UBC Faculty Research and Publications

Fitness-valley crossing with generalized parent-offspring transmission Osmond, M. M.; Otto, Sarah P., 1967- Nov 30, 2015

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
52383-Osmond_et_al_Fitness-valley_crossing.pdf [ 609.21kB ]
Metadata
JSON: 52383-1.0042086.json
JSON-LD: 52383-1.0042086-ld.json
RDF/XML (Pretty): 52383-1.0042086-rdf.xml
RDF/JSON: 52383-1.0042086-rdf.json
Turtle: 52383-1.0042086-turtle.txt
N-Triples: 52383-1.0042086-rdf-ntriples.txt
Original Record: 52383-1.0042086-source.json
Full Text
52383-1.0042086-fulltext.txt
Citation
52383-1.0042086.ris

Full Text

Fitness-valley crossing with generalized parent-offspring transmission1M.M. Osmonda,∗, S.P. Ottoa2aDepartment of Zoology, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada3Abstract4Simple and ubiquitous gene interactions create rugged fitness landscapes composed of coadapted gene com-plexes separated by “valleys” of low fitness. Crossing such fitness valleys allows a population to escapesuboptimal local fitness peaks to become better adapted. This is the premise of Sewall Wright’s shiftingbalance process. Here we generalize the theory of fitness-valley crossing in the two-locus, bi-allelic case byallowing bias in parent-offspring transmission. This generalization extends the existing mathematical frame-work to genetic systems with segregation distortion and uniparental inheritance. Our results are also flexibleenough to provide insight into shifts between alternate stable states in cultural systems with “transmissionvalleys”. Using a semi-deterministic analysis and a stochastic diffusion approximation, we focus on the lim-iting step in valley crossing: the first appearance of the genotype on the new fitness peak whose lineage willeventually fix. We then apply our results to specific cases of segregation distortion, uniparental inheritance,and cultural transmission. Segregation distortion favouring mutant alleles facilitates crossing most whenrecombination and mutation are rare, i.e., scenarios where crossing is otherwise unlikely. Interactions withmore mutable genes (e.g., uniparental inherited cytoplasmic elements) substantially reduce crossing times.Despite component traits being passed on poorly in the previous cultural background, small advantages inthe transmission of a new combination of cultural traits can greatly facilitate a cultural transition. Whilepeak shifts are unlikely under many of the common assumptions of population genetic theory, relaxing someof these assumptions can promote fitness-valley crossing.Keywords: cultural evolution, cytonuclear inheritance, meiotic drive, peak shift, population genetics,5valley crossing61. Introduction7Epistasis and underdominance create rugged fitness landscapes on which adaptation may require a pop-8ulation to acquire multiple, individually-deleterious mutations that are collectively advantageous. Using the9adaptive landscape metaphor, we say the population faces a fitness “valley” (Wright, 1932). Such valleys10∗Corresponding authorEmail addresses: mmosmond@zoology.ubc.ca (M.M. Osmond), otto@zoology.ubc.ca (S.P. Otto)Preprint submitted to Elsevier August 5, 2015. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; appear to be common in nature (Weinreich et al., 2005; Szendro et al., 2013, but see Carneiro and Hartl,112010) and affect, among other things, speciation by reproductive isolation, the evolution of sex, the evolv-12ability of populations, and the predictability of evolution (Szendro et al., 2013). Here we are interested in13the speed and likelihood of fitness-valley crossing, which we determine by examining the first appearance14of an individual with the collectively advantageous set of mutations whose lineage will eventually spread to15fixation.16Believing epistasis to be ubiquitous, Sewall Wright (1931; 1932) formulated his “shifting balance theory”,17which describes evolution as a series of fitness-valley crossings. In phase one of the shifting balance process,18small, partially-isolated subpopulations (demes) descend into fitness valleys by genetic drift. The new19mutations are selected against when rare, as they will tend to occur alone as single deleterious alleles.20Eventually drift may allow the deleterious mutations to reach appreciable frequencies in at least one deme.21Once multiple synergistically-acting mutations arise together, they begin to be locally favoured by selection.22In phase two, these favoured combinations of mutations sweep to fixation, and those demes ascend the new23“fitness peak”. Finally, in phase three, the demes that reach the new fitness peak send out migrants whose24genes invade and fix in the remaining demes, eventually “pulling” the entire population up to the new fitness25peak. Our focus here is in the first appearance of a genotype on the new fitness peak whose lineage will26eventually fix, considering a single isolated deme. This is typically the longest stage of phases one and two27(Stephan, 1996) and hence is likely the limiting step in fitness-valley crossing.28Fitness-valley crossing has been investigated in a large number of theoretical studies. In the context of29multiple loci with reciprocal sign epistasis, the first appearance of the genotype with the best combination30of alleles has been the focus of a few studies (Phillips, 1996; Christiansen et al., 1998; Hadany, 2003; Hadany31et al., 2004; Weissman et al., 2009, 2010). Many authors have gone on to examine the remainder of phases32one and two (Crow and Kimura, 1965; Eshel and Feldman, 1970; Karlin and McGregor, 1971; Kimura,331985; Barton and Rouhani, 1987; Kimura, 1990; Phillips, 1996; Michalakis and Slatkin, 1996; Stephan,341996; Weinreich and Chao, 2005; Weissman et al., 2009, 2010), as well as phase three (Kimura, 1990; Crow35et al., 1990; Barton, 1992; Kondrashov, 1992; Phillips, 1993; Gavrilets, 1996; Hadany, 2003; Hadany et al.,362004). Similar attention has been given to situations with a single underdominant locus (Slatkin, 1981;37Gillespie, 1984; Barton and Rouhani, 1993; Peck et al., 1998) or a quantitative trait (Lande, 1985a; Barton38and Rouhani, 1987; Rouhani and Barton, 1987a,b; Charlesworth and Rouhani, 1988; Barton and Rouhani,391993). The theoretical and empirical support for Wright’s shifting balance process has been summarized and40debated (Coyne et al., 1997; Wade and Goodnight, 1998; Coyne et al., 2000; Whitlock and Phillips, 2000;41Coyne et al., 2000; Goodnight and Wade, 2000; Goodnight, 2013), and the general consensus appears to be42that, unless the valley is shallow (weakly deleterious intermediates), crossing a fitness valley is unlikely.43Despite the abundance of literature on fitness-valley crossing, the above studies all assume perfect44Mendelian inheritance. The question therefore remains: how robust are our ideas of fitness-valley crossing452. