population graphs: the graph theoretic shape of genetic structure

Molecular Ecology (2004)

13

, 1713–1727 doi: 10.1111/j.1365-294X.2004.02177.x

© 2004 Blackwell Publishing Ltd

Blackwell Publishing, Ltd.

Population Graphs: the graph theoretic shape of genetic structure

RODNEY J . DYER and JOHN D. NASON

Department of Ecology, Evolution, and Organismal Biology, Iowa State University, Ames, Iowa 50011, USA

Abstract

Patterns of intraspecific genetic variation result from interactions among both historicaland contemporary evolutionary processes. Traditionally, population geneticists have usedmethods such as

F

-statistics, pairwise isolation by distance models, spatial autocorrelationand coalescent models to analyse this variation and to gain insight about causal evolutionaryprocesses. Here we introduce a novel approach (Population Graphs) that focuses on theanalysis of marker-based population genetic data within a graph theoretic framework. Thismethod can be used to estimate traditional population genetic summary statistics, but itsprimary focus is on characterizing the complex topology resulting from historical and con-temporary genetic interactions among populations. We introduce the application of Popu-lation Graphs by examining the range-wide population genetic structure of a Sonoran Desertcactus (

Lophocereus schottii

). With this data set, we evaluate hypotheses regarding historicalvicariance, isolation by distance, population-level assignment and the importance of specificpopulations to species-wide genetic connectivity. We close by discussing the applicabilityof Population Graphs for addressing a wide range of population genetic and phylogeo-graphical problems.

Keywords

: genetic structure, graph theory, networks, phylogeography

Received 10 September 2003; revision received 10 January 2004; accepted 13 February 2004

Introduction

The amount and geographical patterning of genetic vari-ation within species is an evolutionary consequence ofseveral historical and contemporary processes includingvicariance, range expansion, gene flow and fragmentation(Slatkin 1985; Riddle 1996; Taberlet

et al

. 1998; Hewitt 2001;Sork

et al

. 2001). Quantifying this variation and in turn theextent to which these processes have acted in shapinggenetic structure is a chief concern of the field of evolution-ary biology. Numerous theoretical models and methodo-logical procedures have been developed to address thisgoal, many of which are integral to the foundation ofpopulation genetic theory. A common conceptual featureof these approaches is an a priori definition of a hierarch-ically nested population model (for example in whichpopulations are nested within geographical regions) thatin turn guides the distillation of genetic differences amongpopulations into single or few measures of average genetic

differentiation. Although commonly employed for theanalysis of genetic marker data, this general approachsuffers from potential several shortcomings that are perhapsnot well recognized.

With the goal of stimulating the development of new,hopefully effective, solutions we begin by considering thepotential problems associated with an a priori statisticalmodel-based approach characteristic of many traditionalpopulation genetic procedures. We then describe thedevelopment of a potential solution, a new graph-theoreticapproach to the analysis of genetic marker data, which wecall Population Graphs.

Population models and summary statistics

Wright’s

F

-statistics (Wright 1951),

amova

(Excoffier

et al

.1992) and Nei’s D (Nei 1972, 1978) are commonly usedprocedures describing genetic structure in terms of summarystatistics. Even approaches relying upon pairwise measuresof genetic differentiation, such as isolation by distance(Slatkin 1993; Rousset 1997), summarize relationships interms of averaging statistics such as a regression slope or a

Correspondence: Rodney J. Dyer. Fax: (515) 294 1337; E-mail:[email protected]

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, 13, 1713–1727

correlation coefficient. In contrast, evidence from popu-lation genetics, phylogeography and ecology indicates thatthe processes influencing the evolution of genetic structurevary in both time and space (e.g. Rhodes

et al

. 1996) leadingto the expectation that the complexity of interpopulationrelationships may not be captured adequately by a single orfew averaging statistics. Both simplification and generalityoften lead to a better understanding of underlying processes,but they do not guarantee that we end up with a moreconcise interpretation of how evolution has shaped the data.

Even in the presence of a significant summary statistic,it is unclear that the associated a priori statistical model isone that best describes the patterns of variation in the popu-lation genetic data and, hence, reveals the correct signatureof underlying evolutionary processes. A significant statis-tic signifies only that the proposed statistical model is suf-ficiently correct to reject the null hypothesis, often one ofno differentiation. However, rejecting the null does notmean that the statistical model is defined precisely or cor-rect. For any given value of a mean measure of differenti-ation (e.g.

F

ST

,

Φ

ST

) there is an infinite number of ways inwhich the differentiation between populations may bestructured. We cannot distinguish, for example, betweenall populations being equally differentiated vs. subsets ofpopulations being categorically different. The currentrepertoire of statistical genetic models lack the ability toinspect both the ‘lack of fit’ of our models to the observeddata and the resulting response surfaces (e.g. Box

et al

.1978; Box & Draper 1987). As a result, additional informa-tion is likely to be present in our genetic data sets thatremain quantified inadequately.

Due of the overall complexity of evolutionary processesoperating within and among populations of a species, ana-lytical approaches that focus specifically upon the detailsof genetic interactions and relationships among popula-tions, as opposed to an overall average effect, will providea more integrated description of observed populationgenetic structure. Indeed, the extent to which we embracethis, often multivariate, complexity can impact the degreeto which we can quantify successfully both intraspecificgenetic variability (e.g. Smouse

et al

. 1982; Westfall &Conkle 1992) and the effects of specific evolutionary pro-cesses (e.g. Gavrilets 1997). The majority of methodologicalapproaches employed thus far have allowed us to depictevolutionary affects on intraspecific genetic variation witha broad-brush stroke. If we are interested in examininghow evolution has structured genetic variation at a finerlevel of granularity, however, we may often need to look atthe problem using alternate perspectives.

Several models have been suggested recently thatmove beyond averaging summary statistics. These analy-tical approaches fall, roughly, into two broad categories. Inthe first category are model-based approaches that eitherextend or attempt to increase the precision of standard

F

ST

-

based approaches. Often these models posit an a priorihierarchical model of population relationships. Examplesof these include multivariate ordinations of pairwise dif-ferentiation statistics using principal coordinates and multi-dimensional scaling (e.g. Lessa 1990; Edwards & Sharitz2000; Zhivotovsky

et al

. 2003) and methods aimed at maxi-mizing among strata genetic variance such as

samova

(Dupanloup

et al

. 2002). The second, more recent categoryincludes models based on the coalescence, such as

genetree

(Bahlo & Griffiths 2000) and

migrate

(Beerli & Felsenstein2001). In this study we present a third category of modelsfor examining the distribution of intraspecific geneticstructure using a multivariate graph-theoretic approach.This method, which we call Population Graphs, is free ofan a priori model of population arrangement (e.g. we donot assume that populations are nested within either ahierarchical or bifurcating statistical model). Further, theserelationships are not quantified in terms of averagingstatistics or coalescence parameters, but in the form of agraphical topology (e.g. the pattern of genetic covariancestructures among all populations), which captures the highdimensional genetic covariance relationships among allpopulations simultaneously rather than in a pairwisefashion. It is with this topology, which is easily visualized,that we address both a priori and

post-hoc

hypothesesregarding the distribution of intraspecific genetic variation.

