Continuum limits of pattern formation in hexagonal-cell monolayers

Intercellular signalling is key in determining cell fate. In closely packed tissues such as epithelia, juxtacrine signalling is thought to be a mechanism for the generation of fine-grained spatial patterns in cell differentiation commonly observed in early development. Theoretical studies of such signalling processes have shown that negative feedback between receptor activation and ligand production is a robust mechanism for fine-grained pattern generation and that cell shape is an important factor in the resulting pattern type. It has previously been assumed that such patterns can be analysed only with discrete models since significant variation occurs over a lengthscale concomitant with an individual cell; however, considering a generic juxtacrine signalling model in square cells, in O’Dea and King (Math Biosci 231(2):172–185 2011), a systematic method for the derivation of a continuum model capturing such phenomena due to variations in a model parameter associated with signalling feedback strength was presented. Here, we extend this work to derive continuum models of the more complex fine-grained patterning in hexagonal cells, constructing individual models for the generation of patterns from the homogeneous state and for the transition between patterning modes. In addition, by considering patterning behaviour under the influence of simultaneous variation of feedback parameters, we construct a more general continuum representation, capturing the emergence of the patterning bifurcation structure. Comparison with the steady-state and dynamic behaviour of the underlying discrete system is made; in particular, we consider pattern-generating travelling waves and the competition between various stable patterning modes, through which we highlight an important deficiency in the ability of continuum representations to accommodate certain dynamics associated with discrete systems.     

Intra-membrane ligand diffusion and cell shape modulate juxtacrine patterning

A key problem in developmental biology is how pattern and planar polarity are transmitted in epithelial structures. Examples include Drosophila neuronal differentiation, ommatidia formation in the compound eye, and wing hair polarization. A key component for the generation of such patterns is direct cell-cell signalling by transmembrane ligands, called juxtacrine signalling. Previous models for this mode of communication have considered homogeneous distributions in the cell membrane, and the role of polarity has been largely ignored. In this paper we determine the role of inhomogeneous protein and receptor distributions in juxtacrine signalling. We explicitly include individual membrane segments, diffusive transport of proteins and receptors between these segments, and production terms with a combination of local and global responses to ligand binding. Our analysis shows that intra-membrane ligand transport is vital for the generation of long wavelength patterns. Moreover, with no ligand transport, there is no pattern formation for lateral induction, a process in which receptor activation up-regulates ligand production. Biased production of ligand also modulates patterning bifurcations and predicted wavelengths. In addition, biased ligand and receptor trafficking can lead to regular polarity across a lattice, in which each cell has the same orientation-directly analogous to patterns of hairs in the Drosophila wing. We confirm the trends in pattern wavelengths previously observed for patterns with cellular homogeneity-lateral inhibition tends to give short-range patterns, while lateral induction can give patterns with much longer wavelengths. Moreover, the original model can be recovered if intra-membrane bound receptor diffusion is included and rapid equilibriation between the sides is considered. Finally, we consider the role of irregular cell shapes and waves in such networks, including wave propagation past clones of non-signalling cells.     

Multiscale analysis of pattern formation via intercellular signalling

Lateral inhibition, a juxtacrine signalling mechanism by which a cell adopting a particular fate inhibits neighbouring cells from doing likewise, has been shown to be a robust mechanism for the formation of fine-grained spatial patterns (in which adjacent cells in developing tissues diverge to achieve contrasting states of differentiation), provided that there is sufficiently strong feedback. The fine-grained nature of these patterns poses problems for analysis via traditional continuum methods since these require that significant variation takes place only over lengthscales much larger than an individual cell and such systems have therefore been investigated primarily using discrete methods. Here, however, we apply a multiscale method to derive systematically a continuum model from the discrete Delta-Notch signalling model of Collier et al. (J.R. Collier, N.A.M. Monk, P.K. Maini, J.H. Lewis, Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling, J. Theor. Biol., 183, 1996, 429-446) under particular assumptions on the parameters, which we use to analyse the generation of fine-grained patterns. We show that, on the macroscale, the contact-dependent juxtacrine signalling interaction manifests itself as linear diffusion, motivating the use of reaction-diffusion-based models for such cell-signalling systems. We also analyse the travelling-wave behaviour of our system, obtaining good quantitative agreement with the discrete system.     

