Evolution of reduced mutation under frequency-dependent selection

Theoretical Population Biology 112 (2016) 52–59

Contents lists available at ScienceDirect

Theoretical Population Biology journal homepage: www.elsevier.com/locate/tpb

Evolution of reduced mutation under frequency-dependent selection Uri Liberman a , Hilla Behar b , Marcus W. Feldman b,∗ a

School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

b

Department of Biology, Stanford University, Stanford, CA 94305-5020, United States

article

info

Article history: Received 19 April 2016 Available online 25 August 2016 Keywords: Reduction principle Frequency dependence Multiple modifier alleles Diploid model

abstract Most models for the evolution of mutation under frequency-dependent selection involve some form of host–parasite interaction. These generally involve cyclic dynamics under which mutation may increase. Here we show that the reduction principle for the evolution of mutation, which is generally true for frequency-independent selection, also holds under frequency-dependent selection on haploids and diploids that does not involve cyclic dynamics. © 2016 Elsevier Inc. All rights reserved.

1. Introduction In large populations, when the selection on one or more genetic loci is constant over time (and hence frequency-independent), and there is mutation (or recombination) among the alleles at these loci, the Reduction Principle applies (Feldman and Liberman, 1986). The Reduction Principle states that a new allele introduced at a locus that controls the mutation (or recombination) rate among the alleles under selection at other (linked or unlinked) loci near a stable equilibrium of the mutation–selection system (or recombination–selection system with linkage disequilibrium) will invade the population if it reduces the mutation (or recombination) rate (Liberman and Feldman, 1986a,b). The same reduction result holds for a gene that modifies migration in a migration–selection system (Liberman and Feldman, 1989). The gene or genes on which selection occurs are usually called ‘‘major’’, while the locus controlling the parameter of interest (rate of mutation, recombination, or migration) is called the ‘‘modifier’’ locus (Feldman et al., 1997). The Reduction Principle applies to alleles that reduce mutation, recombination, and migration rates under any constant selection regime that allows an appropriate stable equilibrium at which the modifier allele is introduced. These evolutionary genetic models suggest an alternative to the phenotypic dynamic approach, usually called ‘‘evolutionarily stable strategy’’ (ESS), that Eshel and Feldman (1982) termed ‘‘evolutionary genetic stability’’ (EGS). The latter applies to both haploid and diploid evolutionary dynamics. In the cases of

∗

Corresponding author. E-mail addresses: [email protected] (U. Liberman), [email protected] (H. Behar), [email protected] (M.W. Feldman). http://dx.doi.org/10.1016/j.tpb.2016.07.004 0040-5809/© 2016 Elsevier Inc. All rights reserved.

mutation, recombination, and migration rates, the value zero has the property of EGS. For the case of mutation rate, Rosenbloom and Allen (2014) use the terminology ‘‘evolutionary stable mutation rate’’ (ESMR) to mean exactly the same thing. That the value zero has the property of EGS also entails that when the relevant rate is zero, a new modifier allele that increases the rate to a positive value cannot increase in frequency at a geometric rate. Violations of the Reduction Principle have been demonstrated under a number of model scenarios that violate the conditions stated above. For example, increased recombination may evolve if the recombination-increasing allele arises while the major loci are proceeding towards fixation (Maynard Smith, 1980, 1988; Bergman and Feldman, 1990). Increased recombination may also evolve when the major loci are under cyclically fluctuating selection, either exogenously caused (Charlesworth, 1976) or induced by host–parasite dynamics (Hamilton, 1980; Nee, 1989; Gandon and Otto, 2007). Mutation rates may also increase under some patterns of fluctuating selection. This is often studied in the context of phenotypic switching between phenotypes, represented in population genetic models as different haploid genotypes. Experimental (Acar et al., 2005, 2008) and theoretical analyses (Leigh, 1970; Ishii et al., 1989; Lachmann and Jablonka, 1996; Thattai and van Oudenaarden, 2004; Kussell and Leibler, 2005; Gaal et al., 2010; Liberman et al., 2011) have shown that cyclically (and some forms of stochastically) fluctuating selection can select for alleles that increase mutation rates. However, the direction of change in mutation rates can be very sensitive to the form of selection on the major loci, for example, whether it is symmetric (Salathe et al., 2009; Liberman et al., 2011; Carja et al., 2014). The fluctuations in the selection regime assumed in these studies are exogenous. Most studies of mutation rate evolution with endogenously changing selection have involved some form of host–parasite (or

U. Liberman et al. / Theoretical Population Biology 112 (2016) 52–59

host–virus) interaction (Haraguchi and Sasaki, 1996; Kamp et al., 2003; Pal et al., 2007; M’Gonigle et al., 2009). Mutation rates can increase under host–parasite cycling. However, M’Gonigle et al. show that the stable mutation rate decreases as the recombination between the modifier and major genes increases. A different kind of cyclical trait dynamics was used by Rosenbloom and Allen (2014) in their analysis of the effect of frequencydependent selection on the evolution of the mutation rate. In their model, the cycling was generated by ‘‘rock–paper–scissors’’ competition. The analysis of the invasion process for this model was carried out using adaptive dynamics, namely comparison of the marginal fitnesses of different mutation-modifying alleles. This analysis must be distinguished from the formal multi-dimensional analysis of modifier evolution that has become standard in population genetics (Feldman et al., 1997). Rosenbloom and Allen (loc. cit.) found an evolutionarily stable mutation rate (ESMR) that was non-zero. They also found, as did M’Gonigle et al. in the host–parasite case, that recombination (as modeled in their scheme, which is somewhat different from the way recombination is incorporated into population genetic models) reduced the stable mutation rate. Although the model of Rosenbloom and Allen (2014) and its predecessor Allen and Rosenbloom (2012) were developed in terms of continuous time, the stability analyses that determined invasion made use of the Perron–Frobenius structure of the local stability matrix near the invaded equilibrium. This structure of the positive matrix whose leading eigenvalue determined whether or not a new modifier allele would invade also formed the basis of the analyses by Liberman and Feldman (1986a,b) of the evolution of mutation and recombination. Our objective in this note is to explore how the evolution of mutation rate is affected by frequency-dependent selection. We use classical models of haploid and diploid selection on a single (major) diallelic locus. The modifier locus affects the symmetric (i.e., equal in both directions) mutation rate between the alleles at the major gene. Our model of frequency-dependent selection is classical—in the sense that the fitness is a function of the allele frequency at the major locus (see, e.g., Wright, 1969, Chapter 2). We show that if there is a stable mutation–selection equilibrium then a new mutation-modifying allele introduced near this equilibrium will invade if it reduces the mutation rate. As a result, zero mutation has the property of evolutionary genetic stability (Eshel and Feldman, 1982). We also show that this result does not depend on the rate of recombination between the major and modifier loci and that the zero mutation rate cannot be invaded at a geometric rate. 2. The model

