Wright-Fisher diffusion bridges

The trajectory of the frequency of an allele which begins at $x$ at time $0$ and is known to have frequency $z$ at time $T$ can be modelled by the bridge process of the Wright-Fisher diffusion. Bridges when $x=z=0$ are particularly interesting because they model the trajectory of the frequency of an allele which appears at a time, then is lost by random drift or mutation after a time $T$. The coalescent genealogy back in time of a population in a neutral Wright-Fisher diffusion process is well understood. In this paper we obtain a new interpretation of the coalescent genealogy of the population in a bridge from a time $t\in (0,T)$. In a bridge with allele frequencies of 0 at times 0 and $T$ the coalescence structure is that the population coalesces in two directions from $t$ to $0$ and $t$ to $T$ such that there is just one lineage of the allele under consideration at times $0$ and $T$. The genealogy in Wright-Fisher diffusion bridges with selection is more complex than in the neutral model, but still with the property of the population branching and coalescing in two directions from time $t\in (0,T)$. The density of the frequency of an allele at time $t$ is expressed in a way that shows coalescence in the two directions. A new algorithm for exact simulation of a neutral Wright-Fisher bridge is derived. This follows from knowing the density of the frequency in a bridge and exact simulation from the Wright-Fisher diffusion. The genealogy of the neutral Wright-Fisher bridge is also modelled by branching P\'olya urns, extending a representation in a Wright-Fisher diffusion. This is a new very interesting representation that relates Wright-Fisher bridges to classical urn models in a Bayesian setting. This paper is dedicated to the memory of Paul Joyce

Cuticle permeability of the NILs.

<p>Cuticle permeability was evaluated by air drying at room temperature (a and c) and chlorophyll leaching in 80% ethanol (b and d). The numbers on the x-axes represent hours of treatment. Water loss or chlorophyll leaching at each time point is represented on the y-axes as percentages of the total water content or total chlorophyll content in the tissue. Measurements taken from four individuals were averaged.</p

Wax composition of the six NILs.

<p>(a) Total wax load of the flag leaf sheaths was measured by GC-MS. (b) β-diketone, fatty acid, aldehyde, primary (1°) alcohol, alkane, and sterol and triterpene (ST&TP) content. The numbers on the y-axes indicate average content expressed as µg per g dried tissue (dry weight, DW). The bars indicate standard deviation of the mean estimated from six biological replicates. (c) The percentage of wax species in each genotype was calculated from the means.</p

The expression of cuticular wax- and cutin-related genes in the wheat flag leaf sheath of <i>W1</i>w2, <i>w1W2,</i> and <i>W1W2</i> plants compared to <i>w1w2</i>.

<p>Genes with two- or higher-fold changes are depicted and expression data for all genes analyzed are listed in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0054129#pone.0054129.s007" target="_blank">Table S3</a>. The bars represent standard deviation of the mean fold-change of mRNA levels. Asterisks indicate that the difference is significant at <i>P</i><0.05 (*) or at <i>P</i><0.01 (**).</p

<i>CER4-6</i> expression in sheaths of <i>Iw1iw2</i>, <i>iw1Iw2,</i> and <i>W1W2</i> at different developmental stages.

<p>(a) Transcription levels of <i>CER4-6</i> at stage F4.0 compared to that of the reference gene <i>TaRPII36</i>. (b) Fold changes of <i>CER4-6</i> transcription at stage F9.0 compared to that at stage F4.0. The bars represent standard deviation of the mean fold-change of mRNA levels. Asterisks indicate that the difference is significant at <i>P</i><0.01 (**).</p

Genetic Interactions Underlying the Biosynthesis and Inhibition of β-Diketones in Wheat and Their Impact on Glaucousness and Cuticle Permeability

<div><p>Cuticular wax composition greatly impacts plant responses to dehydration. Two parallel pathways exist in Triticeae for manipulating wax composition: the acyl elongation, reduction, and decarbonylation pathway that is active at the vegetative stage and yields primary alcohols and alkanes, and the β-diketone pathway that predominates at the reproductive stage and synthesizes β-diketones. Variation in glaucousness during the reproductive stage of wheat is mainly controlled by the wax production genes, <em>W1</em> and <em>W2</em>, and wax inhibitor genes, <em>Iw1</em> and <em>Iw2</em>. Little is known about the metabolic and physiological effects of the genetic interactions among these genes and their roles in shifting wax composition during plant development. We characterized the effect of <em>W1, W2, Iw1,</em> and <em>Iw2</em> and analyzed their interaction using a set of six near-isogenic lines (NILs) by metabolic, molecular and physiological approaches. Loss of functional alleles of both <em>W</em> genes or the presence of either <em>Iw</em> gene depletes β-diketones and results in the nonglaucous phenotype. Elimination of β-diketones is compensated for by an increase in aldehydes and primary alcohols in the <em>Iw</em> NILs. Accordingly, transcription of <em>CER4-6</em>, which encodes an alcohol-forming fatty acyl-CoA reductase, was elevated 120-fold in <em>iw1Iw2</em>. <em>CER4-6</em> was transcribed at much higher levels in seedlings than in adult plants, and showed little difference between the glaucous and nonglaucous NILs, suggesting that <em>Iw2</em> counteracts the developmental repression of <em>CER4-6</em> at the reproductive stage. While <em>W1</em> and <em>W2</em> redundantly function in β-diketone biosynthesis, a combination of both functional alleles led to the β-diketone hydroxylation. Consistent with this, transcription of <em>MAH1-9</em>, which encodes a mid-chain alkane hydroxylase, increased seven-fold only in <em>W1W2</em>. In parallel with the hydroxyl-β-diketone production patterns, glaucousness was intensified and cuticle permeability was reduced significantly in <em>W1W2</em> compared to the other NILs. This suggests that both <em>W1</em> and <em>W2</em> are required for enhancing drought tolerance.</p> </div

Expression of cuticular wax genes in the wheat flag leaf sheath of the <i>Iw</i> NILs compared to <i>W1W2</i>.

<p>Genes with two- or higher-fold changes are depicted and expression data for all genes analyzed are listed in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0054129#pone.0054129.s006" target="_blank">Table S2</a>. The bars represent standard deviation of the mean fold-change of mRNA levels. Asterisks indicate that the difference is significant at <i>P</i><0.05 (*) or at <i>P</i><0.01 (**).</p

Flag leaf sheaths and peduncles of the NILs examined in this study.

<p>NIL designations are indicated beneath each peduncle and genotypes are specified in parentheses. The introduced alleles in the genotypes are underlined. The bar indicates 1 cm.</p

Homolog variation of major wax species.

<p>Carbon atom numbers of aldehydes (a) and primary (1°) alcohols (b), and β-diketones (c) are indicated on the x-axes. Their contents are indicated on y-axes as µg per g dried tissue. The bars indicate standard deviation of the mean calculated from six biological replicates. β-D, β-diketone; and OH-β, hydroxyl-β-diketones.</p

Electron micrographs of the cuticle surfaces of flag leaf blades, sheaths, peduncles, and glumes of the NILS.

<p>The tissues are indicated on the top and the NIL designations at the left. The bars indicate 2.5 µm.</p