1 Mate choice for optimal (k)inbreeding
2 Mikael Puurtinen
3 Centre of Excellence in Evolutionary Research, P.O. Box 35, FI-40014 University of
4 Jyväskylä, Finland
5 *E-mail: [email protected]
6
7 Abstract.—Mating between related individuals results in inbreeding depression, and
8 this has been thought to select against incestuous matings. However, theory predicts
9 that inbreeding can also be adaptive if it increases the representation of genes
10 identical by descent in future generations. Here, I recapitulate the theory of
11 inclusive fitness benefits of incest, and extend the existing theory by deriving the
12 stable level of inbreeding in populations practicing mate choice for optimal
13 inbreeding. The parsimonious assumptions of the model are that selection
14 maximizes inclusive fitness, and that inbreeding depression is a linear function of
15 homozygosity of offspring. The stable level of inbreeding that maximizes inclusive
16 fitness, and is expected to evolve by natural selection, is shown to be less than
17 previous theory suggests. For wide range of realistic inbreeding depression
18 strengths, mating with intermediately related individuals maximizes inclusive fitness.
19 The predicted preference for intermediately related individuals as reproductive
20 partners is in qualitative agreement with empirical evidence from mate choice
21 experiments and reproductive patterns in nature.
22 23 Key words. – Inbreeding, incest avoidance, inclusive fitness, kin selection, relatedness,
24 mate choice.
25
26 Adaptive explanations for mate choice concentrate on the benefits that can be accrued by
27 discriminating among potential reproductive partners. The benefits of mate choice can be
28 classified as direct benefits, genetic benefits, and inclusive fitness benefits. Direct
29 benefits of mate choice include factors that affect the number of surviving offspring, e.g.
30 mate’s ability to provide shelter and nutrition for the offspring. Genetic benefits are
31 factors affecting offspring genetic quality, i.e. heritable differences in fitness (‘good
32 genes’) or differences in genetic compatibility between mates (‘compatible genes’)
33 (Puurtinen et al. 2009). Inclusive fitness benefits of mate choice refer to the increased
34 genetic representation in future generations that can be gained by increasing the
35 reproductive success of relatives via mate choice (e.g. Parker 1979).
36
37 Previous theoretical work has shown that inclusive fitness benefits can favor close
38 inbreeding even when this results in substantial reduction in offspring fitness. These
39 models have identified the boundary level of inbreeding depression limiting the evolution
40 of inbreeding among first-order relatives, i.e. between full sibs, or between parents and
41 offspring (Parker 1979, Smith 1979, Waser et al. 1986, Lehmann & Perrin 2003, Kokko
42 & Ots 2006). While these models can easily be extended to study the boundary level of
43 inbreeding depression limiting incest among any pair of relatives, they do not address the
44 important question: What level of inbreeding maximizes fitness? As natural selection
45 ought to maximize fitness, deriving the optimal relatedness for adaptive inbreeding is 46 warranted. Apart from the cursory note in the appendix D of Lehmann & Perrin (2003),
47 optimal inbreeding for inclusive fitness benefits has not been studied before. Here, I’ll
48 first recapitulate the calculation of threshold inbreeding depression that limits the
49 evolution of incest, and the calculation of optimal level of inbreeding under the
50 simplifying assumption of an outbred population. Then I extend the analysis to find the
51 long-term evolutionarily stable level of inbreeding in populations practicing optimal mate
52 choice for inbreeding. In the end, I’ll discuss the implications of the model to
53 understanding mate choice, population structure, and mating system evolution.
54
55 The model is developed primarily for species with separate sexes. For convenience of
56 writing, I’ll use the term female for the sex that has high parental investment and whose
57 reproduction is not limited by mate availability but by resource availability, and male for
58 the sex that is primarily limited by mate availability. For mathematical convenience, let
59 fitness from outcrossing be one. We can then write the net inclusive fitness effect for a
60 female from incestuous mating with a male with relatedness r as
61
62 Ifemale = (1 – δ) + r (1 – δ) – 1 – r c (1)
63
64 where δ is inbreeding depression and c denotes the opportunity cost for a male from
65 incestuous mating (when c = 0 incest does not affect male’s opportunities for outcrossing,
66 and when c = 1 incest precludes outcrossing).
