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The subcollection of c-vascular cells may be thought of as an early prepattern from which veins derive. To keep matters tractable, a cell will be referred to as “c-vascular” (cascade vascular) immediately after this cascade is initiated. We focus on early leaf development and concentrate on establishing signals to initiate the cascade of events that change ground cells into vascular cells within an expanding areole. We distinguish between “ground cells,” those that give rise to all others, and “vascular cells,” those that comprise the venation pattern. Stage 2 ( C and F) was initialized using predictions (in blue in F) from stage 1.Ī leaf is a collection of cells. Stage 1 ( B and E) was initialized by tracing veins from A (shown in blue in E). Predictions start at a maximum of Δ c, follow vectors (e.g., see D), and stop at minimum Δ c (see Supporting Text, which is published as supporting information on the PNAS web site). Veins are in blue, and predictions are in green. ( E and F) Color-coded magnitude of greatest possible Δ c with immediate neighbors. Color-coded magnitudes of vectors elsewhere in the leaf are shown. ( D) Gradient vectors of concentration in an areole developed further in the work. Observe how local peaks “move” after creation of new strands. (Scale bar: 100 μm.) ( B and C) Color-coded concentration levels at stages 1 ( B) and 2 ( C). ( A) Image of stained leaf of Arabidopsis from ref. Developmental time is quantized into two discrete stages for this illustration. As developed in the text, the bottom of the petiole (base of figure) acts as sink, and the hormone diffuses with coefficient D v through vascular cells and D everywhere else ( D v > D). Illustration that global signaling information for vein formation can be obtained from constant hormonal production in every cell. Signals for initiating differentiation are readily available locally, removing the need for complex intercellular communication.įig. The concentration together with the gradient of concentration have substantial predictive power about vein formation ( Fig.
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This information can be interpreted locally as a signal at the cell level, thereby providing global cues. Mathematical analysis of a schematic reaction diffusion model indicates that high concentration sites emerge, which agrees with observations ( 1) but also shows that distributions carry rich information about the geometry of the leaf and its venation. Formally, our “constant production hypothesis” holds that auxin is produced in all cells at the same constant rate. We demonstrate that discrete regulated sites of synthesis need not exist to develop a spatial pattern of discrete responses. However, evidence ( 1, 2) is emerging that production may be present at other sites. Isolated sites of high auxin concentration have been observed in developing leaves ( 1), and it has been assumed that auxin production is dominant there. We predict angiosperm areolation patterns in simulation, and our model further implies the Sachs Canalization Hypothesis and resolves a dilemma regarding the role of auxin in cell growth. Neither complex interactions nor predetermination are necessary. Unlike other models, a single substance suffices for the reaction–diffusion at early, but not initial, stages of development. Because the global information is encoded as auxin concentration and its gradient, those signals provide individual cells with sufficient information to determine their own fate. High concentration sites for auxin emerge naturally in a reaction–diffusion model, together with global information about leaf shape and existing venation. In contrast to the more standard view that a signal (e.g., auxin) is produced at isolated sites to stimulate growth, we determine the consequences of the hypothesis that auxin is produced at a constant rate in every cell. We propose a theoretical mechanism that enables the elaboration of veins to supply distant cells during leaf development.