Split Diversity: Measuring and Optimizing Biodiversity Using Phylogenetic Split Networks
Abstract
About 20 years ago the concepts of phylogenetic diversity and phylogenetic split networks were separately introduced in conservation biology and evolutionary biology, respectively. While it has been widely recognized that biodiversity assessment should better take into account the phylogenetic tree of life, it has also been widely acknowledged that phylogenetic networks are more appropriate for phylogenetic analysis in the presence of hybridization, horizontal gene transfer, or contradicting trees among genomic loci. Here, we aim to combine phylogenetic diversity and networks into one concept, split diversity (SD), which properly measures biodiversity for conflicting phylogenetic signals. Moreover, we reformulate well-known conservation questions under the SD framework and present computational methods to solve these, in general, computationally intractable questions. Notably, integer programming, a technique widely used to solve many real-life problems, serves as a general and efficient strategy that delivers optimal solutions to many biodiversity optimization problems. We finally discuss future directions for the new concept.
Keywords
Biodiversity optimization Phylogenetic diversity Phylogenetic split networks Split diversity Integer programmingIntroduction
The previous book chapters show that in the presence of phylogenetic information it is more appropriate to assess biodiversity based on phylogenetic trees than on the concept of species richness (see also May 1990; Vane-Wright et al. 1991). Phylogenetic diversity (PD; Faith 1992) is a popular measure of the amount of evolutionary history encompassed by the species under consideration. Given a phylogenetic tree for a set of taxa, PD of a taxon subset is defined as the sum of the branch lengths of the minimal subtree connecting those taxa. The definition of PD per se requires “a reliable estimate of phylogenetic relationships among the taxa” (Faith 1992). However, such a reliable estimate is sometimes hard to obtain due to, for example, model misspecification (Jermiin et al. 2008) or even intrinsically non-treelike evolutionary patterns. More recently, phylogenomic studies often revealed conflicting phylogenetic signals among genomic loci, adding the complication how to compute PD from multiple trees.
If one is interested in selecting four species maximizing PD, then one indeed ends up with two different sets of species (highlighted in bold-face, Fig. 1) and only P. emphanum occurs in both subsets.
To resolve this issue, we introduced the concept of Split Diversity (SD), which generalizes PD by combining information from multiple trees (Minh et al. 2009). For example, SD of a taxon set can be defined as the average PD of the two trees. By maximizing SD one then simultaneously maximizes PDs over all trees, which captures conflicting phylogenetic signals between the trees. Moreover, computing SD this way is equivalent to computing “phylogenetic diversity” from the so-called phylogenetic split networks (Bandelt and Dress 1992a; Huson et al. 2010). SD has also been recently applied to prioritize populations for conservation (Volkmann et al. 2014). In the following we formalize the concept of split networks and the measure of split diversity. Further, we reformulate well-known biodiversity optimization problems under the framework of SD, present algorithmic solutions and computational tools to these problems. Finally conclude the chapter with future perspectives.
Phylogenetic Split Networks
Rooted phylogenetic trees as shown in Fig. 1 are well understood. Here, both trees show that the common ancestor of the taxa considered has the ancestors of the two genera as direct descendants. In general, interior nodes indicate ancestral taxa of the leaf nodes, and the edge lengths give an estimate of the amount of change observed between nodes. However, if one wishes to combine the information in both trees, it becomes difficult to identify clear ancestors. For example, T_{CYB} and T_{DCoH3} disagree whether G. sonneratii or G. varius is the basal Gallus species. In order to visualize these conflicts phylogenetic split networks have been devised.
This split set is visualized in a phylogenetic split network (Fig. 2b). The major difference to trees is that the interior nodes of a split network cannot be regarded as representing ancestral taxa. Instead, the weight of a split A|B indicates the amount of difference between the taxon set A and B. A split is visualized by a single edge or a set of parallel edges. The former indicates that the split does not conflict any other splits, while the latter indicates at least one conflict. Therefore, two conflicting splits are visualized by a parallelogram. For example, σ_{14} (in cyan, Fig. 2) and σ_{15} (in pink) contradict each other on the placement of P. emphanum and P. malacense. This disagreement generates a narrow parallelogram at the basal Polyplectron.
