Figure 1

Tissue cell networks. (a) Vascular endothelial cells during vasculogenesis, visualized in a bird embryo according to Ref. 7. (b) Mouse vascular endothelial cells within an in vitro explant [22], after 24 h of culturing. (c) Astroglia-related rat C6 cells and (d) muscle-related mouse C2C12 cells cultured on a rigid tissue culture substrate as described in Ref. 23. The inset depicts the motion of latex beads at the tissue culture surface. To measure convection currents, $10\mu \mathrm{m}$ wide and $250\mu \mathrm{m}$ long rectangles were selected in the images, directed parallel to the flow. For each frame, pixel intensities were averaged across the rectangle, resulting in the horizontal lines in the inset. (e) C6 cells (arrows) migrate along extended projections of adjacent cells (arrowheads). (f) Endothelial cells (arrows) move along a vascular sprout, visualized in the mouse explant by differential interference contrast microscopy.Reuse & Permissions

Figure 2

Network formation in the model. Randomly placed $N=500$ particles assemble into linear structures, detectable already within ${30}^{\prime }$ (a). At a sufficiently high particle density, a characteristic pattern size develops in five hours (b) with a combination of sprouting (branch extension) and coarsening (merger of adjacent branches). Connected dots represent Voronoi neighbor particles. Darkening gray levels indicate increasing local anisotropy. The simulation covered an area of $L=0.7\mathrm{mm}$.Reuse & Permissions

Figure 3

Power spectrum $S(q)$ of the particle configurations, at various time points and model parameters. The dotted, dashed, and solid curves indicated by the arrows were obtained at $t=25$, 75, 125, and 500 h, respectively. The asymmetric model (a) reaches a steady state while the symmetric model (b) continues to coarsen into droplets (inset). The shifted curves in (a) demonstrate that $q_{*}$ is independent of the system size and cell density in the stationary phase. For low cell densities, the peak falls off as $1/q$, in good agreement with data obtained for Fig. 1a (solid symbols). The solid lines represent power-law decays with exponents $-1$ and $-2$.Reuse & Permissions

Figure 4

Network connectivity depends on particle density. Configurations of $N=2000$ particles are shown in the stationary phase for various values of $N/L^2$ particle density: (a) 0.31, (b) 0.44, and (c) 1.54. The maximal cluster size (d) shows a percolation transition at a volume fraction below 0.2 (solid symbols), much less than the critical volume fraction of randomly placed disks (open symbols). The density-density autocorrelation of the critical cluster (e) is biphasic and well approximated by $\varrho (r)\sim r^{-0.9}$ for $r<r_c$ and $\varrho (r)\sim r^{-0.2}$ for $r>r_c$ (solid lines). For comparison, the dashed line represents $\varrho (r)\sim r^{-0.5}$.Reuse & Permissions