67 68 In (1), the first term denotes genes passed directly from the female to the offspring, the
69 second term denotes genes identical by descent (IBD) with the female passed to the
70 offspring from the related male, the third term denotes the loss of one unit of direct
71 fitness that would have been obtained if the female outcrossed, and the fourth term
72 denotes loss of inclusive fitness from lost outcrossing opportunity by the related male.
73 Analogically, we can write the effect of incestuous mating on male fitness as
74
75 Imale = (1 – δ) + r (1 – δ) – c – r (2)
76
77 where the first term denotes genes passed directly from the male to the offspring, the
78 second term denotes genes IBD passed to the offspring from the related female, the third
79 term denotes the lost outcrossing opportunities for the male, and the fourth term denotes
80 the lost indirect fitness that could have been obtained if the related female had outcrossed.
81
82 Solving I = 0 with respect to δ, we get the strength of inbreeding depression that will
83 limit evolution of incest. For females this value is
84
85 δfemale = (r – r c) / (1 + r) (3)
86
87 and for males
88
89 δmale = (1 – c) / (1 + r) (4)
90 91 Equivalent equations have been derived previously by Waser et al. (1986), Lehmann &
92 Perrin (2003) and Kokko & Ots (2006). When c = 0 and relatedness between partners is
93 0.5 (e.g. full sibs in an outbred population), the equations yield the familiar inbreeding
94 tolerances of 2/3 for males and 1/3 for females (Parker 1979, Smith 1979, Waser et al.
95 1986, Taylor & Getz 1994, Lehmann & Perrin 2003) (Figure 1). Waser et al. (1986) also
96 give full discussion of the implications of varying the opportunity cost (c) to benefits of
97 incest. The conflict of interests between sexes over inbreeding has been extensively
98 discussed in the studies cited directly above. Here, I’ll focus on mate choice that
99 maximizes the inclusive fitness of the choosing sex (generally females).
100
101 To find the optimal relatedness that maximizes inclusive fitness, inbreeding depression
102 needs to be defined as a function of parental relatedness. Assuming that the effects of
103 different loci combine additively, inbreeding depression is a linear function of offspring
104 inbreeding coefficient F (Falconer & Mackay 1996). Linear inbreeding deperession is the
105 most typical pattern found in empirical studies. Noting that in an outbred population
106 offspring inbreeding coefficient is one half of parental relatedness (F = r/2), and denoting
107 the slope of the relationship between F and inbreeding depression as b (as in Crnokrak &
108 Roff 1999), inbreeding depression can be written as a function of relatedness of the
109 parents:
110
111 δ = b F = b (r / 2) (5)
112
113 Substituting (5) to equations (1) and (2) yields after rearrangement 114
2 115 Ifemale = 1/2 (2 r – b r – 2 c r – b r ) (6)
116
117 and
118
2 119 Imale = 1/2 (2 – 2 c – b r – b r ) (7)
120
121 Figure 2 shows the effect of incest on female and male fitness for various values of b
122 when the opportunity cost c = 0. From the male perspective, outcrossing always yields
123 the highest fitness, but incestuous matings also yield positive fitness returns unless
124 relatedness is very high (r is close to 1) and inbreeding depression is very severe. For
125 females, there often exists an optimum level of relatedness that maximizes fitness, and
126 this optimum depends on the strength of inbreeding depression. The optimum relatedness
127 for female mate choice in an outbred population is found by solving = 0 with 휕퐼푓푒푚푎푙푒 128 respect to r, which yields 휕푟
129
130 = . (8) ∗ 2−푏−2푐 푓푒푚푎푙푒 2푏 131 푟
132 When male opportunity cost c = 0, female fitness in an initially outcrossed population is
133 optimized by outbreeding when 2, by full-sib mating when b = 1, and in
푏 ≥ 2 2 134 simultaneous hermaphrodites by self-fertilization when 3. Thus, when 3 < <
135 2, there exists an intermediate r* that maximizes female푏 fitness.≤ � As Figure 2 shows,� the푏
136 stronger the inbreeding depression, the less closely related optimal mates are. For 137 example, when b = 1.6, female fitness is maximized by reproducing with first cousin (r =
138 0.125) and reproducing with full brother (r = 0.5) yields lower fitness than full
139 outbreeding.
