Turing's theory of morphogenesis of 1952 and the subsequent discovery of the crucial role of local self-enhancement and long-range inhibition

Preparation of surgical meshes using self-regulating technology based on reaction-diffusion processes

While reaction-diffusion processes are utilized in multiple scientific fields, these phenomena have seen limited practical application in the polymer industry. Although self-regulating processes driven by parallel reaction and diffusion can lead to patterned structures, most polymeric products with repeating subunits are still prepared by methods that require complex and expensive instrumentation. A notable, high-added-value example is surgical mesh, which is often manufactured by weaving or knitting. In our present work, we demonstrate how the polymer and the biomedical industry can benefit from the pattern-forming capabilities of reaction-diffusion. We would like to propose a self-regulating method that facilitates the creation of surgical meshes from biocompatible polymers. Since the control of the process assumes a thorough understanding of the underlying phenomena, the theoretical background, as well as a mathematical model that can accurately describe the empirical data, is also introduced and explained. Our method offers the benefits of conventional techniques while introducing additional advantages not attainable with them. Most importantly, the method proposed in this paper enables the rapid creation of meshes with an average pore size that can be adjusted easily and tailored to fit the intended area of application.

Rho of Plants patterning: linking mathematical models and molecular diversity

ROPs (Rho of Plants) are plant specific small GTPases involved in many membrane patterning processes and play important roles in the establishment and communication of cell polarity. These small GTPases can produce a wide variety of patterns, ranging from a single cluster in tip-growing root hairs and pollen tubes to an oriented stripe pattern controlling protoxylem cell wall deposition. For an understanding of what controls these various patterns, models are indispensable. Consequently, many modelling studies on small GTPase patterning exist, often focusing on yeast or animal cells. Multiple patterns occurring in plants, however, require the stable co-existence of multiple active ROP clusters, which does not occur with the most common yeast/animal models. The possibility of such patterns critically depends on the precise model formulation. Additionally, different small GTPases are usually treated interchangeably in models, even though plants possess two types of ROPs with distinct molecular properties, one of which is unique to plants. Furthermore, the shape and even the type of ROP patterns may be affected by the cortical cytoskeleton, and cortex composition and anisotropy differ dramatically between plants and animals. Here, we review insights into ROP patterning from modelling efforts across kingdoms, as well as some outstanding questions arising from these models and recent experimental findings.

Periodic pattern formation during embryonic development

During embryonic development many organs and structures require the formation of series of repeating elements known as periodic patterns. Ranging from the digits of the limb to the feathers of the avian skin, the correct formation of these embryonic patterns is essential for the future form and function of these tissues. However, the mechanisms that produce these patterns are not fully understood due to the existence of several modes of pattern generation which often differ between organs and species. Here, we review the current state of the field and provide a perspective on future approaches to studying this fundamental process of embryonic development.

Forceful patterning: theoretical principles of mechanochemical pattern formation

Biological pattern formation is essential for generating and maintaining spatial structures from the scale of a single cell to tissues and even collections of organisms. Besides biochemical interactions, there is an important role for mechanical and geometrical features in the generation of patterns. We review the theoretical principles underlying different types of mechanochemical pattern formation across spatial scales and levels of biological organization.

A Multigenerational Turing Model Reproduces Transgressive Petal Spot Phenotypes in Hybrid Mimulus

The origin of phenotypic novelty is a perennial question of genetics and evolution. To date, few studies of biological pattern formation specifically address multi-generational aspects of inheritance and phenotypic novelty. For quantitative traits influenced by many segregating alleles, offspring phenotypes are often intermediate to parental values. In other cases, offspring phenotypes can be transgressive to parental values. For example, in the model organism Mimulus (monkeyflower), the offspring of parents with solid-colored petals exhibit novel spotted petal phenotypes. These patterns are controlled by an activator-inhibitor gene regulatory network with a small number of loci. Here we develop and analyze a model of hybridization and pattern formation that accounts for the inheritance of a diploid gene regulatory network composed of either homozygous or heterozygous alleles. We find that the resulting model of multi-generational Turing-type pattern formation can reproduce transgressive petal phenotypes similar to those observed in Mimulus. The model gives insight into how non-patterned parent phenotypes can yield phenotypically transgressive, patterned offspring, aiding in the development of empirically testable hypotheses.

Closing the loop on morphogenesis: a mathematical model of morphogenesis by closed-loop reaction-diffusion

Morphogenesis, the establishment and repair of emergent complex anatomy by groups of cells, is a fascinating and biomedically-relevant problem. One of its most fascinating aspects is that a developing embryo can reliably recover from disturbances, such as splitting into twins. While this reliability implies some type of goal-seeking error minimization over a morphogenic field, there are many gaps with respect to detailed, constructive models of such a process. A common way to achieve reliability is negative feedback, which requires characterizing the existing body shape to create an error signal-but measuring properties of a shape may not be simple. We show how cells communicating in a wave-like pattern could analyze properties of the current body shape. We then describe a closed-loop negative-feedback system for creating reaction-diffusion (RD) patterns with high reliability. Specifically, we use a wave to count the number of peaks in a RD pattern, letting us use a negative-feedback controller to create a pattern with N repetitions, where N can be altered over a wide range. Furthermore, the individual repetitions of the RD pattern can be easily stretched or shrunk under genetic control to create, e.g., some morphological features larger than others. This work contributes to the exciting effort of understanding design principles of morphological computation, which can be used to understand evolved developmental mechanisms, manipulate them in regenerative-medicine settings, or engineer novel synthetic morphology constructs with desired robust behavior.

