A mathematical model of the role of aggregation in sonic hedgehog signalling

Effective regulation of the sonic hedgehog (Shh) signalling pathway is essential for normal development in a wide variety of species. Correct Shh signalling requires the formation of Shh aggregates on the surface of producing cells. Shh aggregates subsequently diffuse away and are recognised in receiving cells located elsewhere in the developing embryo. Various mechanisms have been postulated regarding how these aggregates form and what their precise role is in the overall signalling process. To understand the role of these mechanisms in the overall signalling process, we formulate and analyse a mathematical model of Shh aggregation using nonlinear ordinary differential equations. We consider Shh aggregate formation to comprise of multimerisation, association with heparan sulfate proteoglycans (HSPG) and binding with lipoproteins. We show that the size distribution of the Shh aggregates formed on the producing cell surface resembles an exponential distribution, a result in agreement with experimental data. A detailed sensitivity analysis of our model reveals that this exponential distribution is robust to parameter changes, and subsequently, also to variations in the processes by which Shh is recruited by HSPGs and lipoproteins. The work demonstrates the time taken for different sized Shh aggregates to form and the important role this likely plays in Shh diffusion

Mapping of electrical signal propagation in E12.5 hearts.

<p>(<b>A</b>–<b>C</b>) Derivatives of the signal recorded in wt and homozygous <i>l11Jus8</i> E12.5 hearts. Black arrowheads point to the present peaks used for map generation, white arrowheads point to the missing peaks. Asterisk on (<b>C</b>) depicts the missing derivative for signal present on <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0107041#pone-0107041-g003" target="_blank"><b>Figure 3F</b></a>. The derivatives used in each map image are indicated on the electrical traces. (<b>D</b>–<b>F</b>) Optical maps showing spatio-temporal propagation of the electrical signal through the atria. Asterisks indicate gaps in normal electrical activity. White arrow shows the direction of the signal propagation in wild type and homozygous <i>l11Jus8</i> atria. (<b>G</b>–<b>I</b>) Optical maps showing electrical signal propagation through the ventricles. Maps are similar for all genotypes. Each colour (<b>D</b>–<b>I</b>) represents a one-millisecond interval with black as the initial time point and fuchsia as the final time point. RA, right atrium; LA, left atrium; LV, left ventricle; RV, right ventricle.</p

Comparison <i>Erbb2</i> null [22], [25] and <i>l11Jus8</i> phenotypes.

<p>Comparison <i>Erbb2</i> null <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0107041#pone.0107041-Lee1" target="_blank">[22]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0107041#pone.0107041-Erickson1" target="_blank">[25]</a> and <i>l11Jus8</i> phenotypes.</p

Numbers of distended hearts, hearts with atrioventricular block (AVb), atrial block (Ab), and atrial signal conduction gaps in optical mapping experiments.

<p>Numbers of distended hearts, hearts with atrioventricular block (AVb), atrial block (Ab), and atrial signal conduction gaps in optical mapping experiments.</p

Optical mapping of the cardiac electrical activity obtained with di-4-ANEPPS voltage sensitive dye.

<p>(<b>A</b>–<b>B</b>) and (<b>C</b>–<b>F</b>) show reading of E11.5 and E12.5 hearts, respectively. The location from which the electrical reading was taken is shown with lines onto the cardiac dissection image. Black arrowheads point to matched atrial and ventricular signals. (<b>D</b>) White arrowhead indicates absent signal in left ventricle corresponding to a present signal in left atria (black arrowhead), indicative of AV block. (<b>E</b>–<b>F</b>) White arrowheads indicate absent atrial signals corresponding to present ventricular signals (A block), a phenotype only present in mutant samples. Asterisks on RA and LA traces in (<b>F</b>) indicate signal described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0107041#pone-0107041-g005" target="_blank"><b>Figure 5C</b></a>. RA, right atrium; LA, left atrium; LV, left ventricle; RV, right ventricle.</p

Analysis of DNA, protein, and morphological defects in the <i>l11Jus8</i> mutant mouse.

