a) and b) show the mean gradient of Wnt concentration (100 simulations) with standard deviation in transparent over the expanding tissue at different times (normalized to the maximum at t = 180 min) with respect to the distance from the embryonic margin, the green area indicates Wnt producing cells. c) and d) show the corresponding 2D cell distribution at the final stage (t = t_TRS = 180 min). Cell fates are determined by thresholds on the local single cell Wnt-concentration. The thresholds are set at t = 180 min so that the cell-numbers in forebrain-, midbrain- and hindbrain-fate are equal. Besides the Wnt producing cells shown in green, the colors of the cells represent different cellular fates: forebrain fate is indicated in red, midbrain fate in white and hindbrain fate in blue.

a) schematic illustration of the experimental procedure: 150ng mRNA injection of β-catenin effector Gsk3β into one cell at the eight blastomere stage. At sphere stage (prior induction of Wnt ligand expression), embryos were subjected to an image-based approach to analyse the distribution of cell clones in the animal tissue. b) expression of Gsk3β-GFP (Wnt-OFF) leads to a dispersed clone at the sphere stage. c) However, expression of dominant-negative Gsk3β-GFP (Gsk3β-DN-GFP; Wnt-ON) leads to clustering of the clonal cells. d) Gsk3β-DN expressing cells (Wnt-ON) show large clusters: demonstrated by an increased cluster surface and a reduced number of total clusters/cells compare to cells expressing WT Gsk3β (Wnt-OFF). The expression levels of Gsk3β and Gsk3β-DN are kept at a similar level shown by comparable total GFP-fluorescence in the clones. e)-g), embryos were injected with the indicated constructs and subjected to in situ hybridization against pax6a at 26hpf. The yellow dotted line illustrates the position of the neural plate boundaries. Insets show expression of the GFP tagged constructs prior fixation. h)-j), schematics illustrate the posterior shift of the borders after the reduction of β-catenin activity in the neural plate and the disruption of the boundaries after clonal decrease of β-catenin activity.

Distributions of the final cell fate at t = t_TRS = 180 min in the tissue for different parameters of the cell dynamics and different transport mechanisms. Directed migration with the sorting parameter p_DirMig = 0.02 is shown in a) and b), and with p_DirMig = 0.2 in c) and d). Cell fates and coloring are as in Fig 1.

To determine the number of apoptotic cells in the developing embryo, immunohistochemical staining against Caspase3 is performed at 5 hpf, 7 hpf, and 9 hpf. a), After antibody staining, 10 areas of ca 20 by 20 cells are placed randomly over the dorsal hemisphere of the embryos (marked by gsc-GFP expression) and within these squares of ca 400 cells, the number of apoptotic cells is counted. b) At 5 hpf, no apoptotic cells are detected. At 7 hpf 2% and at 9 hpf 3% apoptotic cells are observed. c) Induced apoptosis of a similar relative number of cells as experimentally determined are implemented into our simulations. The simulation of cytoneme based Wnt-transport with weak sorting and no induced apoptosis (left) is compared to the simulation with additionally enabled induced apoptosis (right) at t = t_TRS = 180 min. Cell fates and coloring are as in Fig 1. d) Increasing the sorting parameter p_DirMig during the directed transport simulations leads to decreased mixing of cell fates (red with, blue without apoptosis). Even large values for p_DirMig, however, cannot remove truly isolated cells (cf. c)). Adding limited apoptosis (130 cells over entire simulation) can eliminate these cells and leads to an additional improvement. Displayed is the development after t = 180 min averaged over 10 simulations for each value of p_DirMig, with p_DirMig = 0.02 and p_DirMig = 0.2 (cf. Fig 4) highlighted.

In a) and b) the development of the pattern boundaries between brain areas during the tissue growth is shown. Colors indicate the respective brain primordium. The thresholds are set to split the tissue into thirds by number at t_TRS = 90 min. The green area indicates the layers of the morphogen producing cells. c) and d) show the thresholds for splitting the total number of cells into thirds at each respective point in time. The dashed lines are the averages of 100 simulations with the transparent regions showing the standard deviation σ. The lower threshold (red dashed line) separates forebrain and midbrain, the upper threshold (blue dashed line) separates midbrain and hindbrain. The difference between both thresholds (black line) indicates the distinctness between the thresholds separating cell fates.

Comparison of the establishment of the three brain primordia based on cytoneme and diffusion-based transport. The percentage of cells that reach sufficient morphogen concentration to adopt their final cell fate (which is in this case determined by thresholding at t_TRS = 90 min so that the tissue is split into thirds by cell numbers) is plotted over time. This is shown for forebrain (red solid line), midbrain (gray solid line), and hindbrain (blue solid line). The gray dashed lines mark the points in time where 75% of the midbrain and hindbrain cells adopted their final fate, respectively. After about 45 min, the majority of cells (75%) acquire their final fate if the ligand is transported on cytonemes. A diffusion-based distribution requires considerably longer time (about 80 min) to determine the cellular fate in the target tissue. The values are averaged over 100 simulations.

To determine the temporal function Wnt/β-signaling on patterning, zebrafish embryos are treated with 5 μM of the Porcupine inhibitor IWP12 to block Wnt secretion. a) Treatment started at different time points (3–7 hpf) until 10 hpf. At 10 hpf and 20 hpf, embryos are fixed and subjected to ISH against otx2 and pax6a, respectively. b) Embryos treated from 3 hpf and 4 hpf to 10 hpf showed a significant increase of the forebrain and midbrain tissue at 10 hpf and an enlarged area of the eye vesicle at 20 hpf. c) Embryos treated after this time point showed only marginal changes in patterning, suggesting that Wnt signaling is required between 3–5 hpf to pattern the neural plate.

The field in which the tissue is modeled consists of precomputed non-overlapping potential cell positions (“fixed irregular lattice sites”). To initialize a simulation one margin of the field is filled with non-dividing morphogen producing cells (here shown in green), as well as morphogen receiving cells (shown in blue) with no initial morphogen concentration. After initialization (up to) four processes are sequentially executed for one timestep. In the signaling step morphogen is deposited from the producing into the receiving cells, either by diffusive transport or by direct cell to cell contacts via cytonemes, also some morphogen decays here in the recipient cells. Insertion drives the tissue growth and consists of two causes which cause the same effect, namely cell division and intercalation from overlying and underlying cell sheets. Here, if an insertion occurs at one cell, a path to the nearest empty cell position is determined and all cells are subsequently moved along this path, the emerging empty spot is than filled by a cell with its properties chosen randomly from all cells with similar distance to the producing cells. Apoptosis can be induced if the Wnt concentration of a single cell differs strongly from its neighbors, this process is only activated in the last third of the simulation and only the 130 cells with the strongest discrepancy of Wnt contents to neighbors are removed and replaced similarly to insertion. Migration models nearest neighbor interactions by a random swapping of positions of nearest neighbor cells. Directed migration can be introduced if the swapping probabilities are changed so that gradient enhancing swaps become more likely.