Contact-mediated pulling version of the model.

<p>(<b>A</b>) <b>Cells</b> only pulls <b>neighbors</b> (here, 3) that share a common surface area (shown in red) and that lie inside a maximum angle with respect to the convergence plane (here the horizontal axis). (<b>B</b>) Dependence of τ and κ with λ_force is qualitatively the same as before (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004952#pcbi.1004952.g004" target="_blank">Fig 4</a>). (<b>C</b>) Dependence with n_max is reversed, with the speed of intercalation (τ^-1) saturating after n_max = 3 and κ still decreasing. (<b>D</b>) The (κ τ,) x ϑ_max curves are more symmetric, but the <b>tissue</b> still elongates more and faster at lower angles.</p

Simulation results for different levels of polarization misalignment.

<p>(<b>A</b>) Semi-log graph of τ and κ with the variance (σ²). Both metrics are exponential functions of the variance. (<b>B-D</b>) Snapshots of 3 simulations with different levels of misalignment (σ = 40°, 50° and 70°). Each cell is represented by a white vector showing the direction of its polarization. The bigger vectors on (<b>C</b>) and (<b>D</b>) are due to zoom.</p

Types of convergent extension.

<p>In <i>active</i> convergent-extension, the cells in the tissue generate deforming forces due to anisotropic adhesion or pulling forces between cells (red arrows), while in <i>passive</i> convergent-extension, the surrounding environment deforms the tissue (blue arrows). Cell intercalation occurs in types of CE, but the axis of cell elongation is typically perpendicular to the axis of elongation in active CE and parallel in passive CE.</p

3D filopodial tension model versions.

<p>(<b>A</b>) Rotation around the polarization vector produces the 3D equatorial model. (<b>B</b>) Rotation around the convergence line results in the 3D bipolar model. (<b>A’-A”</b>) Initial and final states of a simulation of the equatorial model with all <b>cells’</b> polarization vectors pointing up. (<b>B’-B”</b>) Initial and final states of a simulation of the bipolar model with all <b>cells’</b> convergence axis lying vertically.</p

List of reference parameters values used in the simulations.

<p>Parameter sweeps vary one of the first 7 parameters, while keeping all the others constant. Key: λ_force, pulling strength; t_interval, time interval between link formation/breakage (MCS); r_max, maximum distance between <b>cells</b> (<b>cell</b> diameters); n_max, maximum number of links per <b>cell</b>; ϑ_max, maximum angle (radians); N, number of <b>cells</b>; cd, <b>cell</b> diameter (lattice sites); T, temperature or level of noise in the simulations; λ_volume, <b>cell</b> stiffness; n_orders, neighboring orders for lattice site flip and contact energy; J_c,M, adhesion energy between <b>cells</b> and <b>medium</b>; J_c,c, adhesion energy between <b>cells</b>.</p

Simulation snapshots and metrics.

<p>(<b>A</b>) Snapshots of a 2D simulation with reference parameter values, showing the initial configuration (left) and the configuration when the length of the major axis (L₊, red lines) increases to twice the length of the minor axis (L_-, blue lines), i.e., κ = 0.5. The simulation contains N = 109 <b>cells</b> (in green) with the tension forces shown by the white segments connecting their centers-of-mass. (<b>B</b>) Graph of L_-/L₊ versus time for the reference 2D simulation. For all simulations we measured the final value of the ratio between the length of minor and major axes of the tissue κ (shown in red), and the time τ (shown in blue) when the length of the major axis doubles the length of the minor axis (L_-/L₊ = 0.5).</p

Dependence of τ and κ with ϑ_max in the 3D versions.

<p>(<b>A</b>) 3D extensional model and (<b>B</b>) 3D bipolar model dependence of κ and τ parameters with ϑ_max. The range of best values for both κ and τ lies at much shorter angles in the 3D extension model (<b>A</b>) than the 2D model (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004952#pcbi.1004952.g005" target="_blank">Fig 5D</a>), which in turn has a range of optimal values slightly lower than in the 3D convergence model (<b>B</b>).</p

Competition between filopodial tension and surface tension in the 2D filopodial tension model.

<p>(<b>A</b>) The elongation time (τ) till the <b>tissue’s</b> inverse aspect ratio decreases to 0.5 as a function of the filopodial tension (λ_force) of the cells for different surface tensions (γ). (<b>B</b>) Insert: Degree of tissue deformation (κ) as a function of λ_force. An increase in the surface tension of the tissue reduces the final degree of CE (larger κ) shifting the κ vs. λ_force curve to the right. The opposite effect happens when the surface tension is decreased. Main: The κ vs. λ_force curves collapse when we rescale with the tension force by the surface tension plotting κ vs. λ_force/γ.</p

Simulation results for heterogeneous tissues.

<p>(<b>A</b>) Dependency of parameter κ with the percentage of passive (red dots) and refractory (blue squares) <b>cells</b>. (<b>B</b>) Dependency of parameter τ with the percentage of passive (red dots) and refractory (blue squares) <b>cells</b>. For both graphs, the measured value for the homogeneous <b>tissue</b> is represented by the green open square (<b>A</b>) or dot (<b>B</b>). Values of κ are measured for the whole <b>tissue</b> (active and non-active <b>cells</b>). (<b>C</b>-<b>D</b>) Simulations with passive <b>cells</b>; (<b>E</b>-<b>F</b>) simulations with refractory <b>cells</b>. (<b>C</b>) On a simulation with 95% of passive <b>cells</b> (in red) the remaining 5% of active <b>cells</b> (green) are still able to induce some degree of CE. (<b>D</b>) In a typical simulation with a higher percentage of active <b>cells</b> (here 33%) the active <b>cells</b> align at the center line of the extending <b>tissue</b>. (<b>E</b>) A failed CE for a tissue with less than 20% of active <b>cells</b> (here, 82% of refractory <b>cells</b>, blue). (<b>F</b>) When the percentage of active <b>cells</b> is above 20% (here 54%) the two populations sort out, with the active <b>cells</b> forming an elongated <b>tissue</b> and the refractory <b>cells</b> lying on each side of the structure. Panels (<b>D</b>) and (<b>F</b>) were rotated 90° for visualization purposes.</p

Cell intercalation model.

<p>(<b>A</b>,<b>B</b>) Given a planar-polarization vector (red) and convergence axis (blue), a <b>cell</b> forms links with up to n_max <b>cells</b> that lies within the interaction range r_max from its center-of-mass and within an angle ±ϑ_max, of the convergence axis. Each link exerts a tensions force λ_force on both of the <b>cells</b> it connects. (<b>A</b>) Image of a bipolar cell in chicken limb-bud mesenchyme overlaid with model parameters. (<b>B</b>) Snapshot of a GGH/CPM computer simulation of the filopodial-tension model, overlaid with model parameters. Dark yellow lines represent simulated filopodial links between a <b>cell</b> (light green) and its currently interacting neighbors (dark green). Experimental image courtesy of Gaja Lesnicar-Pucko and James Sharpe, CRG, Barcelona.</p