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; to deviations from Mendelian inheritance? Specifically, how does transmission bias (e.g., meiotic drive or46uniparental inheritance) affect the speed and likelihood of valley crossing? Departing from strict Mendelian47inheritance also allows us to consider the idea of valley crossing in cultures, considering the spread of memes48(Dawkins, 1976) rather than genes. This simultaneously adds a level of complexity to current mathematical49models of cultural transmission, which typically consider only one cultural trait at a time (e.g., Cavalli-Sforza50and Feldman, 1981; but see, e.g., Ihara and Feldman, 2004; Creanza et al., 2012).51Transmission bias in the form of segregation distortion is likely to have a large effect on valley cross-52ing, as distortion represents a second level of selection (Sandler and Novitski, 1957; Hartl, 1970). Insight53into how segregation distortion affects valley crossing comes from models of underdominant chromosomal54rearrangements (mathematically equivalent to models with one diploid biallelic locus), which often find55meiotic drive to be a mechanism allowing fixation of a new mutant homokaryotype (Bengtsson and Bod-56mer, 1976; Hedrick, 1981; Walsh, 1982). Populations that have fixed alternate homokaryotypes produce57heterokaryotype hybrids, which have low viability and/or fertility; thus gene flow between these populations58is reduced. Segregation distortion is therefore thought to be a mechanism that promotes rapid speciation59(stasipatric speciation; White, 1978). Although the role of underdominance in chromosomal speciation has60recently been questioned (reveiwed in Rieseberg, 2001; Hoffmann and Rieseberg, 2008; Faria and Navarro,612010; Kirkpatrick, 2010), it is hypothesized to be relevant in annual plants (Hoffmann and Rieseberg, 2008)62and appears to play a dominant role in maintaining reproductive isolation in sunflowers (Lai et al., 2005)63and monkey flowers (Stathos and Fishman, 2014).64Another common form of transmission bias is sex specific, with the extreme case being uniparental65inheritance. In genetic transmission, strict uniparental inheritance is common for organelle genomes, such66as the mitochondria, which is typically inherited from the mother. Uniparental inheritance will tend to67imply further asymmetries. For instance, the mutation rate of mitochondrial genes is estimated to be two68orders of magnitude larger than the mutation rate of nuclear genes in many animals (e.g., Linnane et al.,691989). Higher mutation rates will likely facilitate crossing. That said, higher mutation rates in only one gene70may have limited effect because the production of double mutants by recombination will be constrained by71the availability of the rarer single mutant. Previous models of fitness-valley crossing have tended to ignore72asymmetries (but see Appendix C of Weissman et al., 2010).73Transmission bias is an integral characteristic of cultural transmission, where it is referred to as “cultural74selection” (Cavalli-Sforza and Feldman, 1981; Boyd and Richerson, 1985). However, to the best of our75knowledge, no attempts have been made to examine the evolution of cultural traits (memes) in the presence76of a “fitness” valley. Boyd (2001) reviews the genetic theory of the shifting balance, and notes that it could77be applied to culture, but no explicit cultural models were presented. Meanwhile, instances such as the78so-called “demographic transition” in 19th century western Europe, where societies transitioned from less79educated, large families to more educated, small families (Borgerhoff Mulder, 1998), suggest that alternate803. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; combinations of cultural traits (e.g., ‘value of education’ and ‘family-size preference’) can be stable and that81peak shifts may occur in cultural evolution. In fact, alternate stable cultural states may be pervasive (Boyd82and Richerson, 2010), as alluded to by the common saying that people are “stuck in their ways.” Paradigm83shifts in the history of science (Kuhn, 1962) may provide further examples (Fog, 1999). Cultural peak shifts84can also be relatively trivial; for instance, changing the unit of time from seconds, minutes, and hours to85a decimal system is only advantageous if we also change units that are based on seconds, such as the joule86and volt (Fog, 1999).87Here we focus on a population genetic model with two bi-allelic loci under haploid selection in a randomly-88mating, finite population. This model can easily be reduced to a single-locus model with two alleles and89diploid selection, which is formally equivalent to a model of chromosomal rearrangements (e.g., a chromosome90has an inversion or not). Interpreting genes as memes produces a model of vertically-transmitted cultural91evolution. Our model incorporates both transmission bias and asymmetries in mutation and initial numbers92of single mutants. We first give a rough semi-deterministic sketch to develop some intuition, then follow93with a stochastic analysis using a diffusion approximation. Our analysis corresponds to the stochastic94simultaneous fixation regime of Weinreich and Chao (2005), and the neutral stochastic tunnelling and95deleterious tunnelling regimes of Weissman et al. (2010), where the appearance of the new, favourable,96and eventually successful “double mutant” occurs before the fixation of the neutral or deleterious “single97mutants”. Finally, we apply our results to the specific cases of segregation distortion, uniparental inheritance,98and cultural transmission.99We derive the expected time until the appearance of a double mutant whose lineage will fix when single100mutants are continuously generated by mutation from residents (the stochastic model assumes neutral101single mutants). We also use the stochastic model to derive the probability that a double mutant appears102and fixes given an initial stock of deleterious single mutants that is not replenished by mutation. Given103typical per-locus mutation rates, valley crossing is generally found to be a slow and unlikely outcome under104fair Mendelian transmission, even when single mutants are selectively neutral. Segregation distortion, in105favour of wild-type or mutant alleles, affects crossing most when recombination and mutation are rare,106the scenarios where crossing is otherwise unlikely. Cytonuclear inheritance allows increased mutational107asymmetries between the two loci; higher mutation rates lead to more single mutants and hence faster108valley crossing, but, when holding the average mutation rate constant, asymmetries hinder crossing by109reducing the probability that the single mutants recombine to produce double mutants. Finally, we show110that, when new cultural ideas or practices are not too poorly transmitted when arising individually within the111previous cultural background, a transmission advantage of the new combination greatly facilitates cultural112transitions.1134. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; 2. Model and Results114Consider two loci, A and B, with xij the current frequency of genotype AiBj , where i, j ∈ {1, 2, ..., p} are115the alleles carried by the individual. When an AiBj individual mates with an AkBl individual, they produce116an AmBn offspring with probability bklij (mn). Summing over all possible offspring types,∑pm,n=1 bklij (mn) =1171. We can specify that the bottom index (here ij) denotes the genotype of the mother, while the top index118(here kl) denotes the genotype of the father. As a consequence, transmission biases according to parental119sex [bklij (mn) 6= bijkl(mn)] are allowed. When considering sex-biased transmission we assume the frequencies120xij are the same in females and males (i.e., no sex linkage and no sex-based differences in selection), which121is automatically the case in hermaphrodites.122Random mating and offspring production is followed by haploid viability selection, which occurs immedi-123ately before censusing. The population size, N , is constant and discrete, and generations are non-overlapping.124Then the expected frequency of AmBn in the next generation, x′mn, solves125V x′mn = wmnp∑i,j,k,l=1xij xkl bklij (mn), (1)where wmn ≥ 0 is the relative viability of AmBn and V is the sum of the right hand side of Equation (1)126over all genotypes, which keeps the frequencies summed to one.127Denote the probability that a mating between an AiBj mother and an AkBl father produces an AmBn128offspring that survives one round of viability selection by bklij (mn)∗ = wmnbklij (mn), where the asterisk129indicates “after selection”. And let the average probability that a mating produces AmBn, regardless of130which parent was which, be b¯klij (mn)∗ = 12wmn[bklij (mn) + bijkl(mn)]. Then (as we will see below) selection on131AiBj in a population of “residents” (A1B1) is described by sij = 2b¯ij11(ij)∗ − 1. Letting wij = 1 + dij > 0132describe viability selection and 2b¯ij11(ij) = 1 + kij describe transmission bias (−1 ≤ kij ≤ 1), then sij =133(1 +dij)(1 +kij)− 1. Here we define the relative fitness of genotype AiBj as 1 + sij , which is determined by134both viability and transmission. Thus defined, fitness is a measure of the “transmissibility” of a genotype as135it includes several processes (e.g., viability, meiotic drive, recombination, mutation) that affect the number136of offspring of a given genotype produced by a parent of that genotype. We will see that it is transmissibility137that determines the dynamics of valley crossing.138Without mutation or recombination, fair transmission implies kij = 0, or b¯ij11(ij) = 1/2 ∀ i 6= 1, j 6= 1.139In words, with fair transmission we expect half of all offspring from matings between A1B1 and AiBj to be140of parental type AiBj . Sex-based inheritance is expected to arise in the form of bij11(ij) = 1 − b11ij (ij) [e.g.,141maternal inheritance implies bij11(ij) = 1 and b11ij (ij) = 0], which does not directly impose selection as kij = 0.142Segregation distortion can, however, impose selection. For example, ignoring mutations, if the A2 allele is143more likely to be transmitted than the A1 allele (in a B1 background) we would have b¯2111(21) > 1/2, giving144k21 > 0. Interpreting genes as memes, transmission bias kij determines the strength of “cultural selection”1455. