article: How a particular attribute of an organism changes or scales with its body size is known as an allometry. Biological allometries, such as metabolic scaling, have been hypothesized to result from selection to maximize how vascular networks fill space yet minimize internal transport distances and resistances. The West, Brown, Enquist (WBE) model argues that these two principles (space-filling and energy minimization) are (i) general principles underlying the evolution of the diversity of biological networks across plants and animals and (ii) can be used to predict how the resulting geometry of biological networks then governs their allometric scaling. Perhaps the most central biological allometry is how metabolic rate scales with body size. A core assumption of the WBE model is that networks are symmetric with respect to their geometric properties. That is, any two given branches within the same generation in the network are assumed to have identical lengths and radii. However, biological networks are rarely if ever symmetric. An open question is: Does incorporating asymmetric branching change or influence the predictions of the WBE model? We derive a general network model that relaxes the symmetric assumption and define two classes of asymmetrically bifurcating networks. We show that asymmetric branching can be incorporated into the WBE model. This asymmetric version of the WBE model results in several theoretical predictions for the structure, physiology, and metabolism of organisms, specifically in the case for the cardiovascular system. We show how network asymmetry can now be incorporated in the many allometric scaling relationships via total network volume. Most importantly, we show that the 3/4 metabolic scaling exponent from Kleiber’s Law can still be attained within many asymmetric networks.
  One of the pervasive characteristics of biology is that the metabolic rate, B, of an organism scales with its body mass, M. When viewed across several orders of magnitude this relationship is approximated by the power law, B = B 0 M θ (1) where B0 is a normalization constant, and θ, the allometric scaling exponent, tends to cluster around the value of 3/4 [1–5]. This metabolic scaling relationship spans more than twenty five orders of magnitude in mass, from respiratory complexes at 10-18 grams to the largest mammals at 107 grams [6]. The mechanistic origin of the 3/4 scaling exponent has been one of the longest running debates in biology [1,7–11]. The West, Brown, and Enquist (WBE) model offers an alternative hypothesis for the origin of allometric scaling exponents in biology, in particular the 3/4 exponent of metabolic scaling [9]. The WBE model shows theoretically how numerous allometric scaling exponents are the result of a repeated, or self-similar, branching architecture. The pattern to be repeated consists of one parent branch and two or more child branches and is shown in Fig 1A. This pattern is sometimes referred to as a branching node, or generator, and depending on the geometric properties constitutes space-filling fractal [9,12]. One of the appeals of the WBE model is that, with relatively few starting assumptions, it predicts numerous biological allometric scaling relationships within and across diverse organisms (such as plants and animals). The WBE model for biological resource distribution networks has seen extensive application across a wide range of scientific disciplines. Within the original context of cardiovascular networks it has been used to further understanding regarding tumor growth [13], the temperature dependence of basic metabolic function on allometric scaling [14], and the limits of its applicability under the constraint of finite-size networks [15]. Beyond cardiovascular networks, the WBE model has aided in the advancement of understanding respiratory networks, plant vascular networks, ecosystem dynamics, and has even served as motivation for the study of other observed allometries, such as the form and function of major metropolitan cities [9,12,16–20]. Of note in all of these extensions is the underlying principle that the network is geometrically symmetric. While the WBE model predicts the metabolic scaling exponent of θ = 3/4, several have questioned its assumptions and ability to explain the observed variation in metabolic scaling exponents as well as capture the biological variation in organism form and function [10,21,22]. In response, several studies have pointed to extensions of the WBE model to explain variation in biological allometries and have shown how relaxation of some of its assumptions can indeed account for observed variation in biological scaling [14,15,23,24]. Nonetheless, it still is unclear if the assumptions of how the WBE model characterizes hierarchical branching networks in biology hold across the diversity of life. A fundamental assumption of the theory is that vascular networks exhibit symmetric branching, where every branch in a given generation has identical geometric properties (namely radius and length). Branching symmetry is rarely, if ever, the case within biology where vascular networks are observed to branch asymmetrically [25–31]. Existing in nature are several biological examples of observed asymmetric branching patterns that help to motivate our approach. In an early study focusing on casts of the human heart, Zamir introduced a characterization of two branching strategies based on differences in the scaling of vessel diameters. By measuring the differences in sibling branch radius (Fig 1) with respect to the frequency of branching events, one can distinguish between so-called distributing vessels and delivering vessels. The former, which act to transport blood toward myocardial zones, correspond to large differences in sibling radii; whereas the latter type, responsible for transporting blood into the myocardial zones, correspond to small differences in radius [26]. Recently, studies of the interplay