation

We link continuum models of reaction-diffusion systems that exhibit diffusion-driven instability to constraints on the particle-scale interactions underpinning this instability. While innumerable biological, chemical and physical patterns have been studied through the lens of Alan Turing's reaction-diffusion pattern-forming mechanism, the connections between models of pattern formation and the nature of the particle interactions generating them have been relatively understudied in comparison with the substantial efforts that have been focused on understanding proposed continuum systems. To derive the necessary reactant combinations for the most parsimonious reaction schemes, we analyse the emergent continuum models in terms of possible generating elementary reaction schemes. This analysis results in the complete list of such schemes containing the fewest reactions; these are the simplest possible hypothetical mass-action models for a pattern-forming system of two interacting species.

Mechanisms Underlying the Formation and Evolution of Vertebrate Color Patterns

Vertebrates exhibit a wide range of color patterns, which play critical roles in mediating intra- and interspecific communication. Because of their diversity and visual accessibility, color patterns offer a unique and fascinating window into the processes underlying biological organization. In this review, we focus on describing many of the general principles governing the formation and evolution of color patterns in different vertebrate groups. We characterize the types of patterns, review the molecular and developmental mechanisms by which they originate, and discuss their role in constraining or facilitating evolutionary change. Lastly, we outline outstanding questions in the field and discuss different approaches that can be used to address them. Overall, we provide a unifying conceptual framework among vertebrate systems that may guide research into naturally evolved mechanisms underlying color pattern formation and evolution.

Patterning of nonlocal transport models in biology: The impact of spatial dimension

Throughout developmental biology and ecology, transport can be driven by nonlocal interactions. Examples include cells that migrate based on contact with pseudopodia extended from other cells, and animals that move based on their awareness of other animals. Nonlocal integro-PDE models have been used to investigate contact attraction and repulsion in cell populations in 1D. In this paper, we generalise the analysis of pattern formation in such a model from 1D to higher spatial dimensions. Numerical simulations in 2D demonstrate complex behaviour in the model, including spatio-temporal patterns, multi-stability, and patterns with wavelength and shape that differ significantly depending on whether interactions are attractive or repulsive. Through linear stability analysis in N dimensions, we demonstrate how, unlike in local Turing reaction-diffusion models, the capacity for pattern formation fundamentally changes with dimensionality for this nonlocal model. Most notably, pattern formation is possible only in higher than one spatial dimension for both the single species system with repulsive interactions, and the two species system with 'run-and-chase' interactions. The latter case may be relevant to zebrafish stripe formation, which has been shown to be driven by run-and-chase dynamics between melanophore and xanthophore pigment cells.

The Unreasonable Effectiveness of Reaction Diffusion in Vertebrate Skin Color Patterning

In 1952, Alan Turing published the reaction-diffusion (RD) mathematical framework, laying the foundations of morphogenesis as a self-organized process emerging from physicochemical first principles. Regrettably, this approach has been widely doubted in the field of developmental biology. First, we summarize Turing's line of thoughts to alleviate the misconception that RD is an artificial mathematical construct. Second, we discuss why phenomenological RD models are particularly effective for understanding skin color patterning at the meso/macroscopic scales, without the need to parameterize the profusion of variables at lower scales. More specifically, we discuss how RD models (a) recapitulate the diversity of actual skin patterns, (b) capture the underlying dynamics of cellular interactions, (c) interact with tissue size and shape, (d) can lead to ordered sequential patterning, (e) generate cellular automaton dynamics in lizards and snakes, (f) predict actual patterns beyond their statistical features, and (g) are robust to model variations. Third, we discuss the utility of linear stability analysis and perform numerical simulations to demonstrate how deterministic RD emerges from the underlying chaotic microscopic agents.

Coupling dynamics of 2D Notch-Delta signalling

Understanding pattern formation driven by cell-cell interactions has been a significant theme in cellular biology for many years. In particular, due to their implications within many biological contexts, lateral-inhibition mechanisms present in the Notch-Delta signalling pathway led to an extensive discussion between biologists and mathematicians. Deterministic and stochastic models have been developed as a consequence of this discussion, some of which address long-range signalling by considering cell protrusions reaching non-neighbouring cells. The dynamics of such signalling systems reveal intricate properties of the coupling terms involved in these models. In this work, we investigate the advantages and drawbacks of a single-parameter long-range signalling model across diverse scenarios. By employing linear and multi-scale analyses, we discover that pattern selection is not only partially explained but also depends on nonlinear effects that extend beyond the scope of these analytical techniques.

Directing Min protein patterns with advective bulk flow

The Min proteins constitute the best-studied model system for pattern formation in cell biology. We theoretically predict and experimentally show that the propagation direction of in vitro Min protein patterns can be controlled by a hydrodynamic flow of the bulk solution. We find downstream propagation of Min wave patterns for low MinE:MinD concentration ratios, upstream propagation for large ratios, but multistability of both propagation directions in between. Whereas downstream propagation can be described by a minimal model that disregards MinE conformational switching, upstream propagation can be reproduced by a reduced switch model, where increased MinD bulk concentrations on the upstream side promote protein attachment. Our study demonstrates that a differential flow, where bulk flow advects protein concentrations in the bulk, but not on the surface, can control surface-pattern propagation. This suggests that flow can be used to probe molecular features and to constrain mathematical models for pattern-forming systems.

