Results are robust to additional complexities including cell division.

<p>(A, B) Green solid curves show slices from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.g002" target="_blank">Fig 2</a> with <i>K</i> = 10 while black dashed line shows unregulated limit . We see that regulation can reduce timing variance even with bursts in activator production of mean size <i>b</i> (A, cyan and magenta dashed), initial Poisson noise in repressor number (B, green dashed), or steady state <i>k</i>/<i>μ</i> in regulator dynamics (blue) unless it approaches regulation threshold <i>K</i> (red). (C) Mean dynamics of activator model (solid) and repressor model (dashed) in which cell division occurs at time on average. Abrupt reductions in molecule numbers are smoothed by noise in <i>t</i>_<i>d</i> and by binomial partitioning of molecules. (D) Timing variance approaches that with no division (dashed) within experimental division region (gray). In A and B, parameters are as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.g002" target="_blank">Fig 2</a>. In C and D, parameters are <i>x</i>* = 15, 〈<i>a</i>〉/<i>x</i>_* = 〈<i>r</i>〉/<i>x</i>_* = 10, and <i>H</i> = 3, with <i>kt</i>_*, <i>μt</i>_*, and <i>K</i> set to optimal values (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.g002" target="_blank">Fig 2</a>) and and <i>σ</i>_<i>d</i> set to experimental values. In all cases, <i>α</i> is set to ensure that mean threshold crossing time equals <i>t</i>_*.</p

Threshold crossing of a regulated molecular species.

<p>(A) A target species <i>X</i> is regulated by either an accumulating activator <i>A</i> or a degrading repressor <i>R</i>. (B) Temporal precision is quantified by the variance of the first-passage time, at which the stochastic molecule number <i>x</i> first crosses the threshold <i>x</i>_*. (C, D) Deterministic dynamics illustrate the effects of regulation. Parameters are <i>kt</i>_* = 20 and <i>K</i> = 15 in C; <i>μt</i>_* = 2.75, <i>K</i> = 2.6, and <i>N</i> = 15 in B and D; and <i>x</i>_* = 15 and <i>H</i> = 1 throughout. <i>t</i>₀ is defined by in C and in D.</p

Optimal regulatory strategies.

<p>Timing variance as a function of the regulatory parameters reveals (A) a trajectory along which the variance decreases in the case of the activator and (B) a global minimum in the case of the repressor. White dashed line in A and white dot in B show the analytic approximations in Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.e029" target="_blank">9</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.e031" target="_blank">11</a>, respectively. Parameters are <i>N</i> = 15 in B, and <i>x</i>_* = 15 and <i>H</i> = 3 in both.</p

Model predictions agree with neuroblast migration data.

<p>(A) Number of <i>mig-1</i> mRNA molecules per cell as a function of time <i>t</i>, obtained by single-molecule fluorescent in situ hybridization, from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.ref005" target="_blank">5</a>]. Magenta shows approximate range of times when cell migration terminates. Black lines show mean (dashed) and standard deviation <i>σ</i>_<i>d</i> of cell division times (black points). (B) Timing variance vs. linearity of <i>x</i>(<i>t</i>), both for experimental data in A (blue circle) and our model (curves, Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.e049" target="_blank">16</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006201#pcbi.1006201.e050" target="_blank">17</a>). Data analyzed using ranges of threshold 10 ≤ <i>x</i>_* ≤ 25 and bin size 3 ≤ Δ<i>x</i> ≤ 12; error bars show standard deviations of these results. We see that for sufficiently large cost 〈<i>a</i>〉/<i>x</i>_* or 〈<i>r</i>〉/<i>x</i>_*, model predictions agree with experimental data point.</p

Targeting strategy to generate a conditional <i>Vps35</i> allele.

<p>(A) LoxP sites were introduced in the third and fourth introns of the <i>Vps35</i> gene by homologous recombination. (B) Southern blot and PCR analysis showing correct targeting of selected ES cell clone. </p

<i>Vps35</i>^{<i>∆/∆</i>} organoids show a growth defect but are competent to respond to Wnt signaling.

<p>(A) Growth was quantified by scoring the organoids in categories based on the number of buds the organoids had produced 5 days after passaging. (B) <i>Vps35</i>^{<i>∆/∆</i>} organoids show reduced proliferation compared to control organoids. This could not be completely rescued by Wnt3a supplemented in the medium (ERN: small intestine organoid medium, containing EGF, R-Spondin, Noggin, WENR: ERN medium supplemented with 30% Wnt3a conditioned medium). (C) <i>Vps35</i>^{<i>∆/∆</i>} organoids can respond to Wnt signaling as assayed by Axin2 qPCR (data are represented as mean ± SD, n=3). (D) Percentage of growing organoids cultured in ENR medium with varying R-spondin concentrations. </p

Knockout of <i>Vps35 in vivo</i>.

