The effect of dominance on polymorphism in Müllerian mimic species

V. Llaurens a,n, S. Billiard b, M. Joron a

a Origine Structure et Evolution de la Biodiversité, CNRS UMR 7205, Muséum National d'Histoire Naturelle, CP50, 45, rue Buffon, 75005 Paris, Franceb Laboratoire de Génétique et évolution des populations végétales, CNRS UMR 8198, Université des Sciences et Technologies de Lille 1, Bâtiment SN2, 59655Villeneuve d'Ascq Cedex, France

H I G H L I G H T S

� We investigate the influence of dominance on the maintenance of polymorphism in a Müllerian mimicry system in a spatially heterogeneousenvironment.

� Complete dominance was shown to extend the parameter space, and thus the breadth of ecological situations, where Müllerian mimicrypolymorphism was maintained.

� Overdominance, which promotes polymorphism, can arise given certain levels of toxicity and predator discrimination accuracy.� Dominant alleles were shown to reach lower frequencies than recessive alleles when selection on both homozygotes was symmetrical.

a r t i c l e i n f o

Article history:Received 17 May 2013Received in revised form7 August 2013Accepted 9 August 2013Available online 21 August 2013

Keywords:MimicryBalancing selectionSpatial heterogeneityAposematism

a b s t r a c t

Dominance controls the phenotype of heterozygous individuals, and plays an important role in themaintenance of polymorphism. Here we focus on the dominance acting on warning-pattern polymorph-ism in species engaged in Müllerian mimicry. Müllerian mimics are toxic species which display brightcolour patterns used as a warning signal to predators and are subject to local positive density-dependentselection. Some Müllerian mimics are polymorphic due to a selection/migration balance in spatiallyheterogeneous communities of prey. Since heterozygotes at a locus controlling warning pattern mightexhibit intermediate, non-mimetic heterozygous morphs, dominance is likely to influence the poly-morphism at this locus. Using a deterministic model describing migration, density-dependent predationand reproduction, we investigated the influence of dominance on the dynamics of alleles at locusdetermining mimetic phenotype. Our results suggest dominance may interact with migration andselection and plays an important role in shaping the conditions of polymorphism persistence and thefrequency of alleles at this locus. Our results thus highlight the important role of dominance in thedynamics of polymorphism at loci under balancing selection due to environmental heterogeneity.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Dominance among alleles at the same genetic locus is a wide-spread phenomenon, described by Mendel in his famous study ofpea crosses in the 19th century (Mendel, 1895). By modifying thephenotype of heterozygotes, dominance can have an importantimpact on the fitness of alleles. For instance, alleles with deleter-ious properties are generally found recessive or partially recessiveto wild-type alleles (Orr, 1991). Dominance has been shown toinfluence the invasion of a new mutation arising in a population,more dominant mutations tending to get fixed more oftenthan more recessive ones (i.e. Haldane ‘s sieve (Haldane, 1927)).

Dominance also plays a significant role in the shape of allelefrequency clines maintained by frequency-dependent selection:Mallet and Barton (Mallet and Barton, 1989) demonstrated thatallelic dominance may result in the formation of asymmetricalclines and thus favour cline movement. Yeaman and Otto (Yeamanand Otto, 2011) also confirmed the impact of dominance in theinvasion of new alleles in a classical two-population migration/selection model.

Since dominance only plays a role in the expression of thephenotype in heterozygous individuals, it is expected to play animportant role in the evolution of polymorphic loci where hetero-zygotes are at high frequency. However, the influence of dominanceon the persistence of polymorphism and on allele-frequencydistribution has received little attention. Balanced polymorphismis maintained in well-documented regimes of selection such asheterozygote advantage, or negative frequency-dependent selection

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n Corresponding author. Tel.: þ33 01 40 79 38 61; fax: þ33 01 40 79 33 42.E-mail addresses: llaurens@mnhn.fr, llaurens.violaine@laposte.net (V. Llaurens).

Journal of Theoretical Biology 337 (2013) 101–110

(Charlesworth, 2006). However, the persistence of polymorphismdue to spatial heterogeneity might also represent a widespreadsituation where balanced polymorphism is maintained (Spichtigand Kawecki, 2004).

In previous theoretical studies on the effect of dominance onpolymorphisms maintained by a migration/selection balance,dominance was modelled using a coefficient which directlymodulates the fitness of heterozygotes (Yeaman and Otto, 2011;Otto and Bourguet, 1999; Orr and Betancourt, 2001), and, in turn,differences in invasion capacities for alleles with a positive vs. anegative effect on the fitness of heterozygotes (i.e. overdominancevs. underdominance). With this specific mathematical formalisa-tion of dominance, the biological mechanism of dominance actingon the fitness of heterozygotes is not straightforward and thismakes it difficult to confront theoretical predictions with empiricaldata. More importantly, predictions obtained by these modelscannot be readily applied to cases (such as Müllerian mimicry)where fitness is frequency and density-dependent, i.e. when thefitness of heterozygotes depends both on the phenotype expressedby heterozygotes and on the overall distribution of the phenotypesin the population (and in the larger community). To address thislimitation, we here consider a dominance coefficient acting on thephenotype and not directly on fitness.