To present the Population Graph framework, we intro-duce briefly some graph theoretic notation as well as thestatistical methods of conditional independence. We thendemonstrate how one utilizes the Population Graph frame-work by focusing on a set of general population geneticand phylogeographical hypotheses using an empiricalexample from the Sonoran Desert cactus,

Lophocereusschottii

, presented recently in Nason

et al

. (2002). With thisnuclear marker data set, we show specifically how Popula-tion Graphs can be used to address hypotheses amenableto traditional analytical approaches, such as the effects ofhistorical vicariance and isolation by distance. Moreover,we show how the graph-theoretic approach on whichPopulation Graphs is based allows us to identify and testhypotheses regarding population genetic history thatwould not have been apparent from traditional analyses.We close by discussing the utility of Population Graphs foraddressing a wide range of population genetic questionsfocusing on the dynamic nature of co-occurring evolutionaryprocesses.

The model

The essential goal of Population Graphs is to allow thegenetic data to describe the statistical relationships amongall populations simultaneously. Our approach to definingthese sets of relationships relies upon a graph theoreticinterpretation of population genetic structure. We begin by

P O P U L A T I O N G R A P H S

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introducing some graph theoretic notations to describe thePopulation Graph framework.

Graph theory

Graph theory is based upon the association amongmathematical sets. A set,

S

, is simply an unordered collectionof objects. The number of objects in

S

, denoted as |

S

|, iscalled the

order

of

S

. A set that has no elements is called anull set and is written as

∅

. We write

x

∈

S

and

y

∉

S

toindicate that

x

is a member of

S

and

y

is not a member of

S

.If all the elements in

S

are also in another set,

T

, then

S

is asubset of

T

, which is denoted as

S

⊂

T

. On the other hand,when none of the elements of

S

are not in

T

, we write

S

⊄

T

(i.e.

S

and

T

are disjoint). Finally, when examining therelationship between two sets, we can write the

intersection

of the two sets as

S

∩

T

, representing the elements sharedbetween both sets.

A graph (

G

) is a mathematical mapping of two disjointsets, a set of nodes (

V

) and a set of edges (

E

), and is denotedas

G

= {

V, E

} (Bollobás 2001). Within the Population Graphframework, the set of nodes,

v

i

∈

V

;

i

= {1, 2, … ,

N

}, repre-sent

N

sample populations and the set of edges,

e

i

∈

E; i

= {1, 2, … ,

K

}, the set of multivariate measures ofgenetic covariance between populations. A saturated graph(Fig. 1A) is one in which all nodes are connected to all othernodes and has

K

total

= N

(

N

−

1)/2 edges. Graphs can berepresented algebraically by an

incidence matrix

, whichdescribes the

topology

of the graph. The topology, in itsmost general sense, is simply the pattern of node connec-tions. The following incidence matrix,

A

, describes thegraph in Fig. 1B. This incidence matrix has four rows andcolumns; that is, the order of the graph is |

V

| = 4. Each ele-ment of the incidence matrix

A

= {

e

ij

};

i, j

= 1, 2, 3, 4 denotesthe presence (a nonzero value) or absence (a zero value) ofan edge,

e

ij

, connecting nodes

i

and

j

.

(1)

The strength of edges (i.e. the elements of

A

) can be vari-able, resulting in a weighted graph, as is used for Popula-tion Graphs. Incidence matrices for weighted graphs onlydiffer from eqn 1 in that the presence of an edge is rep-resented by a real number rather than a ‘1’. Furthermore, theincidence matrix,

A

, in this example is symmetric aroundthe diagonal, although neither graphs in general nor Popu-lation Graphs require this. A nonsymmetrical PopulationGraph incidence matrix would represent anisotropy ingenetic covariance, as may arise, for example, under isola-tion by distance in linear spatial arrangements of popula-tions (such as along a river).

The topology of a graph has several prominent charac-teristics salient to the characterization of intraspecificgenetic variation. First, a subgraph,

G

1

, of a larger graph

G

is one in which the entire edge and node sets of

G

1

areincluded in

G

; that is,

G

1

= {

V

1

,

E

1

} is a subgraph of

G

if andonly if

V

1

⊂

V

and

E

1

⊂

E

. A graph can be partitioned intomany subgraphs and we will return to this topic later whendiscussing the interpretation of Population Graph. Second,the

degree

of a node,

d

(

v

i

), is simply the number of edgesconnected to it. If

d

(

v

i

) = 0 then the node

i

is called

isolated

.From a genetic perspective, graphically isolated popula-tions would be independently evolving entities. Next, agraph may contain

cycles

, which are sequences of non-repeating node–node paths that form a closed loop. Suchcycles in Population Graphs may result from reticulategene flow. The subgraph,

G

1

, consisting of the nodes

V

= {

V

1

, V2, V3} and the edges E = {{V1, V2}, {V2, V3}, {V3,V1}} in Fig. 1B is a cycle. A tree is simply a sequence of con-nected nodes and edges that have no cycles. Again, fromFig. 1B, the subgraph G2 is a tree when the node set isV = {V1, V3, V4} and the edge set E = {{V1,V3}, {V3,V4}}. Withthis general introduction to graphs, we now describe howone can use graphs to represent population genetic struc-ture using multilocus marker data.

Genetic coding

We assume that genetic markers have been assayed fora relatively large number of individuals spread acrossseveral populations. For the purposes of this study we focusspecifically on codominant markers, such as allozymesand microsatellites, and for the sake of brevity address theissues of alternate genetic markers and sampling strategyin a subsequent manuscript (Dyer and Nason, in prep).Multilocus genotypes are translated into multivariatecoding vectors following the mapping described in Smouseet al. (1982). This coding scheme produces a data vector, pj,

A =

0 1 1 01 0 1 01 1 0 10 0 1 0

Fig. 1 Two simple graphs. (A) A saturated graph consisting offour nodes (circles) and six edges (lines connecting the nodes).(B) A nonsaturated graph with the same order as (A) but with areduced edge set.

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© 2004 Blackwell Publishing Ltd, Molecular Ecology, 13, 1713–1727

representing the multivariate genotype for the jthindividual. The full data matrix, X, will have m columns,where m is the number of independent genetic variables(e.g. the number of independently assorting alleles acrossall loci), and ntotal rows, representing all the individualswithin the N populations. Even though this methodproduces a sparse matrix, with a reasonable number of lociand individuals it approaches multivariate normality(see Kempthorne 1969; Westfall & Conkle 1992, for a morecomplete discussion). The set of individuals within the ithpopulation defines a multidimensional population centroid,pi, in m-dimensional genetic space. The centroid defines aunique m-dimensional coordinate representing the averagegenetic individual within the ith population. This distancemetric has been chosen as it is identical to that used in theamova framework (Excoffier et al. 1992), with which manyresearchers are familiar.

Once the N population centroids are defined from the setof all individuals, we then determine the contribution tothe overall genetic variation due to differences among allpairs of populations. These pairwise distances define a dis-tance matrix, D, whose off-diagonal elements, dij, define astatistical distance between populations in m-dimensionalgenetic space. This matrix can be used to describe a satur-ated Population Graph in which all nodes (populations)are connected to all other nodes by edges of weight dij. InAppendix I we demonstrate the general equivalence of Dto the distance matrix in amova (Excoffier et al. 1992) andshow how Φ-statistics can be estimated directly fromthe saturated Population Graph. Because all nodes areinterconnected, the topology of the saturated graph is notimmediately informative as to interpopulation relation-ships. An informative topology is obtained from the mini-mal incidence matrix containing the smallest edge set thatsufficiently describes the among population genetic covari-ance structure. The translation of the population distancematrix, D, to a minimal incidence matrix (as in eqn 1) reliesupon the techniques of conditional independence.