Appearance and suppression of Turing patterns under a periodically forced feed

Turing instability is a general and straightforward mechanism of pattern formation in reaction-diffusion systems, and its relevance has been demonstrated in different biological phenomena. Still, there are many open questions, especially on the robustness of the Turing mechanism. Robust patterns must survive some variation in the environmental conditions. Experiments on pattern formation using chemical systems have shown many reaction-diffusion patterns and serve as relatively simple test tools to study general aspects of these phenomena. Here, we present a study of sinusoidal variation of the input feed concentrations on chemical Turing patterns. Our experimental, numerical and theoretical analysis demonstrates that patterns may appear even at significant amplitude variation of the input feed concentrations. Furthermore, using time-dependent feeding opens a way to control pattern formation. The patterns settled at constant feed may disappear, or new patterns may appear from a homogeneous steady state due to the periodic forcing.

Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems

Turing’s theory of morphogenesis is a generic mechanism to produce spatial patterning from near homogeneity. Although widely studied, we are still able to generate new results by returning to common dogmas. One such widely reported belief is that the Turing bifurcation occurs through a pitchfork bifurcation, which is true under zero-flux boundary conditions. However, under fixed boundary conditions, the Turing bifurcation becomes generically transcritical. We derive these algebraic results through weakly nonlinear analysis and apply them to the Schnakenberg kinetics. We observe that the combination of kinetics and boundary conditions produce their own uncommon boundary complexities that we explore numerically. Overall, this work demonstrates that it is not enough to only consider parameter perturbations in a sensitivity analysis of a specific application. Variations in boundary conditions should also be considered.

Spatiotemporal Patterns in a General Networked Hindmarsh-Rose Model

Neuron modelling helps to understand the brain behavior through the interaction between neurons, but its mechanism remains unclear. In this paper, the spatiotemporal patterns is investigated in a general networked Hindmarsh-Rose (HR) model. The stability of the network-organized system without delay is analyzed to show the effect of the network on Turing instability through the Hurwitz criterion, and the conditions of Turing instability are obtained. Once the analysis of the zero-delayed system is completed, the critical value of the delay is derived to illustrate the profound impact of the given network on the collected behaviors. It is found that the difference between the collected current and the outgoing current plays a crucial role in neuronal activity, which can be used to explain the generation mechanism of the short-term memory. Finally, the numerical simulation is presented to verify the proposed theoretical results.

Patterning, From Conifers to Consciousness: Turing's Theory and Order From Fluctuations

This is a brief account of Turing's ideas on biological pattern and the events that led to their wider acceptance by biologists as a valid way to investigate developmental pattern, and of the value of theory more generally in biology. Periodic patterns have played a key role in this process, especially 2D arrays of oriented stripes, which proved a disappointment in theoretical terms in the case of Drosophila segmentation, but a boost to theory as applied to skin patterns in fish and model chemical reactions. The concept of "order from fluctuations" is a key component of Turing's theory, wherein pattern arises by selective amplification of spatial components concealed in the random disorder of molecular and/or cellular processes. For biological examples, a crucial point from an analytical standpoint is knowing the nature of the fluctuations, where the amplifier resides, and the timescale over which selective amplification occurs. The answer clarifies the difference between "inelegant" examples such as Drosophila segmentation, which is perhaps better understood as a programmatic assembly process, and "elegant" ones expressible in equations like Turing's: that the fluctuations and selection process occur predominantly in evolutionary time for the former, but in real time for the latter, and likewise for error suppression, which for Drosophila is historical, in being lodged firmly in past evolutionary events. The prospects for a further extension of Turing's ideas to the complexities of brain development and consciousness is discussed, where a case can be made that it could well be in neuroscience that his ideas find their most important application.

Introduction to 'Recent progress and open frontiers in Turing's theory of morphogenesis'

Elucidating pattern forming processes is an important problem in the physical, chemical and biological sciences. Turing's contribution, after being initially neglected, eventually catalysed a huge amount of work from mathematicians, physicists, chemists and biologists aimed towards understanding how steady spatial patterns can emerge from homogeneous chemical mixtures due to the reaction and diffusion of different chemical species. While this theory has been developed mathematically and investigated experimentally for over half a century, many questions still remain unresolved. This theme issue places Turing's theory of pattern formation in a modern context, discussing the current frontiers in foundational aspects of pattern formation in reaction-diffusion and related systems. It highlights ongoing work in chemical, synthetic and developmental settings which is helping to elucidate how important Turing's mechanism is for real morphogenesis, while highlighting gaps that remain in matching theory to reality. The theme issue also surveys a variety of recent mathematical research pushing the boundaries of Turing's original theory to more realistic and complicated settings, as well as discussing open theoretical challenges in the analysis of such models. It aims to consolidate current research frontiers and highlight some of the most promising future directions. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.