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Rausher, M. D. and S.-M. Chang.  1999.  Stabilization of mixed-mating
        systems by differences in the magnitude of inbreeding depression for
        male and female fitness components. American Naturalist (in press).

Synopsis:

Introduction

    Although a majority of plant species exhibit either complete selfing or complete outcrossing a distinct minority exhibit a mixed mating system, in which both selfing and outcrossing occur (Schemske and Lande 1985).  Moreover, in at least some species with mixed mating systems, selfing rates are genetically variable (Brown and Clegg 1984; Holtsford and Ellstrand 1992; Carr and Fenster 1994; Chang and Rausher 1998b), suggesting that these mixed mating systems are evolutionarily stable.  Theoretical attempts to account for such stability have generally focused on assessing the conditions under which two opposing sets of evolutionary processes can generate polymorphisms for selfing rates.  One of these processes, originally described by Fisher (1941), arises because an allele that increases the selfing rate automatically enjoys a transmission advantage due to the extra pathway (via self pollen) by which it may transmit copies of itself to the next generation.  Various authors have demonstrated that, under some circumstances, two other processesinbreeding depression and pollen discounting may prevent fixation of such a selfing allele and stabilize a selfing-rate polymorphism (Holsinger 1988, 1991; Charlesworth and Charlesworth 1990; Uyenoyama and Waller 1991b).  In particular, under some circumstances the identity disequilibrium that is expected to arise between loci affecting selfing rate and loci affecting fitness through inbreeding depression may be strong enough to stabilize polymorphisms at both sets of loci (Holsinger 1988; Uyenoyama and Waller 1991b).  Alternatively, mass-action models of pollen discounting may produce a similar stabilization (Holsinger 1991).

     In the common morning glory, Ipomoea purpurea L. (Roth), experimental investigations of associations between characters affecting selfing and the magnitude of inbreeding depression (Chang and Rausher, 1998b) and of pollen discounting (Rausher et al. 1993; Fry and Rausher 1997; Mojonnier and Rausher 1997; Chang and Rausher 1998a) indicate that neither of these processes individually, or acting together, can account for apparently stable polymorphisms in two different traits affecting selfing rates: flower color determined by the W locus (Rausher et al 1993; Fry and Rausher 1997; Mojonnier and Rausher 1997) and anther-stigma separation (Chang and Rausher 1998, 1999).  These observations have led us to seek other possible explanations for maintenance of these polymorphisms.  We report here an additional mechanism by which a polymorphism for selfing rates may be maintained: differences among male and female fitness components in the magnitude of inbreeding depression.  More generally, this mechanism can yield a stable ESS for a mixed mating system.

     In virtually all prior models of the evolution of selfing (but see Damgaard et al. 1994), inbreeding depression, when it is included, is assumed to affect viability, and thus affect the overall male and female components of fitness equally (Maynard Smith 1977; Charlesworth 1980; Feldman and Christiansen 1984; Holsinger et al 1984; Lande and Schemske 1985; Holsinger 1988; Charlesworth and Charlesworth 1990; Charlesworth et al 1990; Uyenoyama and Waller 1991 a, b, c).  This assumption has perhaps been warranted because of the virtual absence of information on the relative effects of inbreeding on male and female components of fitness.  However, Husband and Schemske (1996) have recently demonstrated that in plants, inbreeding depression is often most strongly expressed during the reproductive phase of the life cycle, and there is no a priori reason to believe that it would necessarily affect male and female fitness components equally.  In fact, Chang and Rausher (1999) provide evidence suggesting that in I. purpurea female fitness is reduced more than male fitness by selfing.

     To examine the consequences of this type of difference, we numerically analyzed a model in which selfing rate is controlled by a single locus having two alleles, A and a.  The selfing rates of genotypes AA, Aa, and aa were designated s, sand s, where, without loss of generality, it is assumed that s s s (with one inequality strict).  The male and female components of fitness of an individual that is not inbred are each assumed to be 1, while for inbred individuals, these components are designated M and F, respectively, with M,F  1.  Biologically, F is the expected number of fertilized ovules produced by an inbred individual, relative to outbred individuals, calculated from the time of fertilization.  It thus represents the standard relative cumulative (multiplicative) fitness of inbred individuals (Molina-Freaner and Jain 1993; Willis 1993; Johnston and Schoen 1996), calculated as

        F = A V R_f                                 (1a) ,

where A is relative survivorship from fertilization of a selfed ovule to seed maturation, V is relative survivorship (including germination probability) from seed maturation to commencement of reproduction, and R_f  is the relative number of fertilized ovules produced by a surviving inbred individual.  Similarly,  is the relative male outcross success of inbred individuals, i.e. it is the expected number of ovules on other plants sired by an inbred individual, relative to the number sired by outbred individuals, calculated from the time of fertilization, which is given by

         M  = A V R_m  (1b) ,

where R_m  is the relative outcross success (number of ovules sired on other plants) of an inbred individual, which includes effects of inbreeding not only on pollen number and viability, but also on success at transporting pollen to stigmas on other plants.  In this formulation, selfing pollen is included in the female fitness component.  The magnitude of inbreeding depression for the two components of fitness are thus 1-M and 1-F.

Model

    The model used to analyze the evolution of selfing rate is explicitly genetic and considers a single locus with two alleles affecting selfing rate.  For each genotype, the proportions of individuals that are selfed and outcrossed are explicitly accounted for, yielding a set of six recursion equations.  We deliberately incorporate no pollen discounting in the model because we are interested in whether the differential effects of inbreeding on male and female fitness can, by themselves, lead to evolutionarily stable mixed mating systems.  We restrict our analysis to the case in which there is no overdominance for selfing rate because overdominance is intuitively expected to lead to stabilization of a selfing-rate polymorphism under many circumstances, but determination of whether such polymorphisms are evolutionarily stable to invasion by alleles conferring complete selfing or complete oucrossing is beyond the scope of our current analyses.