If more than two splits are in disagreement, the split network will show multiple connected parallelograms. For example, σ_{17} (in red, Fig. 2) conflicts with σ_{19} (in green) and σ_{20} (in yellow). σ_{19} also contradicts σ_{18} (in blue). Therefore, σ_{17}, σ_{18}, σ_{19} and σ_{20} are visualized by three red, two blue, three green, and two yellow parallel edges, respectively. This generates three parallelograms within Gallus (Fig. 2b).
Not every split set can be visualized in two dimensions. For example, assuming that we had a third tree that places G. gallus at the basal Gallus lineage. This would introduce one split contradicting with both σ_{17} and σ_{19}. These triple-wise conflicting splits are depicted by a three dimensional parallelepiped. The resulting split network is not easily visualized anymore. However, for the following it suffices to directly work on the split set (Fig. 2a).
The Measure of Split Diversity
To illustrate, given Σ in Fig. 2, for Y={P. malacense, P. germaini, P. emphanum, G. lafayetii} we have \( SD(Y)={\lambda}_3+{\lambda}_4+{\lambda}_6+{\lambda}_8+{\lambda}_{13}+{\lambda}_{14}+\dots +{\lambda}_{19} \), where \( {\lambda}_i={\lambda}_{\sigma_i} \) is defined as the average of the corresponding branch lengths in T_{CYB} and T_{DCoH3}. Here, contradicting splits such as σ_{17} and σ_{19} are considered in the SD computation.
If the split set Σ corresponds to a tree (i.e. no conflicting splits exist in Σ), then SD is equivalent to PD. The definition of SD therefore generalizes PD. For this reason we focus on SD for the remaining of the chapter.
Biodiversity Optimization Problems
Taxon Selection Problems
We start with the simplest taxon selection problem formally defined as:
Problem 1 (Taxon Selection)
Given a phylogenetic split set for n taxa, find a subset of k taxa that maximizes SD over all subsets of k taxa.
As an illustration, given the split set for ten pheasants (Fig. 2) we want to select four taxa maximizing SD. By doing so we yield an optimal subset (highlighted in bold-face; Fig. 2b), which shares three taxa (P. emphanum, P. malacense, and G. lafayetii) with the CYB-based subset (left panel of Fig. 1) and only two taxa (P. emphanum and P. germaini) with DCoH3-based subset (right panel of Fig. 1). The SD approach therefore provides a “consensus” solution over the two independent PD analyses. Problem 1 is known to be NP-hard (Spillner et al. 2008), which means that to find an optimal set it may, in the worst case, necessary to compute the SD for the exponentially many subsets n.
Problem 1 implicitly assumes that each taxon requires the same amount of resources for conservation. If we knew the preservation costs for each taxon and were provided with a finite budget, then a more realistic scenario is to allocate this budget among the taxa so as to obtain the highest diversity. This process is known as conservation triage (Bottrill et al. 2008) and formally defined as:
Problem 2 (Budgeted Taxon Selection)
Given a split set and conservation costs for each taxon, find a subset of taxa whose total conservation costs do not exceed a predefined budget while maximizing SD.
A taxon is called viable in a subset of taxa if this taxon does not depend on any other taxon, or if it does depend on some taxa, then at least one of them is also present in the subset. For example, G.sonneratii is viable in a subset if this subset also contains P.malacense or P.germaini. P.emphanum and G.gallus are viable in any (sub)set since they do not depend on any other species.
A subset is called viable if all its taxa are viable in this set. For example, {P. emphanum, P. bicalcaratum, P. germaini, G. sonneratii} is a viable subset, whereas {P.emphanum, P.bicalcaratum, G.lafayetii, G.sonneratii} is not viable.
We now formally define the viable taxon selection problem as
Problem 3 (Viable Taxon Selection)
Given a split set and a dependency network, find a viable subset of k taxa, which maximizes SD.
Reserve Selection Problems
For reserve selection we define the SD of a subset of areas as the SD of the union set of taxa present in these areas. The reserve selection is formalized as:
Problem 4 (Reserve Selection)
Given a split set for n taxa distributed in m areas, find a subset of k areas that maximizes SD over all subsets of k areas.