In this study, we focus on the role of dominance in shapingphenotypic variation and the maintenance of polymorphism inprey species engaged in Müllerian mimicry. Müllerian mimicry isthe adaptive resemblance of multiple noxious prey species whoseshared phenotype functions as a common warning signal topredators (Ruxton et al., 2004). Müllerian mimicry is a widespreadphenomenon documented in many organisms from arthropods(Millipedes (Marek and Bond, 2009), Hemiptera, (Zrzavy andNedved, 1999)) to vertebrates (amphibians (Symula et al., 2001),snakes(Sanders et al., 2006), birds (Dumbacher and Fleischer,2001)…). Predators learn to avoid chemically defended, warninglycoloured prey based on previous experience (Chouteau andAngers, 2011; Mappes et al., 2005) thus favouring the survival ofprey bearing the commonest (most frequent) patterns in a localcommunity (Pinheiro, 2003), a well-known example of positivefrequency-dependent selection (Speed and Turner, 1999).Frequency-dependence operates locally and favours resemblanceamong co-occurring species, favouring the locally commonestphenotypes and selecting away rarer variants. However, selectionon warning patterns may act in opposing directions in distinctlocalities, and mimetic communities indeed vary widely in theirwarning patterns across geographic areas or across habitats.

The evolution and maintenance of warning-pattern poly-morphism would not be predicted in the situation where a single,best-protected phenotype is expected to reach fixation, but poly-morphisms may be maintained in the case of spatially variableselection, e.g. when the composition of the community of toxicprey changes spatially, as found for instance in the poison frogRanitomeya imitator (Chouteau et al., 2011) or in the butterflyHeliconius numata (Joron et al., 1999). In Heliconius numata, multi-ple wing-pattern forms co-occur, each one being a precise mimicof a distinct species in the distantly related genus Melinaea. Fine-scale spatial variations in the relative abundances of the Melinaeaspecies are positively correlated with the frequency of the match-ing wing colour pattern in Heliconius numata (Joron et al., 1999).The fine spatial heterogeneity in the mimetic community compo-sition is thought to translate into variations in the direction ofselection for warning pattern resemblance in distinct populations.This variable selection generates balancing selection on wingcolour patterns at a larger spatial scale.

In the fitness landscape of mimicry, shaped by frequency-dependent selection, peaks correspond to common, mimetic phe-notypes, and fitness valleys correspond to all rare, non-mimetic,

and intermediate phenotypes which are strongly selected against.The persistence of polymorphisms will be influenced by thestrength of selection acting on heterozygotes. By controlling themimicry of heterozygous genotypes, dominance relationshipsamong alleles at a locus controlling mimetic phenotypic elementsare of particular relevance. For instance complete dominance ofmutant alleles would allow full resemblance of heterozygotegenotypes with this mutant allele to one of the mimetic homo-zygotes. Moreover, under strong balancing selection, heterozygotefrequency is expected to be high, and dominance may play animportant role in the selection on alleles, and on polymorphismmaintenance.

The evolutionary dynamics and genetics of mimicry have beenwell studied in butterflies. The genetic loci determining mimicryvariation in the genera Papilio and Heliconius have receivedsustained attention, and both dominance and co-dominancerelationships are observed among alleles coding for alternativemimetic phenotypes (see (Clarke and Sheppard, 1960) and (Clarkeet al., 1985)). In Heliconius, alleles at homologous loci can havedifferently ordered dominance relationships in distinct species(e.g. Heliconius melpomene vs. Heliconius erato (Nijhout, 1991);Heliconius cydno vs. Heliconius melpomene (Naisbit et al., 2003)).Dominance can also vary within species: in Papilio dardanus,dominance tends to be stronger in crosses between individualsdrawn from the same geographical area than in crosses involvingindividuals from different areas (Nijhout, 2003). Such variation indominance level in natural populations underlines the lack ofunderstanding of the role of dominance in the evolution andmaintenance polymorphisms of these adaptive patterns.

To fill this gap, we built a model based on a previouslydescribed theoretical haploid model (Joron and Iwasa, 2005)proposed to depict the evolution of warning-signal mimicry inunpalatable prey. We extended this model to a diploid species, inorder to investigate the influence of dominance on warning-colourpolymorphism.

2. Material and methods

The model presented here is based on the previous model ofJoron and Iwasa (Joron and Iwasa, 2005) which was built todescribe Müllerian mimicry in a spatially distributed community.We extended this model to describe a diploid species withdominance relationships among alleles at the locus controllingmimetic colour pattern.

We considered a system with two patches, 1 and 2, each onecontaining a single mimicry community (i.e. a number of speciesall mimicking each other and bearing a similar wing patternsignalling their toxicity to predators). The two patches differedin the mimetic pattern adopted by local species: the mimicrycommunity exhibited morph A in patch 1 and morph B in patch 2.The two communities constituted the mimetic environment andwere assumed to have a fixed abundance and a fixed warningcolour pattern (A or B).

We studied the polymorphism dynamics of a focal speciesevolving in this spatially distributed system. The focal species wasa Müllerian mimic, mimicking morph A or B, bearing its owntoxicity, and forming two distinct populations exchangingmigrants between patches 1 and 2. This mimetic species wasdiploid and the mimetic morph A or B was assumed to bedetermined by a single locus with two segregating alleles a and b.

2.1. Modelling dominance

Individuals of genotypes aa and bb displayed phenotype A andB respectively. The phenotype of the heterozygote ab depended on

V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110102

the dominance coefficient h. The similarity between the homo-zygotes aa and bb and heterozygote ab morphs was described by ageneralization function with a Gaussian shape, g(h), followingequation:

gðhÞ ¼ e�ðh�1Þ2=2γ2

with γ determining the width of the Gaussian function: ð1ÞThis shape corresponded to the function generally assumed for

predator generalization, i.e. the range for which patterns are toosimilar to be discriminated by predators (Ruxton et al., 2008). Thefunctions g(h) and g(1�h) represented the resemblance of theheterozygote ab to the homozygote aa and to the homozygote bbrespectively. When h tended to 1, the heterozygote ab displayedexactly the same phenotype as the homozygote aa. Fig. 1 showsthe effect of γ on the shape of the function with decreasing γleading to steeper shapes, in which co-dominance (h¼0.5) leadedto high dissimilarity of the heterozygote ab with respect to bothhomozygotes aa and bb.