genotype fitness frequency with have

4

i=1

w1 = w2

AM

Am

aM

w3

w4

x1

x2

x3

x4

w1

In the next generation, after selection, random union of gametes, recombination, and segregation (in that order), the new genotypic frequencies are x′1 , x′2 , x′3 , x′4 :

w x w1 w4  1 1 −r D 2 w x w w w w  3 3 1 4 + µM +r D w w2 w 2 x2 w1 w4  x′2 = (1 − µm ) +r D 2 w x w w w w  4 4 1 4 −r D + µm w w2 w3 x3 w1 w4  x′3 = (1 − µM ) +r D 2 w x w w w w  1 1 1 4 + µM −r D w w2 w 4 x4 w1 w4  x′4 = (1 − µm ) −r D 2 w x w w w w  2 2 1 4 + µm +r D , w w2 x′1 = (1 − µM )

w2

am (1)

xi = 1. As the modifier locus is selectively neutral, we

w3 = w4 .

(2)

(3)

where r is the recombination fraction between the major and modifier loci, D the linkage disequilibrium: D = x1 x4 − x2 x3 ,

(4)

and w is the mean fitness

w=

4 

w i xi .

(5)

i=1

In what follows we will assume that the selection is frequencydependent; that is, wi = wi (x) for i = 1, 2, 3, 4 where x = (x1 , x2 , x3 , x4 ) is the frequency vector. We proceed to evaluate the effect of the frequency-dependent selection on the evolution of mutation. Specifically, our goal is to determine whether the Reduction Principle for mutation rates seen in the case of constant selection (Feldman and Liberman, 1986) holds when selection is frequencydependent. 3. Equilibria In the absence of the modifier allele m (in which case recombination is irrelevant) x2 = x4 = 0, and the transformation (3) reduces to

wx′1 = (1 − µM ) w1 x1 + µM w3 x3 wx′3 = (1 − µM ) w3 x3 + µM w1 x1 ,

(6)

with x1 + x3 = x′1 + x′3 = 1. Let u = x1 /x3 , u′ = x′1 /x′3 and write the frequency dependent fitness parameters as

w1 = w1 (u),

Consider a large population of haploids evolving under the influence of selection, recombination, and mutation. The fitness of an individual is determined by its genotype at one locus, with alleles A and a, which is linked to a modifier locus with alleles M and m that produce mutation rates µM and µm , respectively. The mutation rates from A to a and from a to A at the major locus are the same, and the modifier locus is selectively neutral so it does not affect the fitness parameters for the major locus. Thus there are four genotypes:

53

w3 = w3 (u).

(7)

Then in terms of u, the transformation (6) is equivalent to

(1 − µM ) w1 u + µM w3 . (8) µM w1 u + (1 − µM ) w3 At equilibrium, f (u) = u and the equilibria are the solutions of Q (u) = 0, where u′ = f (u) =

Q (u) = µM w1 u2 + (1 − µM ) (w3 − w1 ) u − µM w3

(9)

with w1 = w1 (u), w3 = w3 (u). We will assume that Q (u) = 0 has a root u∗ that determines a unique equilibrium x∗ = (x∗1 , 0, x∗3 , 0) with x∗1 = u∗ /(1 + u∗ ), x∗3 = 1/(1 + u∗ ) while x2 = x4 = 0. Remark 1. The kind of frequency-dependent selection on the major loci included here precludes the kind of cycling in the absence of mutation that is treated in Rosenbloom and Allen (2014). That kind of cycling requires an analysis different from

54

U. Liberman et al. / Theoretical Population Biology 112 (2016) 52–59

  w∗ w ∗ x∗ w ∗ w ∗ x∗ 1 − µm ) 1∗ 1 − r 3 ∗ 3 + r µm 1∗ 3 ∗ 3 (  w w w w   L∗ex =   w1∗ w1∗ w3∗ x∗3 w3∗ x∗3 + µm ∗ 1 − r ∗ r (1 − µm ) ∗ w w∗ w w 

  w3∗ w1∗ x∗1 w3∗ w1∗ x∗1 + µ 1 − r m w∗ w∗ w∗ w∗  , ∗  ∗ ∗ ∗ w3 w3 w1∗ x∗1  w1 x1 ( 1 − µm ) ∗ 1 − r ∗ + r µm ∗ ∗ w w w w

r ( 1 − µm )

(12)

Box I.

what follows below (see also Salathe et al., 2009; Liberman et al., 2011). 4. Stability of equilibria The local stability of x∗ is determined by the linear approximation L∗ of the transformation (3) ‘‘near’’ x∗ . Specifically, let x = x∗ + ε where ε = (ε1 , ε2 , ε3 , ε4 ) with εi ‘‘small’’ and ε1 +ε2 +ε3 +ε4 = 0 so that x′ = x∗ + ε ′ where ε ′ = (ε1′ , ε2′ , ε3′ , ε4′ ) with εi′ ‘‘small’’ and ε1′ + ε2′ + ε3′ + ε4′ = 0. Then x = x + ε = x + L ε. ′

∗

′

∗

∗

(10)

The stability of the equilibrium x∗ to the introduction of allele m is determined by the eigenvalues of the matrix L∗ . This scenario is well known from classical modifier theory (e.g., Feldman and Liberman, 1986), and the matrix L∗ is known to have the structure 1

 ∗

2

L∗in

 L = 0 0

0 0

3

4

⊕ ⊕

 ⊕ ⊕ 

L∗ex

1 2 3 4

,

(11)

where we have swapped columns 2 and 3 and rows 2 and 3 to show the structure. The entries marked ⊕ do not affect the eigenvalues of L∗ . The eigenvalues of L∗ are therefore those of the sub-matrices L∗in and L∗ex , where L∗in determines the internal stability of x∗ confined to the boundary with only M present. We will assume that x∗ is internally stable, so these eigenvalues are less than one in magnitude. L∗ex is the linear approximation to the evolution near x∗ involving only the gametes Am and am. From (3) we have L∗ex given in Box I, where

w1∗ = w1 (u∗ ), w3∗ = w3 (u∗ ), w∗ = w1 (u∗ )x∗1 + w3 (u∗ )x∗3 .

(13)

Observe that Lex is a positive matrix and by the Perron–Frobenius theory, L∗ex has a positive eigenvalue which is the largest in magnitude. Let M (z ) = det(L∗ex − zI) be the characteristic polynomial of L∗ex . M (z ) is a quadratic polynomial that we claim has the following properties. Claim 1.