Restricted-range gradients and travelling fronts in a model of juxtacrine cell relay

Patterning events in development often depend on the transmission over a range of several cell diameters of signals emanating from a localized source. Experimental studies of such long-range signalling by members of the TGF-β family of growth factors suggests that a cell-relay mechanism in which cells signal only with their immediate neighbours (i.e., juxtacrine signalling) may be operating in some tissues. Here, this possibility is investigated through the analysis of a model of juxtacrine signalling. Depending on the strength of the signal relay between cells, a localized signal source can generate either stable gradients or travelling fronts of cell activation. Both of these behaviors could in principle be involved in the long-range transmission of signals and patterning of cell fates by cell relays. There are significant and surprising differences between the gradients generated by the mechanism studied here, and those generated by the diffusion of a morphogen. In particular, there is an upper limit on the distance over which any given level of cell activation can be attained in a relay-mediated gradient, irrespective of the strength of signal source.     

Mutual inactivation of Notch receptors and ligands facilitates developmental patterning

Developmental patterning requires juxtacrine signaling in order to tightly coordinate the fates of neighboring cells. Recent work has shown that Notch and Delta, the canonical metazoan juxtacrine signaling receptor and ligand, mutually inactivate each other in the same cell. This cis-interaction generates mutually exclusive sending and receiving states in individual cells. It generally remains unclear, however, how this mutual inactivation and the resulting switching behavior can impact developmental patterning circuits. Here we address this question using mathematical modeling in the context of two canonical pattern formation processes: boundary formation and lateral inhibition. For boundary formation, in a model motivated by Drosophila wing vein patterning, we find that mutual inactivation allows sharp boundary formation across a broader range of parameters than models lacking mutual inactivation. This model with mutual inactivation also exhibits robustness to correlated gene expression perturbations. For lateral inhibition, we find that mutual inactivation speeds up patterning dynamics, relieves the need for cooperative regulatory interactions, and expands the range of parameter values that permit pattern formation, compared to canonical models. Furthermore, mutual inactivation enables a simple lateral inhibition circuit architecture which requires only a single downstream regulatory step. Both model systems show how mutual inactivation can facilitate robust fine-grained patterning processes that would be difficult to implement without it, by encoding a difference-promoting feedback within the signaling system itself. Together, these results provide a framework for analysis of more complex Notch-dependent developmental systems.     

Lateral induction by juxtacrine signaling is a new mechanism for pattern formation

Many signaling molecules in epithelia are now known to function in a membrane-bound form, binding to receptors on immediately neighbouring cells. This "juxtacrine" mode of communication has been well studied in the case of lateral inhibition, where ligand binding at the cell surface downregulates ligand and receptor expression, and is known to generate spatial patterns with a wavelength of exactly two cells. However, recent evidence shows that a number of juxtacrine signals can lead to the opposite phenomenon of lateral induction. Here, we use mathematical modeling to show that such positive feedback, in combination with juxtacrine communication, provides a novel mechanism for the generation of spatial patterns, with wavelengths that vary with parameters and can be many cell lengths.     

Accommodating natural and sexual selection in butterfly wing pattern evolution

Visual patterns in animals may serve different functions, such as attracting mates and deceiving predators. If a signal is used for multiple functions, the opportunity arises for conflict among the different functions, preventing optimization for any one visual signal. Here we investigate the hypothesis that spatial separation of different visual signal functions has occurred in Bicyclus butterflies. Using phylogenetic reconstructions of character evolution and comparisons of evolutionary rates, we found dorsal surface characters to evolve at higher rates than ventral characters. Dorsal characters also displayed sex-based differences in evolutionary rates more often than did ventral characters. Thus, dorsal characters corresponded to our predictions of mate signalling while ventral characters appear to play an important role in predator avoidance. Forewing characters also fit a model of mate signalling, and displayed higher rates of evolution than hindwing characters. Our results, as well as the behavioural and developmental data from previous studies of Bicyclus species, support the hypothesis that spatial separation of visual signal functions has occurred in Bicyclus butterflies. This study is the first to demonstrate, in a phylogenetic framework, that spatial separation of signals used for mate signalling and those used for predator avoidance is a viable strategy to accommodate multiple signal functions. This signalling strategy has important ramifications on the developmental evolution of wing pattern elements and diversification of butterfly species.     