140 The above derivation of r* applies only to an initially outbred population. Obviously, if
141 incestuous matings are common, the population becomes inbred and equation (8) no
142 longer applies. I’ll next derive the stable level of inbreeding in populations where females
143 are practicing mate choice for optimal inbreeding. For simplicity, I also assume
144 negligible male opportunity cost (c = 0).
145
146 In the context of inclusive fitness, relatedness is defined as rJK = FJK/FJJ, where FJK is
147 consanguinity between individuals and FJJ is the consanguinity of an individual with
148 itself, and FJJ = (1 + F)/2, where F is the individual’s inbreeding coefficient (Bulmer
149 1994). Because relatedness does not have the same relation to familial relationships in an
150 inbred population as in an outbred population, deriving solution in terms of relatedness is
151 of little practical or heuristic value. It is more useful to derive the stable level of
152 inbreeding (offspring inbreeding coefficient F which is equal to parental consanguinity
153 FJK), as this is easily measured from real populations as reduction in heterozygosity from
154 that expected with random mating, = . 퐻퐸−퐻푂 퐹 퐻퐸 155
156 Denoting consanguinity of mates in the current generation as FJK and the inbreeding
157 coefficient of the choosing female as F, and assuming c = 0, the net inclusive fitness
158 effect for a female mating with male with consanguinity FJK is 159
= 1 + 1 1 160 ( ) . (9) 2 퐹퐽퐾 퐼푓푒푚푎푙푒 � − 푏 퐹퐽퐾� 1+퐹 � − 푏 퐹퐽퐾� − 161
162 Equation 9 assumes that inbreeding and consanguinity coefficients are at their
163 equilibrium values. The optimal consanguinity is found by solving = 0 with 휕퐼푓푒푚푎푙푒 휕퐹퐽퐾 164 respect to FJK:
165
( ) 166 = . (10) 2−푏 1+퐹 퐽퐾 4푏 167 퐹
168 Equation 10 has been derived earlier by Lehmann & Perrin (2003, equation D1b).
169 Because parental consanguinity (FJK) equals inbreeding coefficient of the following
170 generation (F), equation 10 is a recurrence relation that can be solved for stable level of
171 inbreeding (F*):
172
173 = . (11) ∗ 2−푏 5푏 174 퐹
1 175 Thus, for 3 < < 2, there exists a stable intermediate level of inbreeding (Figure 3).
176 This stable �level 푏of inbreeding is 80% of the predicted from equation (8) which applies to
177 a completely outbred population. For b ≤ 1/3, complete inbreeding is expected. For b > 2,
178 disassortative mating avoiding relatives is expected, leading to stable negative F* values
1 179 approaching 5 as b ∞. − � 180 Discussion
181
182 The analysis of optimal inbreeding reveals that (some degree of) inbreeding is expected
183 to evolve under a wide range of inbreeding depression strengths, even when mate choice
184 is done by the sex with the lower inbreeding tolerance. This result corroborates the earlier
185 findings that inbreeding can be adaptive even when it results in appreciable inbreeding
186 depression (Parker 1979, Smith 1979, Waser et al. 1986, Lehmann & Perrin 2003, Kokko
187 & Ots 2006). However, the analysis also reveals that the expected amount of inbreeding
188 is considerably lower than the previous analyses insinuate. For example, it has been
189 suggested that in an outbred population incest among first-order relatives (r = 0.5) should
1 190 evolve whenever < 3 (assuming c = 0) (Parker 1979, Smith 1979, Waser et al. 1986).
훿 � 4 1 191 The current analysis however suggest that when = 3 (which gives = 3 for full-
192 sib matings in an outcrossed population), female 푏inclusive� fitness in an 훿outcrossed�
193 population is maximized at r = 0.25, and that the stable level of inbreeding in a such a
194 population is only F* = 0.1 (from equation 11), instead of F ≈ 0.25 that might be inferred
195 by extrapolation from the previous models.