<p>(<b>A</b>) Three coding mutations found in the 35 Mb candidate intervals on mouse chromosome 11 are located in the <i>Med31</i>, <i>Nf1</i> and <i>Erbb2</i> genes. Only recombinant embryos with the C57BL/6 homozygous DNA fragment inherited from the mutagenised parent which contained the <i>Erbb2</i> mutation were lethal at weaning, whereas embryos with homozygous <i>Med31</i> and <i>Nf1</i> and heterozygous <i>Erbb2</i> mutations were viable. Straight and dotted lines depict homo- and heterozygous DNA, respectively. (<b>B</b>) Confirmation of the T98,433,986G base change in the <i>Erbb2</i> gene in <i>l11Jus8</i> mutants by Sanger sequencing. (<b>C</b>) Phenotype of wild type, homozygous <i>l11Jus8</i> and <i>Erbb2^M802R</i> mutant embryos and hearts. Scale bars: 2 mm for embryos, 500 µm for hearts. (<b>D</b>) Erbb2 protein structure: FLD - Furin-Like Domain, STKD - Serine/Threonine-Tyrosine Kinase Domain. Partial sequence of the STK domain (60 amino acids around the mutation point) is shown for mouse (M), human (H), chick (C) and zebrafish (Z) Erbb2 protein. Asterisks indicate conserved amino acids, colon and period indicate conservation between groups of strongly and weakly similar properties, respectively. Arrowhead demarcates the location of the amino acid change. (<b>E</b>) Predicted changes in the protein structure of the Erbb2^M802R conserved kinase catalytic domain based on the reported crystal structure of the Erbb2 protein. (<b>F</b>) Top panel: expression of the <i>Erbb2</i> mRNA in atria (A) and ventricles (V) of hetero- and homozygous <i>l11Jus8,</i> and embryonic erythrocyte (blood) samples. <i>Prpf8</i> mRNA expression is used as a positive control for cDNA isolation in blood sample. Bottom panel: <i>Erbb2</i> expression in the yolk sack (YS) of homozygous <i>l11Jus8</i> embryo, “-RT” control, genomic DNA (gen) control. A no template negative control (neg) is shown in both panels. (<b>G</b>) Erbb2 immunohistochemistry in heterozygote and <i>l11Jus8</i> atria (Atr) and ventricles (Ven). Embryonic erythrocytes within the cardiac chambers are indicated with an arrowhead. Scale bar  =  200 µm.</p

Effect of the homozygous <i>Erbb2^M802R</i> mutation on heart morphology in <i>l11Jus8</i> hearts.

<p>(<b>A</b>–<b>L</b>) Representative H and E stained coronal sections of E12.5 hearts. (<b>A</b>–<b>B</b>) Atrioventricular cushions, (<b>C</b>–<b>D</b>) dorsal outflow tract, (<b>E</b>–<b>F</b>) ventricles, (<b>G</b>–<b>H</b>) atria. Arrows in (<b>G</b>–<b>H</b>) point to the atrial wall. AVC, atrioventricular cushion; OFT, outflow tract; Ven, ventricle; Atr, atrium; Ep, epicardium; En, endocardium; My, myocardium. (<b>I</b>–<b>J</b>) Longitudial atrial and (<b>K</b>–<b>L</b>) ventricular sections; homozygous <i>l11Jus8</i> heart had distended atria. Embryos in <b>I</b>–<b>L</b> are littermates. Magnified areas in (<b>I</b>–<b>J</b>) show the developing pectinate muscles (arrows). (<b>M</b>–<b>N</b>) Activated Caspase 3 (Cas3) staining of atrial wall at E11.5. Inset shows positive activated Caspase3 staining from trigeminal ganglion on same embryo section. (<b>O</b>–<b>P</b>) Propidium Iodide (PI) labelling of the necrotic cells in E12.5 hearts. RA, right atrium; LA, left atrium. Arrowheads point to the unspecific PI staining resulting from cells of the OFT damaged during dissection. Scale bars: 200 µm in <b>A</b>–<b>H</b> and <b>M</b>–<b>N</b>, 50 µm in <b>I</b>–<b>L</b>.</p

Mapping of electrical signal propagation in E11.5 hearts.

<p>(<b>A</b>) Pattern of wild type electrical signal propagation in the E11.5-12.5 heart. Different colours encode the time scale of the signal conduction; each colour represents a one-millisecond interval with black as the initial time point and fuchsia as the final time point. (<b>B</b>–<b>C</b>) Derivatives of the signal recorded in wt and homozygous mutant <i>l11Jus8</i> E11.5 hearts. Black arrowheads point to the peaks used for map generation. (<b>D</b>–<b>F</b>) Optical maps showing spatio-temporal propagation of the electrical signal through the atria. Asterisk on (<b>F</b>) points to the gap in normal electrical activity in homozygous mutant <i>l11Jus8</i> atria. (<b>G</b>–<b>I</b>) Optical maps showing electrical signal propagation through the ventricles. Maps are similar for all genotypes. SAN, sinoatrial node; RA, right atrium; LA, left atrium; LV, left ventricle; RV, right ventricle; AVC, atrioventricular canal; IVG, interventricular groove.</p