Spatiotemporal Coordination of Rac1 and Cdc42 at the Whole Cell Level during Cell Ruffling

Rho-GTPases are central regulators within a complex signaling network that controls cytoskeletal organization and cell movement. The network includes multiple GTPases, such as the most studied Rac1, Cdc42, and RhoA, along with their numerous effectors that provide mutual regulation through feedback loops. Here we investigate the temporal and spatial relationship between Rac1 and Cdc42 during membrane ruffling, using a simulation model that couples GTPase signaling with cell morphodynamics and captures the GTPase behavior observed with FRET-based biosensors. We show that membrane velocity is regulated by the kinetic rate of GTPase activation rather than the concentration of active GTPase. Our model captures both uniform and polarized ruffling. We also show that cell-type specific time delays between Rac1 and Cdc42 activation can be reproduced with a single signaling motif, in which the delay is controlled by feedback from Cdc42 to Rac1. The resolution of our simulation output matches those of time-lapsed recordings of cell dynamics and GTPase activity. Our data-driven modeling approach allows us to validate simulation results with quantitative precision using the same pipeline for the analysis of simulated and experimental data.

Robust morphogenesis by chaotic dynamics

This research illustrates that complex dynamics of gene products enable the creation of any prescribed cellular differentiation patterns. These complex dynamics can take the form of chaotic, stochastic, or noisy chaotic dynamics. Based on this outcome and previous research, it is established that a generic open chemical reactor can generate an exceptionally large number of different cellular patterns. The mechanism of pattern generation is robust under perturbations and it is based on a combination of Turing's machines, Turing instability and L. Wolpert's gradients. These results can help us to explain the formidable adaptive capacities of biochemical systems.

Spatiotemporal coordination of Rac1 and Cdc42 at the whole cell level during cell ruffling

Rho-GTPases are central regulators within a complex signaling network that controls the cytoskeletal organization and cell movement. This network includes multiple GTPases, such as the most studied Rac1, Cdc42, and RhoA, and their numerous effectors that provide mutual regulation and feedback loops. Here we investigate the temporal and spatial relationship between Rac1 and Cdc42 during membrane ruffling using a simulation model which couples GTPase signaling with cell morphodynamics to capture the GTPase behavior observed with FRET-based biosensors. We show that membrane velocity is regulated by the kinetic rate of GTPase activation rather than the concentration of active GTPase. Our model captures both uniform and polarized ruffling. We also show that cell-type specific time delays between Rac1 and Cdc42 activation can be reproduced with a single signaling motif, in which the delay is controlled by feedback from Cdc42 to Rac1. The resolution of our simulation output matches those of the time-lapsed recordings of cell dynamics and GTPase activity. This approach allows us to validate simulation results with quantitative precision using the same pipeline for the analysis of simulated and experimental data.

A group theoretic approach to model comparison with simplicial representations

The complexity of biological systems, and the increasingly large amount of associated experimental data, necessitates that we develop mathematical models to further our understanding of these systems. Because biological systems are generally not well understood, most mathematical models of these systems are based on experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system we therefore need to understand how the different models are related to each other, with a view to obtaining a unified mathematical description. This goal is complicated by the fact that a number of distinct mathematical formalisms may be employed to represent the same system, making direct comparison of the models very difficult. A methodology for comparing mathematical models based on their underlying conceptual structure is therefore required. In previous work we developed an appropriate framework for model comparison where we represent models, specifically the conceptual structure of the models, as labelled simplicial complexes and compare them with the two general methodologies of comparison by distance and comparison by equivalence. In this article we continue the development of our model comparison methodology in two directions. First, we present a rigorous and automatable methodology for the core process of comparison by equivalence, namely determining the vertices in a simplicial representation, corresponding to model components, that are conceptually related and the identification of these vertices via simplicial operations. Our methodology is based on considerations of vertex symmetry in the simplicial representation, for which we develop the required mathematical theory of group actions on simplicial complexes. This methodology greatly simplifies and expedites the process of determining model equivalence. Second, we provide an alternative mathematical framework for our model-comparison methodology by representing models as groups, which allows for the direct application of group-theoretic techniques within our model-comparison methodology.

Pattern formation features might explain homoplasy: fertile surfaces in higher fungi as an example

Fungi show a high degree of morphological convergence. Regarded for a long time as an obstacle for phylogenetic studies, homoplasy has also been proposed as a source of information about underlying morphogenetic patterning mechanisms. The "local-activation and long-range inhibition principle" (LALIP), underlying the famous reaction-diffusion model proposed by Alan Turing in 1952, appears to be one of the universal phenomena that can explain the ontogenetic origin of seriate patterns in living organisms. Reproductive structures of fungi in the class Agaricomycetes show a highly periodic structure resulting in, for example, poroid, odontoid, lamellate or labyrinthic hymenophores. In this paper, we claim that self-organized patterns might underlie the basic ontogenetic processes of these structures. Simulations based on LALIP-driven models and covering a wide range of parameters show an absolute mutual correspondence with the morphospace explored by extant agaricomycetes. This could not only explain geometric particularities but could also account for the limited possibilities displayed by hymenial configurations, thus making homoplasy a direct consequence of the limited morphospace resulting from the proposed patterning dynamics.

Quantitative Insights into the Contribution of Nematocysts to the Adaptive Success of Cnidarians Based on Proteomic Analysis

Nematocysts are secretory organelles in cnidarians that play important roles in predation, defense, locomotion, and host invasion. However, the extent to which nematocysts contribute to adaptation and the mechanisms underlying nematocyst evolution are unclear. Here, we investigated the role of the nematocyst in cnidarian evolution based on eight nematocyst proteomes and 110 cnidarian transcriptomes/genomes. We detected extensive species-specific adaptive mutations in nematocyst proteins (NEMs) and evidence for decentralized evolution, in which most evolutionary events involved non-core NEMs, reflecting the rapid diversification of NEMs in cnidarians. Moreover, there was a 33-55 million year macroevolutionary lag between nematocyst evolution and the main phases of cnidarian diversification, suggesting that the nematocyst can act as a driving force in evolution. Quantitative analysis revealed an excess of adaptive changes in NEMs and enrichment for positively selected conserved NEMs. Together, these findings suggest that nematocysts may be key to the adaptive success of cnidarians and provide a reference for quantitative analyses of the roles of phenotypic novelties in adaptation.

Beyond Turing: far-from-equilibrium patterns and mechano-chemical feedback

Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator-inhibitor interaction destabilized by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternatives to the classical Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives vis-à-vis the classical Turing analysis, and give an overview of recent results and future challenges for both approaches. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Insights from chemical systems into Turing-type morphogenesis

In 1952, Alan Turing proposed a theory showing how morphogenesis could occur from a simple two morphogen reaction-diffusion system [Turing, A. M. (1952) Phil. Trans. R. Soc. Lond. A 237, 37-72. (doi:10.1098/rstb.1952.0012)]. While the model is simple, it has found diverse applications in fields such as biology, ecology, behavioural science, mathematics and chemistry. Chemistry in particular has made significant contributions to the study of Turing-type morphogenesis, providing multiple reproducible experimental methods to both predict and study new behaviours and dynamics generated in reaction-diffusion systems. In this review, we highlight the historical role chemistry has played in the study of the Turing mechanism, summarize the numerous insights chemical systems have yielded into both the dynamics and the morphological behaviour of Turing patterns, and suggest future directions for chemical studies into Turing-type morphogenesis. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Model comparison via simplicial complexes and persistent homology

In many scientific and technological contexts, we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models a priori. Here, we develop and illustrate two such approaches that allow us to compare model structures in a systematic way by representing models as simplicial complexes. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent model structure, from which the model distances are also obtained. We then expand on this measure of model distance to study the concept of model equivalence to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.

The regulatory network for petal anthocyanin pigmentation is shaped by the MYB5a/NEGAN transcription factor in Mimulus

Much of the visual diversity of angiosperms is due to the frequent evolution of novel pigmentation patterns in flowers. The gene network responsible for anthocyanin pigmentation, in particular, has become a model for investigating how genetic changes give rise to phenotypic innovation. In the monkeyflower genus Mimulus, an evolutionarily recent gain of petal lobe anthocyanin pigmentation in M. luteus var. variegatus was previously mapped to genomic region pla2. Here, we use sequence and expression analysis, followed by transgenic manipulation of gene expression, to identify MYB5a—orthologous to the NEGAN transcriptional activator from M. lewisii—as the gene responsible for the transition to anthocyanin-pigmented petals in M. l. variegatus. In other monkeyflower taxa, MYB5a/NEGAN is part of a reaction-diffusion network that produces semi-repeating spotting patterns, such as the array of spots in the nectar guides of both M. lewisii and M. guttatus. Its co-option for the evolution of an apparently non-patterned trait—the solid petal lobe pigmentation of M. l. variegatus—illustrates how reaction-diffusion can contribute to evolutionary novelty in non-obvious ways. Transcriptome sequencing of a MYB5a RNAi line of M. l. variegatus reveals that this genetically simple change, which we hypothesize to be a regulatory mutation in cis to MYB5a, has cascading effects on gene expression, not only on the enzyme-encoding genes traditionally thought of as the targets of MYB5a but also on all of its known partners in the anthocyanin regulatory network.

Repressors of anthocyanin biosynthesis

Anthocyanins play a variety of adaptive roles in both vegetative tissues and reproductive organs of plants. The broad functionality of these compounds requires sophisticated regulation of the anthocyanin biosynthesis pathway to allow proper localization, timing, and optimal intensity of pigment deposition. While it is well-established that the committed steps of anthocyanin biosynthesis are activated by a highly conserved MYB-bHLH-WDR (MBW) protein complex in virtually all flowering plants, anthocyanin repression seems to be achieved by a wide variety of protein and small RNA families that function in different tissue types and in response to different developmental, environmental, and hormonal cues. In this review, we survey recent progress in the identification of anthocyanin repressors and the characterization of their molecular mechanisms. We find that these seemingly very different repression modules act through a remarkably similar logic, the so-called 'double-negative logic'. Much of the double-negative regulation of anthocyanin production involves signal-induced degradation or sequestration of the repressors from the MBW protein complex. We discuss the functional and evolutionary advantages of this logic design compared with simple or sequential positive regulation. These advantages provide a plausible explanation as to why plants have evolved so many anthocyanin repressors.

Fish primary embryonic pluripotent cells assemble into retinal tissue mirroring in vivo early eye development

Organoids derived from pluripotent stem cells promise the solution to current challenges in basic and biomedical research. Mammalian organoids are however limited by long developmental time, variable success, and lack of direct comparison to an in vivo reference. To overcome these limitations and address species-specific cellular organization, we derived organoids from rapidly developing teleosts. We demonstrate how primary embryonic pluripotent cells from medaka and zebrafish efficiently assemble into anterior neural structures, particularly retina. Within 4 days, blastula-stage cell aggregates reproducibly execute key steps of eye development: retinal specification, morphogenesis, and differentiation. The number of aggregated cells and genetic factors crucially impacted upon the concomitant morphological changes that were intriguingly reflecting the in vivo situation. High efficiency and rapid development of fish-derived organoids in combination with advanced genome editing techniques immediately allow addressing aspects of development and disease, and systematic probing of impact of the physical environment on morphogenesis and differentiation.

Signaling gradients in surface dynamics as basis for planarian regeneration

Based on experimental data, we introduce and analyze a system of reaction-diffusion equations for the regeneration of planarian flatworms. We model dynamics of head and tail cells expressing positional control genes that translate into localized signals which in turn guide stem cell differentiation. Tissue orientation and positional information are encoded in a long range wnt-related signaling gradient. Our system correctly reproduces typical cut and graft experiments, and improves on previous models by preserving polarity in regeneration over orders of magnitude in body size during growth phases. Key to polarity preservation in our model flatworm is the sensitivity of cell differentiation to gradients of wnt-related signals relative to the tissue surface. This process is particularly relevant in small tissue layers close to cuts during their healing, and modeled in a robust fashion through dynamic boundary conditions.