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; (sensu Cavalli-Sforza and Feldman, 1981) on meme combination AiBj . Previous work on multi-locus peak146shifts has assumed that bias does not influence selection (kij = 0) and that maternal and paternal types are147equally transmitted [bij11(ij) = b11ij (ij) = 1/2 ∀ i 6= 1, j 6= 1].148Here we focus on bi-allelic loci (p = 2). We are specifically interested in the case where, in a popula-149tion composed entirely of residents, “single mutants” (A2B1 and A1B2) are selected against while “double150mutants” (A2B2) are selectively favoured: s21, s12 < 0 < s22.151Given that the population is composed primarily of residents, with no double mutants as of yet, the152population faces a fitness valley. The valley can be created by differences in viability alone, or it can be153created by differences in transmission, or both. Here we focus on the limiting step in the peak-shift process,154the probability and expected time until a double mutant arises whose lineage will eventually fix. Following155the lead of Christiansen et al. (1998), we begin by developing a rough semi-deterministic analysis to gain156intuition. A stochastic analysis follows. Table 1 provides a summary of notation and a supplementary157Mathematica file gives a more detailed derivation of the results.158TABLE 1 HERE1592.1. Semi-deterministic analysis1602.1.1. Single mutant dynamics161Selection against single mutants keeps their frequencies (x21 and x12) small. Let these frequencies be162proportional to some small number  << 1. Let the probability that an offspring inherits an allele that163neither parent possesses [i.e., mutation; e.g., b1111(21)] be of the same small order . Then, for large N , the164frequencies of the single mutants in the next generation are165x′ij ≈wijV[b1111(ij) + 2b¯ij11(ij)xij]+O(2), (2)where i 6= j and O(2) captures terms of order 2 and smaller.166We will write µklij (mn)∗ = b¯klij (mn)∗ when m 6∈ {i, k} or n 6∈ {j, l} to highlight the fact that a mutation167has occurred. Then, ignoring O(2), the frequencies of single mutants, which are initially absent [xij(0) = 0],168in generation t are169xij(t) ≈ µ1111(ij)∗[(1 + sij)t − 1]s−1ij : sij 6= 0µ1111(ij)∗ t : sij = 0(3)Viability and transmission are thus coupled together (in sij) throughout our results, and it is primarily the170total amount of selection on AiBj in a population of residents (sij) that determines the dynamics. [As a171technical aside, this is not true in the first generation that mutants appear, via µklij (mn)∗, but this is simply172because of the order of the life cycle chosen, where these mutants experience viability selection, but not173transmission biases, when they first occur.]174Equation (3) assumes the normalizing factor V remains near 1 over the t generations, which is the1756. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; case when single mutants are rare, as is generally true when single mutants are selected against, sij <1760 ∀ i 6= j. When sij < 0 and there has been a sufficiently long period of selection, t > −1/sij , the177single mutant frequencies approach mutation-selection balance xij(t) ≈ −µ1111(ij)∗/sij . This assumes the178probability of mutation, µ1111(ij)∗, is small relative to the strength of selection, sij . We next derive a semi-179deterministic solution for the crossing time, T , given mutation-selection balance is reached. In Appendix180A we follow Christiansen et al. (1998) to derive the semi-deterministic crossing time when crossing occurs181before mutation-selection balance is reached; this occurs when −sijT << 1, which can only be the case if182the valley is shallow, −sij << 1.1832.1.2. Waiting time for first successful double mutant184We now turn to calculating the waiting time until a double mutant that is able to establish first arises.185Assume the probability two residents mate to produce a double mutant (i.e., a double mutation), b1111(22), is186very rare, on the order of 2. Then the expected frequency of double mutants in the next generation before187selection, assuming single mutant are rare and there are currently no double mutants x22 = 0, is188x′22 =[µ1111(22) + 2µ2111(22)x21 + 2µ1211(22)x12 + 2r1221(22)x21x12]+O(3), (4)where we write r1221(22) = b¯1221(22) to highlight the fact that a double mutant has effectively been produced189by recombination. The expected frequency of double mutants (Equation 4) is measured before viability190selection to avoid artificially adjusting the double mutant frequency by its viability difference before it191appears.192In a truly deterministic model (N → ∞) double mutants are present at frequency x′22 after a single193bout of reproduction. However, assuming no double mutants have yet appeared, we can use x22(t) as a194rough approximation for the probability of a double mutant first arising in generation t (Christiansen et al.,1951998). Summing t from 0 to t′ gives the cumulative probability of observing a double mutant in any of the t′196generations. The generation T ′ at which the cumulative probability reaches 1/N can be used as an estimate197of the time we expect to wait until the first double mutant has arisen (Christiansen et al., 1998).198Here we are more interested in the waiting time until the first successful double mutant appears (i.e.,199one whose lineage will eventually fix). We therefore want to multiply the probability that a double mutant200appears at time t, x22(t), by the probability it will fix before taking the sum over t. Using Kimura’s (1962)201approximation, the probability a double mutant fixes is202u22 =1− e−2s221− e−2Ns22 . (5)With a weak double mutant advantage, 0 < s22 << 1, in a large population, Ns22 >> 1, Equation (5)203simplifies to the familiar 2s22 (Haldane, 1927).204The selection coefficient s22 can be calculated from the number of double mutant offspring a newly2057. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; arisen double mutant is expected to leave in the next generation, given that the mean number of offspring206per individual is one, such that the population size is constant. This expectation, 1 + s22, is the probability207of mating with a given type, multiplied by the probability of producing a double mutant offspring, multiplied208by the probability of surviving to the next generation, summed over all possible matings2091 + s22 =2∑i,j=12b¯22ij (22)∗xij , (6)where x22 = 0 in the remaining population (i.e., the double mutant does not mate with itself). Without210transmission bias, mutation, or recombination, b¯22ij (22) = 1/2 ∀ i, j and Equation (6) reduces to the familiar211s22 = w22− 1. Here we allow bias, mutation, and recombination, and assume single mutants are sufficiently212rare, giving s22 ≈ 2b¯2211(22)∗−1. This implies that selection on the double mutant (including transmission) is213constant over time and that fixation depends only on its dynamics in a population composed almost entirely214of residents. With recombination and otherwise fair transmission we have b¯2211(22) = (1 − r)/2, where r215is the probability of recombination between a double mutant and a resident. Writing w22 = 1 + s and216assuming both s and r are small, recovers the well-known first-order approximation s22 = s− r (Crow and217Kimura, 1965). This expression highlights the fact that recombination can reduce the probability of fixation218by breaking up favourable gene combinations (Crow and Kimura, 1965).219When selection is strong and mutation is rare, relative to the strength of genetic drift, the time to fixation220is dominated by the time to the arrival of a successful mutant (Gillespie, 1984; Weinreich and Chao, 2005;221Weissman et al., 2010). The waiting time until the first successful double mutant, which we derive below,222therefore well approximates the fixation time of a double mutant within a population when double mutants223are advantageous but rarely produced, x′22 << 1/N < s22.224Crossing time given mutation-selection balance. When enough time has passed (t > −1/sij) the single-225mutant frequencies approach mutation-selection balance (MSB), xij(t) ≈ −µ1111(ij)∗/sij . Using these fre-226quencies in Equation (4) gives the expected frequency of double mutants in the next generation, which does227not change until a double mutant arises, i.e., x22(t) = x′22 ∀ t. Summing u22x′22 over TMSB generations,228setting equal to 1/N , and solving for TMSB gives an estimate of the number of generations we expect to229wait for a successful double mutant to arise when beginning from mutation-selection balance2308. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; TMSB ≈ 1u22N[(1 +µ1111(21)∗s21+µ1111(12)∗s12)2µ1111(22)−(1 +µ1111(21)∗s21+µ1111(12)∗s12)(µ1111(21)∗s21)2µ2111(22)−(1 +µ1111(21)∗s21+µ1111(12)∗s12)(µ1111(12)∗s12)2µ1211(22)+(µ1111(21)∗s21)2µ2121(22)∗ +(µ1111(12)∗s12)2µ1212(22)+(µ1111(21)∗s21)(µ1111(12)∗s12)2r1221(22)]−1− 1. (7)In our numerical examples, we will track the waiting time until a successful double mutant arises in a231population that has recently established and is fixed for the resident type (e.g., following a bottleneck or232a founder event). This time can be approximated by the time that it takes to reach mutation-selection233balance, T0, and the establishment time once there234T ≈ T0 + TMSB . (8)Here we use T0 = max{ 1−s21 , 1−s12 }. As the deleterious single mutants approach neutrality (sij → 0− ∀ i 6= j)235the waiting time from mutation-selection balance, TMSB , decreases (because there are more single mutants236segregating), but the waiting time to mutation-selection balance, T0, increases dramatically because it takes237longer to produce the higher segregating frequencies of single mutants. As −sij becomes small enough such238that T < −1/sij the approximation breaks down and we must use the non-equilibrium solution derived in239Appendix A.240With symmetric Mendelian assumptions, weak selection on single mutants (δ = 1 − wij ∀ i 6= j), rare241mutation (µ), and infrequent recombination [such that uf ≈ 2(s − r) ≈ 2s], the rate of production of242successful double mutants from mutation selection balance (Equation 7) is243TMSB−1 ≈ 2sNµ2rδ2, (9)aligning with equation 4 in Weissman et al. (2010, see supplementary Mathematica file). This result preforms244well when TMSB−1 < δ, or, equivalently, when 3√2sNµ2r < δ.2452.2. Stochastic analysis2462.2.1. Markov process247Fitness-valley crossing is naturally a stochastic process. We thus now consider the Wright-Fisher model,248where the next generation is formed by choosing N offspring, with replacement, from a multinomial distri-249bution with frequency parameters x′ij (Equation 1). Let the number of A2B1 and A1B2 single mutants in250generation t be it and jt, respectively. Given that there are currently no double mutants, we have N− it−jt2519. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; resident individuals and we let X(t) = (it, jt) describe the state of the system in generation t. Let the252expected frequencies in the t + 1 generation, conditional on X(t) = (i, j), be x′kl(i, j) = x′kl, with x22 = 0.253The transition probabilities to states without double mutants are then254P klij = P{X(t+ 1) = (k, l) | X(t) = (i, j)} =(Nk, l,N − k − l)(x′21)k (x′12)l (x′11)N−k−l. (10)Note that summing over all k, l ∈ {0, 1, ..., N} gives (1 − x′22)N , the probability that no double mutant is255sampled. Equation (10) describes a sub-stochastic transition matrix for the Markov process.256Next, let H be the state with any positive number of double mutants. We then have the transition257probabilities PHij = 1 − (1 − x′22)N ≈ Nx′22, where the approximation assumes a small expected frequency258of double mutants in the next generation, x′22 << 1. To calculate the waiting time until the first successful259double mutant, we replace PHij with P˜Hij = PHij u22 ≈ Nx′22u22, ignoring the segregation of double mutants260when lost. H is now the state with any positive number of successful double mutants. Dividing each261x′ij in Equation (10) by the probability a double mutant does not arise (1 − x′22) and multiplying by the262probability a double mutant does not arise and fix (1−x′22u22) ensures the columns sum to one. To complete263the transition matrix we make H an absorbing state: PHH = 1 and PijH = 0.264We can describe this process, in part, by the moments for the change in number of single mutants,265conditional on the process not being killed by a successful double mutant. The nth moment for the change266in the number of A2B1 individuals, 4i = it+1 − it, is267E[(4i)n|it = i] =N∑k=0(k − i)n(Nk)( x′211− x′22u22)k( x′12 + x′111− x′22u22)N−k. (11)Similar equations can be computed for the change in the number of A1B2 individuals, 4j = jt+1 − jt.268To make analytic progress we use the moment equations to approximate the Markov chain with a diffusion269process (Karlin and Taylor, 1981, Ch. 15). We do so by taking the large population limit (N → ∞) while270finding the appropriate scalings to ensure finite drift and diffusion terms (Appendix B).2712.2.2. Crossing time with neutral single mutants272The diffusion process yields a partial differential equation describing the expected time until a successful273double mutant arises given that we begin with Nβy individuals of type A2B1 and Nβz individuals of type274A1B2 (Christiansen et al., 1998)27512σ2Y (y)∂2T˜ (y, z)∂y2+12σ2Z(z)∂2T˜ (y, z)∂z2+ µY (y)∂T˜ (y, z)∂y+ µZ(z)∂T˜ (y, z)∂z− κ(y, z)T˜ (y, z) = −1, (12)where T˜ (y, z) refers to time scaled in units of Nβ generations (parameters defined in Table 1 and Appendix276B). In Appendix C we solve Equation (12) under the two scenarios explored in Christiansen et al. (1998):277with and without recombination from neutral single mutants to double mutants when the population begins27810. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; with only residents, but here generalized to allow unequal mutation rates and sex-biased transmission. While279the neutrality assumption precludes the existence of a fitness valley, it provides a minimum for the expected280time to observe a successful double mutant. Previous studies have suggested that fitness valleys will only281be crossed if single mutants are nearly neutral (e.g., Walsh, 1982).2822.2.3. Probability of crossing from standing variation283The diffusion process can also be used to describe the production of successful double mutants from an284initial stock of single mutants (i.e., evolution from standing variation). Specifically, assuming that residents285don’t mutate [b1111(12) = b1111(21) = b1111(22) = 0] the process has two absorbing states, fixation of A1B1 and286fixation of A2B2 (a successful double mutant appears and the process is killed). The probability of fixation287of residents is the solution, u(y, z), of (Karlin and Taylor, 1981)28812σ2Y (y)∂2u(y, z)∂y2+12σ2Z(z)∂2u(y, z)∂z2+ µY (y)∂u(y, z)∂y+ µZ(z)∂u(y, z)∂z− κ(y, z)u(y, z) = 0, (13)with terms defined in Appendix B. The probability that a successful double mutant arises is therefore2891 − u(y, z). Karlin and Tavare´ (1981) used a similar equation to find the probability of detecting a lethal290homozygote in the one locus, diploid case with Mendelian transmission.291Deleterious single mutants without recombination. With no recombination from single mutants to double292mutants [r1221(22) = 0] we have scaling parameter β = 1/2. Then, with equal selection on single mutants and293some mutational symmetry between the two loci (see supplementary Mathematica file), the single mutants294are equivalent and we can concern ourselves with only their sum ξ = y+ z. Equation (13) then collapses to29512ξd2u(ξ)dξ2+ Smξdu(ξ)dξ− u22w22[Bm11(22) +B11m (22)]ξu(ξ) = 0, (14)where Sm = s21Nβ = s12Nβ is scaled selection on single mutants and Bm11(22) + B11m (22) = [b2111(22) =296b1121(22)]Nβ = [b1211(22) = b1112(22)]Nβ is the scaled mutation probability from single mutants to double297mutants.298The boundary conditions are u(0) = 1 and u(∞) = 0. Solving the boundary value problem gives the299probability of a double mutant appearing when starting with n0 = i0 + j0 single mutants3001− u(n0) = 1− exp[n0[− sm −√s2m + 2u222µm11(22)∗]], (15)where sm = s21 = s12 is the total strength of selection on each single mutant type. Setting n0 = 1 gives the301probability a newly arisen single mutant will begin a lineage which eventually produces a successful double302mutant.303Interestingly, Equation (15) does not depend strongly on population size, N . Without recombination304double mutants are primarily produced by mutations from single mutants, which are rare and hence always30511. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; mate with one of the large number of residents. In other words, the production of A2 and B2 alleles does306not rely on the number of residents but only on the dynamics of the rare single mutants.307Deleterious single mutants with recombination. Finally, we examine the probability of a successful double308mutant appearing when there is recombination between deleterious single mutants, r1221(22) > 0. With309sufficiently strong selection against single mutants the single mutant frequencies scale as cny ≈ z when we310begin with initial frequencies cny(0) = z(0) and both single mutants are under the same selection pressure,311S21 = S12. Then, without mutation from residents to single mutants, Equation (13) collapses to31212(1 + cn)ξd2u(ξ)dξ2+ S21ξdu(ξ)dξ− u22cn2r1221(22)∗ξ2u(ξ) = 0, (16)where ξ = cn y = z.313With boundary conditions u(0) = 1 and u(∞) = 0 the probability of valley crossing is3141− u(i0, N) = 1− exp[− (1 + cn)i0s21]Ai[N(1+cn)2(s21)2+i02u22cn(1+cn)2r1221(22)∗N1/3[2u22cn(1+cn)2r1221(22)∗]2/3 ]Ai[N(1+cn)2(s21)2N1/3[2u22cn(1+cn)2r1221(22)∗]2/3] , (17)where Ai is the Airy function. Equation (17) extends the one-locus diploid result with Mendelian trans-315mission (equation 28 in Karlin and Tavare´, 1981) by allowing unequal single mutant frequencies (cn 6= 1)316while also incorporating transmission bias, recombination, and double mutant fitness. Equation (17) well-317approximates the Mendelian simulation results of Michalakis and Slatkin (1996, see supplementary Mathe-318matica file).319When (s21)2 and i0 are small, we have the first order approximation3201− u(i0, N) = i0[(1 + cn)s21 +31/3Γ[2/3]Γ[1/3](2u22cn(1 + cn)2r1221(22)∗N)1/3], (18)which is valid only when the term in the large square brackets is positive. Equation (18) can be used321to show that when holding the initial number of single mutants, (1 + cn)i0, constant, the probability the322double mutant fixes is maximized when there are equal numbers of single mutants, cn = 1. This is because323recombination is most efficient in creating double mutants when single mutants are equally frequent.3242.3. Three scenarios325We next apply our results to three different scenarios: segregation distortion, cytonuclear inheritance,326and cultural transmission.3272.3.1. Segregation distortion328One form of segregation distortion, found in the heterothallic fungi Neurospora intermedia, is autosomal329killing (Burt and Trivers, 2006). In heterozygotes, the presence of a “killer” allele results in the death of a330proportion of the spores that contain the wild-type (“susceptible”) allele, leading to a (1+k)/2 frequency of33112. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; the killing allele at fertilization, 0 < k ≤ 1. Letting A2 and B2 represent the killing alleles, and, for the sake332of exploration, assuming that cells with one killing allele are functionally equivalent to cells with two, the333transmission probabilities are shown in Table 2. The Mendelian case is given by k = 0. When −1 ≤ k < 0334the allele identities are reversed: A1 and B1 are killers and A2 and B2 are susceptibles.335INSERT TABLE 2 HERE336Since segregation distortion imposes selection on single mutants, we can only investigate the effect of seg-337regation distortion on valley crossing with the semi-deterministic crossing time estimates allowing selection338on single mutants (Equations 8 and A3) and with the crossing probability estimates from standing variation339(Equations 15 and 17).340Figure 1 shows the crossing time as a function of the probability of recombination, and how segregation341distortion affects this time. Simulations (X’s) well match the numerical solution (Equation A1; dots) and342the MSB approximation (Equation 8; solid curves in top panel) over the range of parameters tested. When343valley crossing occurs before reaching MSB (bottom panel) a transition occurs between when mutation344drives crossing (dashed line; Equation A2) and when recombination does (solid curves; Equation A3), here345approximately r ≈ 10−4. The largest effect of segregation distortion occurs when the crossing time is long,346the scenario in which single-mutants must persist the longest before a successful double mutant appears.347In addition, observe that as the probability of recombination, r, increases above a critical value such that348s22 < 0, the double mutant is broken apart faster than its selective advantage and valley crossing takes349longer [equations B21 and B25 in Weissman et al., 2010 approximate the crossing times with no segregation350distortion (k = 0) when s22 < 0; see also Lynch, 2010; Altland et al., 2011].351INSERT FIGURE 1 HERE352Figure 2 shows the probability of crossing from standing variation. Again, segregation distortion has a353large effect when mutation (top panel) and recombination (bottom panel) are rare, the conditions under354which single mutants must persist the longest before a successful double mutant is formed. When crossing355occurs by recombination our analytical approximation (Equation 17) overestimates the probability of crossing356(bottom panel), especially when the initial number of single mutants is small and therefore subject to strong357stochasticity (results not shown). This occurs because the assumption that the ratio of single mutant358frequencies in these simulations remains roughly cn = 1 is violated, reducing the probability that double359mutants are formed by recombination.360INSERT FIGURE 2 HERE3612.3.2. Cytonuclear inheritance362We next explore how fitness-valley crossing is affected by uniparental inheritance of one of the traits. This363might occur if, for example, there was reciprocal sign epistasis between cytoplasmic and nuclear loci. Without364loss of generality we assume that the B trait is always inherited from the mother. For simplicity we assume36513. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; individuals are hermaphroditic. Here we can use only those results that allow recombination (Equations3668, A3, C5, and 17), as cytoplasmic and nuclear elements are expected to be inherited independently (i.e.,367r1221(22) + r2112(22) = 1/2).368One likely implication of uniparental transmission is asymmetric mutation probabilities. For instance,369in animals the mitochondrial mutation rate is two orders of magnitude larger than typical nuclear rates370(Linnane et al., 1989). Let µ be the mutation probability in the biparentally inherited A trait and ν be371the mutation probability in the uniparentally inherited B trait, with cµ = ν/µ the ratio of uniparental to372biparental mutation probabilities. The transmission probabilities are shown in Table 3.373INSERT TABLE 3 HERE374The top panel of Figure 3 shows the crossing time as a function of the mutation probability in the B375locus, ν. Increasing ν increases the rate at which single and double mutants are created, aiding valley376crossing. The bottom panel of Figure 3 shows the crossing time as a function of the ratio of the mutation377probabilities at the two loci, cµ, while holding the average mutation probability, (µ + ν)/2 = µ(1 + cµ)/2,378constant. When holding the average mutation probability constant the time to fixation is minimized when379ν = µ because single mutant types are then equally frequent, increasing the chances they mate with one380another to produce a double mutant by recombination. As cµ departs from one, the mutation rate at one381of the loci becomes small, causing those single mutants to become rare. The highly stochastic nature of the382rare single mutant frequencies causes our semi-deterministic (Equation A3) and stochastic (Equation C5)383approximations to underestimate the crossing time and, instead, the single mutants first reach mutation-384selection balance (Equation 8; dashed gray curve).385INSERT FIGURE 3 HERE386Given that crossing occurs by recombination from standing variation (Equation 17), asymmetric mutation387rates have little effect given a particular starting population (i0, j0). However, standing variation will also388tend to vary in proportion to mutation rates, implying that uniparental inheritance will cause differences in389the initial numbers of the two single mutants, which can have a large effect. Let cµ now also determine the390ratio of the initial numbers of single mutants, cµ = cn = j0/i0. Figure 4 shows the probability of crossing391from standing variation as a function of cµ. When we hold i0 and µ constant and increase j0 and ν (grey392curve), the probability of crossing increases with cµ as there are then more single mutants segregating. When393we instead hold the total initial number of single mutants (i0 + j0) and the average mutation probability394[(µ + ν)/2] constant (black curve), the probability of crossing is maximized at cµ = 1 because the single395mutants are then equally frequent and hence more likely to mate with one another and produce a double396mutant through recombination.397INSERT FIGURE 4 HERE39814. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; 2.3.3. Cultural inheritance399Finally, we remove Darwinian selection, such that transmission bias alone determines the dynamics, and400interpret the model in a cultural context. For the sake of exposition we consider only one simplified case of401cultural transmission. Let trait combinations with only one new trait (A2B1 and A1B2) be inherited relative402to the previous combination (A1B1) with probability q. Let the new combination of cultural traits (A2B2)403be inherited relative to the previous combination with probability p. We are most interested in the case of a404“transmission valley”, where the previous combination of traits is transmitted more effectively than mixed405combinations of new and old (q < 1/2), but the all-new combination spreads even more effectively than406the previous combination (p > 1/2). We assume that parental trait combinations can be broken up with407probability r and mutation occurs with probability µ. The transmission probabilities are shown in Table 4.408INSERT TABLE 4 HERE409Figure 5 shows that the crossing time is substantially faster when the new combination of traits has a410stronger transmission advantage (T decreases with p; compare thick curve with thin). Nevertheless, even411combinations that are transmitted very effectively (thick curve) spread very slowly when their component412traits are passed on poorly in the previous cultural background (q << 1/2). In particular, the crossing time413increases most quickly as q decreases from 1/2, demonstrating that slight biases in the transmission of the414new traits when arising within the previous cultural background have a strong influence on the spread of415new combinations of cultural traits, effectively preventing establishment if q << 1/2.416INSERT FIGURE 5 HERE417Figure 6 shows the probability of crossing from standing variation. In this case, with such a large418mutation rate, crossing can be more likely by mutation (Equation 15) than by recombination (Equation 17).419Recombination has the added effect of breaking up the new combination of traits, reducing the probability420of crossing. With a lower mutation rate crossing is most likely with moderate amounts of recombination421(e.g., Figure 1). Figure 6 again shows that the transmission advantage of the new combination of traits422(p; compare thick lines to thin) and slight biases in the transmission of new traits in the previous cultural423background (q ≈ 1/2) greatly influence the probability that a new combination of cultural traits successfully424spreads.425INSERT FIGURE 6 HERE4263. Discussion427Our results support the general consensus that, given reasonable population sizes and per locus per428generation mutation rates, crossing a particular fitness valley by genetic drift is typically a slow and unlikely429event (Crow and Kimura, 1965; Bengtsson and Bodmer, 1976; Lande, 1979; Hedrick, 1981; Walsh, 1982;430Lande, 1985b; Michalakis and Slatkin, 1996; Phillips, 1996; Coyne et al., 1997). For example, with a per locus431per generation mutation probability of µ = 10−8, a double mutant viability of w22 = 1.01, recombination43215. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; between the two loci with probability r = 0.01, and a population size of N = 104, the waiting time for a433successful double mutant, in the best case scenario where single mutants are selectively neutral, is on the434order of 107 generations (Equation C5). As this is the typical age for living animal genera (Van Valen, 1973;435Lande, 1979), we should not expect to see this fitness valley forded. Of course, with many potential fitness436valleys across the genome, the chance that one of them is forded can become substantial.437By broadening previous treatments to allow for non-Mendelian inheritance, we have shown that a small438amount of segregation distortion can greatly impact the chances of fitness-valley crossing. Of course, segre-439gation distortion has a large impact because it provides a second level of selection (Sandler and Novitski,4401957), often acting like gametic selection (but see Hartl, 1970, 1977). When the A2 and B2 alleles are more441likely to be passed down than the A1 and B1 alleles, respectively, in matings between single mutants and442residents, the depth of the valley is effectively reduced and hence crossing is much more likely. For example,443when single mutants have a relative viability of wm = 0.95, the mutation rate is µ = 10−8, double mutants444are weakly favoured (w22 = 1.01), and we begin with one single mutant (n0 = 1) in a population of size445N = 104, in the absence of recombination [r1221(22) = 0] and segregation distortion (kij = 0), the probability446of crossing is on the order of 10−9 (Equation 15). With a 5% distortion in favour of A2 and B2 alleles447(k21 = k12 = 0.05) the single mutants are effectively neutral and the probability increases seven orders of448magnitude to 10−2. And with a 10% distortion the single mutants are selectively favoured and the double449mutant fixes with probability 0.25.450Segregation distortion, in the form of meiotic drive, has often been implicated as a force that could help451fix underdominant chromosomal rearrangements (Sandler and Novitski, 1957; Bengtsson and Bodmer, 1976;452Hedrick, 1981; Walsh, 1982; Faria and Navarro, 2010). Chromosomal rearrangements, such as translocations453and inversions, are often fixed in alternate