(1 − 2µm ) (1 − r ) w1∗ w3∗ (w∗ )2    1 M (1) = (µm − µM ) w1∗ − w3∗ w1∗ x∗1 − w3∗ x∗3 2 ∗ (w ) w∗ + w∗ w∗ w∗ M ′ (1) = 2 − (1 − µm ) 1 ∗ 3 + r (1 − 2µm ) 1 32 . w (w ∗ )

2w ∗ − (w1∗ + w3∗ ) = 2 w1∗ x∗1 + w3∗ x∗3 − w1∗ x∗1 + x∗3









  − w3∗ x∗1 + x∗3     = w1∗ x∗1 − x∗3 + w3∗ x∗3 − x∗1    = w1∗ − w3∗ x∗1 − x∗3 .    To sum up, M ′ (1) > 0 if w1∗ − w3∗ x∗1 − x∗3 > 0. Remark 3. If at equilibrium u∗ = 1, then x∗1 = x∗3 =

(17) (18) (19)

1 2

and

= ( , 0, , 0). In this case, from Q (1) = 0 in (9), we have   w1∗ = w3∗ since Q (1) = (1 − 2µM ) w3∗ − w1∗ and 0 ≤ µM < 21 . Then by (15), M (1) = 0, the largest eigenvalue of L∗ex is 1, and the other eigenvalue is less than 1 and equals (1 − 2µM )(1 − r ) since in this case M (0) = (1 − 2µm )(1 − r ) by (14). In this case the equilibrium does not depend on µM , modification cannot occur

x

∗

1 2

1 2

and we do not discuss this case further. That is, we will assume that u∗ ̸= 1. Claim 2. At any equilibrium x∗ with associated u∗ ̸= 1 we have

 ∗   w1 − w3∗ x∗1 − x∗3 > 0,  ∗   w1 − w3∗ w1∗ x∗1 − w3∗ x∗3 > 0.

(20) (21)

The proof of Claim 2 is given in Appendix B. We are now ready to prove our main result. 5. The reduction principle Result 1. Let x∗ = (x∗1 , 0, x∗3 , 0) be an internally stable equilibrium on the boundary where only the M allele is present. Assume that x∗1 ̸= x∗3 , 0 ≤ µM , µm < 12 , and 0 ≤ r < 1. Then x∗ is stable if µm > µM and unstable if µm < µM . So for all recombination rates, a modifier allele that reduces the mutation rate will invade. Proof. x∗ is stable if the largest positive eigenvalue of L∗ex is less

∗

M (0) =

we have

(14) (15) (16)

than 1. As M (z ) is quadratic in z and both M (0) =

(1−2µm )(1−r )w1∗ w3∗ (w∗ )2 ∗

and M (±∞) are positive, this largest positive eigenvalue of Lex is less than 1 if M (1) > 0 and M ′ (1) > 0, and it is larger than 1 if M (1) < 0. ∗   ∗ 1 and  Claim 2, at ′ the equilibrium x we have  ∗By Remark ∗ ∗ w1 − w3 x1− x3 > 0 and so M (1) >  0. Also, since w1∗ − w3∗ w1∗ x∗1 − w3∗ x∗3 > 0, the sign of M (1) in (15) coincides with the sign of (µm − µM ). Thus if µm > µM both M (1) and M ′ (1) are positive and x∗ is stable. If µm < µM then M (1) < 0 and x∗ is unstable.

We have thus secured the Reduction Principle for the evolution of mutation under the influence of any form of frequency-dependent selection that does not entail cycling of the chromosome frequencies.

These claims are proved in Appendix A.

6. Existence and stability of the boundary equilibria

Remark 2. Thus since 0 ≤ r ≤ 1 and 0 ≤ µm ≤ 21 , M ′ (1) > 0 if 2w ∗ > (w1∗ + w3∗ ). But since w ∗ = w1∗ x∗1 + w3∗ x∗3 and x∗1 + x∗3 = 1

In the above analysis we assumed that on the boundary where x2 = x4 = 0 there exists an equilibrium x∗ = (x∗1 , 0, x∗3 , 0) that

U. Liberman et al. / Theoretical Population Biology 112 (2016) 52–59

is internally stable. It is therefore interesting to find out under what conditions this assumption holds. For existence of such an equilibrium we have the following result. Result 2. On the boundary where x2 = x4 = 0 there exists at least one equilibrium x∗ = (x∗1 , 0, x∗3 , 0). Moreover, if u∗ = x∗1 /x∗3 then

(i) u∗ < 1 if w1 (1) < w3 (1) (ii) u∗ = 1 if w1 (1) = w3 (1) (iii) u∗ > 1 if w1 (1) > w3 (1). Proof. Let u = x1 /x3 . Then an equilibrium exists if Q (u) = 0, where from (9) Q (u) = µM w1 u2 + (1 − µM ) (w3 − w1 ) u − µM w3

(22)

with w1 = w1 (u), w3 = w3 (u). Now Q (0) = −µM w3 (0) < 0 and Q (1) = (1 − 2µM ) [w3 (1) − w1 (1)] .

(23)

Hence, if w3 (1) > w1 (1) there exists u∗ with 0 < u∗ < 1 such that Q (u∗ ) = 0. If w3 (1) = w1 (1) then Q (1) = 0 and u∗ = 1. If w3 (1) < w1 (1) let v = 1/u, then the equilibrium equation (22) is equivalent to R(v) = 0 where R(v) = µM w3 v 2 + (1 − µM ) (w1 − w3 ) v − µM w1 , with w1 = w1 (v), w3 = w3 (v), and v = 1 if and only if u = 1. But R(0) = −µM w1 (0) < 0 and R(1) = (1 − 2µM ) [w1 (1) − w3 (1)]. Therefore when w1 (1) > w3 (1) there exists a 0 < v ∗ < 1 such that R(v ∗ ) = 0. As u∗ = 1/v ∗ we conclude that when w3 (1) < w1 (1) there exists a u∗ > 1 such that Q (u∗ ) = 0. Now that we know that there is always at least one boundary equilibrium x∗ = (x∗1 , 0, x∗3 , 0), it is important to ascertain when it is internally stable. The (internal) stability of x∗ on the boundary where x2 = x4 = 0 is determined by the nature of the transformation u′ = f (u) in (8). Assume that Q (u) = 0 has a finite number of roots all of which are simple roots. That is, when Q (u) = 0 we also have Q ′ (u) ̸= 0. Under these assumptions we have the following result, which is proved in Appendix C. Result 3. Let u∗ = minu {Q (u) = 0}. Then its associated equilibrium x∗ = (x∗1 , 0, x∗3 , 0) with x∗1 = u∗ /(1 + u∗ ), x∗3 = 1/(1 + u∗ ) is internally stable provided u′ = f (u) is an increasing function of u. The final question, of course, is: When is the function f (u) monotone increasing in u? In general, from (8) we have f ′ (u) =

  (1 − 2µM ) w1 w3 + w1′ w3 u − w1 w3′ u [µM w1 u + (1 − µM ) w3 ]2

.