2.2. Modelling allele dynamics

We assumed a continuous time deterministic model where allevents occurred simultaneously. Three types of events occurred inthe populations 1 and 2: migration, viability selection due to deathby predation, and density-dependent reproduction. The produc-tion of genotypes was assumed to follow Mendelian segregationand was modelled explicitly. The change in the density of eachgenotype with time during each event was detailed below.

2.2.1. MigrationThe change by time unit of Nuvi (density of genotype uv, i.e. aa,

bb or ab, in population i) due to migration between populations iand j was given by

dNuvidt

¼mðNuvj�NuviÞ with ia j and m as the migration rate:

ð2Þ

2.2.2. Survival within populationWe assumed daai and dbbi to be the death coefficient of the

homozygotes aa and bb in population i. These death coefficientsdepended on the presence of the model species in the populationconsidered. Hence, daa1rdbb1 and daa2Zdbb2 because the geno-type aa (phenotype A) was favoured in population 1 and not inpopulation 2 due to mimicry with model species at differentabundances.

The survival of the individuals in a population not onlydepended on the community environment represented by modelspecies but also on its own abundance in the community. Predatoravoidance was thus assumed to depend both on the density ofeach morph and on the unpalatibility of the focal species, l. Theshape of this function was similar to the function used in haploidmodel of Müllerian mimicry used by Joron and Iwasa (Joron andIwasa, 2005).

Homozygotes and heterozygotes could display similar colourpatterns, and therefore benefited from their mutual abundance, toan extent which depends on the visual ability of the predator, i.e.depending on the generalization function, g(h), described above.

The change in the density of homozygote genotypes aa and bbin population i due to predation, was given by

dNaai

dt¼� daai

1þ lðNaaiþgðhÞNabiÞNaai; ð3Þ

dNbbi

dt¼� dbbi

1þ lðNbbiþgð1�hÞNabiÞNbbi; ð4Þ

The change of the density of heterozygote genotype ab in popula-tions 1 and 2 after predation was given by

dNab1

dt¼� gðhÞdaa1þð1�gðhÞdbb1Þ

1þ lðNab1þgðhÞNaa1þgð1�hÞNbb1ÞNab1; ð5Þ

dNab2

dt¼� gð1�hÞdaa2þð1�gð1�hÞdbb2Þ

1þ lðNab2þgðhÞNaa2þgð1�hÞNbb2ÞNab2: ð6Þ

For the sake of simplicity, we assumed a symmetrical conditionwhere daa1¼dbb2¼d (1�s) and dbb1¼daa2 ¼d (1þs) where drepresented the mean predation risk and s the spatial hetero-geneity due to the distribution of the two model species.

2.2.3. ReproductionThe parameter r was set as the intrinsic per female capita

growth rate, K the carrying capacity, assumed equal in bothpopulations, and Ni ¼NaaiþNabiþNbbi the total density of indivi-duals in population i. We also assume that the sex-ratio wasbalanced and that only females gave birth to new individuals. Weassumed that reproduction was density-dependent and that theper capita growth rate in population i was for all genotype:ðr=2Þð1�ðNi=KÞÞ.

Since we aimed to explicitly model sexual reproduction, therate at which each genotype was produced depended on the rateat which each type of cross occurs, i.e. on the density of eachgenotype. We assumed Mendelian segregation at the mimicrylocus. The production rate of individuals with genotype uv inpopulation i was then described as follows:

r2

1�Ni

K

� �f uvi; ð7Þ

with

f aai ¼ðNabiþ2NaaiÞ2

4Ni; ð8aÞ

f abi ¼2ðNabiþ2NaaiÞðNabiþ2NbbiÞ

4Ni; ð8bÞ

f bbi ¼ðNabiþ2NbbiÞ2

4Ni: ð8cÞ

Although colour patterns are likely to play a role in sexualselection, its influence on mating success was not included in ourmodel for simplicity.

Fig. 1. Shape of the generalization function according for different values of γ.X-axis exhibits the dominance coefficient h and y-axis the generalization functiong. Dashed lines represent g(h) and plain lines g(1-h), black lines : γ¼0.3, dark grey :γ¼0.1, grey : γ¼0.05, light grey : γ¼0.01.

V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110 103

2.2.4. General equations for the population dynamicsCombining the density change of each genotype with time due

to the combination of migration, viability selection by predationand reproduction gave the total density change in a continuoustime model. The dynamics were described by six ordinary differ-ential equations, three for each population. The equations were

for ia j:

dNaai

dt¼mðNaaj�NaaiÞ�

daai1þ lðNaaiþgðhÞNabiÞ

Naai

þ r2

1�Ni

K

� �ðNabiþ2NaaiÞ24Ni

; ð9aÞ

dNbbi

dt¼m Nbbj�Nbbi

� �� dbbi1þ l Nbbiþg 1�hð ÞNabið ÞNbbi

þ r2

1�Ni

K

� �Nabiþ2Nbbið Þ2

4Ni; ð9bÞ

and

dNab1

dt¼m Nab2�Nab1ð Þ� g hð Þdaa1þð1�g hð Þdbb1Þ

1þ l Nab1þg hð ÞNaa1þg 1�hð ÞNbb1ð ÞNab1

þ r2

1�N1

K

� �2ðNab1þ2Naa1ÞðNab1þ2Nbb1Þ

4N1; ð9cÞ

dNab2

dt¼mðNab1�Nab2Þ�

gð1�hÞdaa2þð1�gð1�hÞdbb2Þ1þ lðNab2þgðhÞNaa2þgð1�hÞNbb2Þ

Nab2

þ r2

1�N2

K

� �2ðNab2þ2Naa2ÞðNab2þ2Nbb2Þ

4N2: ð9dÞ

2.3. Analysis of the model

2.3.1. Stability analysesWe first performed a stability analysis of these equations (Otto

and Day, 2007). For a given equilibrium, eigenvalues were com-puted, the sign of which determined the conditions of its stability.Combining the analysis of several equilibria and their stabilitiesrevealed the conditions for the persistence of polymorphism. Wewere able to obtain explicit expressions of the equilibria and theirstability for two simple cases regarding migration: (1) when thereis no migration, and (2) when migration rate is very high. In case(2), we assumed that the two populations behave as a single largepopulation, where Naa1¼Naa2¼Naa, Nab1¼Nab2¼Nab and Nbb1¼Nbb2¼Nbb. For all other value of the migration, we investigated thepolymorphism maintenance by a numerical analysis detailedhereafter.

2.3.2. Numerical analyses of equilibria for intermediatemigration rates

To find the equilibria for any value of the migration rate m, weperformed a numerical analysis using the FindRoot function inMathematica. The first investigated equilibrium was for m¼0.01,specifying in the FindRoot function that the starting point wasnear the equilibrium values for m¼0, as it had been computed inthe previous section. We computed the second equilibrium form¼0.02 using the equilibrium found for m¼0.01 as the startingvalue, and so on until m¼0.4 with an increment of 0.01 betweeneach computation.

We found that the maintenance of polymorphism can dependon the initial state of the population. To approximately determinethe size of the attraction basin, we performed additional numericalanalyses to investigate the conditions for the maintenance ofpolymorphism for 1000 different initial conditions where thenumber of individuals of a given genotype was randomly drawnin a uniform distribution between 0 and 100. We iteratedequations Eqs. (9a),(9b),(9c) and (9d) until the frequency change

between two iterations was below 10�3 for both alleles. Threepossible outcomes were possible: maintenance of polymorphism(both alleles a and b were present in the population at the end ofthe computation); allele a is lost; or allele b is lost.

In the same way, we also computed the equilibrium for varyingvalues of the dominance coefficient h for a given value of themigration rate m using the FindRoot function.

Finally, we also investigated the effect of the toxicity, l, and theheterogeneity of the environment, s, on the maintenance ofpolymorphism for different dominance levels, h. For this, weperformed a numerical analysis, starting with the initial conditionsgiven by the equations from the equilibrium when m¼0, weintroduced a small quantity of heterozygotes to each patch so thatthe initial frequency of the heterozygotes was 0.001. We per-formed 1000 iterative computations of Eqs. (9a)–(9d) and checkedif the densities of all genotypes were higher than 1 in bothpopulations, which corresponds to a frequency higher than 10�3

in a 1000 individuals population. If so, we considered that thepolymorphism was maintained.

2.3.3. Parameter valuesUnless otherwise stated, we used the following values for the

parameters. The generalization function was chosen to be steep(γ¼0.01, see Fig. 1) in the numerical analyses to simulate clearphenotypic distinction of the co-dominant heterozygote ab fromeither homozygote (aa and bb). We assumed a low value, becausethe few studies testing the ability of predators to discriminate thedifferent morphs of Müllerian mimic species show that they wereable to discriminate ‘local' versus ‘exotic' morphs quite accurately(Chouteau and Angers, 2011; Merrill et al., 2012).

Computations were performed assuming a strict symmetrybetween the population 1 and 2, using an intermediate generalpredation risk d¼0.5 and a high spatial heterogeneity parameters¼0.9, thus simulating a strong predation difference betweenpopulations. This assumption of a greater advantage provided bythe model species with respect to the within-species densitydependence is biologically relevant: several species can beinvolved in a mimicry ring leading the total of individuals togenerally outnumber individuals from the focal species only.

We also assumed toxicity l¼0.0025, growth rate r¼1 andcarrying capacity K¼1000. Note that since model species did notevolve, we used the Kl product as an estimate of the relativecontribution of the mimetic species to the community toxicity.

3. Results

3.1. Polymorphism without migration

First we analysed the case of two isolated populations exchan-ging no migrants (m¼0). Four possible equilibria were detected(Table 1). A first equilibrium corresponded to the extinction ofboth populations and the second equilibrium to two mono-morphic populations, each one being fixed for the allele matchingthe local community. The other two equilibria described a mono-morphic mimic species with the same allele invading bothpopulations. However the latter two equilibria were alwaysunstable when predation risk and spatial heterogeneity werepositive (d40 and s40), meaning that these equilibria cannotoccur with predation. In the absence of migration, polymorphismcannot be maintained within each population here, regardless ofthe level of dominance (h).

V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110104

3.2. Polymorphism with infinite migration

Secondly, we explored analytically the case where migrationwas high. There, the two patches were still occupied by differentmimicry communities, but the focal species showed unlimitedmigration between the two populations, approaching the beha-viour of a single population. Three equilibria were detected: thefirst one was the fixation of allele a, the second one the fixation ofallele b, and the third one the extinction of both populations(supplementary Table 1). For each equilibrium, the sign of theleading eigenvalue could be examined to infer equilibrium stabi-lity and therefore the conditions under which polymorphism wasmaintained. However, these conditions were complex becausemany parameters were involved (for more detail seesupplementary Table 2).