A model of visual perception

In this paper we propose a model of visual perception in which a positive feedback mechanism can reproduce the pattern stimulus on a neurons screen. The pattern stimulus reproduction is based on informations coming from the spatial derivatives of visual pattern. This information together with the response of the feature extractors provides to the reproduction of the visual pattern as neuron screen electric activity. We simulate several input patterns and prove that the model reproduces the percept.     

Cell cycle dynamics in a response/signalling feedback system with a gap

We consider a dynamical model of cell cycles of n cells in a culture in which cells in one specific phase (S for signalling) of the cell cycle produce chemical agents that influence the growth/cell cycle progression of cells in another phase (R for responsive). In the case that the feedback is negative, it is known that subpopulations of cells tend to become clustered in the cell cycle; while for a positive feedback, all the cells tend to become synchronized. In this paper, we suppose that there is a gap between the two phases. The gap can be thought of as modelling the physical reality of a time delay in the production and action of the signalling agents. We completely analyse the dynamics of this system when the cells are arranged into two cell cycle clusters. We also consider the stability of certain important periodic solutions in which clusters of cells have a cyclic arrangement and there are just enough clusters to allow interactions between them. We find that the inclusion of a small gap does not greatly alter the global dynamics of the system; there are still large open sets of parameters for which clustered solutions are stable. Thus, we add to the evidence that clustering can be a robust phenomenon in biological systems. However, the gap does effect the system by enhancing the stability of the stable clustered solutions. We explain this phenomenon in terms of contraction rates (Floquet exponents) in various invariant subspaces of the system. We conclude that in systems for which these models are reasonable, a delay in signalling is advantageous to the emergence of clustering.     

The tardigrade Hypsibius dujardini, a new model for studying the evolution of development

Studying development in diverse taxa can address a central issue in evolutionary biology: how morphological diversity arises through the evolution of developmental mechanisms. Two of the best-studied developmental model organisms, the arthropod Drosophila and the nematode Caenorhabditis elegans, have been found to belong to a single protostome superclade, the Ecdysozoa. This finding suggests that a closely related ecdysozoan phylum could serve as a valuable model for studying how developmental mechanisms evolve in ways that can produce diverse body plans. Tardigrades, also called water bears, make up a phylum of microscopic ecdysozoan animals. Tardigrades share many characteristics with C. elegans and Drosophila that could make them useful laboratory models, but long-term culturing of tardigrades historically has been a challenge, and there have been few studies of tardigrade development. Here, we show that the tardigrade Hypsibius dujardini can be cultured continuously for decades and can be cryopreserved. We report that H. dujardini has a compact genome, a little smaller than that of C. elegans or Drosophila, and that sequence evolution has occurred at a typical rate. H. dujardini has a short generation time, 13–14 days at room temperature. We have found that the embryos of H. dujardini have a stereotyped cleavage pattern with asymmetric cell divisions, nuclear migrations, and cell migrations occurring in reproducible patterns. We present a cell lineage of the early embryo and an embryonic staging series. We expect that these data can serve as a platform for using H. dujardini as a model for studying the evolution of developmental mechanisms.     

Involvement of NtERF3 in the cell death signalling pathway mediated by SIPK/WIPK and WRKY1 in tobacco plants

We previously reported that one of the ethylene response factors (ERFs), NtERF3, and other members of the subgroup VIII‐a ERFs of the AP2/ERF family exhibit cell death‐inducing ability in tobacco leaves. In this study, we focused on the involvement of NtERF3 in a cell death signalling pathway in tobacco plants, particularly downstream of NtSIPK/NtWIPK and NtWRKY1, which are mitogen‐activated protein kinases and a phosphorylation substrate of NtSIPK, respectively. An ERF‐associated amphiphilic repression (EAR) motif‐deficient NtERF3b mutant (NtERF3bΔEAR) that lacked cell death‐inducing ability suppressed the induction of cell death caused by NtERF3a. The transient co‐expression of NtERF3bΔEAR suppressed the hypersensitive reaction (HR)‐like cell death induced by NtSIPK and NtWRKY1. The induction of cell death by NtSIPK and NtWRKY1 was also inhibited in transgenic plants expressing NtERF3bΔEAR. Analysis of gene expression, ethylene production and cell death symptoms in salicylic acid‐deficient tobacco plants suggested the existence of some feedback regulation in the HR cell death signalling pathway mediated by SIPK/WIPK and WRKY1. Overall, these results suggest that NtERF3 functions downstream of NtSIPK/NtWIPK and NtWRKY1 in a cell death signalling pathway, with some feedback regulation.     