between genetics and vascular morphology have led to additional branching strategy classifications. In particular, a clear hierarchical pattern is formed during the development of the mouse lung that starts with domain-type branching (“scaffolding”) and transitions to either planar-type branching (“edge” features), or orthogonal-type branching (“surfaces” and “interiors”). The qualitatively defining characteristics of these branching types are as follows: in domain-type branching there are large differences between sibling branch lengths (Fig 1); in planar and orthogonal there are little to no differences between sibling branch characteristics, but in the former all branches exist in the same geometric plane, whereas in the latter a rotation of roughly π/2 occurs at every branching event [28]. At the whole-organism scale, the Zebrafish has become a remarkable standard for the study of vascular form and development, owing to the translucent nature of its skin. A particularly stark feature of the Zebrafish vasculature is the presence of patterned asymmetric branching in the intersegmental vessels originating from the dorsal vein [29]. The rarity of symmetric branching networks in biology violates a core assumption of the WBE model and questions its validity. As all of the predictions of the WBE model are derived from a symmetric network, an important question is, does incorporating asymmetry impact the predictions of the model? However, the fact that approximate 3/4 scaling is observed within and across organisms characterized by asymmetric branching suggests that perhaps the 3/4 scaling may not depend on the degree of asymmetry exhibited [32]. The study of asymmetric branching patterns itself has a long history. At its earliest, asymmetric branching was included in some of the first uses of energy minimization procedures in studying the cardiovascular system [25,33–35]. In these early studies, the energy minimization procedures were imposed at the nodal level instead of the whole network level, where a node is defined by the occurrence of a branching. Thus, the influence of branching asymmetry on biological scaling was not explored. More recently, asymmetric branching has been studied within the context of leaf venation patterns, river branching, and both cardio- and plant vascular metabolic rates [32,36]. Nonetheless, these studies do not provide detailed analytic treatments of the role of asymmetry in governing the scaling behavior of biological networks. Here, we present a more general model for the origin of allometric scaling laws. We present a theoretical foundation for incorporating asymmetric branching in the WBE framework for the cardiovascular system. As a result, this model can better capture the diversity of network morphologies observed in biology by allowing for generational variation in the values of branch radii and lengths between sibling branches. Our model shows how different scaling exponents, such as 2/3 and 1, can be associated with different levels of asymmetry, where 2/3 is the value associated with the idea that metabolic rate is limited by the ability of an organism to dissipate heat [37], and 1 is the value associated with isometric scaling. We also make predictions regarding a generational transition in asymmetry type that is associated with the transition in fluid flow from pulsatile to constant. Lastly, we show that there exists a wide range of asymmetric network morphologies that result in 3/4 metabolic scaling.
  Our work starts with the same assumptions outlined in Ref. [9,15]. However, we relax the assumption that child branches arising from a parent branch have equivalent lengths and radii (Fig 1A). As shown in Fig 1 there are two possible asymmetric network types. A positive asymmetry network is one in which the larger radius is paired with the larger length and the smaller radius with the smaller length (Fig 1B). A negative asymmetry network is one in which the larger radius is paired with the smaller length and the smaller radius is paired with the larger length (Fig 1C). For any pair of two sibling branches, the asymmetric radii (rj, μ, rj, ν) and lengths (lj, μ, lj, ν) can be expressed in terms of their averages (rj, lj) and differences (Δrj, Δlj) as, lj, μ=lj+Δljlj, ν=lj−Δljrj, μ=rj+Δrjrj, ν=rj−Δrj (2) where, lj=lj, μ+lj, ν2Δlj=lj, μ−lj, ν2rj=rj, μ+rj, ν2Δrj=rj, μ−rj, ν2 (3) Here the subscript j denotes the branching generation within the network, and the subscripts μ and ν distinguish the two child branches, as shown in Fig 1. In this formalism, the differences Δlj and Δrj can vary from positive to negative, and in so doing allow for smooth transitions between the two different asymmetry types. For example, in Fig 1B, Δrj > 0 and Δlj > 0, which corresponds to positive asymmetric branching, whereas in Fig 1C, Δrj > 0 but Δlj < 0, which corresponds to negative asymmetric branching. Note that, due to the definitions in Eq (2), the described examples are effectively identical to the scenarios in which Δrj < 0 and Δlj < 0 for positive asymmetry, and Δrj < 0 but Δlj > 0 for negative asymmetry. Next, we define the scale factors between the physical child and parent dimensions, γ j, μ = l j + 1, μ l jγ j, ν = l j + 1, ν l jβ j, μ = r j + 1, μ r jβ j, ν = r j + 1, ν r j (4) Using our definitions for the big and small radii and lengths from Eq (2), and introducing the average scale factors βj = rj+1/rj and γj = lj+1/lj, and the difference scale factors Δβj = Δrj+1/rj and Δγj = Δlj+1/lj, we can express the quantities in Eq (4) as, γ j, μ = γ j + Δ γ jγ j, ν = γ j - Δ γ jβ j, μ = β j + Δ β jβ j, ν = β j - Δ β j (5) The difference scale factors will come to represent the measures of asymmetry exhibited within a network. Specifically, Δβj = Δγj = 0 represents maximally symmetric branching (see Fig 2A and 2B), and Δβj ≈ ±βj and Δγj ≈ ±γj represent maximally asymmetric branching (see Fig 2C and 2D). The benefits of defining asymmetry in the manner