Read-out of morphogen gradients with increasing amplitude <i>C</i>₀ on a growing domain.

<p>Gradient read-out for a gradient with an increasing amplitude, <i>c</i>₀ ∼ <i>L</i>(<i>t</i>)^2<i>β</i> with <i>β</i> = 0.6, in case of (left column) perfect scaling with <i>λ</i> = <i>k</i>₁ ⋅ <i>L</i>(<i>t</i>); <i>k</i>₁ = 0.11 (ref [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143226#pone.0143226.ref010" target="_blank">10</a>]), (center column) absence of scaling with <i>λ</i> = 20.2 <i>μm</i> (ref [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143226#pone.0143226.ref009" target="_blank">9</a>]), and (right column) imperfect scaling with <math><mrow><mi>λ</mi><mo>=</mo><msqrt><mrow><mi>D</mi><mrow><mn>2</mn><msub><mi>v</mi><mi>g</mi></msub></mrow><mi>L</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>D</mi><mrow><mn>2</mn><msub><mi>v</mi><mi>g</mi></msub></mrow><msub><mi>L</mi><mn>0</mn></msub></mrow></msqrt></mrow></math>; <math><mrow><mi>D</mi><mrow><mn>2</mn><msub><mi>v</mi><mi>g</mi></msub></mrow><mo>=</mo><mn>3</mn><mi>μ</mi><mi>m</mi></mrow></math> and <i>L</i>₀ = 46 <i>μm</i> (ref [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143226#pone.0143226.ref020" target="_blank">20</a>]). (a-c) A morphogen gradient at three timepoints <i>L</i>(<i>t</i>) = 100 μm (light grey), <i>L</i>(<i>t</i>) = 200 μm (dark grey), <i>L</i>(<i>t</i>) = 300 μm (black)) on a rescaled domain. Differently colored lines represent different concentration thresholds. (d-f) Trajectories of the positions, <i>ζ</i>_<i>θ</i>, where the different threshold concentration, shown in panels a-c, are attained. The dashed horizontal lines mark the time interval that is further analysed in panels m-o. (g-i) The derivative of the relative read-out position with respect to the domain length, <math><mrow><mrow><mi>d</mi><msub><mi>ζ</mi><mi>θ</mi></msub></mrow><mrow><mi>d</mi><mi>L</mi></mrow></mrow></math>. (j-l) The derivative <math><mrow><mrow><mi>d</mi><msub><mi>ζ</mi><mi>θ</mi></msub></mrow><mrow><mi>d</mi><mi>L</mi></mrow></mrow></math> not only depends on the relative position <i>ζ</i>_<i>θ</i> (red lines), but also on the gradient amplitude, <i>β</i> (blue lines); see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143226#sec008" target="_blank">Methods</a> section. (m-o) The maximal deviation of the relative read-out positions, Δ<i>ζ</i>, in the interval from 125 μm to 300 μm (marked by dashed horizontal lines in panels d-f) for different final read-out positions. The colored dots indicate the final read-out position of the thresholds shown in panels a-c.