Genetic covariance and conditional independence

Conditional independence forms the foundation of severalcommonly used parametric statistical models. For example,in multiple regression the inclusion of a variable requiresthe examination of its conditional independence to thosevariables already entered into the model (Whittaker 1990).In this case, we examine the Type III sums of squares forthe newly entered variable and compare that against thesums of squares accounted for by the set of previouslyincluded predictor variables (e.g. Draper & Smith 1981). Inaddition to multiple regression, conditional independenceis also used in such statistical methods as logistic regres-sion and contingency tables (Whittaker 1990). In the contextof Population Graphs, we are interested in determining

the minimal edge set that sufficiently describes the totalamong population covariance structure. Following fromthe regression analogy, each edge in the Population Graphis analogous to a predictor variable. Our goal is to identifyedges that do not aid in sufficiently describing the totalamong population genetic covariance structure. These edgescan be ‘pruned’ from the graph without significantlydecreasing the fit of the Population Graph model to thepopulation genetic data.

There are several methods amenable to calculatingconditional independence, including edge deviance,covariance selection and vanishing partial correlations(Dempster 1972; Whittaker 1990). For the sake of brevity,we shall use the method of edge deviance in this paper asit has recently been described in an evolutionary contextby Magwene (2001) and defer the comparison of alternatemethods to a subsequent manuscript. Briefly, the methodproceeds as follows. First, we must translate our pairwisepopulation distance matrix, D, into a covariance matrix, C.Gower (1966) showed the duality of distance and covari-ance matrices whereby the covariance between the ithand j th element of D is given by cij = −0.5(dij − di. − d.j + d..),where the subscripts i and j index the elements of D and theperiod subscript, ‘.’, indexes the mean of the rows and/orcolumn(s) in D. Following directly from Magwene’sdiscussion of edge deviance, we proceed by inverting thecovariance matrix producing what is called a precision matrix(Cox & Wermuth 1996). The ith diagonal element of theprecision matrix is equal to 1/(1 − ), where is the multiplecorrelation coefficient between the ith and all remainingpopulations (Whittaker 1990). is also known as thecoefficient of multiple determination (Sokal & Rohlf 1995),which is a measure of the proportion of variation in the ithpopulation jointly accounted for by the remaining popu-lations. To aid in interpretation, the precision matrix isstandardized to a correlation matrix, R, using normalmatrix routines (see Johnson & Wichern 1992).

The ijth off-diagonal element of the standardized preci-sion matrix, rij, is simply the partial correlation coefficientsbetween the ith and jth populations. Absolute values of rijwhich are zero denote pairs of populations whose covari-ance structure are conditionally independent, given all theother populations in the data set. That is, within the edgeset E there is a sufficient number of alternate paths throughthe graph, whose presence explain the overall pattern ofpopulation covariation such that the removal of eij does notinfluence the total genetic covariance.

Determination of how small rij must be to be consideredzero is the final step in estimating the conditional independ-ence structure of the Population Graph. Whittaker (1990)showed that a statistic of information divergence, callededge exclusion deviance (EED), could be used to determineif values of rij are not significantly greater than zero. Edgeexclusion deviance is calculated as: EED = −ntotal ln[1 − (rij)2],

Ri2 Ri

2

Ri2

P O P U L A T I O N G R A P H S 1717


where ntotal is the number of individuals in the entire dataset. EED has an asymptotic χ2 distribution with one degreeof freedom (Whittaker 1990). Using the EED values, wedetermine the minimal edge set Emin describing the hypo-thesized topology of the Population Graph. The goodness-of-fit for this topology can be evaluated analyticallyby estimating the model deviance, D = ntotal ln(|ΣΣΣΣ|/|S|),where |ΣΣΣΣ| is the determinant of a MLE estimate of thecovariance matrix (see Edwards 1995 for its calculation)and |S| is the determinant of observed sample covariancematrix (Whittaker 1990). The model deviance, D, also hasan asymptotic χ2 distribution with the degrees of freedomequal to the number of excluded edges. Significant valuesof D suggest that the topology does not fit the data. For amore complete discussion of model testing, see Magwene(2001).

Hypothesis testing in Population Graphs

In general, hypotheses are evaluated in Population Graphsby examining either the overall topology or specific topo-logical features of the graph itself. For example, a commona priori hypothesis in many population genetic studiesis one specifying a restriction in gene flow between twogroups of populations (e.g. vicariance), which is evaluatedtypically by examining the significance of an averagedifferentiation statistic between groups such as Φ or θ. Thesame hypothesis can be evaluated using Population Graphsby focusing on topological features of edge connectivitybetween specified groups of populations. While PopulationGraphs are not limited to testing only this type of hypo-thesis, the practice of assessing the significance of topologicalfeatures within a graph is so common we will focus first onhow these hypotheses are evaluated. Later, using the L.schottii data set, we will demonstrate methods for testingother categories of hypotheses in Population Graphs.

The null hypothesis regarding the topology of a Popula-tion Graph under restricted gene flow, for instance due tovicariance, states that the presence or absence of an edgeconnecting nodes should be independent of the hypo-thesized geographical barrier to gene flow. If the null istrue then the number of edges connecting nodes across thehypothesized barrier to gene flow should be as numerousas the number of edges in other portions of the graph. Asit turns out, as every edge is connected to only two nodes,by definition, the pattern of edge/node connections has abinomial expectation (Bollobás 2001). Given a PopulationGraph, G, with parameters K = |E| edges and N = |V|nodes the average probability of observing an edge connect-ing any two nodes can be modelled as a binomial randomvariable, B(p, Ktotal), where the parameter p is:

(3)

where

(4)

enumerates the number of edges in a saturated graph(following Bollobás 2001).

By way of an example, consider the graph G = {V, E} withparameters N and K as above. Further, we are interested inpartitioning G into two complete subgraphs, G1 and G2,with node sets, V1 and V2, where V1 ∪ V2 = V and V1 ∩ V2 =∅. We assume that these subgraphs are not disconnected(the trivial solution) and share a set of edges, Kbtw, connectingthe two subgraphs. The subgraph G1 has N1 = |V1| nodesand K1 = |E1| edges. Similarly, the subgraph G2 has N2 =|V2| = N – N1 nodes and K2 = |E2| = K – K1 – Kbtw edges.The probability of finding an edge connecting G1 to G2 is:

(5)

and the probability of observing Kbtw or fewer such edgesis:

(6)

If we are not testing an a priori hypothesis regardingthe significance of G1 and G2 we need to correct p (x ≤ Kbtw)to take into account the number of ways we can obtain a

subgraph of size N1 by multiplying eqn 6 by . This

general binomial approach facilitates the testing of a widerange of both a priori as well as post-hoc topological hypo-theses within the Population Graph framework.

In addition to the binomial approach, one may also usepermutation to evaluate the significance of topologicalfeatures. Following from above, we can easily simulate thenull distribution of the number of between subgraph edgecounts within the set of graphs, ̋ NULL, of the same order asG. The node set of each graph in ̋ NULL will be partitioned intopopulations belonging to subgraphs as in ̋ NULL the observedgraph. From the set of graphs in, we extract the null distri-bution of between subgraph edges, btw, on which we evaluatethe significance of the observed value. General permuta-tion approaches are covered in depth in Manly (1997) thatcan be adapted easily to graph theoretic hypotheses.