described above are twofold. At the theoretical level this will allow us to express all results in terms of contributions due to symmetrically (β and γ) and asymmetrically (Δβ and Δγ) defined quantities. Furthermore, we are not restricted to any particular pair of values for the scale factors (e. g. γμ and γν). That is, any pair of scale factors can be described by their average and difference scale factors. At the experimental level this allows for an easy way to measure asymmetry. Traditionally, in studies conducted under the symmetric paradigm, the values of the scale factors are calculated as nodal averages, or alternatively, distributions of the scale factors for entire networks are collected, where the means of the distributions are meant to represent the average value of the scale factor for that network [30,36,38,39]. An alternative formulation for characterizing asymmetry can also be made. In this approach, the physical scale factors are expressed in terms of the symmetric WBE scale factors and perturbations from those values. Starting with an analogous form of Eq (2), we can write the physical lengths and radii of a pair of sibling branches as, l j, μ = l ˜ j + Δ l ˜ j, μl j, ν = l ˜ j - Δ l ˜ j, νr j, μ = r ˜ j + Δ r ˜ j, μr j, ν = r ˜ j - Δ r ˜ j, ν (6) We can express the above in terms of scale factors using Eq (4), to arrive at, γ j, μ = γ ˜ WBE + Δ γ ˜ j, μγ j, ν = γ ˜ WBE - Δ γ ˜ j, νβ j, μ = β ˜ WBE + Δ β ˜ j, μβ j, ν = β ˜ WBE - Δ β ˜ j, ν (7) where β ˜ WBE and γ ˜ WBE are the symmetric WBE scale factors and Δ γ ˜ j, μ, Δ γ ˜ j, ν, Δ β ˜ j, μ and Δ β ˜ j, ν are subsequently called the symmetric-difference scale factors. Furthermore, β ˜ WBE = (1/2) 1/2 for pulsatile flow and (1/2) 1/3 for constant laminar flow, and γ ˜ WBE = (1 / 2) 1 / 3 for both types of flow, while the symmetric-difference scale factors are free to vary. While this approach is beneficial in that it allows for results to be expressed strictly in terms of deviations from the symmetric WBE results, it does not as easily distinguish between positive and negative type asymmetry, and in certain circumstances may even obscure the presence (or absence) of asymmetry all together. For example, should both child branches have physical length scale factors of 0. 8, then the symmetric-difference length scale factors would have non-zero values of Δ γ ˜ j, μ = 0. 8 - γ ˜ WBE and Δ γ ˜ j, ν = γ ˜ WBE - 0. 8. For a further exploration of this approach, see S3 Text.
  Ensuring that the networks are characterized by space-filling fractal branching architectures that minimize dissipative effects associated with resource transport results in nodal level constraints between the scale factors. These constraints are both specific to, and shared between, the two fluid flow regimes. A summary of these results are presented in Fig 3.
  Our work highlights some important caveats and areas for future work. First, we have modeled asymmetric networks as consisting of strict self-similarity in branching rules. Specifically, the scale factors βj, γj, Δβj, and Δγj are constant throughout the networks. This also assumes that the network is characterized by only one set of branching rules. It would be instructive to explore how relaxing the self-similarity assumption as well as allowing for the mixing of branching rules (asymmetry types) both across and within generations influences our results. Second, finite-size effects in an asymmetric network are also of particular interest to explore. Some of the current (symmetric) allometric relationships rely on assumptions regarding the countability of the number of terminal network tips. Two examples of such assumptions are that all terminal tips are identical, and the relationship N T I P S = 2 N G E N for a strictly bifurcating, where NTIPS represents the total number of terminal tips (capillaries) and NGEN represents the total number of branching generations within the network. In an asymmetric network, these assumptions do not in general hold, as high degrees of asymmetry can lead to some branches reaching the terminal tip size sooner than others. Third, asymmetric branching may provide a novel avenue for studying the flow transition regime within the cardiovascular system. Previous studies have modeled the transition as either a discrete step or a continuous shift, but in both cases symmetric branching was assumed [9,15]. Asymmetric branching provides for transitions such that within a single generation there exists branches with both pulsatile and constant laminar flow. This level of mixing in flow types may provide for a more dynamical analysis of the flow transition regime. The WBE model provides a basis to understand the linkages between organism form and function. It has served as a platform for broad application to various other fields [6,13,16–18]. However, in its original form, the WBE model is built around a questionable assumption of symmetrical branching networks. As a consequence, it does not capture the great extent of variation in biological form. In biology, symmetric networks are rarely observed. Thus, a core assumption of the theory is violated. In this paper, we investigated if including asymmetry into the WBE model yields similar conclusions and predictions as the original model based on symmetric branching. Specifically, do optimal asymmetric networks follow 3/4 metabolic scaling? We have derived a more general form of the WBE model. It incorporates different branching geometries reflected in differences in branching asymmetry. We believe that our approach can offer a more general theory that can better relate variation in organismal form and function than the original WBE model. Our definition of asymmetry in a strictly bifurcating network allows for a more accurate analysis of biological branching networks. In addition, the theory makes a set of novel predictions for the type of branching asymmetry favored under different fluid flow types/transfer regimes.
