Introduction

In the past decades, the use of molecular data has allowed evolutionary, ecological, and conservation questions to be applied to non-model organisms in natural settings [1]. One way of illustrating molecular data for phylogeographic or intraspecific studies is the use of haplotype networks. These networks help to visualize relationships between individuals, populations, and species intuitively, revealing insights about migration, population structure, and speciation [2–4]. While estimating genetic diversity from haplotype networks is common, authors often rely on unquantified topological patterns (such as distributions of haplotypes across topologies) to make additional inferences since the combined genetic and topological components of haplotype network make quantitative comparisons difficult [2–4]. Thus, the implementation of standardized, quantitative methods of comparing haplotype networks that include topological features, will be constructive at a time when comparative phylogeography is becoming an increasingly useful tool to analyze complex geographic patterns of populations from multiple species in an ever-changing environment [4–7].

A variety of network features explained by graph theory such as node degrees, clustering coefficient, centralities, link prediction, and network density, among others, can be exploited to study the evolutionary inter-relationships between individuals illustrated in haplotype networks [8]. However, due to the paucity of interdisciplinary practices, such information is often overlooked in evolutionary studies even when this can be useful to quantifiably compare populations [8]. The introduction of the haplotype network diversity (HNd) metric, henceforth referred to as haplotype network node diversity, was a recent attempt to utilize network properties to quickly compare haplotype networks [3]. In that study, values of HNd (which incorporate haplotype and topological diversity of networks) were used to explore the correlation of endemism and genetic signatures of Galápagos fishes and test predictions of population structure in endemic, insular, and widely-distributed species [3]. The premise for the introduction of the HNd metric was a need for a single value to describe the complexity of haplotype networks in terms of their genetic and topological diversity, that would be simple and intuitive [3]. Since most scientists that use haplotype networks are usually also familiar with the concept and values of haplotype diversity (Hd), the intent was to produce a value similar in concept to Hd that would, in addition, capture the topological diversity of the haplotype network.

The Hd metric was first introduced by Nei in 1987, as the probability that two randomly sampled haplotypes are different [9]. While a number of approaches have refined the theoretical framework and the actual implementation on haplotype networks [10, 11], Hd, which varies between zero (all haplotypes are identical) and one (all haplotypes are different), has remained the metric of choice, and has been universally used to quickly and simply describe how genetically diverse populations are. In fact, the original description has been cited nearly 10,000 times since its publication according to the Web of Science (WoS) and remains current to this day [12]. Yet, Hd does not take network topology into consideration and its use for comparing networks is limited because very different haplotype networks can have the same Hd. The goal of the original description of HNd was therefore to provide a metric akin to Hd that would, in a single value, incorporate genetic and topological diversities and be familiar and intuitive to the users.

Nevertheless, the original description did not elaborate on the HNd approach itself, describe the method in detail, nor discuss the major differences between the methodology calculating Hd and the component of HNd intended to emulate Hd. In this study, we explain in detail the method used to obtain HNd, its principles, and pitfalls. We then introduce branch diversity (Bd), a new index that mirrors the logic used by Hd to estimate topological diversity of networks. Finally, we combined Bd with Hd into a single complexity metric, haplotype network branch diversity (HBd), which provides probabilistic values useful for comparing networks in phylogeographic studies. In addition, we include the R script called HapNetComplexity in the Supporting Information for ease of computing the metrics. Branch and haplotype network branch diversities can be calculated for any network regardless of the method and graphical tool used for construction [10, 13–21]. However, we illustrate the behavior of the new metrics using distance-based networks built with the package pegas [19] in R [22], as it remains a commonly used approach in population genetic studies [13, 23].

Materials and methods

Separate datasets that cover a range of network topologies and number of haplotypes were used to illustrate the behavior of new metrics in comparison to Hd (See Box 1 for metric and variable definitions). The first dataset was generated manually to show results from simple to more complex hypothetical networks (Fig 1). The second dataset, taken from GenBank and compiled in fasta format, used empirical networks also representing a range of complexities from previously published data [24–26] (Fig 2). Our first objective with these two datasets was to depict the variability of network complexity by displaying how different configurations can affect genetic and topological diversity estimates alike. We also used these datasets to showcase the ability of our new metric, Bd, to quantitatively discriminate simple and complex networks even when Hd remains constant. A third and final dataset was created manually to demonstrate the various properties of Bd (see Box 2 and Supporting information). All data files are available as (S1–S29 Files for the first, second and final dataset, respectively). We begin exploring the presented networks by computing Hd and Bd. Then, we combine these indices to calculate our other new metric, HBd, in order to compare the complexity of each haplotype network. All metrics were computed with the provided R script HapNetComplexity (S30 File), and range from zero to one making comparisons and replication of results straightforward. We would like to note that π (nucleotide diversity), another standard metric that captures the genetic distance between sequences, was intentionally not included in our calculations because it does not relate to network topology, and in some cases negates, or overwhelms other factors. However, π is also included in the R script for comparison purposes.