Presence/absence of ten pheasants in eight countries obtained from the global biodiversity information facility (http://www.gbif.org)
Species | Bhutan, BT | Indonesia, ID | India, IN | Sri Lanka, LK | Philippines, PH | Malaysia, MY | Thailand, TH | Vietnam, VN |
---|---|---|---|---|---|---|---|---|
P. bicalcaratum | x | x | ||||||
P. chalcurum | x | |||||||
P. emphanum | x | |||||||
P. germaini | x | |||||||
P. inopinatum | x | |||||||
P. malacense | x | |||||||
G. gallus | x | x | x | x | x | x | x | |
G. lafayetii | x | |||||||
G. sonneratii | x | |||||||
G. varius | x |
Four countries maximizing PD on the CYB tree (first column), PD on the DCoH3 tree (second column), and SD on the split network (third column)
PD – CYB | PD – DCoH3 | SD |
---|---|---|
Malaysia (3) | Indonesia (3) | Malaysia (3) |
Philippines (2) | Malaysia (3) | Philippines (2) |
Sri Lanka (1) | Philippines (2) | Indonesia (3) |
India (2) | Vietnam (2) | India (2) |
If budget data is available, then we have a budgeted reserve selection problem. Here, preserving these species in each country comes at a cost and we need to select those countries that maximize SD within an allocated budget.
Problem 5 (Budgeted Reserve Selection)
Given a split set for n taxa distributed on m areas and conservation costs for each area, find a subset of areas whose total conservation costs do not exceed a predefined budget while maximizing SD.
Computational Methods in Conservation Planning
The algorithms to solve the aforementioned Problems 1–5 are those that are guaranteed to produce an optimal solution, often referred to as exact algorithms, and those that are not. The former includes algorithms that are based on integer programming and dynamic programming, whereas the latter comprise greedy algorithms, approximation algorithms and algorithms based on simulated annealing. We will start with greedy algorithms, as they are simple and probably most widely applied in conservation planning.
Greedy Algorithms
Greedy algorithms are a simple and general heuristic strategy but, usually, do not guarantee optimal solutions. Kirkpatrick (1983) was probably the first to apply a greedy algorithm to find a solution to Problem 4, the simple reserve selection, but under the species richnessconcept. His greedy algorithm coined “complementarity principle” first identifies the most species-rich area. In the second step, it finds the area, which “adds” the most numbers of new species to the firstly chosen area. This is repeated until k areas are obtained. Such a complementarity principle has been applied to maximize PD (Faith 1992) and also applied elsewhere (e.g., Vane-Wright et al. 1991; Pressey et al. 1997). Recently, Bordewich and Semple (2008) have proven that the greedy algorithm applied to Problem 5 under PD will generate a solution that has at least ~63 % of the PD of the optimal solution, which is the best possible approximation ratio.
The only case, where a greedy algorithm delivers the optimal solution is the taxon selection (Problem 1) under PD on trees (Pardi and Goldman 2005; Steel 2005). An efficient implementation of such a greedy algorithm (Minh et al. 2006) finds a solution for trees with millions of taxa within seconds on a standard PC. Greedy algorithms have been further examined in conservation biology (Moulton et al. 2007; Bordewich et al. 2008).
Obviously greedy algorithms can be applied for Problems 1–5 to maximize SD. The general idea is to start with one target (either taxon or area) having the highest SD. We then choose the second target “adding” the most SD while still satisfying the constraints (budget or viability constraints). We repeat this step until no further target can be added (e.g., exceeding k targets for Problem 1, 3, and 4 or exceeding the budget for Problem 2 and 5). As an illustration the greedy algorithm is applied for Problem 4 to find four countries showing the highest pheasant SD for the split network (Fig. 2b) and known geographical distribution (Table 1) as follows. Malaysia is first selected as it contains the highest SD. Philippines, Indonesia, and India are selected in the next steps. In this particular example the greedy algorithm happens to obtain the optimal set of four countries (Table 2).
