Complex pattern formation in reaction diffusion systems with spatially-varying parameters

Spontaneous pattern formation in reaction–diffusion systems on a spatially homogeneous domain has been well studied. However, in embryonic development and elsewhere, pattern formation often takes place on a spatially heterogeneous background. We explore the effects of spatially varying parameters on pattern formation in one and two dimensions using the Gierer–Meinhardt reaction–diffusion model. We investigate the effect of the wavelength of a pre-pattern and demonstrate a novel form of moving pattern. We find that spatially heterogeneous parameters can both increase the range and complexity of possible patterns and enhance the robustness of pattern selection

Speed of reaction diffusion in embryogenesis

Reaction diffusion systems have been proposed as mechanisms for patterning during many stages of embryonic development. While much attention has been focused on the study of the steady state patterns formed and the robustness of pattern selection, much less is known about the time scales required for pattern formation. Studies of gradient formation by the diffusion of a single morphogen from a localized source have shown that patterning can occur on realistic time scales over distances of a millimeter or less. Reaction diffusion has the potential to give rise to patterns on a faster time scale, since all points in the domain can act as sources of morphogen. However, the speed at which patterning can occur has hitherto not been explored in depth. In this paper, we investigate this issue in specific reaction diffusion models and address the question of whether patterning via reaction diffusion is fast enough to be applicable to morphogenesis

Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling

In many developing tissues, adjacent cells diverge in character so as to create a fine-grained pattern of cells in contrasting states of differentiation. It has been proposed that such patterns can be generated through lateral inhibition—a type cells–cell interaction whereby a cell that adopts a particular fate inhibits its immediate neighbours from doing likewise. Lateral inhibition is well documented in flies, worms and vertebrates. In all of these organisms, the transmembrane proteins Notch and Delta (or their homologues) have been identified as mediators of the interaction—Notch as receptor, Delta as its ligand on adjacent cells. However, it is not clear under precisely what conditions the Delta-Notch mechanism of lateral inhibition can generate the observed types of pattern, or indeed whether this mechanism is capable of generating such patterns by itself. Here we construct and analyse a simple and general mathematical model of such contact-mediated lateral inhibition. In accordance with experimental data, the model postulates that receipt of inhibition (i.e. activation of Notch) diminishes the ability to deliver inhibition (i.e. to produce active Delta). This gives rise to a feedback loop that can amplify differences between adjacent cells. We investigate the pattern-forming potential and temporal behavior of this model both analytically and through numerical simulation. Inhomogeneities are self-amplifying and develop without need of any other machinery, provided the feedback is sufficiently strong. For a wide range of initial and boundary conditions, the model generates fine-grained patterns similar to those observed in living systems

A model of primitive streak initiation in the chick embryo

Initiation of the primitive streak in avian embryos provides a well-studied example of a pattern-forming event that displays a striking capacity for regulation. The mechanisms underlying the regulative properties are, however, poorly understood and are not easily accounted for by traditional models of pattern formation, such as reaction–diffusion models. In this paper, we propose a new activator–inhibitor model for streak initiation. We show that the model is consistent with experimental observations, both in its pattern-forming properties and in its ability to form these patterns on the correct time-scales for biologically realistic parameter values. A key component of the model is a travelling wave of inhibition. We present a mathematical analysis of the speed of such waves in both diffusive and juxtacrine relay systems. We use the streak initiation model to make testable predictions. By varying parameters of the model, two very different types of patterning can be obtained, suggesting that our model may be applicable to other processes in addition to streak initiation

Auxin influx importers modulate serration along the leaf margin

Leaf shape in Arabidopsis is modulated by patterning events in the margin that utilize a PIN-based auxin exporter/CUC2 transcription factor system to define regions of promotion and retardation of growth, leading to morphogenesis. In addition to auxin exporters, leaves also express auxin importers, notably members of the AUX1/LAX family. In contrast to their established roles in embryogenesis, lateral root and leaf initiation, the function of these transporters in leaf development is poorly understood. We report that three of these genes (AUX1, LAX1 and LAX2) show specific and dynamic patterns of expression during early leaf development in Arabidopsis, and that loss of expression of all three genes is required for observation of a phenotype in which morphogenesis (serration) is decreased. We used these expression patterns and mutant phenotypes to develop a margin-patterning model that incorporates an AUX1/LAX1/LAX2 auxin import module that influences the extent of leaf serration. Testing of this model by margin-localized expression of axr3–1 (AXR17) provides further insight into the role of auxin in leaf morphogenesis

Neon and Sulfur Abundances of Planetary Nebulae in the Magellanic Clouds

The chemical abundances of neon and sulfur for 25 planetary nebulae (PNe) in the Magellanic Clouds are presented. These abundances have been derived using mainly infrared data from the Spitzer Space Telescope. The implications for the chemical evolution of these elements are discussed. A comparison with similarly obtained abundances of Galactic PNe and HII regions and Magellanic Clouds HII regions is also given. The average neon abundances are 6.0x10(-5) and 2.7x10(-5) for the PNe in the Large and Small Magellanic Clouds respectively. These are ~1/3 and 1/6 of the average abundances of Galactic planetary nebulae to which we compare. The average sulfur abundances for the LMC and SMC are respectively 2.7x10(-6) and 1.0x10(-6). The Ne/S ratio (23.5) is on average higher than the ratio found in Galactic PNe (16) but the range of values in both data sets is similar for most of the objects. The neon abundances found in PNe and HII regions agree with each other. It is possible that a few (3-4) of the PNe in the sample have experienced some neon enrichment, but for two of these objects the high Ne/S ratio can be explained by their very low sulfur abundances. The neon and sulfur abundances derived in this paper are also compared to previously published abundances using optical data and photo-ionization models.Comment: 13 pages, 4 tables, 5 figures. Accepted for publication in Ap

Large Magellanic Cloud Planetary Nebula Morphology: Probing Stellar Populations and Evolution

Planetary Nebulae (PNe) in the Large Magellanic Cloud (LMC) offer the unique opportunity to study both the Population and evolution of low- and intermediate-mass stars, by means of the morphological type of the nebula. Using observations from our LMC PN morphological survey, and including images available in the HST Data Archive, and published chemical abundances, we find that asymmetry in PNe is strongly correlated with a younger stellar Population, as indicated by the abundance of elements that are unaltered by stellar evolution (Ne, Ar, S). While similar results have been obtained for Galactic PNe, this is the first demonstration of the relationship for extra-galactic PNe. We also examine the relation between morphology and abundance of the products of stellar evolution. We found that asymmetric PNe have higher nitrogen and lower carbon abundances than symmetric PNe. Our two main results are broadly consistent with the predictions of stellar evolution if the progenitors of asymmetric PNe have on average larger masses than the progenitors of symmetric PNe. The results bear on the question of formation mechanisms for asymmetric PNe, specifically, that the genesis of PNe structure should relate strongly to the Population type, and by inference the mass, of the progenitor star, and less strongly on whether the central star is a member of a close binary system.Comment: The Astrophysical Journal Letters, in press 4 figure

Gene expression time delays & Turing pattern formation systems

The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this paper, we investigate the effects of incorporating gene expression time delays into a one-dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. While oscillatory behaviour is rare, we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning