Studies of Turing pattern formation in zebrafish skin

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.

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!

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.

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'.