A model of the spatio-temporal dynamics of drosophila eye disc development

Patterning and growth are linked during early development and have to be tightly controlled to result in a functional tissue or organ. During the development of the Drosophila eye, this linkage is particularly clear: the growth of the eye primordium mainly results from proliferating cells ahead of the morphogenetic furrow (MF), a moving signaling wave that sweeps across the tissue from the posterior to the anterior side, that induces proliferating cells anterior to it to differentiate and become cell cycle quiescent in its wake. Therefore, final eye disc size depends on the proliferation rate of undifferentiated cells and on the speed with which the MF sweeps across the eye disc. We developed a spatio-temporal model of the growing eye disc based on the regulatory interactions controlled by the signals Decapentaplegic (Dpp), Hedgehog (Hh) and the transcription factor Homothorax (Hth) and explored how the signaling patterns affect the movement of the MF and impact on eye disc growth. We used published and new quantitative data to parameterize the model. In particular, two crucial parameter values, the degradation rate of Hth and the diffusion coefficient of Hh, were measured. The model is able to reproduce the linear movement of the MF and the termination of growth of the primordium. We further show that the model can explain several mutant phenotypes, but fails to reproduce the previously observed scaling of the Dpp gradient in the anterior compartmen

Read-Out of Dynamic Morphogen Gradients on Growing Domains

Quantitative data from the Drosophila wing imaginal disc reveals that the amplitude of the Decapentaplegic (Dpp) morphogen gradient increases continuously. It is an open question how cells can determine their relative position within a domain based on a continuously increasing gradient. Here we show that pre-steady state diffusion-based dispersal of morphogens results in a zone within the growing domain where the concentration remains constant over the patterning period. The position of the zone that is predicted based on quantitative data for the Dpp morphogen corresponds to where the Dpp-dependent gene expression boundaries of spalt (sal) and daughters against dpp (dad) emerge. The model also suggests that genes that are scaling and are expressed at lateral positions are either under the control of a different read-out mechanism or under the control of a different morphogen. The patterning mechanism explains the extraordinary robustness that is observed for variations in Dpp production, and offers an explanation for the dual role of Dpp in controlling patterning and growth. Pre-steady-state dynamics are pervasive in morphogen-controlled systems, thus making this a probable general mechanism for the scaled read-out of morphogen gradients in growing developmental systems.ISSN:1932-620

The read-out position is robust to parallel changes in the Dpp production rate and the growth rate in the pre-steady state wing disc model, but not in the steady-state model.

<p>(a-b) Impact of a 2-fold increase in the Dpp production rate and the growth rate (green line) on the Dpp gradient compared to the standard parameterization (black line) in the detailed wing disc model on a linear scale (a) and a log-scale (b). (c) The deviation in the final relative readout position between the black and the red gradient in the detailed wing disc model. Gray dashed lines indicate absolute deviations of 0.05. Colored dashed lines correspond to the predicted upper and lower final readout positions of the data shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143226#pone.0143226.g005" target="_blank">Fig 5</a>. (d,e) Impact of a 2-fold increase in the Dpp production rate and the growth rate (green line) on the Dpp gradient compared to the standard parameterization (black line) in a steady-state model on a linear scale (d) and a log-scale (e). (f) The deviation in the final relative readout position between the black and the red gradient in the steady-state model. Gray dashed lines indicate absolute deviations of 0.05. Colored dashed lines correspond to the predicted upper and lower final readout positions of the data shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143226#pone.0143226.g005" target="_blank">Fig 5</a>.</p

Read out of the Dpp gradient in the <i>Drosophila</i> wing disc.

<p>(a) A cartoon of the model for Dpp spreading in the wing disc. For details see main text and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143226#sec008" target="_blank">Methods</a>. (b, c) The simulated gradient profiles of total Dpp both increase in amplitude and expand with the growing domain as shown (b) on the absolute domain, and (c) on a rescaled domain. Time points shown (light to dark grey): 24 h, 46 h, 68 h, 90 h. (d) Total ligand concentration of the simulated profiles shown on a logarithmic scale. Different colors represent different concentration thresholds. (e) Depending on the concentration thresholds, the read-out positions follow different trajectories as the domain length expands over time. The dashed, black horizontal lines indicate the domain length interval used to calculate the positional inaccuracy in the read-out position, Δ<i>ζ</i>, shown in panels f,g. (f) The inaccuracy in the read-out position, Δ<i>ζ</i>, for different concentration thresholds, plotted on a logarithmic scale. (g) The inaccuracy in the read-out position, Δ<i>ζ</i>, has a minimum at around 25% of the domain length. A rather broad range of possible final boundary positions exhibits small positional inaccuracies. Panels a and b were reprinted from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143226#pone.0143226.ref020" target="_blank">20</a>] under a CC BY license, with permission from Nature Publishing Group & Palgrave Macmillan, original copyright 2014.</p

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.</p

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

<p>Gradient read-out for a gradient with constant amplitude, <i>c</i>₀, 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> with <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 domain that is scaled with respect to the current length of the domain, <i>L</i>(<i>t</i>). Five different concentration thresholds are shown as differently colored lines. (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, black horizontal lines mark the time interval that is further analysed in panels j-l. (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>, evaluated at different final read-out positions. (j-l) 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, black 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.</p