Numerical Analysis of Model

     Numerical iteration of the recursion equations indicates that for any pair of alleles A
and a, which differ in selfing rate, the M-F parameter space is divided into four regions (Fig. 1).  The boundaries of these regions are formed by two "isoclines", each corresponding to one of the homozygote genotypes.

     Numerical iterations indicate that the equilibrium frequences of the two alleles are determined by which region of Fig. 1a the actual value of (M, F) lies in, as follows:

     1.  In Region I, increased outcrossing (fixation of A) is favored.
     2.  In Region II, increased selfing (fixation of a) is favored.
     3.  In Region III, a stable polymorphism is maintained.
     4.  In Region IV, either complete outcrossing or complete selfing is favored, depending
              on initial gene frequencies.

Fig. 1   Stability properties of selfing-rate variants for different combinations of male, M, and female, F, fitness components.  a.  For a single locus with two alleles differing in selfing rates of their corresponding homozygotes, the M-F parameter space is divided into four stability regions: Region I--allele conferring higher outcrossing favored; Region II--allele conferring lower outcrossing favored; Region III--stable polymorphism; Region IV--disruptive selection, i.e. either allele favored depending on initial frequencies.  The isoclines portrayed correspond to s=0.2  and s=0.9.  b. Parameter space is divided into two regions, within which the point (M,F) acts either as an attractor or a repeller of isoclines.  c. Illustration of attractor and repeller properties (See text for explanation). d.  Regions corresponding to different types of evolutionary stability: Region A--ESS is complete outcrossing; Region B--ESS is complete selfing; Region C--ESS is mixed mating system; Region D--there are two ESS's, corresponding to complete selfing and complete outcrossing.
 
 
 
 
 
 
 
 

    While this analysis indicates that stabilization of a selfing rate-polymorphism is possible if (M,F)  lies in Region III, long-term evolutionary stability will be determined by whether a stable polymorphism of this type can be invaded by other alleles with different selfing rates.  The stability properties described above imply that the long-term evolutionary equilibrium for outcrossing rate can be described by a very simple model, which we have confirmed by extensive numerical analysis.  In particular, the stability properties of the two-allele system imply that the the M-F parameter space is divided into two portions by the line M=F (Fig. 1b).  In the region below this line, corresponding to combinations of M and F for which M>F, the point (M,F) acts as an attractor of the isoclines.  Specifically, in a population segregating for two alleles affecting selfing rate, when  (M,F) is on the same side of both isoclines corresponding to the two homozygote genotypes, selection will fix the allele corresponding to the genotype with the isocline that passes closer to the point (M,F).

    This explanation for this behavior is illustrated in Fig. 1c, in which the solid circle  represents a particular combination of M and F within the attractor region.  Consider a population that is initially completely outcrossing (isocline d) and into which a mutation causing a small degree of selfing is introduced (isocline d').  The phase diagram then corresponds to Fig. 1a, in which regions III and IV are defined by the isoclines d and d'.  Because the (M,F) point lies in region II, the mutant allele, which increases selfing, is favored and will go to fixation.  In other words, the original homozygote is replaced with a new homozygote having an isocline closer to the attractor (M,F).  Subsequently, another mutation may arise that confers a slightly higher selfing rate, corresponding to isocline d''.  The boundaries of Regions III and IV are then the lines d' and d''.  The  (M,F) point still lies in region II, and hence the new mutant, conferring increased selfing, is again favored.  Once again, the isocline of the fixed allele lies closer to (M,F).  This process will continue until selfing rate is sufficiently high that a new mutation causes the boundaries of Regions III and IV to encompass the (M,F) point, at which a stable polymorphism is achieved. By similar reasoning, introduction, into a completely selfing population, of a mutation that causes a small degree of outcrossing yields regions III and IV corresponding to lines isoclines d* and d**.  The  (M,F) point then lies in region I, which leads to fixation of the mutation conferring increased outcrossing.  Thus, from either extreme, selection will cause outcrossing rates to evolve to an intermediate value, thus producing an evolutionarily stable mixed mating system.  Numerical iteration indicates that the allele corresponding to the homozygote with a isocline passing through (M,F) is stable to invasion by any other allele, and thus represents an ESS.

    The qualitative evolutionary stability associated with a particular set of male and female inbreeding depression values, (M,F), as deduced by the above considerations, is summarized in Fig. 1d.  Regions A and B correspond to complete outcrossing and complete selfing, respectively, being evolutionarily stable.  Region C corresponds to a stable mixed mating system, which may or may not exhibit genetic variation for selfing rate, while in Region D both complete selfing and complete outcrossing are locally evolutionarily stable.  Moreover, the conditions for a stable mixed mating system have a straightforward relationship to total fitness (defined as the average of male and female fitness components: W = 0.5(M+F) ) and total inbreeding depression (1-W):  W>0.5 and  F< 0.5, i.e. total inbreeding depression is less than 0.5, while inbreeding depressions is greater than 0.5 for the female component of fitness.

 The primary implication of this analysis is that differences in the magnitude of inbreeding depression for male and female fitness components may favor the evolution of a mixed mating system.  In particular, in species in which the value of (M,F) falls within Region C of Fig. 1d, a mixed mating system is an evolutionarily stable state.  Moreover, for such species, complete outcrossing or complete selfing are evolutionarily unstable states, in the sense that any mutation causing selfing rates that deviate from one of these fixed states will increase in frequency.  Such a mutation may become fixed or may equilibrate in a polymorphic state, depending on the exact values of (M,F) and of the mutant's selfing rate, but in either case a mixed mating system results.
 

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