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Fig 1. Haplotype networks and diversity indices for manually generated sequence datasets.

Values for haplotype diversity (Hd), branch diversity (Bd), and haplotype network branch diversity (HBd), are given for each haplotype network. Colors represent individuals since each individual was intentionally set to represent a distinct population. For each network, haplotype classes (Hc) are represented in parenthesis with pairs of numbers where the class’ number of branches (nbHc) and individuals (niHc) are presented to the left and right of a colon, respectively. For example, the haplotype network in Panel A is made up of two haplotype classes, thus (1:15, 5:6) represents that there are 15 individuals within the 1-branch haplotype class and 6 individuals within the 5-branch haplotype class. This breakdown indicates the number of haplotype classes and the frequency-evenness among them, components which directly affect Bd and subsequently, HBd. All panels show datasets comprising 21 individuals and ranging from 6 to 21 haplotypes. Top panels (A, B, C, and D) illustrate four simple network configurations with six haplotypes that maintain Hd constant but that can be differentiated by Bd and HBd. Middle panels (E, F, G, and H) show variation in network configuration that maintains Bd constant but increases Hd from left to right. Bottom panels show more complex haplotype networks with 9, 17, 21, and 21 haplotypes, in panels I, J, K, and L, respectively, where Bd provides a larger margin to make comparisons than Hd, particularly between panels with equal Hd values (H and L). Additional dataset information for each panel is given in Table 1. Sequence and site files for all panels can be found in (S1–S13 Files).

https://doi.org/10.1371/journal.pone.0251878.g001

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Fig 2. Haplotype networks and diversity indices for published sequence datasets.

Colors represent sampled populations. All datasets (described in the Methods section) are based on CO1 or d-loop sequences of fish species shown in italics and were chosen to represent real scenarios with different levels of network complexity (A and B [26]; C [25]; and D [24]). Values for haplotype diversity (Hd), branch diversity (Bd) and haplotype network branch diversity (HBd) are given for each haplotype network. Similar to Fig 1, haplotype classes (Hc) are represented in parenthesis with pairs of numbers where the class’ number of branches (nbHc) and individuals (niHc) are presented to the left and right of a colon, respectively. Additional values referring to these networks are given in Table 1. Sequence and site files for all panels can be found in (S14–S21 Files).

https://doi.org/10.1371/journal.pone.0251878.g002

Results and discussion

In this study, we present the new metric haplotype network branch diversity (HBd), as a tool to quantitatively compare and illustrate the variability of haplotype network complexity. Haplotype network branch diversity is computed by combining the commonly quantified haplotype diversity (Hd), with the new index of the topological diversity of haplotype networks, branch diversity (Bd). While Hd calculates the genetic diversity of a population, Bd measures the diversity of the evolutionary interrelationships between the haplotypes observed in a population. All metrics, Hd, Bd, and HBd, vary between zero and one allowing for direct comparisons of diversity and complexity between networks.

The first manually generated model dataset consists of 21 individuals that partition into 6, 8, 9, 11, 14, 17, or 21 haplotypes, and 2, 3, or 4 classes of haplotypes (Fig 1 and Table 1). Haplotype diversity varies between 0.63 and 1.00, Bd varies between 0.43 and 0.71, and HBd varies between 0.3 and 0.67 (Fig 1 and Table 1). The top row of Fig 1 (Panels A-D) shows that even when Hd remains constant, Bd varies depending on the network topology, clearly illustrating its ability to distinguish between simple networks.

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Table 1. Diversity indices for model (Fig 1) and published datasets (Fig 2).

