Tension force and pressure.

<p>Forces applied to the junction vertex of three cells <i>a</i>, <i>b</i>, and <i>c</i>. The <i>tension force</i> <b><i>T</i></b>(<b><i>e</i></b>_{<i>i</i>,<i>j</i>}) exerts along the direction <b><i>e</i></b>_{<i>i</i>,<i>j</i>} of an inner edge (interior cell boundary), or along the tangent direction of outer edge <b><i>e</i></b>_<i>i</i> (free cell boundary), where (<i>i</i>,<i>j</i>) are the two indices of cells <i>a</i>,<i>b</i>, or <i>c</i>. The <i>pressure force</i> <b><i>P</i></b>(<b><i>e</i></b>_{<i>i</i>,<i>j</i>}) acts along the direction normal to the cell boundary, in the direction from the cell with higher pressure to that with lower pressure.</p

Algorithm 1. UpdateCellPattern (<b><i>V</i></b>(<i>t</i>), Δ (<i>t</i>), Δ<i>η</i>(<i>t</i>), <i>σ</i>, <i>k</i>).

<p>Algorithm 1. UpdateCellPattern (<b><i>V</i></b>(<i>t</i>), Δ (<i>t</i>), Δ<i>η</i>(<i>t</i>), <i>σ</i>, <i>k</i>).</p

Model of feedback circuits for tissue size control.

<p>a) Division types of stem cells and progenitor cells. Red sphere labeled with (S) indicates stem cells, blue hexagon (P) indicates progenitor cells, and white diamond (D) indicates differentiated cell. The same color code is used for illustration of resulting tissues. b) Feedback controls of stem cell model. Blue arrows indicate self-renewal or proliferation divisions. Black arrows indicate symmetric differentiation divisions. Red arrows indicate asymmetric divisions. Flat-head arrows extending from differentiated cell with corresponding colors indicate inhibitions to respective type of divisions.</p

Data Structure.

<p>The boundary of a cell is formed by connected edges. It is modeled as an oriented closed curve in the counterclockwise direction. Each physical inner edge is represented twice using two half-edges, once each in opposite directions for each of the two contacting cells. Each physical outer edge is also represented twice with two half-edges, once for the cell, and once for the outside space.</p

Model of tissue development starting from a single cell.

<p>(a) A single cell and its plane of first division; (b) Two daughter cells after the first division, each is slight deformed from its shape in (a); (c) The formation of four cells after two cell divisions.</p

The degree of alignment by an index ϱ.

<p>It is calculated by drawing a vertical line in the middle of each stripe and summing the number of cells that lie between the bristles and the line. This is then normalized with the total number of bristles in the stripe. Here, a) is an example of good alignment with ϱ = 0.5 and b) is an example of bad alignment with ϱ = 2.0</p

Fusion of cells and tissues.

<p>(a) and (b) shows two separate growing cells come into contact and become fused together. A new edge is formed between the two cells. (c) and (d) shows the case of three growing cells fusion. Three new edges and a new vertex are formed after fusion. (e) and (f) shows the fusion of two growing tissues. Two separate tissues contact with each other and become fused together to form a continuous tissue. The new edges and vertices formed are highlighted.</p

Steady state spatial gradients of <i>Dl</i> of different diffusion rates.

<p>The steady state spatial gradients of <i>Dl</i> are due to diffusion and degradation. The red solid line with triangular markers is the steady state gradient formed with the diffusion coefficient 1.2 <i>μm</i>² <i>s</i>⁻¹. The green dash line with square markers is the gradient of the diffusion coefficient 4.8 <i>μm</i>² <i>s</i>⁻¹. The blue dotted line with circle markers is of the diffusion coefficient 9.6 <i>μm</i>² <i>s</i>⁻¹. The black straight lines represents the 0.05 <i>Dl</i> concentration threshold for cellular response.</p

Bristle Plotting Puzzle.

<p>Simulation results of bristle pattern formation using different models. a) The pattern of gene expression used for the stripe models. Green stripes have almost equal expression of <i>Dl</i> and <i>N</i> genes but <i>ac</i> is not expressed. Blue stripes have high expression of <i>ac</i> and <i>Dl</i> genes but low expression of <i>N</i> gene. Red stripes have high expression of <i>ac</i> and <i>N</i> genes but low expression of <i>Dl</i> gene. Bristles only form in the blue stripes. b) Lateral inhibition with stripes does not ensure equal spacing or good alignment. c) Inhibition field with out stripes ensures proper spacing but does not produce a good alignment. d) Inhibition field with stripes produces equal spacing as well as good alignment.</p

Visualization of the growth of tissue from 2 to 4,000 cells.

<p>a) 2 cells; b) 100 cells; c) 500 cells; d) 1,000 cells; e) 4,000 cells, and f) zoomed in view of 4,000 cells.</p