Briefly, polymorphism maintenance under high migrationdepended mainly on spatial heterogeneity s and the shape ofthe generalization function γ: when both parameters were high,polymorphism was maintained. This means that polymorphismpersisted when there was a sufficient spatial heterogeneity in thedistribution of mimicry community (leading to disruptive selec-tion on the mimetic pattern in the focal species) and when thegeneralization function was high enough to allow heterozygotes tobe considered similar to the mimetic patterns. In this case,spatially heterogeneous selection was sufficient to maintainpolymorphism.

The dominance coefficient h played a significant role onpolymorphism only when parameters s and γ are relatively small.This was because dominance has an influence on predation only ifpredators are able to distinguish heterozygotes from homozygotes(γ small). In this case, a tight equilibrium between dominance,demographic parameters (growth rate r and carrying capacity K)and toxicity could allow the maintenance of polymorphism bybalancing the number of different phenotypes matching each preycommunity. In particular, the Kl product (the total contribution ofthe focal species to overall toxicity at the community level)modified the strength of selection exerted by the structure ofpredation given by parameters s and γ (i.e. spatial distribution ofmodel species). When Kl was large, the selection regime switchedfrom balancing selection due to spatial heterogeneity to direc-tional selection due to within-species positive frequency-dependent selection.

Dominance thus plays a complex role in the persistence ofpolymorphism in a heterogeneous environment, by interactingwith toxicity, community structure and demography.

3.3. Persistence of polymorphism with intermediate migration

As previously shown in studies on spatially distributed popula-tions, migration is a key parameter in the persistence of poly-morphism. In this system, migration is balancing the effect of local

directional selection and, therefore, polymorphism is generallymaintained for low migration rates but becomes unstable above acritical value of migration (Joron and Iwasa, 2005).

Numerical simulations were used to analyse the influence ofmigration on polymorphism in our two-population model. Asshown in Fig. 2, polymorphism was maintained in both popula-tions for all dominance levels when migration is relatively low.When m40.3, the two populations started to behave like a singlepopulation, and for any dominance coefficient h, one allele becamefixed while the other went extinct.

However, the interaction between dominance and migrationhad an influence on the persistence of polymorphism. Indeed,dominance determined the strength of migration above whichpolymorphismwas lost, and which allele becomes fixed (a or b). Inthe examples showed on Fig. 2, polymorphism was lost formZ0.22 when allele a and b are co-dominant (h¼0.5) (Fig. 2C),as compared to mZ0.26 when allele a was either recessive (h¼0)or dominant (h¼1) (Fig. 2A and D respectively). Similarly, Fig. 3highlighted that polymorphism started to be lost in some simula-tions for lower migration rates in intermediate dominance condi-tions as compared to complete dominance. For instance whenh¼0.05, polymorphism was lost in some simulations whenm40.08 whereas strictly recessive alleles were lost only whenm40.25. This suggests that complete dominance allowed poly-morphism to remain stable at higher rates of migration comparedto intermediate dominance. The same trend was observed forspatial heterogeneity (s) with polymorphism observed at lowerlevel of heterogeneity when alleles exhibited complete dominanceas compared to co-dominance (see supplementary Figure 1).Altogether, this suggested that complete dominance allowed thepersistence of polymorphism with lower levels of ecological and/or population structure.

3.4. Influence of initial conditions and dominance on allele fixation

For high rates of migration, the fixation of the same allele inboth populations was observed for all dominance coefficients.Here, because of a perfectly symmetrical situation (opposing butidentically scaled mimicry selection in the two populations), theidentity of the allele becoming fixed depended mainly on thedirection of dominance.

In the case of co-dominance (Fig. 2C), roughly half of allsimulations showed the fixation of allele a whereas the other halffixed allele b. There, the identity of the allele dominating at highmigration rates was only determined by the initial frequencies ofthe alleles.

In contrast, when dominance was complete, the dominantallele became fixed in the large majority of simulations wheremigration was high (allele b when h¼0 and allele a when h¼1respectively, see Fig. 2A and D). For migration rate mZ0.26, thestrictly dominant allele reached a frequency close to 1 in 99%

Table 1Description of the four possible equilibria obtained through the analytical analyses, assuming no migration (m¼0).

Equilibrium Frequencies of genotypes Description of equilibrium Stability conditions

1 Nab1 ¼ Nab2 ¼ Naa2 ¼ Nbb1 ¼ Naa1 ¼ Nbb2 ¼ 0 Extinct populations Stable when d4 r2ð1�sÞ

2 Nab1 ¼ Nab2 ¼ Naa2 ¼ Nbb1 ¼ 0 Two monomorphic populations Stable when do r2ð1�sÞ

Naa1 ¼ Nbb2 ¼ðKl�1Þrþ ffiffir

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKlþ1Þ2r�8dKlð1�sÞ

p2lr

3 Naa1 ¼ðKl�1Þrþ ffiffir

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKlþ1Þ2r�8dKlð1�sÞ

p2lr

Allele a fixed in both population Always unstable for realistic range of predation parameters(d 4 0 and s 4 0)

Naa2 ¼ðKl�1Þrþ ffiffir

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKlþ1Þ2r�8dKlð1þsÞ

p2lr

4 Nbb1 ¼ðKl�1Þrþ ffiffir

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKlþ1Þ2r�8dKlð1þsÞ

p2lr

Allele b fixed in both populations Always unstable for realistic range of predation parameters(d 4 0 and s 4 0)

Nbb2 ¼ ðKl�1Þrþ ffiffir

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKlþ1Þ2r�8dKlð1�sÞ

p2lr

V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110 105

of cases. Dominance thus had an important influence on the fate ofthe alleles. In case of partially recessive alleles (as for instancewhen h¼0.005), the most dominant allele became fixed in themajority of simulations although the fixation of the allele was alsoinfluenced to the initial conditions (Fig. 2B). This effect illustratedthe interaction between an initial allelic density advantage and thephenotypic density advantage provided by dominance. This inter-action would be modified in case of asymmetrical selection,

e.g. when the total densities of the different mimicry communitiesare not equal.