Case study: L. schottii

Chief among the historical factors influencing intraspeci-fic genetic variation are deep time geotectonic events overthe past several million years and shallow time glacial–interglacial cycles over the most recent hundreds of

pK

K =

total

K

N Ntotal

( )=

− 12

pN N

K = 1 2

total

p x K Ki

p pi K

i

K

( ) ( )≤ =

− −

=∑btw

btw

1 1

0

NN1

1718 R . J . D Y E R


thousands of years. The relative importance of geologicaland climatic factors on the present day distributions of plantand animal taxa has long been contentious (Cracraft 1988;Riddle 1996; Avise 2000). Molecular data are being broughtincreasingly to bear on this problem, with the strongestinference obtained when testing pre-existing regional hypo-theses concerning dispersal and vicariance as opposedto generating ad hoc explanations from genetic data alone(Cruzan & Templeton 2000; Knowles & Maddison 2002).

The Sonoran Desert of the southwestern United Statesand northwestern Mexico is an area of active ecologicaland evolutionary research due to its diverse habitats, hightaxonomic diversity and well-characterized geotectonicand palaeoclimatic history. In particular, three major geo-tectonic events are predicted sources of vicariance for theregion’s biota (Case et al. 2002): (i) the formation of the Seaof Cortéz (c. 5 Mya) separating peninsular Baja from main-land Mexico; (ii) the Isthmus of La Paz separating thesouthern Cape Region and south-central Baja and resultingfrom the formation of a trans-peninsular seaway connect-ing the Sea of Cortéz and the Pacific Ocean (c. 3 Mya); and(iii) a geologically cryptic mid-peninsula vicariance eventin Central Baja (suggested by vertebrate allozyme andmtDNA data; Riddle et al. 2000), resulting from the forma-tion of a trans-peninsular seaway during the mid Pleis-tocene (c. 1 Mya). Layered onto these deep time events aremore recent Pleistocene climatic cycles driving changes ingeographical distributions that may have eroded or evenerased evidence of older vicariance events.

The geographical concordance of phylogenetic topolo-gies with ancient geological events has led Riddle et al.(2000) to two general conclusions regarding present daySonoran Desert vertebrates: first that lineage diversifica-tion resulting from vicariance events in deep time persistsas important sources of taxic diversity, and second thatQuaternary climatic fluctuations have had relatively littleimpact on biogeographical distributions of taxa in Baja.Sonoran Desert plants, in contrast, are mostly of tropicalorigin and frost sensitive, suggesting that their popula-tions may be more prone to extinction during climaticfluctuations and less likely to persist on opposite sides ofhistorical sources of vicariance.

Using traditional population genetic tools, Nason et al.(2002) tested these hypotheses with respect to a SonoranDesert plant, the endemic columnar cactus, senita (L. schot-tii). A total of 21 Continental Sonoran and Peninsular Bajapopulations of L. schottii were assayed for 29 polymorphicallozyme loci. The distribution of genetic variation wasdecomposed using a hierarchical F-statistic model with thefollowing levels: individuals nested within populations;populations nested within putative Continental Sonoranand Peninsular Baja phylogroups defined by a Sea of Cortézvicariance event; and among phylogroups. Nason et al.(2002) found genetic differentiation among populations to

be exceedingly large for an insect pollinated plant species(FST = 0.431), as was the genetic variance among phylo-groups (FPT = 0.302). Isolation by distance was significantamong Peninsular Baja and among Continental populations(but not between), consistent with the hypothesis of limitedgene flow within regions (the lone exception was a Contin-ental population, SenBas, to which we return shortly). A popu-lation phenogram of all Peninsular Baja populations,constructed using Nei’s genetic distance and neighbourjoining, revealed a clear pattern of consecutive nesting ofpopulations from south to north supporting the hypothesisof recent northward range expansion. The phenogram des-cribing Sonoran populations contained less spatial structure.

From the L. schottii data set, we examine two distinct setsof hypotheses using Population Graphs. The first set isconcerned with more traditional population geneticand phylogeographical questions regarding the historicalvicariance between the putative Continental and Peninsularphylogroups (akin to testing for significant differentiationamong strata or H0: FPT = 0 in Nason et al. 2002) and thepresence of isolation by distance. The second set of hypo-theses focuses on specific topological characteristics ofthe Population Graph, examining the placement of aparticular continental population (SenBas) within thegraph and, more generally, the relative importance of indi-vidual populations in maintaining genetic connectivity, orinformation flow, among all populations.

The L. schottii Population Graph

A total of 948 individuals with 29 multilocus genotypeswere translated into a 21 × 21 population distance matrix,D (Table 1; below diagonal). The Population Graph repres-enting the minimal topology of 21 L. schottii populationshas 50 edges (significant edges shown in Table 1; above thediagonal) with two visually identifiable subgraphs (Fig. 2).This minimal edge set, Emin, depicted in Fig. 2, representsthe best fit model among several alternate topologies (modeldeviance: D = 162.4559, d.f. = 160, P = 0.43). Techniquesfor exploring alternate topologies are beyond the scope ofthis study, as it requires a full mathematical treatment (seeMagwene 2001 for a general overview). We return to thistopic in a later manuscript focusing on global humanpopulation genetic structure (Dyer and Nason, in prep).

The smaller of the subgraphs, hereafter G1 and depictedwith lighter coloured nodes in Fig. 2, contains populationsPL, LF, CP, Seri, SG, SI, SN and TS, all of which are Con-tinental Sonoran populations. The larger subgraph, G2,contains the remaining populations that are, with theexception of SenBas from southern Arizona, all Baja Penin-sular populations. A single connection, or bridge, con-necting SenBas and PL forms the only link between G1 andG2. The connectivity of SenBas within Fig. 2 suggests thateven though it is geographically proximal to Continental

PO

PU

LA

TIO

N G

RA

PH

S1719

© 2004 B

lackwell Publishing Ltd, M

olecular Ecology, 13, 1713–1727

Table 1 Population-wise multivariate statistical distances (dij; below diagonal) and standardized inverse correlations (rij; above diagonal) among peninsular Baja and mainland Sonorapopulations of Lophocereus schottii. Population names in italics denote Continental Populations. Standardized correlations significantly greater than zero are denoted in bold (abovediagonal) and index the minimal edge set in the Population Graph shown in Fig. 2. Significance was evaluated using edge exclusion deviance (see text)