Analysis of the genetic loci of pigment pattern evolution in vertebrates

Vertebrate pigmentation patterns are amongst the best characterised model systems for studying the genetic basis of adaptive evolution. The wealth of available data on the genetic basis for pigmentation evolution allows for analysis of trends and quantitative testing of evolutionary hypotheses. We employed Gephebase, a database of genetic variants associated with natural and domesticated trait variation, to examine trends in how cis-regulatory and coding mutations contribute to vertebrate pigmentation phenotypes, as well as factors that favour one mutation type over the other. We found that studies with lower ascertainment bias identified higher proportions of cis-regulatory mutations, and that cis-regulatory mutations were more common amongst animals harbouring a higher number of pigment cell classes. We classified pigmentation traits firstly according to their physiological basis and secondly according to whether they affect colour or pattern, and identified that carotenoid-based pigmentation and variation in pattern boundaries are preferentially associated with cis-regulatory change. We also classified genes according to their developmental, cellular, and molecular functions. We found a greater proportion of cis-regulatory mutations in genes implicated in upstream developmental processes compared to those involved in downstream cellular functions, and that ligands were associated with a higher proportion of cis-regulatory mutations than their respective receptors. Based on these trends, we discuss future directions for research in vertebrate pigmentation evolution.     

Predicting the Distribution of Spiral Waves from Cell Properties in a Developmental-Path Model of Dictyostelium Pattern Formation

The slime mold Dictyostelium discoideum is one of the model systems of biological pattern formation. One of the most successful answers to the challenge of establishing a spiral wave pattern in a colony of homogeneously distributed D. discoideum cells has been the suggestion of a developmental path the cells follow (Lauzeral and coworkers). This is a well-defined change in properties each cell undergoes on a longer time scale than the typical dynamics of the cell. Here we show that this concept leads to an inhomogeneous and systematic spatial distribution of spiral waves, which can be predicted from the distribution of cells on the developmental path. We propose specific experiments for checking whether such systematics are also found in data and thus, indirectly, provide evidence of a developmental path.     

Overgrowth of a mouse model of Simpson– Golabi–Behmel syndrome is partly mediated by Indian Hedgehog

Loss‐of‐function mutations of Glypican 3 (Gpc3) cause the Simpson–Golabi–Behmel overgrowth syndrome (SGBS), and developmental overgrowth is observed in Gpc3‐null mice, a mouse model for SGBS. We recently reported that GPC3 inhibits Hedgehog (Hh) signalling by inducing its endocytosis and degradation. Here, we show that the developmental overgrowth observed in Gpc3‐null mice is, at least in part, a consequence of the hyperactivation of the Hh pathway. We bred Gpc3‐null mice with mice that are Hh signalling‐deficient owing to the lack of Indian Hh (Ihh), one of the three mammalian Hhs. We found that the Gpc3‐null mice showed a 29.9% overgrowth in an Ihh wild‐type background, whereas an Ihh‐null background partly rescues the overgrowth caused by the lack of Gpc3 as the double mutants were 19.8% bigger than the Ihh‐null mice. Consistent with the role of GPC3 in Hh endocytosis and degradation, the Gpc3‐null mice show increased levels of Ihh protein and signalling, but similar levels of Ihh messenger RNA.     