3.5. Allele frequencies in polymorphic populations

Fig. 3 showed that the interaction between migration anddominance was shaping the allele frequency in polymorphicpopulations. Allele frequencies were shown to depend closely on

Fig. 3. Frequency of allele a in population 1 (a) and 2 (b) most commonly observed out of 1000 simulations for a realistic range of migration rates [0.01; 0.3]. To simplify thevisualisation, frequencies were plotted only in cases of polymorphism. The different lines represent values of the dominance coefficient h: red line: h ¼ 0 (the allele a isstrictly recessive), Orange: h ¼ 0.005, Purple: h ¼ 0.01, Green: h ¼ 0.5 (the allele a is exactly co-dominant), Blue: h ¼ 1 (the allele a is strictly dominant). Note that the linesare interrupted as soon as polymorphism was lost. The shape of the generalization was chosen to be steep (γ ¼ 0.01), the global death risk was d ¼ 0.5, the predation risklinked to the different abundance of the model species in the two populations was s ¼ 0.9, toxicity was l ¼ 0.0025, growth rate in each population was r ¼ 1 and carryingcapacity in each population was K ¼ 1000.

Fig. 2. Simulations for recessive (h ¼ 0) (Fig. 2A), mildly recessive (h ¼ 0.005) (Fig. 2B), co-dominant (h ¼ 0.5) (Fig. 2C) and dominant (h ¼ 1) allele a (Fig. 2D), for a range ofa realistic migration rates [0.01; 0.4] with an increment of 0.01. Each parameter set was replicated 1000 times. Grey: simulations where polymorphism was maintained, i.e.both alleles had a frequency Z 0.01. White: simulations where allele a was fixed, i.e. the frequency of a overall both populations was Z 0.99. Black: simulations where alleleb was fixed, i.e. the frequency of allele a overall both populations was r 0.01. In all simulations, the shape of the generalization was chosen to be steep (γ ¼ 0.01), the globaldeath risk was d ¼ 0.5, the predation risk linked to the different abundance of the model species in the two populations was s ¼ 0.9, toxicity was l ¼ 0.0025, growth rate ineach population was r ¼ 1 and carrying capacity in each population was K ¼ 1000.

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the level of dominance, with recessive alleles (0rho0.5) exhibit-ing higher frequencies than dominant alleles, resulting in a totalfrequency greater than 50%. This negative relationship betweendominance and frequency, leading to a higher frequency for morerecessive alleles held for wider shapes of the generalizationfunction (see supplementary Figure 2).

However, for very wide generalisation function (γZ0.3), thegeneralisation curves, g (h) and g (1�h), overlapped (see Fig. 1)and intermediate heterozygotes ab (hE0.5) were perceived bypredators as similar to both model species (and both homozy-gotes). Fig. 4 showed that the frequency of the intermediateheterozygote genotype ab increased with the width of the general-isation function, γ, leading to the persistence of the polymorphismin both populations, even for high migration rates. This over-dominance effect depended on the discrimination capacities ofpredators with respect to the variable aposematic signal.

3.6. Migration load

It was notable that, for all dominance levels, the total numberof individuals overall both populations decreased when migrationrate increased (for more details see supplementary Figure 3). Thiscould be explained by a migration load which limits local adapta-tion (Lenormand, 2002). When there is no migration, all indivi-duals in each population exhibit the matching phenotype of thelocal mimicry community, but as soon as migration increases,individuals are exchanged between populations, introducing non-matching phenotypes which suffer higher predation and cause adecrease in the total number of individuals overall bothpopulations.

More interestingly, our diploid model showed that thisdecrease in population size depended on dominance: when hwas close to 0.5 (co-dominance), the decrease was larger, pre-sumably because heterozygotes did not match the mimetic patternin any of the two patches, and suffered more predation in bothpopulations. In the case of high dominance, heterozygotesmatched the mimetic pattern of one of the two mimetic commu-nities, leading to a higher total population size. Dominance thusalso influenced the demography of both populations.

3.7. Influence of toxicity (l) and spatial heterogeneity (s)

As previously described in the haploid model (Joron and Iwasa,2005), the level of toxicity (l) and spatial heterogeneity (s) are key

parameters for local adaptation to spatially distributed mimicrycommunities. Briefly, under a haploid model polymorphism couldbe maintained when spatial heterogeneity was high due to theimportance of local selection. Similarly, low values of toxicity wereshown to favour polymorphism, because mildly defended preygain high benefits from resemblance to the local mimicry com-munity whereas the dynamics of highly defended prey are lesssensitive to local mimicry. Here, we investigated more preciselythe roles of parameters l and s in the interaction betweendominance and migration (Fig. 5). Overall, our diploid modelconfirms the positive effect of spatial heterogeneity on thepersistence of polymorphism whatever the dominance coefficient;it also highlights that for mildly defended preys, complete dom-inance tends to promote polymorphism.

