Introduction

The Tree of Life is a synoptic depiction of the pathways of evolutionary differentiation between Earth life forms [1], and contains valuable clues on the key issue of understanding the diversification of life in the planet [2]. The branching pattern of the Tree of Life, which is being captured at increasing resolution by the advent of molecular tools [3], can be examined to investigate fundamental questions, such as whether it follows universal rules, and at what extent random differentiation mechanisms explain the shape of phylogenetic trees. The examination of the structure of the Tree of Life can also help to infer whether evolution acts at intraspecific scales in a way different from the action of evolution at the interspecific scale. Here we address these fundamental questions on the basis of a comprehensive comparative analysis of phylogenetic trees representing different fractions and domains of the Tree of Life, from interspecific to intraspecific scales. We draw from previous analyses of the geometry of the Tree of Life [4], the characterization of other branching systems [5], [6], and using tools derived from modern network theory [7]–[10] to examine the scaling of the branching in the Tree of Life [11], [12]. Our analysis is based on a thorough data set of more than 5000 interspecific phylogenies and a sample of 67 intraspecific phylogenies (see Text S1), thereby testing the universality of the results derived across scales.

A phylogenetic tree is a set of nodes, each node representing a diversification event, connected by branches (links). For each node i, a subtree S_i is made up of a root at node i and all the descendant nodes stemming from this root. The subtree size A_i gives the number of subtaxa that diversify from node i (including itself). Beyond this measure of the diversity degree, the characterization of how the diversity is arranged through the phylogenies can be achieved through the cumulative branch size, C_i, a measure of the subtree shape. It is defined [13] as the sum of the branch sizes associated to all the nodes in the subtree S_i, C_i = ΣA_j. For the same tree size, and restricting to binary branching events, the smallest value of the cumulative branch size is obtained for a completely symmetric, balanced tree, whereas the most asymmetric, the pectinate or comb-like tree in which all branches split successively from a single one, yields the largest C_i value [13]. To be clearer, we show in Figure 1 the analysis of A_i and C_i for a completely balanced tree (Figure 1A) and for a completely imbalanced tree (Figure 1B). A portion of a real phylogenetic tree is also shown (Figure 1C). How the shape of the tree (i.e., the distribution of the biological diversification) does change with tree size (i.e., with the number of taxa it contains) is given by the scaling of the subtree shape C vs. the subtree size A, as described by the allometric scaling relation C∼A^η. We quantitatively characterize the shape of each tree in our data set by calculating the functions F(A) and F(C), which are the complementary cumulative distribution functions (CCDF) of A_i and C_i values in the tree, respectively, and the value of the allometric scaling exponent, η. We compare the results derived from the analyses of inter- and intra-specific phylogenetic trees among them, to test for the preservation of branching patterns across evolutionary scales, and against those derived from the analyses of randomly-generated trees to test whether the allometric scaling derived can be modeled using simple, random branching rules.

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Figure 1. Branch size and cumulative branch size examples.

The values of the branch size (A) and of the cumulative branch size (C) are shown (in brackets, as (A,C)) at each node of three small example trees. A: a completely balanced tree of 15 nodes; B: a completely imbalanced tree of 15 nodes; C: a subtree of 15 nodes of a real phylogenetic tree, the intraspecific Vibrio vulnificus phylogeny presented in full in Fig. S2A. Note that the value of C at the root is maximum for the fully imbalanced tree, and minimum for the balanced one.

https://doi.org/10.1371/journal.pone.0002757.g001

Results

The branch-size CCDF displays power-law tails of the form for large branch size A (Figure 2A). The power-law exponents τ_A are remarkably similar for the data sets analyzed: τ_A = 1.76±0.03, and 1.74±0.02 for intra- and interspecific phylogenies, respectively. Similarly, the cumulative-branch-size CCDF also displays a power-law tail of the form at large C, with a similar agreement between the exponents of the intra- and interspecific data sets: τ_C = 1.53±0.02 and 1.53±0.02, respectively (Figure 2B). The discrepancy observed between the two data sets at the tail of the distributions can be explained by the different sizes of the typical trees on them: each tree contributes a natural cutoff to the overall distribution, and since the intraspecific trees are smaller in average, their cutoff appears at smaller tree sizes.

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Figure 2. Average distributions.

Cumulative complementary distribution functions (CCDFs) averaged and logarithmically binned over all phylogenetic trees in the interspecific (empty squares) and intraspecific (solid circles) data sets. A: CCDF of branch size, F(A). Solid line corresponds to a power law with the exponent given by the best fit to the interspecific data set τ_A = 1.74. B: CCDF of the cumulative branch size, F(C). The line corresponds to a power law with the exponent given by the best fit to the interspecific data set τ_C = 1.53.

https://doi.org/10.1371/journal.pone.0002757.g002

The allometric exponent, η, that characterizes the scaling of tree shape with tree size (Figure 3A), is also remarkably similar for the intraspecific (η = 1.43±0.01) and the interspecific (η = 1.44±0.01) phylogenies. This constancy of the exponents is still more remarkable when realizing (inset of Figure 3A) that it does not only apply to average properties of sets of intraspecific and interspecific trees, but also to individual phylogenies of groups of organisms pertaining to different kingdoms and living across widely contrasting environments, as it is reflected by the very narrow range of η obtained from different phylogenies (〈η〉 = 1.47, σ = 0.03, Figure 3A). The scaling exponents for our large interspecific data set are also matched almost perfectly (Figure S1) by those derived from a set of 67 interspecific phylogenies randomly drawn from the published literature thereby validating the uniformity of the scaling rules of the broad interspecific phylogenies and the smaller set of intraspecific ones used here. The later was also derived from a similar random sample taken from the published literature (see Text S1).