(24)

As 0 ≤ µM < 21 there are several sufficient conditions for f ′ (u) > 0 for all u. These are: (i) w1 (u), w3 (u) are both monotone in u, the first increasing and the second decreasing; w (u) (ii) the ratio w1 (u) is a monotone increasing function of u; 3

(iii) w1 (u) is constant, and of u.

w3 (u) u

is a monotone decreasing function

55

fitness parameters were assumed to be constant, the same analysis works in the case where selection is frequency-dependent. We summarize the ingredients of the model and the results. Consider a diploid population and a major locus with two alleles A and a; the three genotypes are AA, Aa, and aa with associated fitness parameters w11 , w12 , w22 , respectively, which can be frequency-dependent. At the modifier locus there are n possible alleles M1 , M2 , . . . , Mn that determine the mutation rates at the major locus. Assuming the same mutation rates from A to a as from a to A, the genotype Mi Mj at the modifier locus produces the mutation rate µij at the major locus where µij = µji for i, j = 1, 2, . . . , n. The recombination rate between the two loci is r, and it assumed that 0 ≤ r ≤ 1 and 0 ≤ µij ≤ 21 for all i and j. Let x1 and x2 = 1 − x1 be the frequencies of A and a, respectively, at the present generation. Then the frequencies of the AMi and aMi gametes can be represented as freq(AMi ) = x1 pi freq(aMi ) = x2 qi

i = 1, 2, . . . , n,

(25)

n n i=1 pi = i=1 qi = 1.  n Let M = µij i,j=1 be the n × n matrix of mutation rates, E the n × n matrix all of whose entries are 1, p = (p1 , p2 , . . . , pn ), q = (q1 , q2 , . . . , qn ). Then the frequencies x′1 , x′2 , p′ = (p′1 , p′2 , . . . , p′n ), q′ = (q′1 , q′2 , . . . , q′n ) at the next generation are given as: where 0 ≤ pi , qi ≤ 1, and

C p′ = w11 x21 p ◦ (E − M) p + w22 x22 q ◦ Mq

  + w12 x1 x2 (1 − r ) p ◦ (E − M) q + q ◦ Mp   + w12 x1 x2 r p ◦ Mq + q ◦ (E − M) p , Dq = w ′

◦ (E − M) q + w ◦ Mp   + w12 x1 x2 (1 − r ) q ◦ (E − M) p + p ◦ Mq   + w12 x1 x2 r q ◦ Mp + p ◦ (E − M) q , 2 22 x2 q

(26)

2 11 x1 p

(27)

where C = w x′1 = w11 x21 (1 − µ) ¯ + w22 x22 ν¯ + w12 x1 x2 (1 − µ ˜ + ν˜ ) , D = w x′2 = w22 x22 (1 − ν¯ ) + w11 x21 µ ¯ + w12 x1 x2 (1 − ν˜ + µ) ˜ , (28)

w = w11 x21 + 2w12 x1 x2 + w22 x22 , and

µ ¯ = (p, Mp),

ν¯ = (q, Mq),

µ ˜ = ν˜ = (p, Mq).

(29)

In the above, if a = (a1 , a2 , . . . , an ), b = (b1 , b2 , . . . , bn ), then a ◦ b = ( a1 b1 , a2 b2 , . . . , an bn ) is the Schur product of vectors and (a, b) = ni=1 ai bi is the scalar product. Also, a ⊗ b is the Kronecker product of the vectors a and b, where (a ⊗ b)ij = ai bj . Although Liberman and Feldman (1986a) assumed that the fitness parameters wij are fixed, we assume here that wij = wij (x1 , x2 ). That is, selection on the major locus can be frequency dependent. It turns out that exactly the same results as in the haploid model hold in this diploid model. These results are as follows. Result 4. At equilibrium, either p = q or the equilibrium frequencies of A and a do not depend on the mutation rates of M.

7. Extensions to the diploid case

As we are interested in the evolution of mutation, we concentrate on equilibria where p = q and we have the following result.

Our analysis so far assumed a haploid population with a major diallelic locus under selection, linked to a modifier locus with alleles M and m, whose sole function is to determine the mutation rates at the major locus. Our results can be extended to a diploid population and also generalized to the case of an arbitrary number of alleles at the modifier locus. This extension follows Liberman and Feldman (1986a). Although in that paper the

Result 5. Suppose there exists a frequency vector p∗ for which p∗ ◦ Mp∗ = µ∗ p∗ , where µ∗ = (p∗ , Mp∗ ). Then corresponding to p∗ there is at least one Hardy–Weinberg equilibria x∗ ⊗ p∗ of the twolocus system, where x∗ = (x∗1 , x∗2 ) and x∗1 , x∗2 solve Eqs. (28) at equilibrium with µ ¯ = ν¯ = µ ˜ = ν˜ = µ∗ = (p∗ , Mp∗ ) and ∗ ∗ ∗ ∗ wij = wij (x1 , x2 ), w = w(x1 , x2 ). We call these equilibria ‘‘viability analogous Hardy–Weinberg’’ (VAHW) equilibria.

56

U. Liberman et al. / Theoretical Population Biology 112 (2016) 52–59

Observe that when p = q the equilibrium equations for x1 , x2 resulting from (28) are

w x1 = w11 x21 (1 − µ) + w22 x22 µ + w12 x1 x2 w x2 = w

2 22 x2

(1 − µ) + w

2 11 x1

(30)

µ + w12 x1 x2 ,

where p ◦ Mp = µp, µ = (p, Mp), and wij = wij (x1 , x2 ). Let u = x1 /x2 , then (30) is equivalent to u=

w11 (1 − µ) u2 + w12 u + w22 µ , w11 µu2 + w12 u + w22 (1 − µ)

(31)

or Q (u) = w11 µu3 + w12 − w11 (1 − µ) u2





  − w12 − w22 (1 − µ) u − w22 µ = 0. Since wij = wij (x1 , x2 ), we can write wij = wij (u). Now Q (0) = −µw22 (0) < 0, while Q (∞) has the sign of µw11 (∞) since wij (∞) is finite (because u = ∞ corresponds to x1 = 1, x2 = 0). Thus Q (0) < 0, Q (∞) > 0, and hence Q (u) = 0 has at least one root u∗ > 0 where Q (u∗ ) = 0. This proves that there is at least one x∗ for which x∗ ⊗ p∗ is an equilibrium of the two-locus system. Suppose that a stable VAHW polymorphic  equilibrium {p∗ , x∗ } n ∗ ∗ ∗ ∗ ∗ ∗ exists and let µ = (p , Mp ). Then µ = i,j=1 µij pi pj is the average mutation rate among all A (or a) carrying gametes. We investigate the external stability of this equilibrium to the invasion of a new allele Mn+1 at the modifier locus. Let µi,n+1 be the mutation rate between A and a if the modifier locus is heterozygous Mi Mn+1 . At the VAHW equilibrium with the frequencies p∗1 , p∗2 , . . . , p∗n of M1 , M2 , . . . , Mn we define

