The zebrafish is a model organism for pattern formation in vertebrates. Understanding what drives the formation of its coloured skin motifs could reveal pivotal to comprehend the mechanisms behind morphogenesis. The motifs look and behave like reaction–diffusion Turing patterns, but the nature of the underlying physico-chemical processes is very different, and the origin of the patterns is still unclear. Here we propose a minimal model for such pattern formation based on a regulatory mechanism deduced from experimental observations. This model is able to produce patterns with intrinsic wavelength, closely resembling the experimental ones. We mathematically prove that their origin is a Turing bifurcation occurring despite the absence of cell motion, through an effect that we call differential growth. This mechanism is qualitatively different from the reaction–diffusion originally proposed by Turing, although they both generate the short-range activation and the long-range inhibition required to form Turing patterns.
Recent studies have demonstrated that enzyme-instructed self-assembly (EISA) in extra- or intracellular environments can serve as a multistep process for controlling cell fate. There is little knowledge, however, about the kinetics of EISA in the complex environments in or around cells. Here, we design and synthesize three dipeptidic precursors (ld-1-SO, dl-1-SO, dd-1-SO), consisting of diphenylalanine (l-Phe-d-Phe, d-Phe-l-Phe, d-Phe-d-Phe, respectively) as the backbone, which are capped by 2-(naphthalen-2-yl)acetic acid at the N-terminal and by 2-(4-(2-aminoethoxy)-4-oxobutanamido)ethane-1-sulfonic acid at the C-terminal. On hydrolysis by carboxylesterases (CES), these precursors result in hydrogelators, which self-assemble in water at different rates. Whereas all three precursors selectively kill cancer cells, especially high-grade serous ovarian carcinoma cells, by undergoing intracellular EISA, dl-1-SO and dd-1-SO exhibit the lowest and the highest activities, respectively, against the cancer cells. This trend inversely correlates with the rates of converting the precursors to the hydrogelators in phosphate-buffered saline. Because CES exists both extra- and intracellularly, we use kinetic modeling to analyze the kinetics of EISA inside cells and to calculate the cytotoxicity of each precursor for killing cancer cells. Our results indicate that (i) the stereochemistry of the precursors affects the morphology of the nanostructures formed by the hydrogelators, as well as the rate of enzymatic conversion; (ii) decreased extracellular hydrolysis of precursors favors intracellular EISA inside the cells; (iii) the inherent features ( e.g., self-assembling ability and morphology) of the EISA molecules largely dictate the cytotoxicity of intracellular EISA. As the kinetic analysis of intracellular EISA, this work elucidates how the stereochemistry modulates EISA in the complex extra- and/or intracellular environment for developing anticancer molecular processes. Moreover, it provides insights for understanding the kinetics and cytotoxicity of aggregates of aberrant proteins or peptides formed inside and outside cells.
Reactive systems are known to give birth to complex spatiotemporal phenomena, when they are maintained far enough from their equilibrium state. There are literally hundreds of experimental evidences showing the emergence of such self-organized behaviors at the macroscopic scale. Examples include the appearance of regular oscillations of concentration in both space and time, the formation of stationary spatial organization of reactants and products, and the emergence of spatiotemporal chaos, to cite but a few examples.The theoretical understanding of these phenomena can be considered as being well established. Chemical reactions play a central role in the appearance of complex behaviors because they are nonlinear processes. Indeed, the rates of reactions are typically polynomials of the concentrations and moreover include constants that depend exponentially on the temperature. Because of this, the equations ruling the spatiotemporal development of chemical reactions, which often take the form of reaction-diffusion equations, admit complicated (and even sometimes multiple) solutions. The number and the type of solutions change abruptly for some precise combinations of the parameters of the system, which are known as bifurcation points. This feature explains why new dynamical behaviors are observed only whenever a
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