BaC CP Ctv LaV LF Lig PL PtC PtP SenBas Seri SG SI SLG SN SnE SnF SnI StR TS TsS

BaC −0.04 0.05 0.15 –0.01 0.07 0.00 0.05 0.10 0.07 0.00 −0.02 0.02 −0.01 0.01 0.11 –0.01 0.07 0.23 0.06 −0.01CP 10.93 −0.03 0.04 0.07 0.02 −0.04 0.01 −0.03 0.01 0.41 0.21 –0.05 0.04 0.15 0.03 0.03 0.00 0.00 0.20 –0.01Ctv 6.84 10.43 −0.02 0.01 0.02 0.00 0.00 0.25 0.10 0.00 0.00 −0.02 0.64 0.01 0.00 0.20 0.05 −0.04 0.00 0.00LaV 9.04 13.23 10.76 0.01 0.08 –0.02 0.05 0.02 0.01 −0.02 0.04 0.00 −0.02 −0.01 0.23 0.09 –0.03 0.03 0.02 0.38LF 10.76 4.29 10.12 13.44 0.02 0.39 –0.01 −0.02 −0.01 0.04 0.28 0.17 0.01 0.06 −0.02 −0.01 0.01 −0.03 0.06 0.01Lig 9.75 12.35 9.82 12.18 12.40 0.01 0.09 0.03 0.05 0.02 0.02 0.01 0.01 0.03 0.06 0.04 0.10 0.19 0.03 0.05PL 10.77 5.24 10.23 13.67 2.46 12.60 0.00 0.00 0.07 0.00 0.15 0.34 0.03 0.06 0.02 −0.02 −0.02 0.00 0.05 0.04PtC 10.75 13.84 11.96 10.65 14.01 12.88 14.07 0.01 0.04 0.01 0.03 0.02 −0.01 0.03 0.09 0.04 −0.01 0.12 0.05 0.49PtP 6.46 10.44 2.65 10.33 10.21 9.62 10.25 11.64 0.11 –0.01 0.00 0.02 0.03 0.01 0.01 0.30 0.15 0.01 0.00 −0.01SenBas 7.67 9.05 5.73 11.41 8.95 10.40 8.87 12.07 5.82 0.06 0.00 −0.01 0.03 0.05 0.00 0.02 0.05 0.07 0.01 0.00Seri 10.57 2.67 10.23 13.37 3.97 12.13 4.71 13.71 10.16 8.58 0.19 0.05 −0.01 0.25 0.01 −0.02 0.03 0.01 0.06 −0.02SG 11.37 3.70 11.04 13.67 2.86 12.85 3.60 14.20 10.95 9.48 3.46 0.13 –0.04 0.05 −0.01 0.02 0.00 0.01 −0.01 0.00SI 10.70 5.29 10.37 13.74 3.19 12.65 2.95 14.12 10.24 9.21 4.67 3.91 0.01 0.07 0.00 0.02 0.00 0.02 0.22 –0.01SLG 7.22 10.32 1.39 10.94 10.09 10.02 10.17 12.12 3.26 6.13 10.24 11.07 10.32 −0.03 0.03 0.18 0.10 0.00 0.03 0.04SN 10.60 4.08 10.47 13.49 4.49 12.32 4.98 13.73 10.36 8.91 3.57 4.53 4.96 10.56 0.04 −0.01 −0.01 0.02 0.17 0.00SnE 7.76 11.75 8.74 8.44 12.02 10.81 12.05 10.47 8.55 9.68 11.61 12.39 12.18 8.85 11.66 0.03 0.05 0.19 –0.05 0.09SnF 7.27 10.69 2.70 10.45 10.63 9.97 10.77 11.99 2.99 6.54 10.62 11.34 10.77 3.01 10.90 8.83 0.10 0.03 −0.02 −0.05SnI 6.88 9.82 4.46 10.68 9.73 9.31 9.93 11.58 4.45 6.66 9.49 10.38 9.93 4.61 9.91 8.50 4.86 0.09 0.02 0.05StR 6.64 11.39 8.58 9.82 11.52 9.25 11.49 10.06 8.07 8.63 11.08 11.84 11.48 8.74 11.18 7.62 8.53 7.74 −0.03 0.08TS 10.22 4.36 9.91 13.31 4.44 12.20 4.81 13.63 9.91 8.91 4.43 4.88 4.34 9.84 4.60 12.05 10.43 9.49 11.37 −0.01TsS 9.83 13.42 10.86 7.48 13.24 12.08 13.25 6.91 10.70 11.42 13.38 13.67 13.56 10.88 13.39 9.11 11.17 10.40 9.31 13.29

1720 R . J . D Y E R


populations (see Fig. 1 in Nason et al. 2002), it shares a higherdegree of genetic covariance with Peninsular Baja popula-tions. In contrast, the original hierarchical analysis ofContinental vs. Peninsular populations in Nason et al. (2002)grouped SenBas with the remaining Continental popula-tions due to its geographical proximity. The authors didsingle out SenBas as unusual because it alone exhibited nopattern of isolation by distance with other populations (seeFig. 3B in Nason et al. 2002). Given palaeoecolgical evid-ence of the absence of L. schottii from northern Baja andSonora prior to 2000 years ago (Van Devender et al. 1994;Peñalba & Van Devender 1998), the Population Graph sug-gests a new hypothesis for the origin of SenBas, that it is theresult of a relatively recent long-distance dispersal eventout of Peninsular Baja. This hypothesis would explain theclustering of SenBas within the Peninsular subgraph, G2, aswell as the lack of isolation by distance between thispopulation and the remaining continental populations.

Vicariance and isolation by distance in Population Graphs

The presence of vicariance and isolation by distance aretwo hypotheses commonly addressed in population geneticanalyses. We begin our examination of the PopulationGraph in Fig. 2 by evaluating the significance of thesetwo hypotheses. As outlined above, vicariance within aPopulation Graph should be visually evident by relativelyfew edges between populations spanning the hypothe-sized source of vicariance. In Fig. 2, it is immediatelyobvious that subgraphs G1 and G2 may be separated by apotential source of vicariance, which in this case corre-sponds to the Sea of Cortéz. Nason et al. (2002) tested thisa priori hypothesis by examining the genetic differenti-ation between these two putative phylogroups and found asignificantly large amount of among phylogroup variation;FPT = 0.302; CI 95% = (0.177–0.435). Here we test the same

hypothesis within the Population Graph framework usingboth the binomial representation outlined above as well asa permutation-based approach.

From Fig. 2, we find that subgraph G1 has N1 = 8 nodesand K1 = 16 edges and subgraph G2 has N2 = 13 nodes andK2 = 33 edges (assuming for the time being that the SenBaspopulation is part of the Peninsular subgraph). From eqn 5above, the null hypothesis states that the probability ofobtaining an edge connecting G1 to G2 in the graph is:

Further, within the entire graph, containing K = 50 edges,the probability of observing a single edge (a bridge) con-necting G1 to G2 is (Kbtw = 1 from eqn 6 above):

a highly significant result. This significant result suggeststhat the Sea of Cortéz, as represented in the topology ofFig. 2, has indeed acted as a significant source of vicariancebetween the two subgraphs (or putative phylogroupsfollowing Nason et al. 2002). Because we are focusing hereon demonstrating the use of Population Graphs, we referthe interested reader to the original analysis of thesedata for discussion of the evolutionary significance ofthis vicariance.

In addition to the binomial test for vicariance, we alsoperformed a permutation test to highlight the dualityof these two analyses. Again, under the null hypothesis,the Sea of Cortéz has no influence on the topology ofrepresented in Fig. 2. As a result, we can simulate a largenumber of random graphs of the same size and order asthat show in Fig. 2. From these graphs, we can build the

Fig. 2 Population Graph representing thegenetic relationships among Peninsular (darknodes) and Continental (light nodes) popu-lations of Lophocereus schottii. The differencesin node size reflect differences in withinpopulation genetic variability, whereas theedge lengths represent the among populationcomponent of genetic variation due to theconnecting nodes. Both node sizes andedge lengths are projected within a three-dimensional drawing space.

p

N NK

.= =1 2 0 4952total

p x K K

ip p ei K

i

K

( ) ( ) . ≤ =

− ≈ −−

=∑btw

btw

1 7 17 141

0



null distribution of how many edges are predicted, underthe null, to connect Peninsular and Continental populations.We show in Fig. 3 the probability densities for both thebinomial and permutation tests for the graph parametersoutlined above. Both approaches yield similar results; namelythat the probability of observing a single bridge connectingthese two phylogroups is exceedingly unlikely if the Sea ofCortéz is not acting as a source of vicariance. The binomialand permutation approaches are not the only methods totest the significance of topological features; however, theydo are relatively intuitive and follow from the work onrandom graph theory. We return to discussing otheranalytical approaches in a subsequent manuscript.