µ ˆ =

n 

µi,n+1 p∗i

(32)

as the average mutation rate from A to a (or from a to A) over all gametes carrying the allele Mn+1 at the modifier locus. Then we have the following result. Result 6. The VAHW polymorphism is externally stable if µ∗ < µ ˆ and unstable if µ∗ > µ ˆ . Thus with frequency-dependent selection there is selection in favor of an allele which reduces the mutation rate. Near the VAHW equilibrium {p , x } with associated mutation rate µ∗ = (p∗ , Mp∗ ), the linear approximation for pn+1 and qn+1 , the frequencies of AMn+1 and aMn+1 , respectively, that determine the external stability of {p∗ , x∗ }, is ∗

∗

  ∗ ∗ w∗ p′n+1 = w1∗ (1 − µ) ˆ − r w12 x2 (1 − 2µ) ˆ pn+1   ∗ ∗ w 2 x2 ∗ ∗ µ ˆ + r w12 x2 (1 − 2µ) ˆ qn+1 + x∗1  ∗ ∗  w1 x1 ∗ ∗ w∗ q′n+1 = µ ˆ + r w12 x1 (1 − 2µ) ˆ pn+1 x∗2  ∗  ∗ ∗ + w2 (1 − µ) ˆ − r w12 x1 (1 − 2µ) ˆ qn+1 . ∗ ∗ ∗ ∗ w1∗ = w11 x1 + w12 x2 , ∗ ∗ ∗ ∗ ∗ w = w 1 x1 + w 2 x2 ,

wij = wij (u ), ∗

∗

(33)

∗ ∗ ∗ ∗ w2∗ = w12 x1 + w22 x2 ,

x∗1

p′n+1 q′n+1



∗

.

(34)

u = ∗ x2

= Lex



(w∗ )2 x∗ x∗

 µ ˆ − µ∗ ,

(36)

1 2



µ ˆ −µ

∗



> 0 implies Q (1) > 0, ′

and Result 6 follows. Therefore the Reduction Principle for mutation rates holds in the diploid case where selection can be frequency-dependent. If n = 1, only M1 is initially present at the modifier locus and {p∗ , x∗ } is such that p∗ = (1, 0, 0, . . . , 0). Then the model reduces to the diploid case with only two alleles at the modifier locus. Thus our original result for haploids also holds in the diploid case. 8. Existence and stability of boundary equilibria for diploids As in the haploid case, we would like to know when x∗ is internally stable on the boundary with M1 , M2 , . . . , Mn present and when p∗ = q∗ and p∗ ◦ Mp∗ = µ∗ p∗ with µ∗ = (p∗ , Mp∗ ). On this boundary, as µ ˆ =µ ˜ = µ∗ ,

w x′1 = w11 x21 (1 − µ∗ ) + w22 x22 µ∗ + w12 x1 x2

(37)

w x′2 = w22 x22 (1 − µ∗ ) + w11 x21 µ∗ + w12 x1 x2 , where wij = wij (x). Let u = x1 /x2 , then wij = wij (u) and

w11 u2 (1 − µ∗ ) + w12 u + w22 µ∗ = f (u). (38) w11 u2 µ∗ + w12 u + w11 µ∗ It is important to know when f (u) is an increasing function of u, as then x∗ is internally stable. Since f (u) = g (u)/h(u),

u′ =

h(u)g ′ (u) − g (u)h′ (u) [h(u)]2

.

(39)

We write g ′ (u) and h′ (u) as sums: g ′ (u) = g1′ (u) + g2′ (u), h′ (u) = h′1 (u) + h′2 (u), where in g1′ , h′1 we differentiate u2 , u, 1 and in g2′ , h′2 we differentiate w11 , w12 , w22 . Thus we can write [h(u)]2 f ′ (u) = F1 (u) + F2 (u), Fi (u) = h(u)gi′ (u) − g (u)h′i (u),

i = 1, 2.

(40)

We have F1 (u) =

 w11 µ∗ u2 + w12 u + w22 (1 − µ∗ )   × 2w11 (1 − µ∗ )u + w12   − w11 (1 − µ∗ )u2 + w12 u + w22 µ∗   × 2w11 µ∗ u + w12 ,   F2 (u) = w11 µ∗ u2 + w12 u + w22 (1 − µ∗ )  ′  ′ ′ × w11 (1 − µ∗ )u2 + w12 u + w22 µ∗   − w11 (1 − µ∗ )u2 + w12 u + w22 µ∗  ′  ′ ′ × w11 (µ∗ )u2 + w12 u + w22 ( 1 − µ∗ ) . 

(41)

F1 (u) = 1 − 2µ∗

  w11 w12 u2 + 2w11 w22 u + w22 w12 ,   ′ ′ F2 (u) = 1 − 2µ∗ (w11 w12 − w11 w12 )u3 

(42)

 + (w22 w11 − w22 w11 )u + (w22 w12 − w22 w12 )u . ′



pn+1 , qn+1

 ∗ ∗ 2 w1 x1 − w2∗ x∗2 

On simplification,

Observe that (33) can be written as



Q (1) =

f ′ ( u) =

i=1

∗

eigenvalue with largest magnitude is less than 1 if Q (1) > 0, Q ′ (1) > 0, and it is larger than 1 if Q (1) < 0. It is easily seen (see Liberman and Feldman, 1986a) that

(35)

where L∗ex determines the external stability of the VAHW equilibrium. Let Q (z ) be the characteristic polynomial of L∗ex , namely Q (z ) = det(L∗ex − zI). Q (z ) is a quadratic polynomial and, as L∗ex is a positive matrix, by the Perron–Frobenius theory its positive

Since 0 ≤ µ∗ <

1 2

′

2

′

′

and u > 0, F1 (u) > 0 for all u > 0. Also sufw (u)

ficient conditions for F2 (u) > 0 are that the three functions w11 (u) , 12 w11 (u) w12 (u) , are monotone increasing. Without loss of generality w22 (u) w22 (u) we can assume w12 (u) = 1; hence sufficient conditions are that w11 (u) is a monotone increasing function and w22 (u) is a monotone decreasing function.