Integer Programming
Integer Programming (IP; Dantzig et al. 1954; Gomory 1958) is a widely used and powerful optimization technique to solve a variety of decision-making problems (Wolsey 1998; Jünger et al. 2010). IP methods maximize or minimize a linear objective function subject to linear constraints (equalities or inequalities) when one or more variables are restricted to be integers. Theoretically solving IP is NP-hard. However, thanks to powerful solvers like CPLEX (2012) and GUROBI (Gurobi Optimization Inc. 2013), problems with thousands of variables and constraints can be solved optimally within reasonable time (Jünger et al. 2010; and references therein).
The first application of IP in conservation problems goes back to (Cocks and Baird 1989), who solved the reserve selection (Problem 4) under species richness. Such IP formulations have been extended to more realistic scenarios (Underhill 1994; Church et al. 1996; Possingham et al. 2000), to maximize PD (Rodrigues and Gaston 2002; Rodrigues et al. 2005), and to maximize SD (Minh et al. 2010).
Here, we show how to model biodiversity optimization problems 1–5 in IP parlance, which allows available IP software packages to solve the problem. We first introduce some notations and definitions and further exemplify IP formulations for Problems 1–5 using the pheasant data set.
IP for Taxon Selection Problems
Given a set of n taxa, we encode a subset S by an n-element binary vector with entries of 0 and 1 indicating the absence and presence of the corresponding taxa in S. The elements of this vector are called taxon variables. For the pheasant data set there are ten taxon variables x_{PB}, x_{PC}, x_{PE}, x_{PG}, x_{PI}, x_{PM}, x_{GG}, x_{GL}, x_{GS}, x_{GV} (indices follow initials of species names). We say that a split σ = A|B is preserved in S if A and B each contain at least one taxon from S. For each split σ we introduce a binary split variabley_{σ}, where y_{σ} = 1 if σ is preserved in S and 0 otherwise.
IP Formulation of Problem 1
Maximize objective function (1), subject to subset size constraint (2), binary constraints (3, 4), split constraints (5) (see Table 3).
IP Formulation for Problem 2
Maximize objective function (1), subject to budget constraint (6), binary constraints (3, 4), and split constraints (5) (Table 3).
IP Formulation of Problem 3
Maximize objective function (1), subject to subset size constraint (2), binary constraints (3, 4), split constraints (5), and viability constraints (7) (Table 3).
IP for Reserve Selection Problems
For reserve selection we encode a subset W of m areas by a binary vector (z_{1},z_{2},…,z_{m}), where z_{r} is 1 if arear is present in W, and 0 otherwise. We call z_{r} area variables. For the pheasant habitat (Table 1) we have eight area variables z_{ID}, z_{LK}, z_{BT}, z_{IN}, z_{PH}, z_{MY}, z_{TH}, z_{VN} (indices follow two-letter country codes). We now redefine split constraints in terms of area variables instead of taxon variables as follows.
IP Formulation of Problem 4
Maximize objective function (1), subject to subset size constraint (8), binary constraints (4, 9), and area-split constraints (10) (Table 4).
IP Formulation of Problem 5
Maximize objective function (1), subject to budgetary constraint (11), binary constraints (4, 9), and area-split constraints (10) (Table 4).
Other Algorithms
While greedy algorithms and IP are general strategies for all Problems 1–5, other algorithms have been applied to solve special cases. For example, simulated annealing algorithms (Possingham et al. 2000) were introduced to solve the reserve selection Problems 4 and 5 under species richness with an opportunity to minimize the connectivity between the areas such as the boundary lengths. Dynamic programming algorithms (DPA) have been applied to solve Problem 2 under PD (Pardi and Goldman 2007). DPA was further extended to maximize SD on circular split networks (Minh et al. 2009a, b). Other special types of split networks were exploited to solve Problem 1 (Spillner et al. 2008; Bordewich et al. 2009).
Computer Software
Conservation planning software like Marxan (Ball et al. 2009) and Zonation (Moilanen et al. 2009) mainly focus on species richness. However, both programs can indirectly account for phylogenetic diversity (see also Silvano, Valdujo and Colli, chapter “Priorities for Conservation of the Evolutionary History of Amphibians in the Cerrado” and Arponen and Zupan, chapter “Representing Hotspots of Evolutionary History in Systematic Conservation Planning for European Mammals”). Only a few programs explicitly allow to compute phylogenetic diversity (Webb et al. 2008; Kembel et al. 2010). In the following we describe two programs relevant for the SD analysis.