The next hypothesis we test concerns the spatial arrange-ment of populations with respect to their genetic differ-ences under a model of isolation by distance (IBD). IBD isa non-random association of genetic similarity resultingfrom limited gene exchange among geographically separ-ated populations (e.g. Slatkin 1993). Several approacheshave been used to test for isolation by distance, rangingfrom simple regression of pairwise FST and genetic dis-tances on physical distance to more sophisticated evolu-tionary models (e.g. Slatkin 1993; Roussett 1997). PopulationGraphs can also be used to investigate isolation by distanceby examining the relationship between graph and geograph-ical distances.

By definition, the graph distance between nodes i and jis the shortest path through the graph connecting them, lij.The length of this minimum path can be measured in termsof the minimum number of edges separating nodes (inunweighted graphs), or the minimum path length (inweighted graphs). In a Population Graph, the graph dis-tance, lij, corresponds to the among population componentof genetic variance, (following Appendix I), for all pairs

of populations. As with other IBD methods, these pairwisegraph distances are regressed on the geographical separa-tion (using nonparametric methods due to the lack ofindependence). Under the model of IBD, we expect a pos-itive relationship between pairwise genetic and geograph-ical measures of differentiation. For the set of Peninsularpopulations, G2 in Fig. 2 excluding SenBas, we find a sig-nificant relationship between graph distance and physicaldistance (Fig. 4; P < 0.001, R2 = 0.48). Similar results werefound when testing for IBD among Continental Sonoranpopulations (P = 0.017; R2 = 0.15; figure not shown).Finally, we found no significant relationship betweenthe geographical and graph distances between G1 and G2or between SenBas (a southern Arizona population)and populations from either of these subgraphs. Theseresults concur with Nason et al. (2002), who found a signi-ficant relationship between a pairwise estimator of geneflow (M) and physical distance for the same subset ofpopulations.

Population assignment and graph connectivity

Based upon the topology of the Population Graph, we haveassumed that the SenBas population in southern Arizona,while geographically associated with the remainingCentennial populations, is genetically more similar to existingPeninsular populations. The connectivity of SenBas (Fig. 2)with a single Sonoran population (PL) and four northernand central Baja populations (Ctv, StR, PtP and BaC)supports the hypothesis of a relatively recent long-distancedispersal event out of Baja into southern Arizona. Thisinterpretation of the data provides an explanation for theapparent lack of isolation by distance between SenBasand all remaining Continental populations as shownabove and reported in Nason et al. (2002). The improperassignment of SenBas to the Continental phylogrouphighlights the differences between an a priori allocation ofpopulations within a hierarchical structure vs. allowing

Fig. 3 The probability of observing edges connecting Peninsular andContinental populations of Lophocereus schottii using the binomial(solid lines) and permutation (dashed lines) tests. The observednumber of interphylogroup edges (Kbtw = 1) is indicated by thefilled circle.

σA2

Fig. 4 Relationship between graph distance, l, and physicalseparation between populations (km) as a measure of isolation bydistance (P < 0.001, R2 = 0.48).

1722 R . J . D Y E R


the data to define their own set of relationships as advocatedhere.

We test the hypothesis that SenBas is genetically Contin-ental, despite its high degree of connectivity with severalPeninsular populations, using the binomial properties ofedge connectivity outlined above. The null hypothesis forthis test states that the topology of the Continental sub-graph, consisting of SenBas and the remaining Sonoranpopulations, has a pattern of edge assignment that sup-ports the inclusion of SenBas with the remaining Continen-tal populations. The node set for this subgraph, V3, containsnine populations (SenBas, PL, LF, CP, Seri, SG, SI, SN andTS), whereas the edge set, E3, has 17 edges (Fig. 2). We evalu-ate this subgraph, G3 = {V3, E3}, in terms of how likely itis to observe a graph with N3 = 9 nodes and K3 = 17 edgesthat contains a pendant (a node with a single edge asdepicted by SenBas).

The binomial nature of edge connectivity leads to anexpectation for the degree, or number of edges, for eachnode. The probability that any two nodes are connected,following eqn 3 above, is P = 0.472 for all graphs of the samesize as G3. The expected degree of any node is given byP(N3 − 1) = 3.78. Using the binomial distribution, we find theprobability of observing any graph of this size where at leastone of the nodes has degree one. This probability is given by:

(5)

Notice here, we used the combinatorial in our calcula-

tions of the probability. In this case, it is necessary becausethe hypothesis being tested was not a priori; rather, ittested for the probability of observing a single pendantwithin a graph defined by the parameters p and N3. Fromthese results, it appears that within the set of all graphs ofthis size, it is significantly rare to observe any of themcontaining a pendant. The high degree of connectivitybetween SenBas and Peninsular populations combinedwith the single connection to a Continental Population isconsistent with the hypothesized Baja origin of SenBasindividuals.

Finally, we focus on the overall topology of the Popula-tion Graph in Fig. 2 to address the topic of genetic connec-tivity among all populations. In so doing, we shift ourfocus from population genetic and phylogeographicalhypotheses and towards ones addressed typically in con-servation genetics. However, the issue of genetic connec-tivity is just as salient to the former fields of inquiry as thelatter. From a conservation genetic perspective, popula-tions targeted for preservation are often ones with highgenetic variability. Genetic variability is quantifiedtypic-ally with summary statistics including: the proportion ofpolymorphic loci (P), heterozygosity (H ), and the numberof alleles per locus (A). The size of the nodes in Fig. 2 sum-

marizes these statistics in multivariate space as it is definedas a measure of the within population genetic variance.As indicated in Fig. 2B, populations BaC and TS have thelargest within-population genetic variability for each phylo-group. These populations also have the largest P, H andA (Nason unpubl. data). Nason et al. (2002) showed asignificant reduction in P, H and A when regressed onlatitude (P < 0.001 in all cases) consistent with the hypo-thesis of a recent northward range expansion.

In addition to identifying the differences in within-population genetic variation or heteroscedasticity (a statis-tical test of which is to be addressed in a later manuscript),we can use the topology of the graph to identify populationsimportant to the flow of genetic material across the land-scape. For example, the within-population genetic vari-ability of PL and SenBas is not as large as other populationsin Fig. 2, yet they form the only connection betweenContinental and Peninsular phylogroups; that is, any pathoriginating in one phylogroup and ending in the othermust pass through these two populations. Nodes that havethis characteristic are called articulation points. In general,any subset of nodes whose removal significantly increasesthe length of the minimum spanning tree for the graphare of importance to the overall graph connectivity, andin Population Graph terms the overall flow of geneticmaterial across the graph.

