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Control of diffusion-driven pattern formation behind a wave of competency.
TL;DR: In this paper, the effects of a wave of competency on the patterning outcome is not well-understood, and the authors use Turing's diffusion-driven instability model to study pattern formation behind a wave, under a range of wave speeds.
Abstract: In certain biological contexts, such as the plumage patterns of birds and stripes on certain species of fishes, pattern formation takes place behind a so-called ``wave of competency". Currently, the effects of a wave of competency on the patterning outcome is not well-understood. In this study, we use Turing's diffusion-driven instability model to study pattern formation behind a wave of competency, under a range of wave speeds. Numerical simulations show that in one spatial dimension a slower wave speed drives a sequence of peak splittings in the pattern, whereas a higher wave speed leads to peak insertions. In two spatial dimensions, we observe stripes that are either perpendicular or parallel to the moving boundary under slow or fast wave speeds, respectively. We argue that there is a correspondence between the one- and two-dimensional phenomena, and that pattern formation behind a wave of competency can account for the pattern organization observed in many biological systems.
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TL;DR: In this paper , the Fisher-Stefan model was used to solve the 2D Fisher-KPP problem on a compactly-supported region with a moving boundary and the boundary evolves analogously to the classical one-phase Stefan problem, such that boundary speed is proportional to local population density gradient.
3 citations
19 Mar 2022
TL;DR: In this paper , a general theory of pattern formation in spatio-temporal variations of 'pre-pattern' morphogens, which determine gene-regulatory network parameters, is proposed.
Abstract: Abstract In biological systems, chemical signals termed morphogens self-organise into patterns that are vital for many physiological processes. As observed by Turing in 1952, these patterns are in a state of continual development, and are usually transitioning from one pattern into another. How do cells robustly decode these spatio-temporal patterns into signals in the presence of confounding effects caused by unpredictable or heterogeneous environments? Here, we answer this question by developing a general theory of pattern formation in spatio-temporal variations of ‘pre-pattern’ morphogens, which determine gene-regulatory network parameters. Through mathematical analysis, we identify universal dynamical regimes that apply to wide classes of biological systems. We apply our theory to two paradigmatic pattern-forming systems, and predict that they are robust with respect to non-physiological morphogen variations. More broadly, our theoretical framework provides a general approach to classify the emergent dynamics of pattern-forming systems based on how the bifurcations in their governing equations are traversed.
2 citations
TL;DR: In this paper , a general theory of pattern formation in spatio-temporal variations of 'pre-pattern' morphogens, which determine gene-regulatory network parameters, is proposed.
Abstract: In biological systems, chemical signals termed morphogens self-organise into patterns that are vital for many physiological processes. As observed by Turing in 1952, these patterns are in a state of continual development, and are usually transitioning from one pattern into another. How do cells robustly decode these spatio-temporal patterns into signals in the presence of confounding effects caused by unpredictable or heterogeneous environments? Here, we answer this question by developing a general theory of pattern formation in spatio-temporal variations of ‘pre-pattern’ morphogens, which determine gene-regulatory network parameters. Through mathematical analysis, we identify universal dynamical regimes that apply to wide classes of biological systems. We apply our theory to two paradigmatic pattern-forming systems, and predict that they are robust with respect to non-physiological morphogen variations. More broadly, our theoretical framework provides a general approach to classify the emergent dynamics of pattern-forming systems based on how the bifurcations in their governing equations are traversed.
1 citations
TL;DR: In this article , a general form of one-dimensional domain evolution for reaction-diffusion systems is proposed and extended to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model.
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TL;DR: In this article, it is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.
Abstract: It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
9,015 citations
TL;DR: It is shown that relatively simple molecular mechanisms based on auto- and cross catalysis can account for a primary pattern of morphogens to determine pattern formation of the tissue, and the theory is applied to quantitative data on hydra and is shown to account for activation and inhibition of secondary head formation.
Abstract: One of the elementary processes in morphogenesis is the formation of a spatial pattern of tissue structures, starting from almost homogeneous tissue. It will be shown that relatively simple molecular mechanisms based on auto- and cross catalysis can account for a primary pattern of morphogens to determine pattern formation of the tissue. The theory is based on short range activation, long range inhibition, and a distinction between activator and inhibitor concentrations on one hand, and the densities of their sources on the other. While source density is expected to change slowly, e.g. as an effect of cell differentiation, the concentration of activators and inhibitors can change rapidly to establish the primary pattern; this results from auto- and cross catalytic effects on the sources, spreading by diffusion or other mechanisms, and degradation. Employing an approximative equation, a criterium is derived for models, which lead to a striking pattern, starting from an even distribution of morphogens, and assuming a shallow source gradient. The polarity of the pattern depends on the direction of the source gradient, but can be rather independent of other features of source distribution. Models are proposed which explain size regulation (constant proportion of the parts of the pattern irrespective of total size). Depending on the choice of constants, aperiodic patterns, implying a one-to-one correlation between morphogen concentration and position in the tissue, or nearly periodic patterns can be obtained. The theory can be applied not only to multicellular tissues, but also to intracellular differentiation, e.g. of polar cells. The theory permits various molecular interpretations. One of the simplest models involves bimolecular activation and monomolecular inhibition. Source gradients may be substituted by, or added to, sink gradients, e.g. of degrading enzymes. Inhibitors can be substituted by substances required for, and depleted by activation. Sources may be either synthesizing systems or particulate structures releasing activators and inhibitors. Calculations by computer are presented to exemplify the main features of the theory proposed. The theory is applied to quantitative data on hydra — a suitable one-dimensional model for pattern formation — and is shown to account for activation and inhibition of secondary head formation.
2,832 citations
TL;DR: Quite a lot of limit cycle systems are selected which are altogether simpler than, for example, the “Brusselator” with its number of four reactions.
609 citations
TL;DR: Recent experiments on the chlorite-iodide-malonic acid-starch reaction in a gel reactor give the first evidence of the existence of the symmetry breaking, reaction-diffusion structures predicted by Turing in 1952.
Abstract: Recent experiments on the chlorite-iodide-malonic acid-starch reaction in a gel reactor give the first evidence of the existence of the symmetry breaking, reaction-diffusion structures predicted by Turing in 1952. A five-variable model that describes the temporal behavior of the system is reduced to a two-variable model, and its spatial behavior is analyzed. Structures have been found with wavelengths that are in good agreement with those observed experimentally. The gel plays a key role by binding key iodine species, thereby creating the necessary difference in the effective diffusion coefficients of the activator and inhibitor species, iodide and chlorite ions, respectively.
446 citations
TL;DR: It is shown that the feather tract is initiated by a continuous stripe of Shh, Fgf-4, and Ptc expression in the epithelium, which then segregates into discrete feather primordia that are more strongly Shh and FgF-4 positive.
369 citations