The Wnt-specific astacin proteinase HAS-7 restricts head organizer formation in Hydra

Background

The Hydra head organizer acts as a signaling center that initiates and maintains the primary body axis in steady state polyps and during budding or regeneration. Wnt/beta-Catenin signaling functions as a primary cue controlling this process, but how Wnt ligand activity is locally restricted at the protein level is poorly understood. Here we report a proteomic analysis of Hydra head tissue leading to the identification of an astacin family proteinase as a Wnt processing factor.

Results

Hydra astacin-7 (HAS-7) is expressed from gland cells as an apical-distal gradient in the body column, peaking close beneath the tentacle zone. HAS-7 siRNA knockdown abrogates HyWnt3 proteolysis in the head tissue and induces a robust double axis phenotype, which is rescued by simultaneous HyWnt3 knockdown. Accordingly, double axes are also observed in conditions of increased Wnt activity as in transgenic actin::HyWnt3 and HyDkk1/2/4 siRNA treated animals. HyWnt3-induced double axes in Xenopus embryos could be rescued by coinjection of HAS-7 mRNA. Mathematical modelling combined with experimental promotor analysis indicate an indirect regulation of HAS-7 by beta-Catenin, expanding the classical Turing-type activator-inhibitor model.

Conclusions

We show the astacin family protease HAS-7 maintains a single head organizer through proteolysis of HyWnt3. Our data suggest a negative regulatory function of Wnt processing astacin proteinases in the global patterning of the oral-aboral axis in Hydra.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12915-021-01046-9.

Influence of survival, promotion, and growth on pattern formation in zebrafish skin

The coloring of zebrafish skin is often used as a model system to study biological pattern formation. However, the small number and lack of movement of chromatophores defies traditional Turing-type pattern generating mechanisms. Recent models invoke discrete short-range competition and long-range promotion between different pigment cells as an alternative to a reaction-diffusion scheme. In this work, we propose a lattice-based “Survival model,” which is inspired by recent experimental findings on the nature of long-range chromatophore interactions. The Survival model produces stationary patterns with diffuse stripes and undergoes a Turing instability. We also examine the effect that domain growth, ubiquitous in biological systems, has on the patterns in both the Survival model and an earlier “Promotion” model. In both cases, domain growth alone is capable of orienting Turing patterns above a threshold wavelength and can reorient the stripes in ablated cells, though the wavelength for which the patterns orient is much larger for the Survival model. While the Survival model is a simplified representation of the multifaceted interactions between pigment cells, it reveals complex organizational behavior and may help to guide future studies.

Synthetic Lateral Inhibition in Periodic Pattern Forming Microbial Colonies

Multicellular entities are characterized by intricate spatial patterns, intimately related to the functions they perform. These patterns are often created from isotropic embryonic structures, without external information cues guiding the symmetry breaking process. Mature biological structures also display characteristic scales with repeating distributions of signals or chemical species across space. Many candidate patterning modules have been used to explain processes during development and typically include a set of interacting and diffusing chemicals or agents known as morphogens. Great effort has been put forward to better understand the conditions in which pattern-forming processes can occur in the biological domain. However, evidence and practical knowledge allowing us to engineer symmetry-breaking is still lacking. Here we follow a different approach by designing a synthetic gene circuit in E. coli that implements a local activation long-range inhibition mechanism. The synthetic gene network implements an artificial differentiation process that changes the physicochemical properties of the agents. Using both experimental results and modeling, we show that the proposed system is capable of symmetry-breaking leading to regular spatial patterns during colony growth. Studying how these patterns emerge is fundamental to further our understanding of the evolution of biocomplexity and the role played by self-organization. The artificial system studied here and the engineering perspective on embryogenic processes can help validate developmental theories and identify universal properties underpinning biological pattern formation, with special interest for the area of synthetic developmental biology.

Perturbation analysis of a multi-morphogen Turing reaction-diffusion stripe patterning system reveals key regulatory interactions

Periodic patterning is widespread in development and can be modelled by reaction-diffusion (RD) processes. However, minimal two-component RD descriptions are vastly simpler than the multi-molecular events that actually occur and are often hard to relate to real interactions measured experimentally. Addressing these issues, we investigated the periodic striped patterning of the rugae (transverse ridges) in the mammalian oral palate, focusing on multiple previously implicated pathways: FGF, Hh, Wnt and BMP. For each, we experimentally identified spatial patterns of activity and distinct responses of the system to inhibition. Through numerical and analytical approaches, we were able to constrain substantially the number of network structures consistent with the data. Determination of the dynamics of pattern appearance further revealed its initiation by 'activators' FGF and Wnt, and 'inhibitor' Hh, whereas BMP and mesenchyme-specific-FGF signalling were incorporated once stripes were formed. This further limited the number of possible networks. Experimental constraint thus limited the number of possible minimal networks to 154, just 0.004% of the number of possible diffusion-driven instability networks. Together, these studies articulate the principles of multi-morphogen RD patterning and demonstrate the utility of perturbation analysis for constraining RD systems.This article has an associated 'The people behind the papers' interview.

Robust banded protoxylem pattern formation through microtubule-based directional ROP diffusion restriction

In plant vascular tissue development, different cell wall patterns are formed, offering different mechanical properties optimised for different growth stages. Critical in these patterning processes are Rho of Plants (ROP) proteins, a class of evolutionarily conserved small GTPase proteins responsible for local membrane domain formation in many organisms. While te spotted metaxylem pattern can easily be understood as a result of a Turing-style reaction-diffusion mechanism, it remains an open question how the consistent orientation of evenly spaced bands and spirals as found in protoxylem is achieved. We hypothesise that this orientation results from an interaction between ROPs and an array of transversely oriented cortical microtubules that acts as a directional diffusion barrier. Here, we explore this hypothesis using partial differential equation models with anisotropic ROP diffusion and show that a horizontal microtubule array acting as a vertical diffusion barrier to active ROP can yield a horizontally banded ROP pattern. We then study the underlying mechanism in more detail, finding that it can only orient curved pattern features but not straight lines. This implies that, once formed, banded and spiral patterns cannot be reoriented by this mechanism. Finally, we observe that ROPs and microtubules together only form ultimately static patterns if the interaction is implemented with sufficient biological realism.