U. Liberman et al. / Theoretical Population Biology 112 (2016) 52–59

9. Discussion Our formal population genetic models for the evolution of genetic modifiers of the mutation rate are extended here to include frequency-dependent selection at the major locus. It should be stressed that this approach, following Feldman (1972) and Liberman and Feldman (1986a,b), is entirely based on the properties of the eigenvalues of the local stability matrix that governs the dynamics of the two-locus system near a starting (boundary) equilibrium. As noted in our earlier work on modifiers of mutation, recombination, and migration, these external local stability matrices are strictly positive, and the Perron–Frobenius theory for such matrices is a crucial component of the analysis. In the diploid case, with two major alleles and n modifier alleles, AMn+1 and aMn+1 appear close to the 2n-dimensional equilibrium of A, a with M1 , M2 , . . . , Mn . In both haploid and diploid models, it is somewhat surprising that the external stability matrix leads to the Reduction Principle, which holds for any frequency-dependent selection at the major loci, as long as there exists a starting stable equilibrium. The classical mutation–selection balance permits one such equilibrium, as described by Eq. (9), in the haploid case, while in the diploid case the analysis is carried out near a viability analogous Hardy–Weinberg equilibrium. An example of the latter with two modifier alleles occurs if µ11 > µ12 , µ22 > µ12 , in which case M2 would invade near fixation of M1 , and M1 would invade near fixation of M2 . The nature of resident equilibrium/equilibria under frequency-dependent selection and its stability inside these fixation boundaries become important. One case of frequency-dependence that is easy to describe allows more than a single equilibrium on the boundary with M fixed, say, prior to the appearance of m. Take w1 = 1 and w3 = a − bx1 + bx21 in the haploid model, and take a = 0.7, b = 1.5. Then w3 < w1 when the frequency of AM , x1 , is small or large, but w3 > 1 for an intermediate range of frequencies of AM. In this case there are three equilibria on the AM − aM boundary, two of which are stable in that boundary, and a mutation-reducing allele m increases in frequency when it is introduced near either of these. Thus for our analysis it is not necessary that there be only a single stable equilibrium in the boundary where m is absent. A second example reflects the widely observed advantage to rare genotypes (e.g., Hughes et al., 2013, Salceda and Anderson, 1988), especially to males in mating success. If w3 (x1 ) = 0.8 + 1.2x1 − 2.4x21 + 1.6x31 , for example, then w3 (0) = 0.8 and w3 (1) = 1.2, giving symmetrical advantages to A and a when each is rare. Again in this case reduced mutation will be favored, while if µM = 0, the leading eigenvalue of L∗ex is 1. We have given sufficient conditions for the existence of a boundary equilibrium in the haploid and diploid cases, but these are obviously not necessary conditions. Since the mathematical analysis of invasion (or not) by modifiers in large (deterministic) populations depends on the existence of such equilibria, it would be of interest to derive general necessary and sufficient existence conditions in both haploid and diploid cases. Zero mutation rate was shown by Liberman and Feldman (1986a) to be uninvadable. For diploid population genetic models, Eshel and Feldman (1982) termed this property of the zero mutation rate (recombination rate, migration rate) under constant selection ‘‘evolutionary genetic stability’’, because it derived from formal population genetic analysis rather than game-theoretic or adaptive dynamic analyses, such as those in Allen and Rosenbloom (2012) and Rosenbloom and Allen (2014). In the frequency-dependent case, we can also address whether mutation rate zero is uninvadable. Theorem 3 of Allen and Rosenbloom (2012) provides conditions on the fitnesses and mutation rate due to allele m under which invasion of zero mutation rate

57

can occur at a geometric rate. The following argument shows that in the haploid population genetic framework described here, this cannot occur. Suppose that w1 = 1 and w3 = w3 (x1 ) (recalling that the fitness w1 (x1 ) of A can be set to 1 without loss of generality), and w3 (x1 ) is differentiable. Then a polymorphism of AM and aM (in the absence of allele m), with zero mutation rate produced by M, is possible if there is a solution x∗1 to w3 (x1 ) = 1 with 0 < x∗1 < 1. (Equilibria with x1 = 0 and x1 = 1 also exist.) In the neighborhood of x∗1 , the evolution of allele m is again determined by the properties of the local stability matrix L∗ex . It is straightforward to show that in this case the leading eigenvalue of L∗ex is 1. Thus allele m cannot invade at a geometric rate, and the linear approximation is not able to describe its invasion properties, which seems to contradict Theorem 3 of Allen and Rosenbloom (2012), which reports conditions on w1 , w3 , and µm under which allele m can invade a population fixed on the zero mutation allele M. We should note that in our analyses of invasion we assume that allele m arises at an equilibrium of M that is locally (or, of course, globally) stable in the boundary where M is fixed. For the case described in the previous paragraph, it is straightforward to show that x∗1 is locally stable if 0 < w3′ (x∗1 ) < 2/[x∗1 (1 − x∗1 )]. In general, however, for the population genetic theory of the evolution of mutation, recombination, or migration rates, properties of the point at which the modifier is introduced, restricted to the frequency space prior to the introduction, also demand investigation. Haploid genetic modifier models such as those of Balkau and Feldman (1973), Salathe et al. (2009), Carja et al. (2014), and in many of the references cited therein, combine replicator dynamics and sequences of genetic substitutions in a single evolutionary framework. The role of this framework relative to those encompassed by evolutionarily stable strategies and adaptive dynamics is discussed in Feldman et al. (1997) and Spencer and Feldman (2005). As seen in our analysis of diploids, the scope for polymorphisms is greater with diploids, although some of the important conclusions from modifier theory with diploids, such as the Reduction Principle, also hold in many haploid models. In particular, under a wide set of assumptions on the frequencydependent selection regime, the Reduction Principle remains valid. Forthcoming studies will develop the mathematical underpinnings of the Reduction Principle for multiple major alleles in mutation, migration, and recombination modification. Acknowledgment This research supported in part by the Center for Computational, Evolutionary, and Human Genomics at Stanford University. Appendix A. Proofs of Eqs. (14), (15), (16) Proof of (14). M (0) = det L∗ex and from (12), we have M (0) as given in Box II. Add the second row of (A.1) to the first row and we have M (0) as given in Box III. Hence

   w1∗ x∗1 w1∗ x∗1 w1∗ w3∗ M (0) = ( 1 − µm ) 1 − r + µ r m 2 w∗ w∗ (w ∗ )  ∗ ∗ ∗ ∗  w x w x − r (1 − µm ) 3 ∗ 3 − µm 1 − r 3 ∗ 3 w w   ∗ ∗  ∗ ∗ w1 w3 w1 x1 w3∗ x∗3 = ( 1 − µm ) 1 − r ∗ − r ∗ w w (w ∗ )2  ∗ ∗  ∗ ∗ w x w x + µm r 1 ∗ 1 + r 3 ∗ 3 − 1 . w w

(A.3)

(A.4)

58

U. Liberman et al. / Theoretical Population Biology 112 (2016) 52–59

   w ∗ x∗ w3∗ x∗3   + µm r 3 ∗ 3 ( 1 − µ ) 1 − r m ∗ ∗  ∗ w1 w3  w w   w3∗ x∗3 w3∗ x∗3 (w∗ )2  r (1 − µm ) ∗ + µm 1 − r ∗ w w

  w1∗ x∗1 w1∗ x∗1   + µ 1 − r m w∗ w ∗    . w1∗ x∗1 w1∗ x∗1  ( 1 − µm ) 1 − r ∗ + µm r ∗  w w

r ( 1 − µm )

(A.1)

Box II.

w1∗ w3∗ (w∗ )2

  1     w3∗ x∗3 w3∗ x∗3 r (1 − µm ) 1 − r + µ m  w∗ w∗

1



( 1 − µm ) 1 − r

w1∗ x1 w∗

∗

  ∗ ∗  w x . + µm r 1 ∗ 1  w

(A.2)

Box III.