A particular case emerged for co-dominant alleles wherepolymorphism could persist when toxicity was very high (greyareas in bottom panels of Fig. 5). When hZ0.1, heterozygotes abresembled neither homozygotes (and neither correspondingmimetic communities), but were produced in higher proportionsthan homozygotes due to Mendelian segregation. Since prey werehighly defended, positive frequency-dependent selection was thusmainly influenced by the phenotypic composition of the focalspecies populations (as opposed to the local mimicry community).Since the commonest phenotype was carried by heterozygotes ab,a strong positive selection on heterozygotes promoted polymorph-ism. This overdominance effect explained the level of polymorph-ism observed at high values of toxicity when heterozygotes weredistinct from either homozygote.

Finally, for very high levels of toxicity, polymorphism wasobserved regardless of the spatial heterogeneity or dominance(see bottom of each graph on Fig. 5). In these cases, toxicity was sohigh that all individuals became protected irrespective of theiraposematic pattern.

4. Discussion

4.1. Persistence of polymorphism

As already demonstrated in the haploid model (Joron andIwasa, 2005), the persistence of polymorphism was mainly drivenby a balance between migration (here represented by migrationrate, m) and selection (here represented by the overall predationrisk, d, and spatial heterogeneity, s). However, our diploid modelshowed that complete dominance favoured the persistence ofpolymorphism for a larger range of parameters (m, d, s) than co-dominance. This is in accordance with the general model of Ottoand Bourguet (Otto and Bourguet, 1999) investigating dominancein patchy environments, which shows that stable polymorphism ismaintained for small values of h, i.e. in cases of completedominance. In the case of Müllerian mimics, the generallyobserved positive effect of dominance on polymorphism main-tenance was due to the higher levels of predation experienced byco-dominant alleles in both populations, because of their lack ofresemblance to any mimicry community. In the case of partiallydominant or co-dominant alleles, homozygotes had an importantadvantage over heterozygotes in both populations and selectionagainst non-mimetic heterozygotes might thus favour the fixationof one of the two alleles, leading to the loss of polymorphism.

The positive effect of dominance on the persistence of poly-morphism in Müllerian mimics is supported by empirical data: forinstance the toxic mimic butterfly Heliconius numata exhibits highlocal polymorphism and complete dominance among sympatricalleles controlling the wing-patterns forms (Joron et al., 2006).However, to demonstrate the crucial role of dominance on poly-morphism, empirical estimations of the level of dominance and

Fig. 4. Frequency of heterozygotes ab for a realistic range of migration rates,m : [0.01; 0.3], averaged across both populations. The different lines representshapes of the generalisation function (γ): red line: γ ¼ 0.1, Orange: γ ¼ 0.3, Purple:γ ¼ 0.5, Green: γ ¼ 0.7, Blue: γ ¼ 0.9. Note that the lines are interrupted as soon aspolymorphism is lost. The alleles were strictly co-dominant (h ¼ 0.5), the globaldeath risk was d ¼ 0.5, the predation risk linked to the different abundance of themodel species in the two populations was s ¼ 0.9, toxicity was l ¼ 0.0025, growthrate in each population was r ¼ 1 and carrying capacity in each population wasK ¼ 1000.

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Fig. 5. Conditions for Mullerian polymorphism maintenance as a function of the migration rate (m) (columns, from left to right: m ¼ 0.001, 0.005, 0.01, 0.1), dominancecoefficient (h) (rows, from top to bottom lines: h ¼ 0, 0.005, 0.01, 0.1, 0.5), habitat heterogeneity (s) and unpalatibility (l) (note the logarithmic scale on the y-axis). In dark,polymorphism is maintained in both populations, in grey, polymorphism is also maintained in both populations with a higher frequency of heterozygotes compared tohomozygotes (overdominance), in white polymorphism is lost.

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the level of colour pattern polymorphism in natural populations ofMüllerian mimetic species would be required.

Co-dominance also leads to the persistence of polymorphism intwo special cases: (1) when the discrimination capacities ofpredators were not accurate, providing an advantage to inter-mediate heterozygotes through their resemblance to both mimeticcommunities, providing a multi-model advantage (Edmunds,2000), and (2) when the focal species was highly toxic and thusprotected independently of the mimicry community, favouringheterozygotes through simple density-dependent advantage. Insuch cases, the advantage of heterozygotes (overdominance)provides a balancing selective pressure, allowing the persistenceof polymorphism in natural populations.

Dominance thus appears to magnify the effect of heteroge-neous selection and thus to modulate the dynamics of themigration/selection balance. However, the influence of dominancedepends closely on the mechanism of selection acting on thefunctional locus and more precisely on the heterozygotes in bothpopulations. For instance, behavioural studies have demonstratedthat the physiological state of predators can affect their discrimi-nation vs. generalization capacities (Barnett et al., 2012; Kokkoet al., 2003; Halpin et al., 2012), and may thus limit selection ondominance. Generalization behaviour has also been shown todepend on the diversity of communities (Ihalainen et al., 2012),stressing the need to investigate the discrimination capacities ofpredators in natural populations of polymorphic Müllerian mimicspecies to characterize the selection on dominance more precisely.

4.2. Allele dynamics

In our model without migration, each allele became fixed in thepopulation where it was locally adaptive, and dominance had noinfluence on polymorphism stability. However, as soon as migra-tion was introduced, our model showed the key role of dominancein equilibrium allele frequencies. For moderate migration rates(lower than 0.2 in the example presented here), the equilibriumfrequency was higher for the recessive allele than for the domi-nant allele. Since this corresponds to a two-population situationwith strictly symmetrical selection acting on each homozygote,this might be similar to the situation found in parapatric mimeticcommunities where the relative abundance of mimicry commu-nities could be similar on either sides of a suture zone. Empiricaltests comparing allele frequencies with respect to dominance inlocalities where the abundances of the different mimic commu-nities are similar would allow confirming this prediction ofthe model.