SplitsTree
SplitsTree (Huson and Bryant 2006) is a user-friendly and leading software to reconstruct and visualize phylogenetic networks from multiple sequence alignments, distance matrices, or sets of trees. SplitsTree implements a wide range of split network inference methods such as split decomposition (Bandelt and Dress 1992b) and neighbor-net (Bryant and Moulton 2004). SplisTree has a limited ability to compute PD and SD. It works for all major platforms including Windows, Mac OS X, and Unix. More information about SplitsTree is available at http://www.splitstree.org.
PDA: Phylogenetic Diversity Analyzer
PDA (Minh et al. 2009) is a software tool that computes and maximizes species richness, PD, and SD given a variety of user-defined constraints including budget, ecological, and geographical constraints. PDA can be used in conjunction with SplitsTree to work with SD. It solves all Problems 1–5 by greedy algorithms, dynamic programming, and integer programming methods. Moreover, it supports weighted dependency networks for viable taxon selection and spatial reserve selection problems (Chernomor et al. 2015). Among other features is the computation of PD/SD endemism and complementarity (Faith et al. 2004). PDA is available as a command-line program for Windows, Mac OS X, and Unix as well as an online web service. More information about PDA is available at http://www.cibiv.at/software/pda.
Conclusions and Perspectives
In this chapter we have presented the concept of split diversity, a generalization of PD to account for contradicting phylogenetic information in biodiversity optimization. We demonstrated the new concept with a small pheasant data set. We note that this example is not realistic because neither genera are vulnerable nor the selection of entire countries is reasonable. Moreover, genetic data for galliforms are available for more genera and genomic loci (Wang et al. 2013) and the methodology developed here is well applicable to this new data.
We then presented computational tools to perform the analysis under the SD framework. Both greedy algorithms and IP can be generally applied to solve the same conservation questions, where the former quickly computes a solution and the latter ensures optimal solutions. Moreover, IP works well for data set sizes usually encountered in real data. For example, we have recently applied IP to solve the viable taxon selection (Problem 3) for 242 marine species of Caribbean coral community and the budgeted reserve selection (Problem 5) for the Cape of South Africa with 735 plant genera (Chernomor et al. 2015). IP always returned optimal sets of taxa and areas within seconds to a few minutes.
SD can be extended to include species extinction risks as developed for PD (Weitzman 1992; Witting and Loeschcke 1995). Such a “probabilistic” PD approach (see chapters “The Value of Phylogenetic Diversity” and “Reconsidering the Loss of Evolutionary History: How Does Non-random Extinction Prune the Tree-of-Life?”) predicts future diversity given the fact that some species might become extinct in, say, 20 years. The problem, previously coined the Noah’s Ark Problem (NAP; Weitzman 1998), is then to maximize future PD given limited budgets. The same concept can be applied to SD as follows. One first computes “survival probabilities” for each split in split networks in the same fashion as for branches in phylogenetic trees. The future SD is then defined as the dot product of the split weights and split survival probabilities. This definition of future SD consistently generalizes that of future PD.
From a computational view point, solving the extended NAP under future SD is NP-hard as proven for PD (Hartmann and Steel 2006). Dynamic programming algorithms (DPA) optimally solve the NAP under future PD in a special scenario, where the species extinction probability becomes 0 if it is given enough resources (Pardi and Goldman 2007). For general scenarios Hickey et al. (2008) devised such a DPA that gives an approximation ratio of nearly 1 compared to the optimal solution. More recently, Billionnet (2013) presented an IP approach for the NAP that runs within a few minutes for simulated 4,000-taxon cases and provides near-optimal solutions, which are only 1.2 % away from the optimal solution. It will be interesting to investigate how such DPA and IP approaches can be adapted to solve the NAP under the more general SD framework.
Notes
Acknowledgments
The authors thank Tung Lam Nguyen for developing PDA web service. The authors also thank Roseli Pellens and Philippe Grandcolas for inviting us to write this chapter and three anonymous reviewers for constructive comments. This work was supported by University of Vienna (Initiativkolleg I059-N) to O.C. and the Austrian Science Fund – FWF (I760-B17) to B.Q.M. and A.v.H.
Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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