Figure S7.
Branching and Annihilating Random Walks as a Generic Framework to Understand Pathologies and Other Branched Organs, Related to Figures 6 and 7
(A and B) Simulations using control parameters, but in an anisotropic setting lead to a similar phenomenology of a self-organized pulse of active tips at the edge (section of a simulated tree shown on [A]), although with decreased tree heterogeneity ([B], n = 3 kidneys for each time point).
(C) Section of a E17 murine kidney, displaying collecting epithelial ducts (green, stained for cytokeratin 8) and a nephrogenic zone positioned at the growing periphery of the tissue: glomeruli (used as a proxy for maturing nephrons) stain with both the red (laminin, renal basement membrane) and blue (podocalyxin for podocytes in the glomerulus) channels and appear pink. Bifurcating tips (white arrows) can also be observed at the tissue periphery.
(D) Theoretical number of inactive particle (i.e., nephrons, measured via glomeruli numbers experimentally) as a function of tip (active of inactive) number (data: black squares), both for the control model described in the main text with time-invariant parameters (purple line), and for time-varying parameters (with R′a = 0.125 before E15 and R′a = 0.5 afterward), which an absence of power-law scaling (green curve).
(E) Typical output of a numerical simulation from the phase diagram of Figure 7F, for an annihilating radius Ra = 2.5, with systematic tree survival through an initial excess of branching compared to termination.
(F and G) Two typical outputs of numerical simulations from the phase diagram of Figure 7F, for an annihilating radius Ra = 3.9, i.e., close to the critical point. Simulated trees then stochastically self-annihilate at varying sizes, from rudimentary trees [F]) to complex structures (G).
(H and I) Two typical outputs of numerical simulations of a E15 kidney, using the default model parameter (H) and an averaged branch length halved (I), mirroring Vitamin-A deficient kidneys.
(J) Experimental versus theoretical average number of branches per generation in E15 kidneys for wild-type (purple) and Vitamin-A deficient mice (green, data from Sampogna et al., 2015), along side the theoretical predictions from the simulations shown in (H) and (I).
(K) Two examples of tree topology from the three-dimensional reconstruction of a subunit of human prostate, displaying heterogeneity and numerous early termination events.
(L) Tip termination probability as a function of generation, for n = 5 human prostates, showing a rapid convergence toward a balance between tip termination and tip branching (green horizontal line).
(M) Subtree (defined as starting at generation 6) heterogeneity, assessed via its cumulative size distribution.
(N) Numerical simulation of a four species Turing-Meinhart system (see STAR Methods for details and parameters). We represent a density plot of the concentration of the activator a, which localizes at the growing tip. High concentrations are color-coded in yellow and low concentration in blue. The system performs branching and annihilating persistent random walks, and could therefore serve as a molecular basis for our model.
Error bars in (B) and (H) represent mean and s.e.m and in (I) a confidence interval of 1 s.d. In (A) and (E)–(I), active tips are red, inactive tips yellow, and ducts black. Scale bars: 200 μm.