But w1∗ x∗1 + w3∗ x∗3 = w ∗ . Thus

In fact

 w w M (0) = 1 32 (1 − µm )(1 − r ) + µm (r − 1) (w∗ ) w∗ w∗ = 1 ∗ 32 (1 − r )(1 − 2µm ), (w )

a + d = ( 1 − µm )

∗

∗

w3∗ w1∗ − 1, − 1. ∗ w w∗ But w ∗ = w1∗ x∗1 + w3∗ x∗3 and x∗1 + x∗3 = 1. Therefore

(A.7)

+ r (1 − 2µm )

w1∗ + w3∗ w∗

w1∗ w3∗ (w∗ )2

(A.16)

Appendix B Proof of Claim 2. At equilibrium x∗ with associated u∗ , we have

 2

Q (u∗ ) = µM w1∗ u∗

x w3 − 1 = 1∗ (w3∗ − w1∗ ). ∗ w w ∗

(A.8)

w ∗ −w∗

Therefore M (1) = 1w∗ 3 multiplied by the following determinant, given in Box IV. On expansion, (A.9) reduces to

w ∗ x∗ w∗ x∗ (1 − µm ) 3 ∗ 3 + µm 1 ∗ 1 − x∗3 . (A.10) w w Since x∗ = (x∗1 , 0, x∗3 , 0) is an equilibrium, from (6) we have

  + (1 − µM ) w3∗ − w1∗ u∗

− µM w3∗ = 0.

(B.1)

If u∗ < 1, then from (B.1) 0 = Q (u∗ ) < µM w1∗ u∗ + (1 − µM ) w3∗ − w1∗ u∗



w3∗ x∗3 w ∗ x∗ + µM 1 ∗ 1 . ∗ w w

(A.11)

(µm − µM )(w1∗ x∗1 − w3∗ x∗3 ).

(A.12)



− µM w3∗ u∗ .

(B.2)

Thus u∗ (1 − 2µM ) w3∗ − w1∗ > 0.





(B.3)

But (1 − 2µM ) > 0, u > 0 and so (w3 − w1 ) > 0. Hence, in this    case, x∗1 < x∗3 and w1∗ < w3∗ giving w1∗ − w3∗ x∗1 − x∗3 > 0. If u∗ > 1, then x∗1 > x∗3 , and we must show w1∗ > w3∗ . Assume to the contrary that w1∗ ≤ w3∗ . Then from (B.1) ∗

∗

Thus (A.10) simplifies to

∗

0 = Q (u∗ ) > µM w1∗ + (1 − µM ) w3∗ − w1∗ − µM w3∗ ,





(B.4)

or

To sum up M (1) =

(A.15)

as in (16).

Similarly

w∗

M ′ (1) = 2 − (a + d) = 2 − (1 − µm )

(A.6)

1 x∗3 w1∗ ∗ ∗ ∗ ∗ ∗ − 1 = (w − w x − w x ) = (w ∗ − w3∗ ). 1 1 1 3 3 w∗ w∗ w∗ 1

1

w1∗ x∗3 ∗ (x1 + x∗3 ). (w ∗ )2

But x∗1 + x∗3 = 1, so

Proof of (15). M (1) = det(L∗ex − I). Add the second row of M (1) to the first row and the first row becomes

x∗3 = (1 − µM )

− r ( 1 − µm )

(A.5)

as in (14).

∗

w1∗ + w3∗ w1∗ w3∗ ∗ + r µ (x1 + x∗3 ) m w∗ (w ∗ )2

  (1 − 2µM ) w3∗ − w1∗ < 0,

µm − µM ∗ (w1 − w3∗ )(w1∗ x∗1 − w3∗ x∗3 ), (w∗ )2

(A.13)

which is (15).

 a − z

Proof of (16). Observe that if M (z ) = 

c



b  , d − z

then M ′ (z ) =

2z − (a + d) and M (1) = 2 − (a + d). From (12) we then see that ′

  w1∗ w3∗ x∗3 w1∗ w3∗ x∗3 a + d = ( 1 − µm ) ∗ 1 − r + r µ m w w∗ w∗ w∗ ∗  ∗ ∗ w ∗ w ∗ x∗ w w x + (1 − µm ) 3∗ 1 − r 1 ∗ 1 + r µm 3∗ 1 ∗ 1 . w w w w

(A.14)

(B.5)

a contradiction. Hence when u∗ > 1 (i.e., x∗1 > x∗3 ), w1∗ > w3∗ and    ∗ w1 − w3∗ x∗1 − x∗3 > 0. This proves (20). To prove (21), notice that since u∗ ̸= 1 we also have w1∗ ̸= w3∗ . If w1∗ > w3∗ , then by (20) x∗1 > x∗3 and so

  w1∗ x∗1 − w3∗ x∗3 > w1∗ − w3∗ x∗3 > 0.

(B.6)

If w1 < w3 , then by (20) x1 < x3 and ∗

∗

∗

∗

  w1∗ x∗1 − w3∗ x∗3 < w1∗ − w3∗ x∗3 < 0. (B.7)  ∗  ∗ ∗  ∗ ∗ ∗ Hence w1 − w3 w1 x1 − w3 x3 > 0, and we have proved Claim 2.

U. Liberman et al. / Theoretical Population Biology 112 (2016) 52–59

  x∗3  ∗  ∗ ∗ ∗ ∗ ∗  r (1 − µ ) w1 w3 x3 + µ w1 1 − r w3 x3 m m  w∗ w∗ w∗ w∗

(1 − µm )

w3∗ w∗



  −x∗1   ∗ ∗ ∗ ∗ ∗ . w 1 x1 w3 w1 x1  1−r + µ r − 1 m  w∗ w∗ w∗

59

(A.9)

Box IV.

Appendix C Proof of Result 3. Since u∗ = minu {Q (u) = 0}, we have u∗ = f (u∗ ). From (8), f (0) = µM /(1 − µM ) > 0. Hence, for all 0 < u < u∗ we have f (u) > u. As u∗ is a simple root of Q (u) = 0 it is easily seen that f ′ (u∗ ) ̸= 1 and so for u > u∗ and close to u∗ we have f (u) < u. Suppose now that u′ = f (u) is an increasing function of u, then u < f (u) < u∗

when 0 < u < u∗ ,

u < f (u) < u

when u > u∗ and close to u∗ .