Evo-devo: Hydra raises its Noggin

Noggin, along with other secreted bone morphogenetic protein (BMP) inhibitors, plays a crucial role in neural induction and neural tube patterning as well as in somitogenesis, cardiac morphogenesis and formation of the skeleton in vertebrates. The BMP signalling pathway is one of the seven fundamental pathways that drive embryonic development and pattern formation in animals. Understanding its evolutionary origin and role in pattern formation is, therefore, important to evolutionary developmental biology (evo-devo). We have studied the evolutionary origin of BMP–Noggin antagonism in hydra, which is a powerful diploblastic model to study evolution of pattern-forming mechanisms because of the unusual cellular dynamics during its pattern formation and its remarkable ability to regenerate. We cloned and characterized the noggin gene from hydra and found it to exhibit considerable similarity with its orthologues at the amino acid level. Microinjection of hydra Noggin mRNA led to duplication of the dorsoventral axis in Xenopus embryos, demonstrating its functional conservation across the taxa. Our data, along with those of others, indicate that the evolutionarily conserved antagonism between BMP and its inhibitors predates bilateral divergence. This article reviews the various roles of Noggin in different organisms and some of our recent work on hydra Noggin in the context of evolution of developmental signalling pathways.     

Modelling the dynamics of senescence spread

Cellular senescence is a cell surveillance mechanism that arrests the cell cycle in damaged cells. The senescent phenotype can spread from cell to cell through paracrine and juxtacrine signalling, but the dynamics of this process are not well understood. Although senescent cells are important in ageing, wound healing and cancer, it is unclear how the spread of senescence is contained in senescent lesions. In the absence of the immune system, senescence could theoretically spread infinitely from one cell to another, but this contradicts experimental evidence. To investigate this issue, we developed both a minimal mathematical model and a stochastic simulation of senescence spread. Our results suggest that differences in the number of signalling molecules secreted between subtypes of senescent cells can limit the spread of senescence. We found that dynamic, time-dependent paracrine signalling prevents the uncontrolled spread of senescence, and we demonstrate how model parameters can be determined using Bayesian inference in a proposed experiment.     

Pigment pattern formation in the larval salamander Ambystoma maculatum

As part of an ongoing comparative study of pigment patterns and their formation in embryos and larvae of ambystomatid salamanders, Ambystoma maculatum from two differnt populations, one in the northern (New York) and one in the central (Tennessee) United States, were investigated. Scanning electron microscopy was used to study early neural crest development. Light microscopy in combination with markers for the two pigment cell types (xanthophores and melanophores) made it possible to follow pigment cell migration before the pigment cells were fully differentiated. A bilateral pigment pattern consisting of two horizontal melanophore stripes surrounding an interstripe area populated by xanthophores formed in the larvae. In both populations, some variation was present in the form of a continuum ranging from clear horizontal stripes to extreme cases with a random pattern. Unlike the other ambystomatids that have been investigated, the neural crest cells in A. maculatum do not form aggregates and no vertical bars are formed. Instead, both the pattern and its formation are very similar to what has been reported for salamandrids. If pattern formation mechanisms can act as developmental constraints we would expect the A. maculatum pattern to be the primitive condition in the Ambystomatidae, using the Salamandridae as the outgroup. There is no strong support for this when aggregate formation is used as a character and mapped onto phylogenies for the group. The aggregate formation mechanism, and the pigment pattern that it leads to, have most likely been secondarily lost in A. maculatum. © 1993 Wiley-Liss, Inc.     

Wnt‐3 and Wnt‐3a play different region‐specific roles in neural crest development in avians

Wnt signalling regulates cell proliferation and cell fate determination during embryogenesis. However, little is known about the developmental role of one Wnt family member, Wnt‐3, during avian development. To investigate the possible functions of Wnt‐3, its expression pattern was determined using whole‐mount in situ hybridization. Wnt‐3 is expressed in important signalling centres, including the dorsal neural tube, Hensen's node and the AER (apical ectodermal ridge). Most interestingly, Wnt‐3 is expressed in the dorsal neural tube as a gradient, with the strongest expression anterior in the trunk. Furthermore, this study showed that Wnt‐3 and Wnt‐3a play a different role in neural crest lineages derived from different axial level of neural tube. Wnt‐3 might be involved in proliferation of neural crest lineages, whereas Wnt‐3a plays an important role in melanogenesis in vagal. However, both Wnt‐3 and Wnt‐3a cause a significant increase in melanogenesis in the trunk neural crest lineage.