Extended Data Fig. 12: Numerical simulations of BARE wave model variants.

(a-c) Variants with different assumptions about the spatial flux \({\boldsymbol{J}}(n)\) underlying the wave speed \(c\) (see Supplementary Discussion section 4.3.1). (a) \({\boldsymbol{J}}(n)=-\,nv\hat{{\boldsymbol{r}}}\) with \(\alpha =\) 0.039 h⁻¹, β = 22 μm h⁻¹, v = 200 μm h⁻¹. Simulated wave speed was c = 330 μm h⁻¹. Colour gradient: equally spaced intervals from t = 0 h to t = 120 h. (b) \({\boldsymbol{J}}(n)=D{\rm{\nabla }}n\) with α = 0.039 h⁻¹, β = 22 μm h⁻¹, and D = 0.55 mm² h⁻¹, v = 200 μm h⁻¹. Simulated wave speed was c = 280 μm h⁻¹. Colour gradient: equally spaced intervals from t = 0 h to t = 400 h. (c) \({\boldsymbol{J}}(n)=-\,n{v}_{d}\hat{{\boldsymbol{r}}}+D\nabla n\) with \(\alpha \) = 0.039 h⁻¹, \({v}_{d}\) = 235, β = 22 μm h⁻¹, D = 0.02 mm² h⁻¹ and v = 200 μm h⁻¹. Simulated wave speed was \(c\) = 280 μm h⁻¹. Colour gradient: equally spaced intervals from t = 0 h to t = 90 h. (d) Alternative model with explicit representation of puller hyphae (Eq. (7) of Supplementary Discussion), and \({\boldsymbol{J}}(n)=-\,n{v}_{d}\hat{{\boldsymbol{r}}}\) with \(\alpha \) = 0.039 h⁻¹, v_d = 220 μm h⁻¹, β = 22 μm h⁻¹, v = 220 μm h⁻¹, \({K}_{1}=0.25\,{{\rm{h}}}^{-1},\,{K}_{2}=0.40\,{{\rm{h}}}^{-1}\) and v_p = 300 μm h⁻¹. Simulated wave speed was c = 280 μm h⁻¹. Colour gradient: equally spaced intervals from t = 0 h to t = 90 h. Insets in (a-d): position of simulated tip density peak over time (blue line), with linear fit (Red dashed line) that yields wave speed \(c\). (e-i) Variant with refined density control that accurately predicts saturation density (see Supplementary Discussion 4.3.3). (e-g) Temporal profile in the ring reference frame for key density-dependent quantities (shaded areas: mean \(\pm 2\times \) s.e.m.). Within the same measured network, the (nonlinear) density product \(n{\rho }^{2}\) (purple line) is compared against anastomosis rate \(a\) (e), rate of tip annihilation by stopping \(s\) (f), and the residual \(\alpha n-b\) quantifying the error in the linear form \(b(n)=\alpha n\) (as in Fig. 3b) approximating the observed branching rate \(b\). Good overall agreement between the pairs of curves in (e-g) suggests modelling \(a\), \(s\), and \(\alpha n-b\) each proportional to \({n\rho }^{2}\), with coefficients \({\beta }^{{\prime} }\), \({\beta }_{\text{s}}^{{\prime} }\), \({\alpha }^{{\prime} {\prime} }\), respectively (see Supplementary Discussion 4.3.3). Combining these three processes we obtain refined expressions for the branching and annihilation rates: \(b(n,\rho )=\alpha n+{\alpha }^{{\prime} {\prime} }n{\rho }^{2}\) and \(a(n,\rho )=({\beta }^{{\prime} }+{\beta }_{\text{s}}^{{\prime} })n{\rho }^{2}\), respectively. Plugging into the BARE wave model and solving yields \({\rho }_{\text{sat}}=\sqrt{3\alpha /\gamma }=1.3\,{{\rm{mm}}}^{-1}\) mm⁻¹, where \({\gamma \equiv \beta }^{{\prime} }+{\beta }_{\text{s}}^{{\prime} }-{\alpha }^{{\prime} {\prime} }\). From the data in (e-g), \(\gamma \approx 0.075\) mm² h⁻¹, leading to \({\rho }_{\text{sat}}\approx 1.3\) mm⁻¹ – very close to the experimentally observed \({\rho }_{\text{sat}}\approx 1\) mm⁻¹. (h,i) Simulations of BARE wave model with those refined rate expressions \(b(n,\rho )\) and \(a(n,\rho )\), demonstrating that these refinements do not compromise the existence of travelling-wave solutions with constant wave speed. We used \(\gamma =0.075\) mm² h⁻¹, and all other parameters as in (c). In (a-d,h-i), initial conditions were \(\rho (r,t=0)=0\) and \(n(r,{\rm{t}}=0)={n}_{\max }{e}^{-{\rm{\lambda }}{{\rm{|x}}-{x}_{0}|}^{k}}\), with parameters \({n}_{\max }\)/ \({\rm{\lambda }}\) / \({x}_{0}\)/\(k\) (in units \({{\rm{m}}{\rm{m}}}^{-2}\)/\({{\rm{m}}{\rm{m}}}^{-k}\) /\({\rm{m}}{\rm{m}}\)/dimensionless): (a) 1/0.4/10/1; (b) 0.2/0.015/10/2; (c) 0.8/0.1/7/2; (d) 1/0.2/5/2 (\(n\)) 0.6/0.2/5/2 (\({n}_{\text{puller}}\)); (h) 0.6/0.3/7/2.