Here we focus SenBas and PL because these two popu-lations form a bridge between the two inferred phylo-groups and are the graphs most apparent articulationpoints. This bridge may be present in the graph for one oftwo reasons. First, this bridge may simply represent themost intermediate point of connectivity between phylo-groups in terms of their genetic covariation. In this case, theremoval of SenBas or PL from subsequent graph construc-tion would reveal alternative connections between thetwo phylogroups formed through other population pairs.Conversely, SenBas and PL may represent the only possiblelink between the two phylogroups. In this case, excludingeither population would result in two completely discon-nected subgraphs. We can evaluate these alternate hypo-theses by removing each one of the two populations in turnand reconstructing the overall topology. To accomplishthis we must recalculate the incidence matrix following theremoval of either population because the resulting topologyis based upon the pattern of genetic covariation among allpopulations. The Population Graphs resulting from theexclusion of either SenBas or PL reveal two distinct, dis-connected subgraphs (not shown). The genetic covariancestructure of the remaining populations does not supportthe hypothesis of robust genetic connectivity between Con-tinental and Peninsular populations without either of thearticulation populations. These results suggest that bothSenBas and PL represent the sole source of genetic connect-ivity between Peninsular and Continental populations.

p pendant N p N p p N( , ) ( ) .| 3

3 1 1

11 0 01083=

− =−

N3

1



Discussion

The evolution of population genetic structure is a dynamicprocess influenced by both historical and recurrentevolutionary processes. Vicariance and gene flow, in par-ticular, create a system of interacting populations whosegenetic relationships can be readily investigated within agraph theoretic framework. The methods presented in thisstudy provide an introductory treatment of the use ofgraphical techniques for addressing hypotheses regardingthe genetic connectivity of a set of populations within aphylogeographical context. With the example of L. schottii,we have shown how the information contained within thetopology of a Population Graph can be utilized to addressimportant population genetic questions concerning thenature and significance of genetic separation and isolationby distance. However, if traditional population geneticapproaches provide mechanisms to address the same typesof questions it becomes relevant to ask why one wouldspecifically use Population Graphs.

There is a fundamental distinction between the modelsapplied to the decomposition of genetic variance usingtraditional structure statistics (e.g. Wright’s F-statistics,amova, etc.) and the methods outlined above for Popula-tion Graphs. This distinction lies in the fact that when weapply structure statistics to a set of data we impose pre-defined hierarchical models that we believe reflect the spatialand temporal scales evolutionary processes have operatedon to create the observed distribution of genetic structure.The estimation of a significant summary statistic under apredefined model does not mean that the allocation of popu-lations to specific strata is correct, nor does it signify thatthe hierarchical separation of strata conform to the spatialor temporal scales at which the underlying populationgenetic processes operate. Rather, it simply implies that thecurrent arrangement of strata is sufficient to assume non-random association of genotypes. Short of permuting allstrata to find the ‘best’ fit of the model to the data, there areno methods available to evaluate the assumed hierarchicalpopulation model. With the L. schottii data set, there areover 2.1 × 106 different ways to allocate 21 populationsinto two phylogroups. Clearly, evaluating the fit of suchpopulation models is not feasible given the amount oftime necessary to enumerate all possible arrangementsand the effect it would have on the experiment-wise errorrates.

By allowing the data to define their own topology, basedupon the genetic composition of the entire data set, Popu-lation Graphs are not subject to the problem of incorrectlyassigning populations to higher-level strata. We speci-fically used the example of L. schottii with respect to theSenBas population to highlight the potential pitfalls ofspecifying a particular a priori model. Nason et al. (2002)had no prior information suggesting that one of their sam-

ple populations (SenBas) was the result of a long-distancedispersal event. Only by examining how the all the popu-lations organize themselves within the topology of thePopulation Graph did the Peninsular origin of thisContinental population become apparent. The ability toresolve intrapopulation relationships, within the contextof the genetic covariance structure of all populations, isnot a strength of traditional population genetic methods.Methods such as pairwise analyses allow the elucidationof pairs of relationships, however, we have no a priorireason to assume that evolutionary processes operate in apairwise fashion. Methods such as Population Graphspresented here, which focus on the details of population-level relationships within the context of the covariancestructures from all populations, are likely to be particularlyuseful for population genetic structures more complex thanthat of L. schottii.

Population Graphs also provide a single heuristic approachto addressing a variety of general population genetic questions.As shown above with L. schottii, questions concerninggenetic differentiation, isolation by distance, population assign-ment and genetic connectivity are all addressed under asingle analytical framework. In contrast, traditional popu-lation genetic analyses require different analytical proceduresto address each of these questions, some of which, such aspopulation assignment and genetic connectivity, are stillbeing developed (e.g. Cornuet et al. 1999).

In a more general context, by focusing on an averageeffect, significant summary statistics indicate only broadtrends across an entire data set. The complexity of interactingevolutionary forces shaping intraspecific genetic variationcan easily be generalized by summary statistics. Indeed,most of our understanding of how evolutionary processesoperate is based on just such statistics. However, if significantdetails of the evolutionary process lie in the relationshipsamong sets of individual populations rather than theiraverage effects then we must adopt approaches that do notdistill the complexity of these relationships into a single orsmall set of values. Rather, we must adopt methodologiesthat capture this complexity without undue averagingacross strata. A Population Graph is one such method.While it is possible to extract analogues of the traditionalsummary statistics from this approach, the main focusPopulation Graphs is directed towards a holistic quanti-fication of population interactions across an entire data set.

It must be pointed out that the Population Graph frame-work does not increase the precision of a summarystatistics. In fact, in some cases it appears to decrease theprecision resulting in a larger variance around the teststatistic (Dyer unpubl. data). The degree to which the vari-ance surrounding the test statistic increases depends uponthe order and size of the graph, as well as how among-population components of genetic variation (e.g. the edgelengths) are distributed across the topology. We address

1724 R . J . D Y E R


this issue in greater depth as well as provide an indexrelating to how minimizing the topology influences the vari-ance around summary statistics in a subsequent manuscript.

Finally, we believe that the intrinsic visual nature ofPopulation Graphs facilitates the interpretation of popula-tion genetic data. Several aspects of population geneticstructure are immediately apparent in the L. schottii graph(Fig. 2) including the unique placement of SenBas, theseparation of Continental Sonoran and Peninsular popula-tions, and the difference in within population geneticvariability. While we have chosen to illustrate PopulationGraphs using the L. schottii data set because of its relativesimplicity and obvious patterns, in other species the pat-terns in the data will often be more complex. In these casesespecially, being able to visually inspect the relationshipsamong all populations may prove to be immensely import-ant for exploratory data analysis, data interpretation andthe development of subsequent hypotheses.

Conclusions

The primary focus of Population Graphs is concerned withexamining specific topological characteristics of the graphin terms of the relationships among individual populations.The focus on topological characteristics is essential as thestructure of the graph represents a model of how evolu-tionary processes have acted on interacting populations.Albert & Barabási (2002) argue that when the time scalesgoverning the dynamics on any type of graph are compar-able to those characterizing the graph assembly (e.g. theprocess governing the connectivity of nodes), the dynamicalprocesses influence the resulting topology. In the case ofPopulation Graphs, it is the interaction of historical andrecurrent evolutionary processes that structure geneticvariation within and among populations and as a resulteach of these processes are expected to leave specifictopological features within the overall graph.

