Fig. 2: Side-branching optimizes the space-filling by lymphatic vessel network.
A, B Quantification of the efficiency of space-filling of the network as a function of time, measured by the amount of spatial density fluctuations. Fluctuations at later time points (for P16 n = 8 ear pinna representing 7 mice and P21 n = 9 ear pinna, representing 8 mice) follow the value expected from equilibrium physics (exponent close to \(\alpha=0.5\), dashed black line, see Supplementary Information Theory Note for more details), while spatial fluctuations show a larger exponent at P13 (\(\alpha=0.60,\) n = 5 (ear pinna, representing 5 mice), significantly different from P16, p = 0.0102 and P21, p = 0.0016, with p = 0.95 between P16 and P21). Error bars indicate mean and +/− SD. Two-sided t test was used for measuring statistical significance. C Sketch of the stochastic rules used in simulations of lymphatic branching morphogenesis via branching and annihilating random walks (BARW). Active growing tips (red) can elongate to give rise to ducts (black), as well as a branch (tip-branching, probability pb) or terminate their growth if they come too close to neighboring ducts (tip termination/annihilation). We also consider the effect of repulsion on tip growth (see Supplementary Information for details). Finally, we consider the possibility of side-branching (probability ps), which is the reactivation of growth in a duct. D–F Exemplary simulations of lymphatic network growth. The size of the ear pinna increases linearly via uniform growth (dilating the existing network) with kinetic parameters inputted from experimental measurements (see Fig. 1C and Supplementary Fig. 2B, C). In the absence of side-branching (D, E), network growth terminates once tips have reached the edge of the ear pinna, and fluctuations persist. Upon increasing the probability of side-branching (F, G), active tips are constantly generated. Quantifications of spatial network fluctuations under different model parameters for the tip-branching rate, either without E or with G side-branching. Without side-branching, we generically observe giant fluctuations due to the inability of the system to correct local density inhomogeneities, while with side-branching, the system can converge to small fluctuations robustly, irrespective of other model parameters (see also Supplementary Movies 10 and 11). Source data for Fig. 2A, B, E, G are provided as a Source data file.