196
197 It has been argued that there is a mismatch between theory and data regarding the
198 prevalence of inbreeding in nature: previous models suggest that close inbreeding should
199 be fairly common in nature, yet empirical studies seldom find evidence for preferential
200 close inbreeding (Kokko & Ots 2006). The analysis of mate choice for optimal
201 inbreeding suggests that the expected degree of inbreeding has been overestimated: as
202 explained above, the optimal long-term levels of inbreeding are much lower than the 203 previous studies suggest. Intriguingly, when preference for related individuals has been
204 found in empirical studies, preference is usually not for immediate but for slightly more
205 distant kin. In a review on inbreeding avoidance in animals, Pusey and Wolf (1996) list
206 experimental studies on mate preferences. Five of the seven studies with relevant
207 information found evidence for preference of intermediately related individuals (first or
208 second cousins) while full sibs were avoided. Preference for intermediately related mates
209 may be a common phenomenon also in natural environments. In a recent review of mate
210 choice in birds, Mays et al. (2008) found that in general preference seems to be for
211 genetically similar rather than for genetically dissimilar mates, and this pattern is evident
212 in both choice of social mate (five out of eight studies) and extra-pair mate choice (14 out
213 of 18 studies). It certainly seems that the empirical evidence is in qualitative agreement
214 with the prediction that intermediately related individuals should be preferred as
215 reproductive partners. Inbreeding depression (b) on fitness components has been
216 estimated to range between 0.55 and 0.82 in the wild (Crnokrak & Roff 1999). As
217 inbreeding depression for total fitness is probably considerably higher than inbreeding
218 depression on components of fitness, values of b predicting preference for intermediately
219 related mates (1 < b < 2) are expected to be common in natural populations.
220
221 The analysis of optimal inbreeding predicts that for a range of realistic values of
222 inbreeding depression, positive F values should be observed in populations practicing
223 mate choice for optimal inbreeding. This immediately suggests an empirical test of the
224 model: the level of inbreeding in a population (F) should relate to strength of inbreeding
225 depression (b) as in equation (11). While such analysis comparing inbreeding depression 226 and inbreeding would be very welcome, it must be noted that deciding the proper F
227 statistic for inferring inbreeding may prove difficult. Inferring inbreeding will be easy if
228 the population has clear boundaries and the individuals can easily choose partners from
229 the entire population. In this type of scenario, inbreeding will be detected as positive FIS
230 at conception. If the population is more widespread, and/or dispersal of individuals is low
231 in relation to the distribution of the population, it is more difficult to decide on the correct
232 reference population. If there is isolation by distance, breeding can be random at local
233 scale but non-random at larger scales. Even with random local mating, FIT will be
234 positive because of the spatial structuring of the population, and hence inbreeding is
235 taking place. The intriguing viewpoint emerging from the current analysis is that the
236 isolation by distance may arise as a consequence of mate choice for optimal inbreeding: if
237 mating with kin maximizes inclusive fitness, this will select for short dispersal distances
238 and hence give rise to spatial structuring in the population. Short dispersal distances and
239 population structuring could thus plausibly evolve in the absence of any costs to dispersal,
240 simply as a mechanism to maximize inclusive fitness via mate choice (see also Lehmann
241 & Perrin 2003).
242
243 The analysis also suggests a new approach to the theory of mating system evolution in
244 hermaphroditic organisms. The main question in hermaphrodite mating system evolution
245 has been to understand the forces directing the evolution of self-fertilization versus cross-
246 fertilization. This approach is conceptually identical to the study of evolution of incest
247 among first-order relatives versus complete outcrossing discussed at length above. The
248 very basic theory of hermaphrodite mating system evolution predicts the evolution of full 249 selfing when self-fertilization depression is less than ½, and complete outcrossing when
250 self-fertilization depression is more than ½, though many ecological and genetic forces
251 are known to cause deviations from this prediction (see e.g. Uyenoyama et al. 1993, de
252 Jong & Klinkhamer 2005). The current analysis however suggests that biparental
253 inbreeding, not selfing, is expected for self-fertilization depression values between 1 and
1 254 3, without any assumptions about purging, pollen discounting, or linkage between
255 fitness� loci and mating system modifiers (see Uyenoyama et al. 1993). Unfortunately,
256 most analyses of hermaphrodite mating systems have not distinguished between
257 biparental inbreeding and self-fertilization as sources of inbreeding, leaving open the
258 question of the role of biparental inbreeding on estimated rates of self-fertilization
259 (Goodwillie et al. 2005). New methods employing multilocus genetic estimates of mating
260 system parameters and information about population spatial genetic structure provide
261 tools for simultaneous estimation of selfing and biparental inbreeding, and data are
262 starting to accumulate (Vekemans & Hardy 2004, Herlihy & Eckert 2004, Jarne & Auld
263 2006, Jarne & David 2008). It seems that for population characterized by high F values,
264 selfing is indeed the main source of inbreeding. Because self-fertilization can function as
265 a reproductive assurance mechanism, and because physiological mechanisms of selfing
266 differ from biparental reproduction, it is to be expected that a model derived for optimal
267 biparental inbreeding will not capture all key aspects affecting the evolution of self-
268 fertilization. Nevertheless, considering the possibility that biparental inbreeding may be
269 adaptive might yield important insights to evolution of hermaphrodite mating systems.