https://doi.org/10.1371/journal.pone.0251878.t001

Panels E-H of Fig 1, illustrate how networks with a similar topology, but increasing number of haplotypes, increase Hd and can maintain the same Bd values if the number of haplotype classes and their number of individuals are unchanged. Finally, in Panels I-L of Fig 1, Bd increases as topology becomes more compounded, and the number of haplotype classes increase from 2 to 4. In fact, the two main factors influencing Bd are the number of haplotype classes in a network and how evenly distributed the individuals are among these classes. Branch diversity increases when the number of haplotype classes also increases and with a more even number of individuals among the classes (see Box 2 and Supporting information for all Bd properties). This behavior results in Bd values that are low for simple networks and rise with increasing complexity. Unlike Hd, the multi-property character of Bd allows it to discriminate between networks with equal number of haplotypes or haplotype frequencies, whether these have simple (Fig 1, Panels A-D) or complex (Fig 1, Panels K-L) topologies. In contrast, Hd is able to capture variation in the number of haplotypes in situations when this does not affect Bd. Thus, combining Hd and Bd into HBd allows this metric to distinguish networks with the same number of haplotypes but diverse topologies (which keeps Hd constant but affects Bd; Fig 1, Panels A-D, K, L) and networks with similar topologies and diverging number of haplotypes (which influences Hd but it might not affect Bd; Fig 1, Panels G and H). Given that both Hd and Bd are calculated with frequencies, this comprehensive measurement of network complexity is provided as a single probabilistic value, the main goal of this study.

Similarly, the values of Hd, Bd, and HBd from networks of published sequence data (Fig 2) vary between 0.44–0.99, 0.23–0.81, and 0.1–0.73, respectively. Branch diversity and HBd span a wider range than Hd. The lowest values for all metrics are found within Plectropomus laevis (Fig 2, Panel A), a widespread grouper with a recent population expansion, which show only four haplotypes in its range [26]. The highest value for Hd was found in Chlorurus sordidus (Fig 2, Panel D), which exhibited a complex haplotype network driven by a large distribution associated with a recent history of repeated shifts between isolation and increased migration amongst populations [24]. In comparison, the highest Bd and HBd values are recorded in Atractoscion aequidens due to the combination of having many haplotype classes and these occurring at similar frequencies. This is a pelagic fish species, found along the coast of southwestern Africa, in which an ancient vicariant event has been proposed to explain two very distinct genetic lineages (shown in red and blue, respectively) [25]. It is worth noting that the network created by the C. sordidus dataset actually contains more haplotype classes than that of A. aequidens (10 vs 6), however, it also has a heavily skewed distribution of individuals as two of these classes alone (1-branch and 2-branch classes) hold more than 70% of all the individuals, which keeps Bd relatively low (Fig 2 and Table 1). Furthermore, while Bd is not directly affected by the actual number of branches in the different haplotype classes (just as Hd is not affected by the distance among haplotypes), the fact that more than half of the C. sordidus haplotypes were subtended by only one branch, limits the network from increasing its topological diversity, and ultimately dampens its overall complexity score (HBd).

Ultimately, the complexity estimates of HBd simultaneously provide a measurement of the distinctness of individuals (as calculated by Hd) and the diversity of the interrelationships among them (as calculated by Bd). For instance, Hd is relatively higher than Bd in populations where rare variants are prevalent. This scenario might arise as a result of, among others, a population expansion (as in P. laevis; Fig 2, Panel A), repeated cycles of isolation and secondary contacts (as in C. sordidus; Fig 2, Panel A), or a hypothetical bottleneck that produces a variety of haplotypes by chance (i.e. not heavily dominated by few haplotypes). In contrast, Bd is highest when a population conserves high connectivity among haplotypes (rare variants just represent another haplotype class) and classes have similar or equal frequencies. Yet, the more haplotypes present in a population the harder it becomes to maintain high connectivity and an even distribution of haplotypes across classes (as in C. sordidus). Whereas mechanisms that advocate the conservation of diversity such as random mating and balancing selection should help maintain high Bd values in a population, other processes such as assortative mating or directional selection are likely to decrease this measurement. In this way, Hd and Bd describe distinct properties of a population and can be used independently in evolutionary studies and conservation strategies with different goals. Compared to Hd or Bd alone, HBd provides a more holistic and conservative view of populations where high values indicate populations with high genetic diversity with well interconnected haplotypes that also include rare variants.

Inferring the evolutionary history of populations using haplotype networks presents several challenges including the difficulty comparing networks, network reticulations, alternative links, missing haplotypes, etc. Yet, solutions to these challenges are likely to be developed as interdisciplinary approaches become more frequent. This study provides a useful and simple tool to describe haplotype networks and streamline comparisons between network complexity, a property traditionally overlooked in evolutionary studies. The metric introduced herein simultaneously quantifies genetic and topological diversity of networks while also discriminating situations that are difficult to resolve with simpler available metrics. Furthermore, since every network is treated independently, our metric can be applied across haplotype building methods and alternate networks within a set of sequences. We therefore recommend the use of haplotype network branch diversity (HBd) as a single metric to describe and easily compare the complexity of different haplotype networks.

Supporting information

Acknowledgments

We would like to thank Chris Bird for useful comments on the manuscript. HTP thanks the Hope for Reefs initiative of the California Academy of Sciences.