Scaling of the limit cycle for first evolved model.

<p>(A) Sketch of the model. Parameters and equations are given in Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002585#pcbi.1002585.s006" target="_blank">Text S1</a> (B) Variation of the period with input level demonstrating compensation. (C) limit cycle for different values of the input in space. Limit cycle varies over almost one order of magnitude in while the period changes by . The input values follow the color bar in F. (D) Rescaling the limit cycles to the unit interval in each variable shows almost perfect collapse for different input values. Circles indicate the fixed point. (E) PRC for different input values, represented by different colors. The PRC was computed by adding a degradation term of for for of the period. (F) as a function of phase for the limit cycle at different temperatures. The maximum of is defined as phase for the PRC. There is almost perfect overlap. Panels D–F here and the following figure, demonstrate our contention that the evolved models replicate essential properties of the Goodwin model even though there is no direct parameter rescaling.</p

Absence of scaling for an evolved model with distributed temperature compensation.

<p>(A) Sketch of the model. Parameters and equations are given in Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002585#pcbi.1002585.s006" target="_blank">Text S1</a> (B) Variation of the period as a function of input. (C) limit cycle for different values of the input in space. Limit cycle varies over one order of magnitude in variable 1 and 2 while the period changes at most (D) Rescaling of those limit cycle to the unit interval for each variable. The orbits for different inputs no longer scale. Circles mark the fixed point (E)The PRC for different input values do not scale. The PRC was computed by adding a degradation term of for variable 2 for of the period. (F) Variable as a function of phase for the limit cycles at different temperature. Its rescaled maximum of 1 is defined as phase for the PRC.</p

Scaling of the limit cycle for the Mixed Feedback Loop adaptive model.

<p>(A) Sketch of the model. Parameters and equations are given in Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002585#pcbi.1002585.s006" target="_blank">Text S1</a> (B) Variation of the period as a function of input values. (C) limit cycle for different values of the input in the B-transcript, A-protein plane. Limit cycle varies over almost one order of magnitude in B-transcript level, while the period changes by a few percent. Note that the fixed point for A is adaptive (independent of input). (D) Linear rescaling of the limit cycles to the unit interval in each variable showing almost perfect collapse for different input values. Circles indicate the fixed point. Color code follows bar in panel F (E) PRC for different input values, represented by different colors. The PRC was computed by adding a degradation term of for B transcripts for of the period (see equations in Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002585#pcbi.1002585.s006" target="_blank">Text S1</a>). (F) B as a function of phase for the limit cycle at different temperatures. The maximum of B is defined as phase for the PRC. There is almost perfect overlap.</p

Predicting Ancestral Segmentation Phenotypes from Drosophila to Anopheles Using <i>In Silico</i> Evolution

<div><p>Molecular evolution is an established technique for inferring gene homology but regulatory DNA turns over so rapidly that inference of ancestral networks is often impossible. <i>In silico</i> evolution is used to compute the most parsimonious path in regulatory space for anterior-posterior patterning linking two Dipterian species. The expression pattern of gap genes has evolved between <i>Drosophila</i> (fly) and <i>Anopheles</i> (mosquito), yet one of their targets, <i>eve</i>, has remained invariant. Our model predicts that stripe 5 in fly disappears and a new posterior stripe is created in mosquito, thus <i>eve</i> stripe modules 3+7 and 4+6 in fly are homologous to 3+6 and 4+5 in mosquito. We can place <i>Clogmia</i> on this evolutionary pathway and it shares the mosquito homologies. To account for the evolution of the other pair-rule genes in the posterior we have to assume that the ancestral Dipterian utilized a dynamic method to phase those genes in relation to <i>eve</i>.</p></div

Simulated evolutionary pathway (label <i>ftz</i> on Fig 2A) from <i>Drosophila</i> to <i>Anopheles</i>, including <i>ftz</i>.

<p>Conventions of <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g003" target="_blank">Fig 3</a> for <i>eve</i> and <i>ftz</i> stripes are used.</p

Summary of our predictions.

<p>A Predicted evolutionary pathways from different simulations detailed in <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g003" target="_blank">Fig 3</a> (label F1), <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g004" target="_blank">Fig 4</a> (label F2), <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g005" target="_blank">Fig 5</a> (label Ftz), <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g007" target="_blank">Fig 7</a> (label LC). The times shown for the intermediates are only schematic. Gap and <i>eve</i> patterns in three insect species and the inferred last common ancestor (LCA) are indicated. B Summary of homology between Eve modules in different species predicted by our evolutionary simulations.</p

Simulated evolutionary pathway (label F1 on Fig 2A) from <i>Drosophila</i> to <i>Anopheles</i>, with salient changes discussed in the main text.

<p>For each transcriptional <i>eve</i> module only the gap genes that regulate it are shown with the same color scheme as Figs <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g001" target="_blank">1</a> and <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.g002" target="_blank">2A</a>. <i>eve</i> stripe 1 is not shown and the maximum expression of each module is normalized to 1 except when it dips beneath a threshold equivalent to its loss.</p

Simplified model of <i>Drosophila</i> network.

<p>A-B: simulated maternal and gap gene profiles. C: simulated pair-rule gene profiles. A’-C’ Summary of interactions used to generate these profiles. Equations and references for the interactions are given in the <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.s005" target="_blank">S1 Text</a>. A generic spatially uniform activator is assumed where needed.</p

A model for the LCA that imparts stable phase relations among the pair-rule genes and remains consistent with the evolutionary pathway from fly to mosquito.

<p>(A) Schematic of the model showing gap input to eve only. The intensity of repression among the remaining genes (chosen arbitrarily as the primary pair-rule genes in fly) is shown by the line intensity and defines their relative phase. (B) Behavior of the model in A in response to imposed temporal oscillations of Eve, showing phase relationships between different pair-rule genes. Viewed within a cell, one cycle of temporal oscillations would result from the forward shift of the entire Eve pattern by one period. (C,D) If we implement a forward shift of <i>eve</i> by one stripe (left to right panels), by suitably scaling the maternal gradients, then an arbitrary initial arrangement of the three remaining genes is reset to the proper phasing for fly. We show the gap gene configuration for fly in (C) and for mosquito in (D). <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.s003" target="_blank">S3</a> and <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1006052#pgen.1006052.s004" target="_blank">S4</a> Videos show the evolution from the left to right panels respectively for panels (C) and (D). For simplicity only, the model is applied across the entire embryo, though in reality the anterior gap gene input to the primary pair rule genes can remain invariant.</p