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Figure 3. Allometric scaling.

A: Plot of the logarithmically binned set of values of branch size, A, and cumulative branch size, C, for the interspecific (empty squares) and intraspecific (solid circles) data sets considered. The line corresponds to a power law C∼A^η, with the exponent given by the best fit through all data, η = 1.44. The inset shows probability distributions of the values of η fitted to each individual tree (left: interspecific, right: intraspecific data sets) illustrating the small dispersion in the values. B: Plot of the logarithmically binned set of values of C as a function of A for the interspecific data, normalized by the prediction from the ERM model (the horizontal line). Data systematically deviate from ERM, especially for large size A.

https://doi.org/10.1371/journal.pone.0002757.g003

The allometric scaling of C∼A^1.44 derived from our analysis falls somehow in between those obtained by simulated phylogenies derived from two extreme topologies: The symmetric tree gives C∼A ln A, which corresponds to η = 1 with a logarithmic correction, while the pectinate tree has η = 2. The natural null model for tree construction, the Equal-Rates Markov (ERM) model [14], [15], yields a scaling C∼A ln A similar to the symmetric tree with η = 1 but different from the scaling displayed by empirical inter- and intraspecific phylogenies, particularly for large ones (Figure 3B). Therefore some topological aspects of phylogenetic trees are not adequately reproduced by the ERM model. Our results imply that successful lineages diversify more profusely than expected under random branching, generating the large imbalances that characterize emerging depictions of the Tree of Life [4]. Alternative models introducing correlations, such as the proportional-to-distinguishable-arrangements (PDA) model [4], [16] or the beta splitting model [17], could generate more realistic phylogenies. Guided by previous biological allometric scaling analysis, we have assumed a power-law scaling of the form C∼A^η. However, other ansatz could also fit the data. The important point, however, is that these modeling approaches should give similar scaling properties for intra- as for interspecific branching.

Discussion

Traditionally, microevolutionary and macroevolutionary processes have been studied independently by population geneticists and evolutionary biologists, respectively [18]. The divide between these two levels of generation of biological diversity is an old one, rooted in the controversy between Darwinian gradualism and the saltationism proposed by others, prominently paleontologists, to explain macroevolutionary processes [19]. The debate as to whether macroevolution is more than the accumulation of microevolutionary events remains active [18], [20], [21], although refined paleontological evidence supports the continuum between micro- and macroevolution for some lineages [22]. The results presented here show that the branching and scaling patterns in intraspecific and interspecific phylogenies do not differ significantly for the topological properties we have calculated. Thus, shall saltation processes be a factor at the macroevolutive level, this is not reflected in the topology of phylogenetic branching as examined here. Evidence for possible differences in phylogenetic topologies between the inter- and intraspecific levels may require a detailed analysis of branching times, which we have not attempted.

Processes leading to scaling laws in size distributions in natural systems have been formulated as growth models [23], [24]. Many of the findings carry over to scaling properties found in networks [25] and their description in terms of branching processes [26]. But most of these models predict branching topologies similar to the ERM model. An alternative approach to understand the observed exponent would be to trace analogies with scaling laws in different branching systems [5], [6], [27] which have been explained by invoking a natural optimization criterion based in the fact that the observed trees contain the largest possible number of apices within the smallest number of branching levels. For binary trees of size A, where nodes are restricted to occupy uniformly a D dimensional Euclidean space, the minimum value of C scales as A^η, with η = (D+1)/D. This scaling also describes the D-dimensional tree with the maximum size for a given depth (the average distance between root and leaves). The value of η obtained in our phylogeny analysis, η≅1.44, is achieved only for optimal trees restricted to spaces of D≅2.27 dimensions. Given the apparently unlimited number of variables that may yield differences among taxa, restricting their representation to a space with such a small number of dimensions seems unreasonable. This interpretation suggests that the evolutionary process yielding the observed phylogenies is not the most parsimonious one, which could potentially yield a similar biodiversity with fewer branching levels. In fact, the natural choice D = ∞ gives an optimal exponent η = 1, which correspond to the ERM value and departs from observed scaling. Optimal traffic networks [28] also led to the exponent τ_A = 2 which departs from the empirical scaling exponent reported here for phylogenetic trees.

In summary, the remarkably similar allometric exponents reported here to characterize universally the scaling properties of intra- and inter-specific phylogenies across kingdoms, reproductive systems and environments, strongly suggests the conservation of branching rules, and hence of the evolutionary processes that drive biological diversification, across the entire history of life. Although at short branch sizes the topology of observed phylogenies cannot differ much from that expected under random and symmetric trees, due to the restriction of binary bifurcations in phylogenetic tree reconstruction, significant departures become universally evident as trees become larger, where the null ERM model and real phylogenies differ (Figure 2B). These deviations suggest (a) that the evolution of life leads to less biodiversity than an optimal tree can possibly generate; and (b) the operation of a mechanism generating a correlated branching, where some memory of past evolutionary events is maintained along each branch. This correlated branching pattern implies that entities that diversify faster than average lead to new biological forms that diversify more than average themselves. Invariance across the broad scales considered here indicates that relatively simple rules govern the phylogenetic branching and the unfolding of biodiversity. Their deviation from random models indicates that evolutionary success is a correlated trait within lineages, yielding present asymmetries in the structure of the Tree of Life.

Materials and Methods

Supporting Information