Effective range of non-cell autonomous activator and inhibitor peptides specifying plant stomatal patterning

Stomata are epidermal valves that facilitate gas exchange between plants and their environment. Stomatal patterning is regulated by the EPIDERMAL PATTERING FACTOR (EPF) family of secreted peptides: EPF1 enforces stomatal spacing, whereas EPIDERMAL PATTERNING FACTOR-LIKE9 (EPFL9), also known as Stomagen, promotes stomatal development. It remains unknown, however, how far these signaling peptides act. Utilizing Cre-lox recombination-based mosaic sectors that overexpress either EPF1 or Stomagen in Arabidopsis cotyledons, we reveal a range within the epidermis and across the cell layers in which these peptides influence patterns. To determine their effective ranges quantitatively, we developed a computational pipeline, SPACE (stomata patterning autocorrelation on epidermis), that describes probabilistic two-dimensional stomatal distributions based upon spatial autocorrelation statistics used in astrophysics. The SPACE analysis shows that, whereas both peptides act locally, the inhibitor EPF1 exerts longer range effects than the activator Stomagen. Furthermore, local perturbation of stomatal development has little influence on global two-dimensional stomatal patterning. Our findings conclusively demonstrate the nature and extent of EPF peptides as non-cell autonomous local signals and provide a means for quantitative characterization of complex spatial patterns in development.This article has an associated 'The people behind the papers' interview.

Grading TESI: Crypt and villus formation in tissue-engineered small intestine alters with stem/progenitor cell source

The small intestine has a remarkable ability to enhance its absorptive and digestive surface area through the formation of villi, a process known as villification. We sought to learn whether developing mouse and human tissue-engineered small intestine (TESI) followed known developmental biology routes to villification, such as Sonic hedgehog (SHH)/Indian hedgehog (IHH) and bone morphogenetic protein 4 (BMP4)/forkhead box F1 (FOXF1) signaling to identify targets to enhance the development of TESI. After generating TESI from prenatal and postnatal stem cell sources, we evaluated the effect of cell source derivation on villification with a grading scheme to approximate developmental stage. χ² analysis compared the prevalence of TESI grade from each stem cell source. RNAscope probes detected genes known to direct villification and the development of the crypt-villus axis in mouse and human development. These were compared in TESI derived from various pluripotent and progenitor cell donor cell types as well as native human fetal and postnatal tissues. Prenatal and pluripotent cell sources form mature villus and crypt-like structures more frequently than postnatal donor sources, and there are alternate routes to villus formation. Human TESI recapitulates epithelial to mesenchymal crosstalk of several genes identified in development, with fetal and pluripotent donor-derived TESI arriving at villus formation following described developmental patterns. However, postnatal TESI is much less likely to form complete villus-crypt patterns and demonstrates alternate SHH/IHH and BMP4/FOXF1 signaling patterns. Grading TESI and other cellular constructs may assist discoveries to support future human therapies.NEW & NOTEWORTHY The small intestine can enhance its absorptive and digestive surface area through a process known as villification. Tissue-engineered small intestine achieves mature villification at varying levels of success between differing sources. We have developed a consistent grading schema of morphology and characterized it across multiple developmental pathways, allowing objective comparison between differing constructs and sources.

On the problem of biological form

Embryonic development, which inspired the first theories of biological form, was eventually excluded from the conceptual framework of the Modern Synthesis as irrelevant. A major question during the last decades has centred on understanding whether new advances in developmental biology are compatible with the standard view or whether they compel a new theory. Here, I argue that the answer to this question depends on which concept of morphogenesis is held. Morphogenesis can be conceived as (1) a chemically driven or (2) a mechanically driven process. According to the first option, genetic regulatory networks drive morphogenesis. According to the second, morphogenesis results from an invariant tendency of embryonic tissues to restore changes in mechanical stress. While chemically driven morphogenesis allows an extension of the standard view, mechanically driven morphogenesis would deeply transform it. Which of these hypotheses has wider explanatory power is unknown. At present, the problem of biological form remains unsolved.

Pattern formation mechanisms of self-organizing reaction-diffusion systems

Embryonic development is a largely self-organizing process, in which the adult body plan arises from a ball of cells with initially nearly equal potency. The reaction-diffusion theory first proposed by Alan Turing states that the initial symmetry in embryos can be broken by the interplay between two diffusible molecules, whose interactions lead to the formation of patterns. The reaction-diffusion theory provides a valuable framework for self-organized pattern formation, but it has been difficult to relate simple two-component models to real biological systems with multiple interacting molecular species. Recent studies have addressed this shortcoming and extended the reaction-diffusion theory to realistic multi-component networks. These efforts have challenged the generality of previous central tenets derived from the analysis of simplified systems and guide the way to a new understanding of self-organizing processes. Here, we discuss the challenges in modeling multi-component reaction-diffusion systems and how these have recently been addressed. We present a synthesis of new pattern formation mechanisms derived from these analyses, and we highlight the significance of reaction-diffusion principles for developmental and synthetic pattern formation.

Two MYB Proteins in a Self-Organizing Activator-Inhibitor System Produce Spotted Pigmentation Patterns

Many organisms exhibit visually striking spotted or striped pigmentation patterns. Developmental models predict that such spatial patterns can form when a local autocatalytic feedback loop and a long-range inhibitory feedback loop interact. At its simplest, this self-organizing network only requires one self-activating activator that also activates a repressor, which inhibits the activator and diffuses to neighboring cells. However, the molecular activators and inhibitors fully fitting this versatile model remain elusive in pigmentation systems. Here, we characterize an R2R3-MYB activator and an R3-MYB repressor in monkeyflowers (Mimulus). Through experimental perturbation and mathematical modeling, we demonstrate that the properties of these two proteins correspond to an activator-inhibitor pair in a two-component, reaction-diffusion system, explaining the formation of dispersed anthocyanin spots in monkeyflower petals. Notably, disrupting this pattern impacts pollinator visitation. Thus, subtle changes in simple activator-inhibitor systems are likely essential contributors to the evolution of the remarkable diversity of pigmentation patterns in flowers.

