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Figure 4.

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ZDB-FIG-210927-58

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Song et al., 2021 - Cost-precision trade-off relation determines the optimal morphogen gradient for accurate biological pattern formation

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Figure 4.

Optimal concentration profile of morphogens.

(a) Three possible morphogen profiles (i) (red), (ii) (cyan), (iii) (brown) with different λ’s ( $λ^{(i)} < λ^{(i i)} < λ^{(i i i)}$ $λ^{(i)} < λ^{(i i)} < λ^{(i i i)}$ ), generated with different values of morphogen influxes $j_{i n}$ $j_{i n}$ ( $j_{in}^{(i)} > j_{in}^{(ii)} > j_{in}^{(iii)}$ $j_{in}^{(i)} > j_{in}^{(ii)} > j_{in}^{(iii)}$ ). The morphogen concentration of the three profiles coincide at $x_{b}$ $x_{b}$ , giving rise to the same threshold value $ρ (x_{b})$ $ρ (x_{b})$ but different positional errors ( $ϵ^{2} (x_{b}; λ)$ $ϵ^{2} (x_{b}; λ)$ ). (b) The precision of three possible morphogen profiles (i) (red), (ii) (cyan), (iii) (brown) with different λ’s ( $λ^{(i)} < λ^{(i i)} < λ^{(i i i)}$ $λ^{(i)} < λ^{(i i)} < λ^{(i i i)}$ ), generated with different values of morphogen influxes $j_{i n}$ $j_{i n}$ ( $j_{in}^{(i)} > j_{in}^{(iii)} > j_{in}^{(ii)}$ $j_{in}^{(i)} > j_{in}^{(iii)} > j_{in}^{(ii)}$ ). The λ values are identical to those with matching colors in (a), but the red and brown curves are generated with different $j_{i n}$ $j_{i n}$ values from those in (a). The cost associated with each morphogen profile are shown in units of $α_{o} L / l_{cell}$ $α_{o} L / l_{cell}$ . (c) The diagram of the trade-off product associated with the point measurement, $π_{o} (x; λ)$ $π_{o} (x; λ)$ , plotted with respect to $x$ $x$ and λ. The black line indicates the optimal decay length, $λ_{\min}$ $λ_{\min}$ at position $x$ $x$ . Shown on the diagram are the trade-off product $π_{o}$ $π_{o}$ ’s for the three cases shown in (a) and (b). (d) The value of $π_{o}$ $π_{o}$ as a function of λ at $x = x_{b}$ $x = x_{b}$ . The trade-off product is minimized to $π_{o} = π_{o}^{(ii)} \approx α_{o} (L / l_{cell}) 0.4$ $π_{o} = π_{o}^{(ii)} \approx α_{o} (L / l_{cell}) 0.4$ with $π_{o}^{(ii)} < π_{o}^{(iii)} < π_{o}^{(i)}$ $π_{o}^{(ii)} < π_{o}^{(iii)} < π_{o}^{(i)}$ .

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