We are currently examining several aspects of the evolu-tion of topological features within Population Graphs.What we have presented here is a simply an introductionto the utility of graphical methods for describing popula-tion genetic structure. Currently we are actively expandingupon the framework presented herein. Some of the mostimportant directions include: (i) quantifying the direction-ality of genetic covariation as would arise from periodsof range expansion, anisotropic gene flow or source–sinkdynamics; (ii) a more complete analysis of the relativeimportance of particular populations (articulation points)or edges (bridges) to the graphs overall connectivity; (iii)topological clustering algorithms for objectively partition-ing a graph into completely induced subgraphs; and (iv)the set of topological features characteristic of particularmodels of evolution (e.g. stepping-stone, N-island). Bycharacterizing the topological features that are expected to

emerge due to identifiable evolutionary processes, we arein a better position to interpret correctly the topologies weobserve in natural populations.

AcknowledgementsThe authors thank L. Excoffier, J. Fernandez, P. Smouse, V. Sork,R. Westfall and two anonymous reviewers for insightful com-ments on this manuscript. The authors would also like to thank theOffice of Biotechnology and the Plant Sciences Institute at IowaState University for their support of this research.

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Rodney Dyer is currently a postdoctoral researcher working withJohn Nason at Iowa State University. Rodney’s interests are in theore-tical population genetics with particular emphasis in issues relatingto quantifying the complexity of population genetic processes.John Nason researches the population genetics of plants and theirassociated insects with a special interest in obligate pollinationmutualisms and host-race formation in phytophagous insects.This study reflects our shared interest in developing analyticaltools for the analysis of population genetic data.

1726 R . J . D Y E R


Appendix I

Corollary between Population Graphs and the AMOVA analysis

In this Appendix, we show how the a saturated graph isidentical to the genetic variance decomposition under theamova framework. Our overall goal is to show how thegenetic variance can be partitioned into population wisecomponents in a manner similar to Long et al. (1987). Theamova analysis is a random effects, multivariate analogueof Weir and Cockerham’s θ (Weir & Cockerham 1984) thatrelies upon an elegant use of squared genetic distances, ,measured in a pairwise fashion between all individuals.For a single strata analysis, the sums of squared distances(SSD) allow the partitioning of the total genetic variance,

, into within- (or error) and among-population componentsof genetic variation, and , respectively. The ratioof to supplies the statistic of differentiation, ΦST.

The error variance, , in the amova model is the sum,across populations, of the average genetic distance amongall individuals within populations. From Excoffier et al.(1992), with some simplification due to considering anon-nested model, it is given by (notation changed fromExcoffier et al. 1992 for consistency):

(A1)

where N is the number of populations (or nodes), ni isthe number of individuals within the ith population, and

is the pairwise squared genetic distance between thej th and kth individuals in population i. The constructionof Population Graphs partitions the within populationsums of squares as:

(A2)

where |N| is the number of nodes (populations) in thegraph, pij is the jth individual within the ith population and

is the centroid of the ith population in m-dimensionalspace. Therefore, the average distance between an indi-vidual and its population centroid is defined as the errorvariance attributable to that particular population. We definethe volume of a node within the graph as vi = SSWi. The sumof all node volumes would equal the total sums of squareswithin populations.

The decomposition of the among-population componentof variance, MSA in the amova framework, is given by:

(A3)

where the notation is as above with the addition that Ntotalis the total number of individuals and SSAivs.j representsthe contribution to the overall among population sums ofsquares due to the differences between populations i and j.The quantity SSAivs.j is divided by two because the among-population sums of squares due to individual pairs ofpopulations is calculated twice, once in terms of the distancesbetween populations i and j and again for the calculationof the distance between j and i. Within the distance-basedapproach of the amova framework, SSAivs.j is the distancebetween the centroids of the ith and jth population, exactlyhow we have defined the off-diagonal components ofthe incidence matrix within Population Graphs. Therefore, theamong-population component of genetic variance in theamova model is equal to the sum of the edge lengths inPopulation Graphs.

From a graph theoretic perspective we can define agraph, G, consisting of the node set V = { A, B, C } and theedge set E = {e1, e2, e3} (Fig. A1A). Again, the sum of thevolumes of all nodes is defined as , and the sums ofthe edges are similarly defined as . Therefore, an overallΦST for all three populations is estimated, by the definitionsof how we construct Population Graphs, as:

(A5)

δij

2

σT2

σW2

σA2

σA2

σT2

σW2

σ

δ

W2

2

11

1

1 2

1

11 2

11

( . . . )

= =−

=−

=−

+ + +

==

=

∑∑∑

MSWSSWN

N n

NSSW SSW SSW

jkk

n

j

n

ii

N

K

ii

δij2

σW2

11

1 2

1

11

11

[ ] [ ]

( . . . )

= =−

=−

− −

=−

+ + +

==∑∑

MSWSSWN

N

NSSW SSW SSW

ij iT

ij ij

n

i

N

K

i

p pπ π

MSASSA

N N

N N n n

N NSSA SSA SSA

kll l

n

k

n

i jj l

N

i

N

vs vs N vs N

ii

. . . . . .

=−

=− +

=−

+ + +

==

≠=

−

∑∑∑∑

total

total

total

1

12 2 2

2

1

1

1 2 1 3 1

δ

σW2

σA2

ΦST

A

A W

=

+

=+

=

= =

∑

∑ ∑

e

e v

ii

E

ii

E

ii

V1

1 1

2

2 2

| |

| | | |

σσ σ



where |E| is the number of edges, |V| is the number of nodes,vi is the volume of the ith node and the and termsfollow the definition of random effects estimates of differenti-ation of Weir & Cockerham (1984) and Excoffier et al. (1992).

There are two salient points to make with respect to thegeometry of averaging statistics such as ΦST. First, thecentroid of any population is, by definition, determinedentirely by the genetic identity of the individuals withinthe population. This means that the coordinates of thepopulation centroid are independent of how different aparticular population is from others. The distances fromthe centroids to the individuals within a population define

. So, is independent of where populations arelocated in m-space. The only thing that affects an averagestatistic of genetic differentiation such as ΦST, or its univari-ate analogs such as FST or ΦST, is .

Second, , defined as the sum of the edge lengths, hasno unique solution. The Population Graph in Fig. A1A isportrayed as an equilateral triangle, that is e1 = e2 = e3, and

is defined as the sum of these edge lengths. However,there is an infinite number of other ways one could draw atriangle with the restriction that the sum of the lengths ofall edges equals that shown in Fig. A1A. Figure A1B andA1C also show the arrangement of three populationswithin three unique topologies. However, if the sums ofthe edges all equal the same , then all three will estimatethe exact same ΦST value. In general, for any set of popu-lations, the nature of the pairwise genetic relationshipsamong populations will not be revealed in statistics ofaverage differentiation. However, heterogeneity in thenature of these relationships will be represented explicitlyin the topology of a Population Graph.

Fig. A1 Graphical representation of population genetic differentiation among three populations (A, B and C). Differentiation amongpopulations quantified by edges e1, e2 and e3. All graphs have the same ΦST. (A) All populations equally differentiated from each other.(B) Nonsymmetrical differentiation among three populations under the constraint that the sum of edge lengths is equal to the sum of theedge lengths in (A). (C) Permutation of populations within a graph where the sum of the edge lengths equals the previous two examples.

σA2

σW2

σW2

σW2

σA2

σA2

σA2

σA2

population graphs: the graph theoretic shape of genetic structure

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