<p>(A) PCR analysis using primers that anneal outside <i>Vps35</i> exon 4 shows deletion of exon 4 from genomic DNA isolated from small intestinal epithelium (SI) of 4-OHT induced <i>Vps35</i>^<i>fl/fl</i> (control) or <i>Vps35</i>^<i>fl/fl</i><i>; Villin-CreERT2</i> mice (<i>Vps35^∆/∆</i>). Histological analysis of <i>Vps35</i> knockout intestine showed no defects in crypt-villus morphology. Intestine sections were Periodic Acid Schiff (PAS) stained (B) and immunohistochemistry was performed to stain Lysozyme (C). </p

Knockout of <i>Vps35</i> in intestinal organoids.

<p>Intestinal organoids were obtained from a <i>Vps35</i>^<i>fl/fl</i><i>; Villin-CreERT2</i> mouse and treated with 0.5 μM 4-OHT for 12 hours (<i>Vps35^∆/∆</i>), or control treated (control). (A) PCR analysis of genomic DNA from <i>Vps35</i> knockout organoids shows complete deletion of exon 4 of <i>Vps35 </i><i>in </i><i>vitro</i>. (B) RT-PCR shows absence of <i>Vps35</i> exon 4 from mRNA of <i>Vps35</i>^{<i>∆/∆</i>} organoids. (C) Western blot analysis shows absence of Vps35 protein and reduced Wls protein levels in <i>Vps35</i>^{<i>∆/∆</i>} organoids. (D) <i>Vps35</i>^{<i>∆/∆</i>} organoids show normal morphology, Paneth cells are indicated by arrowheads. (E) RT-PCR analysis of molecular markers of differentiated intestinal cells and intestinal stem cells.</p

HMG and Helper sites contribute differentially to the regulation of <i>end-1</i> during early embryogenesis.

<p>Deconvolved (A–C) and Nomarski (A′–C′) images showing expression of a stably integrated <i>end-1::GFP::H2B</i> reporter in the endodermal (E) and/or mesodermal (MS) daughters of live embryos at the 2E stage. The wild type (WT) reporter shows strong GFP expression in the E cell daughters (A, A′). Mutation of the HMG site leads to a significant reduction of GFP expression in the E daughters and a significant derepression of <i>end-1::GFP::H2B</i> in the MS daughters (B, B′). Mutation of two Helper sites leads to a significant reduction of GFP in the E daughters (C, C′), but little or no depression in the MS daughters (C, C′). (D) Histograms summarizing the results from over 100 embryos from three independent lines for each construct, grouped by strong, weak or no expression in the E (upper graph) and MS (lower graph) cells.</p

Helper sites and the C-clamp are not required for basal repression of Wg targets in <i>Drosophila</i>.

<p>(A–I) Confocal images of stage 16–17 embryos containing a <i>pxb::lacZ</i> WRE reporter immunostained for Wg (green) (A, D & G), lacZ (red) (B, E & H) or merged (C, F & I). The wild-type reporter shows a pattern overlapping with Wg in the second constriction of the midgut, and a non-overlapping pattern in the hindgut (A–C). Mutation of two HMG sites leads to a strong depression through the entire midgut (arrowheads), without affecting lacZ expression in the second constriction (arrow) (D–F). Mutation of two Helper sites leads to a significant decrease in the lacZ expression in the second constriction (arrow) with weak ectopic expression (arrowheads)(G–I). The hindgut expression did not vary in the different constructs and was used as an internal control. All images are representative of at least 20 embryos. (J–M) Images of adult wings containing the wing driver <i>C96-Gal4</i> crossed to wildtype (WT) (J, J′), UAS-TCF/Pan RNAi (K, K′) or UAS-TCF/Pan RNAi plus UAS-LEF1 (L, L′) or UAS-LEF1 plus the C-clamp of TCF/Pan (M, M′). Knockdown of TCF/Pan leads to notches (arrowheads) and ectopic wing margin bristles (block arrows) along the periphery of the wing (where <i>C96-Gal4</i> is active; K, K′). Expression of the human LEF1 transgene significantly rescues the ectopic bristle expression, but not the notches (L, L′). Expression of a LEF1-C-clamp chimera rescues the wing margin defects and prevents ectopic bristle formation, and causes a L5 vein defect (arrow). Details about the penetrance of these phenotypes are listed in <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1004133#pgen-1004133-t001" target="_blank">Table 1</a>.</p