Development: Painting Flowers with MYBs

Color patterns influence how attractive flowers are to bees, butterflies, and birds. By combining experiments and theory, a new study shows how a pair of MYB transcription factors orchestrates the formation of pigmentation patterns on monkeyflowers.

NetLogo agent-based models as tools for understanding the self-organization of cell fate, morphogenesis and collective migration of the zebrafish posterior Lateral Line primordium

Interactions between primordium cells and their environment determines the self-organization of the zebrafish posterior Lateral Line primordium as it migrates under the skin from the ear to the tip of the tail forming and depositing neuromasts to spearhead formation of the posterior Lateral Line sensory system. In this review we describe how the NetLogo agent-based programming environment has been used in our lab to visualize and explore how self-generated chemokine gradients determine collective migration, how the dynamics of Wnt signaling can be used to predict patterns of neuromast deposition, and how previously defined interactions between Wnt and Fgf signaling systems have the potential to determine the periodic formation of center-biased Fgf signaling centers in the wake of a shrinking Wnt system. We also describe how NetLogo was used as a database for storing and visualizing the results of in toto lineage analysis of all cells in the migrating primordium. Together, the models illustrate how this programming environment can be used in diverse ways to integrate what has been learnt from biological experiments about the nature of interactions between cells and their environment, and explore how these interactions could potentially determine emergent patterns of cell fate specification, morphogenesis and collective migration of the zebrafish posterior Lateral Line primordium.

The Morphogenesis of Speech Gestures: From Local Computations to Global Patterns

A subtle property of speech gestures is the fact that they are spatially and temporally extended, meaning that phonological contrasts are expressed using spatially extended constrictions, and have a finite duration. This paper shows how this spatiotemporal particulation of the vocal tract, for the purpose of linguistic signaling, comes about. It is argued that local uniform computations among topographically organized microscopic units that either constrict or relax individual points of the vocal tract yield the global spatiotemporal macroscopic structures we call constrictions, the locus of phonological contrast. The dynamical process is a morphogenetic one, based on the Turing and Hopf patterns of mathematical physics and biology. It is shown that reaction-diffusion equations, which are introduced in a tutorial mathematical style, with simultaneous Turing and Hopf patterns predict the spatiotemporal particulation, as well as concrete properties of speech gestures, namely the pivoting of constrictions, as well as the intermediate value of proportional time to peak velocity, which is well-studied and observed. The goal of the paper is to contribute to Bernstein's program of understanding motor processes as the emergence of low degree of freedom descriptions from high degree of freedom systems by actually pointing to specific, predictive, dynamics that yield speech gestures from a reaction-diffusion morphogenetic process.

The Sialic Acid-Dependent Nematocyst Discharge Process in Relation to Its Physical-Chemical Properties Is A Role Model for Nanomedical Diagnostic and Therapeutic Tools

Formulas derived from theoretical physics provide important insights about the nematocyst discharge process of Cnidaria (Hydra, jellyfishes, box-jellyfishes and sea-anemones). Our model description of the fastest process in living nature raises and answers questions related to the material properties of the cell- and tubule-walls of nematocysts including their polysialic acid (polySia) dependent target function. Since a number of tumor-cells, especially brain-tumor cells such as neuroblastoma tissues carry the polysaccharide chain polySia in similar concentration as fish eggs or fish skin, it makes sense to use these findings for new diagnostic and therapeutic approaches in the field of nanomedicine. Therefore, the nematocyst discharge process can be considered as a bionic blue-print for future nanomedical devices in cancer diagnostics and therapies. This approach is promising because the physical background of this process can be described in a sufficient way with formulas presented here. Additionally, we discuss biophysical and biochemical experiments which will allow us to define proper boundary conditions in order to support our theoretical model approach. PolySia glycans occur in a similar density on malignant tumor cells than on the cell surfaces of Cnidarian predators and preys. The knowledge of the polySia-dependent initiation of the nematocyst discharge process in an intact nematocyte is an essential prerequisite regarding the further development of target-directed nanomedical devices for diagnostic and therapeutic purposes. The theoretical description as well as the computationally and experimentally derived results about the biophysical and biochemical parameters can contribute to a proper design of anti-tumor drug ejecting vessels which use a stylet-tubule system. Especially, the role of nematogalectins is of interest because these bridging proteins contribute as well as special collagen fibers to the elastic band properties. The basic concepts of the nematocyst discharge process inside the tubule cell walls of nematocysts were studied in jellyfishes and in Hydra which are ideal model organisms. Hydra has already been chosen by Alan Turing in order to figure out how the chemical basis of morphogenesis can be described in a fundamental way. This encouraged us to discuss the action of nematocysts in relation to morphological aspects and material requirements. Using these insights, it is now possible to discuss natural and artificial nematocyst-like vessels with optimized properties for a diagnostic and therapeutic use, e.g., in neurooncology. We show here that crucial physical parameters such as pressure thresholds and elasticity properties during the nematocyst discharge process can be described in a consistent and satisfactory way with an impact on the construction of new nanomedical devices.