270 271 The analysis presented in this paper ignores many aspects likely to influence mating
272 system evolution in nature. As a simplified model, the analysis however pins down the
273 finding that in absence of any complicating factors related to ecology or the specifics of
274 genetic systems, selection should favor non-zero inbreeding for wide range of inbreeding
275 depression strengths. In future work, it will be worth studying the consequences of
276 factors like costs of mate choice, sexual conflict over mate choice (Parker 1979, Pizzari et
277 al. 2004), kin recognition mechanisms, purging of genetic load with inbreeding, and
278 possible non-linear inbreeding depression (synergistic epistasis) (Charlesworth et al. 1991)
279 on model predictions. Broadening the scope of adaptive inbreeding from incest between
280 close relatives to inbreeding optimized with respect to relatedness holds promise of a
281 rewarding avenue for both theoretical and empirical research.
282 Acknowledgements
283 I thank Janne S. Kotiaho for inspiring discussions, Risto Paatelainen for help with the
284 recursion solution, and two anonymous referees for constructive comments. This study
285 was funded by Academy of Finland (grant 7121616).
286
287 288 289 Literature Cited 290
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342
343 Figure 1. Threshold inbreeding depression (δ) limiting the evolution of inbreeding as a
344 function of parent relatedness (r) in an outbred population assuming male
345 opportunity cost c = 0.
346
347 Figure 2. The effect of incestuous mating on male and female fitness for various values of
348 inbreeding depression (b), assuming male opportunity cost c = 0 (equations 6 and
349 7). For males, outbreeding is always more beneficial than incest. Incest however
350 yields positive fitness returns for realistic values of inbreeding depression,
351 especially when relatedness to the female is not extreme. For females, there often
352 exists an optimum relatedness that maximizes fitness (see text for details).
353
354 Figure 3. The stable level of inbreeding (F*, equation 11) in populations practicing
355 optimal inbreeding as a function of the strength of linear inbreeding depression
356 (b). When 1/3 < b < 2, population is expected to be partially inbred (F* is
357 between 0 and 1), and when b > 2, dissassortative mating leading to negative F*
358 values is expected. 1 male threshold 0.9 female threshold 0.8 ) δ 0.7
0.6
0.5
0.4
0.3 Inbreeding depression ( depression Inbreeding 0.2
0.1
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
359 r 360 361 Figure 1. Threshold inbreeding depression (δ) limiting the evolution of inbreeding as a
362 function of parental relatedness (r) in an outbred population assuming male opportunity cost
363 c = 0.
364 1 male,male, b b = = 1 1 male,male, b b = = 1.33 1.33 male,male, b b = = 1.6 1.6 0.8 female,female, b b = = 1 1 female,female, b b = = 1.33 1.33 0.6 female,female, b b = = 1.6 1.6
0.4
0.2 Effect of incest on fitness incest of Effect
0
-0.2 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 r 365 366 367 Figure 2. The effect of incestuous mating on male and female fitness for various values of
368 inbreeding depression (b), assuming male opportunity cost c = 0 (equations 6 and 7). For
369 males, outbreeding is always more beneficial than incest. Incest however yields positive
370 fitness returns for realistic values of inbreeding depression, especially when relatedness
371 to the female is not extreme. For females, there often exists an optimum relatedness that
372 maximizes fitness (see text for details).
373 1.0 0.9 0.8 0.7 0.6 0.5 F* 0.4 0.3 0.2 0.1 0.0 -0.1 0.33 0.67 1.00 1.33 1.67 2.00 2.33 2.67 3.00 Inbreeding depression (b) 374
375 Figure 3. The stable level of inbreeding (F*, equation 11) in populations practicing
376 optimal inbreeding as a function of the strength of linear inbreeding depression (b). When
377 1/3 < b < 2, population is expected to be partially inbred (F* is between 0 and 1), and
378 when b > 2, dissassortative mating leading to negative F* values is expected.
379