  We can first write the total volume of the ith generation in terms of the total volume of the kth generation, where i > k, as follows, V i, T O T = V k, T O T ∏ j = k i - 1 β j, μ 2 γ j, μ + β j, ν 2 γ j, ν (24) where the volume of a given branch is expressed as, V j, μ = π r j, μ 2 l j, ν. The above expression becomes more transparent when setting i = 1 and k = 0. Using the definitions in Eq (4), the above reduces to an expression of how the volume in the base is split into two volume fractions for the two child branches. For larger values of i and k, the above expression can then be expanded to express Vi, TOT in terms of volume fractions of Vk, TOT, where the fractions consist of different k to i path-dependent permutations. For example, when i = 2 and k = 0, there are four volume fractions for the four respective paths from the base branch to the four tips. It should be noted that Eq (24) already assumes that the scale factors within a given generation are the same. This does not however imply that all of the big child radii or lengths are equal within a given generation, and similarly for the small child radii and lengths. Solving for, and summing over, the dummy generation-index k, and taking the limit that i goes to the Nth generation, will give us the total volume for an asymmetric network, V T O T = V N, T O T ∑ k = 0 N ∏ j = k N - 1 β j, μ 2 γ j, μ + β j, ν 2 γ j, ν - 1 (25) where VN, TOT represents the combined volume of all of the terminal capillaries, and can be expressed as VN, TOT = NC VC. Imposing the assumption that the scale factors are the same across generations simplifies Eq (25) to, V T O T = N C V C β μ 2 γ μ + β ν 2 γ ν N 1 - β μ 2 γ μ + β ν 2 γ ν N + 1 1 - β μ 2 γ μ + β ν 2 γ ν (26) A consequence of our first assumption, that the networks are space-filling fractals that minimize energy-loss, is that β μ 2 γ μ + β ν 2 γ ν < 1, which allows us to approximate everything in the brackets in the above expression to a value of 1, to leading order. This can be interpreted as stating that the total volume in the network is decreasing from one generation to the next, and is consistent with Eqs (16,17,20) and (21). Upon making this approximation we arrive at, V T O T ≈ N C V C β μ 2 γ μ + β ν 2 γ ν N (27) Having identified two candidates for asymmetric networks, we can impose energy minimization principles while simultaneously requiring that the networks be space-filling fractals for the two different flow regimes of constant laminar flow and pulsatile flow. Doing so allows for constraints on the extent to which asymmetry is exhibited. The underlying concept in these approaches is to determine which values of βj, γj, Δβj, and Δβj have been naturally selected for, through the process of evolution, such that the network transports resources with the least amount of resistance. As there are two different flow regimes, constant laminar and pulsatile, we use two different methods of energy minimization. In employing our different techniques for the different types of flow we are considering the flow regimes as independent from one another in that we do not incorporate transitioning between them. Although in naturally occurring organisms (i. e. ones with beating hearts) pulsatile flow dominates at lower-order generations where the vessel diameters are large, constant flow dominates at higher-order generations where the vessel diameters are small.
summary: We present a model for incorporating geometrically asymmetric branching into biological resource distribution networks. Our work shows how space-filling and fluid flow principles constrain allowed branching morphologies within the context of our model. Simultaneously, we demonstrate that there is a wide range of asymmetrically branching network architectures that still give rise to 3/4 metabolic scaling exponents.
keywords: classical mechanics
  fluid mechanics
  metabolic networks
  radii
  geometry
  mathematics
  network analysis
  molecular biology techniques
  research and analysis methods
  cell labeling
  computer and information sciences
  fractals
  fluid dynamics
  molecular biology
  laminar flow
  continuum mechanics
  fluid flow
  physics
  metabolic labeling
  fluids
  states of matter
  biology and life sciences
  physical sciences
  asymmetry
title: A general model for metabolic scaling in self-similar asymmetric networks