From non-excitable single-cell to multicellular bioelectrical states supported by ion channels and gap junction proteins: Electrical potentials as distributed controllers

Endogenous bioelectric patterns within tissues are an important driver of morphogenesis and a tractable component of a number of disease states. Developing system-level understanding of the dynamics by which non-neural bioelectric circuits regulate complex downstream cascades is a key step towards both, an evolutionary understanding of ion channel genes, and novel strategies in regenerative medicine. An important capability gap is deriving rational modulation strategies targeting individual cells' bioelectric states to achieve global (tissue- or organ-level) outcomes. Here, we develop an ion channel-based model that describes multicellular states on the basis of spatio-temporal patterns of electrical potentials in aggregates of non-excitable cells. The model is of biological interest because modern techniques allow to associate bioelectrical signals with specific ion channel proteins in the cell membrane that are central to embryogenesis, regeneration, and tumorigenesis. As a complementary approach to the usual biochemical description, we have studied four biophysical questions: (i) how can single-cell bioelectrical states be established; (ii) how can a change in the cell potential caused by a transient perturbation of the cell state be maintained after the stimulus is gone (bioelectrical memory); (iii) how can a single-cell contribute to the control of multicellular ensembles based on the spatio-temporal pattern of electrical potentials; and (iv) how can oscillatory patterns arise from the single-cell bioelectrical dynamics. Experimentally, endogenous bioelectric gradients have emerged as instructive agents for morphogenetic processes. In this context, the simulations can guide new procedures that may allow a distributed control of the multicellular ensemble.

Self-organizing hair peg-like structures from dissociated skin progenitor cells: New insights for human hair follicle organoid engineering and Turing patterning in an asymmetric morphogenetic field

Human skin progenitor cells will form new hair follicles, although at a low efficiency, when injected into nude mouse skin. To better study and improve upon this regenerative process, we developed an in vitro system to analyse the morphogenetic cell behaviour in detail and modulate physical-chemical parameters to more effectively generate hair primordia. In this three-dimensional culture, dissociated human neonatal foreskin keratinocytes self-assembled into a planar epidermal layer while fetal scalp dermal cells coalesced into stripes, then large clusters, and finally small clusters resembling dermal condensations. At sites of dermal clustering, subjacent epidermal cells protruded to form hair peg-like structures, molecularly resembling hair pegs within the sequence of follicular development. The hair peg-like structures emerged in a coordinated, formative wave, moving from periphery to centre, suggesting that the droplet culture constitutes a microcosm with an asymmetric morphogenetic field. In vivo, hair follicle populations also form in a progressive wave, implying the summation of local periodic patterning events with an asymmetric global influence. To further understand this global patterning process, we developed a mathematical simulation using Turing activator-inhibitor principles in an asymmetric morphogenetic field. Together, our culture system provides a suitable platform to (a) analyse the self-assembly behaviour of hair progenitor cells into periodically arranged hair primordia and (b) identify parameters that impact the formation of hair primordia in an asymmetric morphogenetic field. This understanding will enhance our future ability to successfully engineer human hair follicle organoids.

Small GTPase patterning: How to stabilise cluster coexistence

Many biological processes have to occur at specific locations on the cell membrane. These locations are often specified by the localised activity of small GTPase proteins. Some processes require the formation of a single cluster of active GTPase, also called unipolar polarisation (here “polarisation”), whereas others need multiple coexisting clusters. Moreover, sometimes the pattern of GTPase clusters is dynamically regulated after its formation. This raises the question how the same interacting protein components can produce such a rich variety of naturally occurring patterns. Most currently used models for GTPase-based patterning inherently yield polarisation. Such models may at best yield transient coexistence of at most a few clusters, and hence fail to explain several important biological phenomena. These existing models are all based on mass conservation of total GTPase and some form of direct or indirect positive feedback. Here, we show that either of two biologically plausible modifications can yield stable coexistence: including explicit GTPase turnover, i.e., breaking mass conservation, or negative feedback by activation of an inhibitor like a GAP. Since we start from two different polarising models our findings seem independent of the precise self-activation mechanism. By studying the net GTPase flows among clusters, we provide insight into how these mechanisms operate. Our coexistence models also allow for dynamical regulation of the final pattern, which we illustrate with examples of pollen tube growth and the branching of fungal hyphae. Together, these results provide a better understanding of how cells can tune a single system to generate a wide variety of biologically relevant patterns.

Post-Turing tissue pattern formation: Advent of mechanochemistry

Chemical and mechanical pattern formation is fundamental during embryogenesis and tissue development. Yet, the underlying molecular and cellular mechanisms are still elusive in many cases. Most current theories assume that tissue development is driven by chemical processes: either as a sequence of chemical patterns each depending on the previous one, or by patterns spontaneously arising from specific chemical interactions (such as “Turing-patterns”). Within both theories, mechanical patterns are usually regarded as passive by-products of chemical pre-patters. However, several experiments question these theories, and an increasing number of studies shows that tissue mechanics can actively influence chemical patterns during development. In this study, we thus focus on the interplay between chemical and mechanical processes during tissue development. On one hand, based on recent experimental data, we develop new mechanochemical simulation models of evolving tissues, in which the full 3D representation of the tissue appears to be critical for obtaining a realistic mechanochemical behaviour. The presented modelling approach is flexible and numerically studied using state of the art finite element methods. Thus, it may serve as a basis to combine simulations with new experimental methods in tissue development. On the other hand, we apply the developed approach and demonstrate that even simple interactions between tissue mechanics and chemistry spontaneously lead to robust and complex mechanochemical patterns. Especially, we demonstrate that the main contradictions arising in the framework of purely chemical theories are naturally and automatically resolved using the mechanochemical patterning theory.

During embryogenesis, biological tissues gradually increase their complexity by self-organised creation of diverse chemical and mechanical patterns. Detailed mechanisms driving and controlling these patterns are not well understood. Previous theories mostly assume that these patterns are driven by chemical processes. Based on these theories, mechanical patterns are usually considered being mainly determined by chemical pre-patterns. However, experimental evidence for these theories is sparse, and several inconsistencies have been discovered. Furthermore, an increasing amount of data shows that tissue mechanics plays an important role in pattern formation. In this study, we present 3D computer simulations of evolving tissues to investigate the capacity of mechanochemical interactions for pattern formation. We show that even simple interactions between tissue mechanics and tissue chemistry spontaneously lead to robust chemical and mechanical pattern formation. We additionally demonstrate that main contradictions arising in the framework of purely chemical theories are naturally and automatically resolved using the mechanochemical patterning theory. The presented modelling approach can be used to combine simulations with recent experimental developments, to help unravel one of the big mysteries in biology: The mechanisms of self-organised pattern formation during embryogenesis.