∗

(C.1)

Let u1 = f (u),

un+1 = f (un )

n = 1, 2, . . . .

(C.2)

When 0 < u < u , as f is an increasing function, 0 < u1 < u∗ and by induction un < un+1 < u∗ for all n = 1, 2, . . . . Thus the ∗ sequence {un }∞ n=1 is increasing, bounded above by u , and its limit, by the continuity of f , is a fixed point v = f (v) with 0 < v ≤ u∗ . But u∗ is the only fixed point in this range so un −−−→ u∗ for all ∗

n→∞

starting u with 0 ≤ u ≤ u∗ . When u > u∗ and close to u∗ , as f ′ (u∗ ) ̸= 1 by the monotonicity of f we have u∗ < u1 < f (u) and by induction u∗ < un+1 < un such that {un }∞ n=1 is a monotone decreasing sequence and again un −−−→ u∗ . n→∞

This proves that x∗ , the equilibrium associated with u∗ , is internally stable. References Acar, M., Becskei, A., van Oudenaarden, A., 2005. Enhancement of cellular memory by reducing stochastic transitions. Nature 435, 228–232. Acar, M., Mettetal, J.T., van Oudenaarden, A., 2008. Stochastic switching as a survival strategy in fluctuating environments. Nat. Genet. 40, 471–475. Allen, B., Rosenbloom, D.I.S., 2012. Mutation rate evolution in replicator dynamics. Bull. Math. Biol. 74, 2650–2675. Balkau, B.J., Feldman, M.W., 1973. Selection for migration modification. Genetics 74, 171–174. Bergman, A., Feldman, M.W., 1990. More on selection for and against recombination. Theor. Popul. Biol. 38, 68–92. Carja, O., Liberman, U., Feldman, M.W., 2014. Evolution in changing environments: modifiers of mutation, recombination, and migration. Proc. Natl. Acad. Sci. USA 111, 17935–17940. Charlesworth, B., 1976. Recombination modification in a fluctuating environment. Genetics 83, 181–195. Eshel, I., Feldman, M.W., 1982. On evolutionary genetic stability of the sex ratio. Theor. Popul. Biol. 21, 430–439. Feldman, M.W., 1972. Selection for linkage modification: I. Random mating populations. Theor. Popul. Biol. 3, 324–346.

Feldman, M.W., Liberman, U., 1986. An evolutionary reduction principle for genetic modifiers. Proc. Natl. Acad. Sci. 83, 4824–4827. Feldman, M.W., Otto, S.P., Christiansen, F.B., 1997. Population genetic perspectives on the evolution of recombination. Annu. Rev. Genet. 30, 261–295. Gaal, B., Pitchford, J.W., Wood, A.J., 2010. Exact results for the evolution of stochastic switching in variable asymmetric environments. Genetics 184, 1113–1119. Gandon, S., Otto, S.P., 2007. The evolution of sex and recombination in response to abiotic or coevolutionary fluctuations in epistasis. Genetics 175, 1835–1853. Hamilton, W.D., 1980. Sex versus non-sex versus parasite. Oikos 35, 282–290. Haraguchi, Y., Sasaki, A., 1996. Host-parasite arms race in mutation modifications: indefinite escalation despite a heavy load? J. Theoret. Biol. 183, 121–137. Hughes, K.A., Houda, A.E., Price, A.C., Rodd, F.H., 2013. Mating advantage for rare males in wild guppy populations. Nature 503, 108–110. Ishii, K., Matsuda, H., Iwasa, Y., Sasaki, A., 1989. Evolutionarily stable mutation rate in a periodically changing environment. Genetics 121, 163–174. Kamp, C., Wilke, C.O., Adami, C., Bornholdt, S., 2003. Viral evolution under the pressure of an adaptive immune system: optimal mutation rates for viral escape. Complexity 8, 28–33. Kussell, E., Leibler, S., 2005. Phenotypic diversity, population growth, and information in fluctuating environments. Science 309, 2075–2078. Lachmann, M., Jablonka, E., 1996. The inheritance of phenotypes: an adaptation to fluctuating environments. J. Theoret. Biol. 181, 1–9. Leigh, E.G., 1970. Natural selection and mutability. Am. Nat. 104, 301–305. Liberman, U., Feldman, M.W., 1986a. Modifiers of mutation rate: A general reduction principle. Theor. Popul. Biol. 30, 125–142. Liberman, U., Feldman, M.W., 1986b. A general reduction principle for genetic modifiers of recombination. Theor. Popul. Biol. 30, 341–371. Liberman, U., Feldman, M.W., 1989. The reduction principle for genetic modifiers of the migration rate. In: Feldman, M.W. (Ed.), Mathematical Evolutionary Theory. Princeton University Press, pp. 111–137. Liberman, U., van Cleve, J., Feldman, M.W., 2011. On the evolution of mutation in changing environments: recombination and phenotypic switching. Genetics 187, 837–851. Maynard Smith, J., 1980. Selection for recombination in a polygenic model. Genet. Res. 35, 269–277. Maynard Smith, J., 1988. Selection for recombination in a polygenic model—the mechanism. Genet. Res. 51, 59–63. M’Gonigle, L.K., Shen, J.J., Otto, S.P., 2009. Mutating away from your enemies: the evolution of mutation rate in a host-parasite system. Theor. Popul. Biol. 75, 301–311. Nee, S., 1989. Antagonistic co-evolution and evolution of genotypic randomization. J. Theoret. Biol. 140, 499–518. Pal, C., Maciá, M.D., Oliver, A., Schachar, I., Buckling, A., 2007. Coevolution with viruses drives the evolution of bacterial mutation rates. Nature 450, 1079–1081. Rosenbloom, D.I.S., Allen, B., 2014. Frequency-dependent selection can lead to evolution of high mutation rates. Am. Nat. 183, E131–E153. Salathe, M., Van Cleve, J., Feldman, M.W., 2009. Evolution of stochastic switching rates in asymmetric fitness landscapes. Genetics 182, 1159–1164. Salceda, V.M., Anderson, W.W., 1988. Rare male mating advantage in a natural population of Drosophila pseudoobscura. Proc. Natl. Acad. Sci. USA 85, 9870–9874. Spencer, H.G., Feldman, M.W., 2005. Adaptive dynamics, game theory and evolutionary population genetics. J. Evol. Biol. 18, 1191–1193. Thattai, M., van Oudenaarden, A., 2004. Stochastic gene expression in fluctuating environments. Genetics 167, 523–530. Wright, S., 1969. Evolution and the Genetics of Populations: Volume 2: The Theory of Gene Frequencies. The University of Chicago Press, Chicago.