Fig 1.
Diagram of symmetric and asymmetric bifurcations.
Symmetric branching (A) is characterized by every branch within a given generation j having equal values of radius and length. Positive asymmetry (B) is such that one child branch is larger than the other in both radius and length. Negative asymmetry (C) is such that one child branch has a larger radius and shorter length while the other has a smaller radius and greater length.
Fig 2.
Rendering of Selected Networks.
An assortment of networks are presented with associated average (β, γ) and difference (Δβ, Δγ) scale factors, and metabolic scaling exponents (θ). Note that in all of these cases there is no switching of asymmetry type either within or across generations, and the scale factors are assumed to be constant both within and across branching generations. Networks (A) and (B) represent the symmetric limits under the constraints associated with pulsatile flow (Eqs (16) and (17)) and constant flow (Eqs (20) and (21)), respectively. Networks (C) and (D) represent the two extreme asymmetric limits. Networks (E) through (J) exhibit varying degrees of branching asymmetry while satisfying the constraints associated with pulsatile flow. Each of these tree networks are represented as points that fall along the 3/4 metabolic scaling contour as shown in Fig 4. Networks (I) through (P) satisfy constant laminar flow, and they also fall along the 3/4 metabolic scaling contour as shown in Fig 5.
Fig 3.
Results of Minimizing Network Energy Loss.
In the pulsatile flow regime (A) either network asymmetry type is allowed, but both must follow the area-preserving and space-filling principles at the nodal level, as per Eqs (16) and (17). In this regime switching between asymmetry types can occur both within and across generations. In the constant laminar flow regime (B) only positive network asymmetry is predicted, and the scale factors must follow the space-filling and Murray’s Law relations, as per Eqs (20) and (21). We also find that the asymmetric version of Murray’s Law allows for an expression of the total cross-sectional area, A, of any two child branches in terms of the parent area and the average and difference scale factors, showing that asymmetric branching allows for a toggling between branching with increasing area and constant area. Lastly, the average radius and length scale factors are predicted to be equal, as well as the difference radius and length scale factors. When combining these two results with the space-filling and Murray’s Law expressions, we can make a strict prediction of the metabolic scaling exponent θ = 1 for the constant laminar flow regime.
Fig 4.
Colormap of Metabolic Scaling Exponent for Pulsatile Flow.
The metabolic scaling exponent is graphed as a function of the difference scale factors Δγ and Δβ, and is shown to range in value from 0 to 1. The scale factors are such that the networks are space-filling fractals that minimize energy-loss from resource transport, as dictated by Eqs (16) and (17). Positive asymmetry is graphed in the first and third quadrants, and negative asymmetry in the second and fourth quadrants. Contours of constant values of the metabolic scaling exponent are plotted in bold. The points labelled A, C, D, and E—J correspond to the rendered trees found in Fig 2.
Fig 5.
Colormap of Metabolic Scaling Exponent for Constant Laminar Flow.
The metabolic scaling exponent is graphed as a function of the difference scale factors Δγ and Δβ, and is shown to range in value from 0 to 1. The scale factors are such that the networks are space-filling fractals that minimize energy-loss from resource transport, as dictated by Eqs (20) and (21). Positive asymmetry is graphed in the first and third quadrants, and negative asymmetry in the second and fourth quadrants. Contours of constant values of the metabolic scaling exponent are plotted in bold. The points labelled B, C, D, and K—P correspond to the rendered trees found in Fig 2.
Fig 6.
Colormap of Metabolic Scaling Exponent for a Network with a Transition in Flow Type.
Here we present colormaps of Eq (15) for the cases when c = 1, c = 0.5, and c = 0. It should be noted that when the transition in flow type occurs within the networks, the same values for the difference scale factors are used, but the equations that determine the average scale factors switch from Eqs (16) and (17) to Eqs (20) and (21). In all three colormaps, the contour lines take on the same values as in Figs 4 and 5. In particular, the bolded contour corresponds to a metabolic scaling exponent value of 3/4.
Fig 7.
Average vs. Difference Scale Factors for Area-Preserving and Space-Filling/Area-Increasing Principles.
The functional dependence between the average scale factors (β, γ) and the difference scale factors (Δβ, Δγ) are graphed. For both cases of either the square-law (area-preserving) or the cube-law (space-filling and area-increasing) the average scale factors decrease from 1/21/2 and 1/21/3, respectively, and converge to 0.5 as the difference scale factors increase from 0 to 0.5.