Axon Regeneration: Antagonistic Signaling Pairs in Neuronal Polarization

Genome-wide screens, proteomics, and candidate-based approaches have identified numerous genes associated with neuronal regeneration following central nervous system (CNS) injury. Despite significant progress, functional recovery remains a challenge, even in model systems. Neuronal function depends on segregation of axonal versus dendritic domains. A key to functional recovery may lie in recapitulating the developmental signals that instruct axon specification and growth in adult neurons post-injury. Theoretically, binary activator-inhibitor elements operating as a Turing-like system within neurons can specify axonal versus dendritic domains and promote axon growth. We review here various molecules implicated in axon specification that function as signaling pairs driving neuronal polarization and axon growth.

Planarians and the History of Animal Regeneration: Paradigm Shifts and Key Concepts in Biology

Regeneration has captured human imagination for much of recorded history. Its sociological influence is evident in ancient and modern folklore, art, politics, and even language. In many ways, the study of regeneration helped establish the field of biology as a legitimate scientific discipline. Furthermore, regeneration research yielded critical insights that challenged flawed scientific models and uncovered fundamental principles underpinning the workings of life on this planet. This chapter details some ways in which the study of animal regeneration-with special emphasis on planarian regeneration-influenced the evolution of thought in biology. This includes contributions to the discovery of stem cells, the nature of heredity, and key concepts in pattern formation.

Self-organizing spots get under your skin

Sixty-five years after Turing first revealed the potential of systems with local activation and long-range inhibition to generate pattern, we have only recently begun to identify the biological elements that operate at many scales to generate periodic patterns in nature. In this Primer, we first review the theoretical framework provided by Turing, Meinhardt, and others that suggests how periodic patterns could self-organize in developing animals. This Primer was developed to provide context for recent studies that reveal how diverse molecular, cellular, and physical mechanisms contribute to the establishment of the periodic pattern of hair or feather buds in the developing skin. From an initial emphasis on trying to disambiguate which specific mechanism plays a primary role in hair or feather bud development, we are beginning to discover that multiple mechanisms may, in at least some contexts, operate together. While the emergence of the diverse mechanisms underlying pattern formation in specific biological contexts probably reflects the contingencies of evolutionary history, an intriguing possibility is that these mechanisms interact and reinforce each other, producing emergent systems that are more robust.

Systems morphodynamics: understanding the development of tissue hardware

Systems morphodynamics describes a multi-level analysis of mechanical morphogenesis that draws on new microscopy and computational technologies and embraces a systems biology-informed scope. We present a selection of articles that illustrate and explain this rapidly progressing field.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

Weakly coupled map lattice models for multicellular patterning and collective normalization of abnormal single-cell states

We present a weakly coupled map lattice model for patterning that explores the effects exerted by weakening the local dynamic rules on model biological and artificial networks composed of two-state building blocks (cells). To this end, we use two cellular automata models based on (i) a smooth majority rule (model I) and (ii) a set of rules similar to those of Conway's Game of Life (model II). The normal and abnormal cell states evolve according to local rules that are modulated by a parameter κ. This parameter quantifies the effective weakening of the prescribed rules due to the limited coupling of each cell to its neighborhood and can be experimentally controlled by appropriate external agents. The emergent spatiotemporal maps of single-cell states should be of significance for positional information processes as well as for intercellular communication in tumorigenesis, where the collective normalization of abnormal single-cell states by a predominantly normal neighborhood may be crucial.

A cell-based computational model of early embryogenesis coupling mechanical behaviour and gene regulation

The study of multicellular development is grounded in two complementary domains: cell biomechanics, which examines how physical forces shape the embryo, and genetic regulation and molecular signalling, which concern how cells determine their states and behaviours. Integrating both sides into a unified framework is crucial to fully understand the self-organized dynamics of morphogenesis. Here we introduce MecaGen, an integrative modelling platform enabling the hypothesis-driven simulation of these dual processes via the coupling between mechanical and chemical variables. Our approach relies upon a minimal 'cell behaviour ontology' comprising mesenchymal and epithelial cells and their associated behaviours. MecaGen enables the specification and control of complex collective movements in 3D space through a biologically relevant gene regulatory network and parameter space exploration. Three case studies investigating pattern formation, epithelial differentiation and tissue tectonics in zebrafish early embryogenesis, the latter with quantitative comparison to live imaging data, demonstrate the validity and usefulness of our framework.

Vertebrate Axial Patterning: From Egg to Asymmetry

The emergence of the bilateral embryonic body axis from a symmetrical egg has been a long-standing question in developmental biology. Historical and modern experiments point to an initial symmetry-breaking event leading to localized Wnt and Nodal growth factor signaling and subsequent induction and formation of a self-regulating dorsal "organizer." This organizer forms at the site of notochord cell internalization and expresses primarily Bone Morphogenetic Protein (BMP) growth factor antagonists that establish a spatiotemporal gradient of BMP signaling across the embryo, directing initial cell differentiation and morphogenesis. Although the basics of this model have been known for some time, many of the molecular and cellular details have only recently been elucidated and the extent that these events remain conserved throughout vertebrate evolution remains unclear. This chapter summarizes historical perspectives as well as recent molecular and genetic advances regarding: (1) the mechanisms that regulate symmetry-breaking in the vertebrate egg and early embryo, (2) the pathways that are activated by these events, in particular the Wnt pathway, and the role of these pathways in the formation and function of the organizer, and (3) how these pathways also mediate anteroposterior patterning and axial morphogenesis. Emphasis is placed on comparative aspects of the egg-to-embryo transition across vertebrates and their evolution. The future prospects for work regarding self-organization and gene regulatory networks in the context of early axis formation are also discussed.