Our diploid model also highlights that dominance can havecomplex interactions with overall prey community structure andwith the toxicity of a species relative to other species in a mimicrycommunity and may therefore modify the dynamics of poly-morphism. In particular, relative toxicity with respect to otherspecies of the mimetic community can lead to heterozygoteadvantage, and thus explain the polymorphism observed inseveral Müllerian mimic species.

Finally, our model also predicted that the spatial distribution ofthe distinct mimicry communities can drive the direction ofdominance. The traditional view of Haldane's sieve predicts thatnew establishing alleles are more likely to be dominant overancestral alleles (Clarke et al., 1985), so that dominance couldreflect the chronology of the evolution of the different alleles. Inthe case of balancing selection in subdivided populations, geneticdrift is more likely to eliminate unexpressed advantageous allelesthan dominant ones (Schierup et al., 1997), thus favouring theinvasion of dominant alleles in a population with homogeneousselection. However, our model shows that recessivity can beadvantageous in case of mimicry in a spatially heterogeneous

environment, because it allows locally non-mimetic alleles topersist. Depending on the ecological context (i.e. the compositionof the butterfly community, migration behaviour and relativetoxicity), recessive alleles can thus reach high frequencies inmimetic species and persist over long evolutionary timescales.The capacity for invasion and persistence of migrant or new alleleswith varying dominance levels thus needs to be investigated todistinguish the effects of population history, ecology andsubdivision.

For high migration rates, when predators had accurate discri-mination capacities, selection led to the fixation of the dominantallele in both populations, assuming symmetrical spatial selectionon homozygous phenotypes. In this situation, since the two modelspecies provided the same advantage, the advantage provided tothe most dominant allele was given by a positive number-dependent advantage because both homozygote and heterozygotedisplay the same phenotype. This is equivalent to the situationwhere two distinct mimicry communities confer unequal advan-tages and lead to the fixation of the morph mimicking the morenumerous community (Joron and Iwasa, 2005). This asymmetricaladvantage due to dominance has previously been described byMallet and Barton (Mallet and Barton, 1989): they showed that themovement of clines between different alleles of genes involved inmimicry is enhanced by dominance because the side of the clinewhere dominant alleles are most common is more likely toexpand. Dominance is thus an important factor to consider tounderstand the evolution of wing colour patterns in mimicspecies.

4.3. Evolution of dominance

Our model suggested that larger population size was observedin cases of complete dominance. This suggests that completedominance may limit the mortality of intermediate non-mimeticheterozygotes. Complete dominance can either be due to (1) posi-tive selection of dominant new or migrant alleles only (i.e.Haldane's sieve) or (2) evolution of expression levels towardsstrict dominance among alleles. This last possibility of evolution ofdominance through natural selection, described by Fisher (Fisher,1928), is still a debated topic. However, Billiard and Castric (Billiardand Castric, 2011) recently suggested that dominance is likely toevolve in many “special cases”, including loci under balancingselection. Indeed, in cases of balanced polymorphism, the fre-quency of heterozygotes is high enough to allow the evolution ofdominance through natural selection (Otto and Bourguet, 1999).The possibility of the evolution of dominance in wing colourpattern genes has been suggested by Clarke and Sheppard(Clarke and Sheppard, 1960) based on the study of dominance incontrolled crosses of sympatric versus allopatric morphs in themimetic butterfly Papilio dardanus.

In our diploid model, we highlighted the importance ofdominance on adaptation of mimic alleles, suggesting that com-plete dominance would be a cost-effective way to express onlymimetic phenotypes. Here, we assumed that each allele has a fixeddominance level, corresponding to a situation where dominance isan intrinsic property of the allele or is encoded by a gene tightlylinked to the locus encoding colour pattern itself. This situationcan arise in a supergene architecture where recombination is lowas is observed in Heliconius numata (Joron et al., 2011). However, ithas also been shown that the level of expression of major genesencoding wing colour pattern might be influenced by modifiergenes which are not necessarily linked to these loci of major effect.For instance, improvement of mimicry has been suggested to relyon modifier genes with relatively small effect on wing colourpattern, as demonstrated in Heliconius numata (Jones et al., 2012)and Heliconius erato (Papa et al., 2013). These data raise the

V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110 109

possibility that dominance at major genes could be modified byindependent loci, allowing the rapid adaptation of heterozygotesto the local community of toxic butterflies. The influence ofrecombination between dominance modifiers and loci controllingaposematic colour pattern should be investigated in the future andcould shed light on the mechanisms of the evolution of dominance.

5. Conclusions

Our theoretical diploid model suggests that dominance canhave an important impact on the dynamics of alleles responsiblefor aposematic pattern in mimetic species and, specifically, on thepersistence of polymorphism. In polymorphic mimetic species,dominance interacts with the structure of the community ofdefended prey, migration capacities and demography, and thusinfluences the adaptation of aposematic patterns. Our resultsstress the importance of investigating the evolution of dominancein adaptive traits in heterogeneous environments.

Acknowledgements

The authors would like to thank A. Whibley for useful discus-sions on the manuscript. This work has been sponsored by theATM Formes possibles, formes realisées from the National Museumof Natural History to VL, by the ERC starting grant MIMEVOL to MJand the ANR BRASSIDOM to SB.

Appendix A. Supporting information

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.jtbi.2013.08.006.

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