forms in closely related species (White, 1978; Coyne, 1989;454Faria and Navarro, 2010). Because heterokaryotypes typically have severely reduced fertility (Sandler and455Novitski, 1957; Lande, 1979), such rearrangements are thought to promote rapid speciation (stasipatric456speciation; White, 1978, but see Faria and Navarro, 2010; Kirkpatrick, 2010). The trouble is explaining457how such rearrangements originally increase in frequency when they are so strongly selected against when458rare (Navarro and Barton, 2003; Kirkpatrick, 2010). Meiotic drive provides one possible answer. Our459results can be used to investigate valley crossing with chromosomal rearrangements by assuming A and460B are homologous chromosomes, with A2 and B2 being the novel chromosomes, and A2B1 and A1B2461interchangeable. For example, with free recombination [r1221(22) = 1/4, r2211(22) = 1/4], a 5% viability462reduction in heterokaryotypes (wm = 0.95), no meiotic drive [b¯m11(m) = 1/2], a very beneficial mutant463homokaryotype (w22 = 2.5), and a spontaneous chromosome mutation rate of µ = 10−3 (Lande, 1979),464when starting with one copy of each mutant chromosome (i0 = 1, cn = 1) in a population of size N = 104,465Equation (17) gives a 0.4% chance of fixing the mutant homokaryotype. When the mutated chromosome has466a 70% chance of being passed down in matings with residents, a relatively weak amount of drive (Sandler46716. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; and Novitski, 1957), the chance of crossing increases two orders of magnitude, to nearly 75%.468Here we have shown that, for a given number of single mutants, the chance of crossing a valley by469recombination is best when the two single mutant types are at equal frequencies. This is an important470factor when the mutation rates in A and B are highly asymmetric. One instance where this asymmetry is471likely is when one locus (say B) is in the mitochondrial genome, and is passed down maternally, while the472other (say A) is in the nuclear genome, and is passed down biparentally. Mutation rates in the mitochondria473can be orders of magnitude higher than in the nucleus (Linnane et al., 1989). With r1221(22) = r/2 = 1/4,474N = 104, w22 = 2, b1111(21) = µ = 10−6, and neutral single mutants, when the mutation rates in A and B475are equal [b1111(12)/b1111(21) = cµ = 1] the crossing time is 40, 000 generations. When the mutation rate in B476is two orders of magnitude larger (cµ = 100) the waiting time is reduced to 2, 500 generations. But when the477average mutation rate (1 + cµ)b1111(21)/2 is held constant, the asymmetrical mutation rates instead hinder478crossing, increasing the crossing time to nearly 120, 000 generations.479By expanding a mathematical model of fitness-valley crossing beyond symmetrical Mendelian inheritance480we gain insight into transitions between alternate stable states in non-genetic systems, such as culture.481As mentioned in the introduction, culture may often exhibit alternate stable states; here valley crossing482corresponds to a shift between alternate combinations of cultural ideas or practices (e.g., the demographic483transition; Borgerhoff Mulder, 1998). The valley is a “transmission valley”, created by new cultural traits484that are transmitted effectively in concert but poorly when arising individually within the previous cultural485background. In this case our simplified example above demonstrates that, given that the component pieces486are not passed on too poorly in the previous cultural background, the probability that a new set of practices487or ideas becomes pervasive in society is greatly improved by its transmission advantage over the previous488set. Valley crossing might also be relevant in the context of gene-culture coevolution, where one trait is489cultural and the other genetic. For instance, the ability to absorb lactose as an adult is largely genetically490determined and is positively correlated with the cultural practice of dairy farming, reaching frequencies over49190% in cultures with dairy farming but typically remaining less than 20% in cultures without (Feldman492and Laland, 1996). If, as seems reasonable, the ability to absorb lactose as an adult has a cost in the493absence of dairy farming and the cultural practice of dairy farming has a cost when adults are unable to494absorb lactose, then the transition from non-pastoralist non-absorbers to pastoralist absorbers may represent495another example of fitness-valley crossing outside the purely genetic arena. We have used our generalized496model to begin to explore cultural transitions, but it should be emphasized that we neglect oblique and497horizontal transmission, common features of cultural evolution (Cavalli-Sforza and Feldman, 1981) and498likely components of the demographic transition (Ihara and Feldman, 2004). Generalizing models of fitness-499valley crossing further to include oblique and horizontal transmission would improve insight into cultural500transitions.501We have incorporated transmission bias in a model of multi-locus fitness-valley crossing. This allows us50217. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; to investigate fitness-valley crossing in new scenarios, such as in genetic systems with segregation distortion503and/or uniparental inheritance. Segregation distortion acts as a second level of selection and therefore can504greatly help or hinder fitness-valley crossing, especially when crossing is otherwise unlikely. Uniparental505inheritance will often imply asymmetric mutation rates, which in turn lead to unequal frequencies of single506mutants, and therefore, all else being equal, a lower probability of fitness-valley crossing by recombination.507However, uniparental-inherited cytoplasmic elements tend to have increased mutation rates, which helps508crossing. Generalizing transmission also allows us to begin to extend the theory of valley crossing to non-509genetic systems, such as culture. Despite component traits being passed on poorly in the previous cultural510background, we find that small advantages in the transmission of the new set of cultural traits will greatly511facilitate a cultural transition. While crossing a deep fitness valley is difficult under Mendelian inheritance,512it can be easier when Mendel is left behind.5134. Acknowledgements514We thank Eva Kisdi, Stefan Geritz, Helene Weigang, and the labs of S.P. Otto, Michael Whitlock,515Michael Doebeli, and Christoph Hauert for helpful discussions. We thank Mark Kirkpatrick for suggesting516the simplicity of considering crossing times from mutation-selection balance. Funding provided by Natural517Sciences and Engineering Research Council (Canada) Discovery (SPO) and CGS-D (MMO) grants.5185. References519Altland, A., Fischer, A., Krug, J., and Szendro, I. G. (2011). Rare events in population genetics: Stochastic tunneling in a520two-locus model with recombination. Physical Review Letters, 106(8):088101.521Barton, N. H. (1992). On the spread of new gene combinations in the third phase of Wright’s shifting-balance. Evolution,52246(2):551–557.523Barton, N. H. and Rouhani, S. (1987). The frequency of shifts between alternative equilibria. Journal of Theoretical Biology,524125(4):397–418.525Barton, N. H. and Rouhani, S. (1993). Adaptation and the shifting balance. Genetical Research, 61:57–74.526Bengtsson, B. and Bodmer, W. (1976). On the increase of chromosome mutations under random mating. Theoretical Population527Biology, 9:260–281.528Borgerhoff Mulder, M. (1998). The demographic transition: are we any closer to an evolutionary explanation? Trends in529Ecology and Evolution, 13(7):266–270.530Boyd, R. (2001). Equilibrium selection and the evolution of norms. In Nicita, A. and Pagano, U., editors, The evolution of531economic diversity. Routledge, New York, NY.532Boyd, R. and Richerson, P. J. (1985). Culture and the evolutionary process. The University of Chicago Press, Chicago, IL.533Boyd, R. and Richerson, P. J. (2010). Transmission coupling mechanisms: cultural group selection. Philosophical Transactions534of the Royal Society B: Biological Sciences, 365(1559):3787–3795.535Burt, A. and Trivers, R. (2006). Genes in conflict. Belknap Press, Cambridge, Massachusetts.53618. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; Carneiro, M. and Hartl, D. L. (2010). Adaptive landscapes and protein evolution. Proceedings of the National Academy of537Sciences of the United States of America, 107:1747–1751.538Cavalli-Sforza, L. L. and Feldman, M. W. (1981). Cultural transmission and evolution: a quantitative approach. Princeton539University Press, Princeton, NJ.540Charlesworth, B. and Rouhani, S. (1988). The probability of peak shifts in a founder population. II. An additive polygenic541trait. Evolution, 42(6):1129–1145.542Christiansen, F. B., Otto, S., Bergman, A., and Feldman, M. W. (1998). Waiting with and without recombination: the time543to production of a double mutant. Theoretical Population Biology, 53(3):199–215.544Coyne, J. A. (1989). A test of the role of meiotic drive in fixing a pericentric inversion. Genetics, 123:241–243.545Coyne, J. A., Barton, N. H., and Turelli, M. (1997). A critique of Sewall Wright’s shifting balance theory of evolution. Evolution,54651(3):643–671.547Coyne, J. A., Barton, N. H., and Turelli, M. (2000). Is Wright’s shifting balance process important in evolution? Evolution,54854(1):306–317.549Creanza, N., Fogarty, L., and Feldman, M. W. (2012). Models of cultural niche construction with selection and assortative550mating. PLoS ONE, 7(8):e42744.551Crow, J., Engels, W., and Denniston, C. (1990). Phase three of Wright’s shifting-balance theory. Evolution, 44(2):233–247.552Crow, J. and Kimura, M. (1965). Evolution in sexual and asexual populations. American Naturalist, 99(909):439–450.553Dawkins, R. (1976). The selfish gene. Oxford University Press, Oxford, UK.554Eshel, I. and Feldman, M. W. (1970). On the evolutionary effect of recombination. Theoretical Population Biology, 1(1):88–100.555Faria, R. and Navarro, A. (2010). Chromosomal speciation revisited: rearranging theory with pieces of evidence. Trends in556Ecology and Evolution, 25(11):660–669.557Feldman, M. W. and Laland, K. N. (1996). Gene-culture coevolutionary theory. Trends in Ecology and Evolution, 5347(96):453–558457.559Fog, A. (1999). Cultural selection. Kluwer Academic Publishers, Dordrecht, Netherlands.560Gavrilets, S. (1996). On phase three of the shifting-balance theory. Evolution, 50(3):1034–1041.561Gillespie, J. (1984). Molecular evolution over the mutational landscape. Evolution, 38(5):1116–1129.562Goodnight, C. and Wade, M. (2000). The ongoing synthesis: a reply to Coyne, Barton, and Turelli. Evolution, 54(1):317–324.563Goodnight, C. J. (2013). Wrights shifting balance theory and factors affecting the probability of peak shifts. In Svensson, E.564and Calsbeek, R., editors, The adaptive landscape in evolutionary biology, pages 74–86. Oxford University Press, Oxford,565UK.566Hadany, L. (2003). Adaptive peak shifts in a heterogenous environment. Theoretical Population Biology, 63(1):41–51.567Hadany, L., Eshel, I., and Motro, U. (2004). No place like home: competition, dispersal and complex adaptation. Journal of568Evolutionary Biology, 17(6):1328–1336.569Haldane, J. B. S. (1927). A mathematical theory of natural and artificial selection, part V: selection and mutation. Mathematical570Proceedings of the Cambridge Philosophical Society, 23:838–844.571Hartl, D. L. (1970). Population consequences of non-Mendelian segregation among multiple alleles. Evolution, 24(2):415–423.572Hartl, D. L. (1977). Applications of meiotic drive in animal breeding and population control. In Pollak, E., Kempthorne, O.,573and Bailey, Jr., T. B., editors, Proceedings of the International Conference on Quantitative Genetics, 1976. The Iowa State574University Press, Ames, IA.57519. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; Hedrick, P. (1981). The establishment of chromosomal variants. Evolution, 35(2):322–332.576Hoffmann, A. A. and Rieseberg, L. H. (2008). Revisiting the impact of inversions in evolution: from population genetic markers577to drivers of adaptive shifts and speciation? Annual Review of Ecology, Evolution, and Systematics, 39:21–42.578Ihara, Y. and Feldman, M. W. (2004). Cultural niche construction and the evolution of small family size. Theoretical Population579Biology, 65(1):105–111.580Karlin, S. and McGregor, J. (1971). On mutation selection balance for two-locus haploid and diploid populations. Theoretical581Population Biology, 70:60–70.582Karlin, S. and Tavare´, S. (1981). The detection of particular genotypes in finite populations. I. Natural selection effects.583Theoretical Population Biology, 19(2):187–214.584Karlin, S. and Taylor, H. M. (1981). A second course in stochastic processes. Academic Press, New York, NY.585Kimura, M. (1962). On the probability of fixation of mutant genes in a population. Genetics, 47:713–719.586Kimura, M. (1985). The role of compensatory neutral mutations in molecular evolution. Journal of Genetics, 64(1):7–19.587Kimura, M. (1990). Some models of neutral evolution, compensatory evolution, and the shifting balance process. Theoretical588Population Biology, 37:150–158.589Kirkpatrick, M. (2010). How and why chromosome inversions evolve. PLoS Biology, 8(9):e1000501.590Kondrashov, A. (1992). The third phase of Wright’s shifting-balance: a simple analysis of the extreme case. Evolution,59146(6):1972–1975.592Kuhn, T. S. (1962). The structure of scientific revolutions. University of Chicago Press, Chicago, IL.593Lai, Z., Nakazato, T., Salmaso, M., Burke, J. M., Tang, S., Knapp, S. J., and Rieseberg, L. H. (2005). Extensive chromosomal594repatterning and the evolution of sterility barriers in hybrid sunflower species. Genetics, 171(1):291–303.595Lande, R. (1979). Effective deme sizes during long-term evolution estimated from rates of chromosomal rearrangement. Evo-596lution, 33(1):234–251.597Lande, R. (1985a). Expected time for random genetic drift of a population between stable phenotypic states. Proceedings of598the National Academy of Sciences of the United States of America, 82(22):7641–7645.599Lande, R. (1985b). The fixation of chromosomal rearrangements in a subdivided population with local extinction and colo-600nization. Heredity, 54:323–332.601Linnane, A., Ozawa, T., Marzuki, S., and Tanaka, M. (1989). Mitochondrial DNA mutations as an important contributor to602ageing and degenerative diseases. The Lancet, 333:642–645.603Lynch, M. (2010). Scaling expectations for the time to establishment of complex adaptations. Proceedings of the National604Academy of Sciences of the United States of America, 107(38):16577–16582.605Michalakis, Y. and Slatkin, M. (1996). Interaction of selection and recombination in the fixation of negative-epistatic genes.606Genetical Research, 67(3):257–269.607Navarro, A. and Barton, N. H. (2003). Chromosomal speciation and molecular divergence-accelerated evolution in rearranged608chromosomes. Science, 300(5617):321–324.609Peck, S., Ellner, S., and Gould, F. (1998). A spatially explicit stochastic model demonstrates the feasibility of Wright’s shifting610balance theory. Evolution, 52(6):1834–1839.611Phillips, P. (1993). Peak shifts and polymorphism during phase three of Wright’s shifting-balance process. Evolution,61247(6):1733–1743.613Phillips, P. (1996). Waiting for a compensatory mutation: phase zero of the shifting-balance process. Genetical Research,61420. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; 67:271–283.615Rieseberg, L. H. (2001). Chromosomal rearrangements and speciation. Trends in Ecology and Evolution, 16(7):351–358.616Rouhani, S. and Barton, N. H. (1987a). Speciation and the “shifting balance” in a continuous population. Theoretical Population617Biology, 31:465–492.618Rouhani, S. and Barton, N. H. (1987b). The probability of peak shifts in a founder population. Journal of Theoretical Biology,619126:51–62.620Sandler, L. and Novitski, E. (1957). Meiotic drive as an evolutionary force. American Naturalist, 91(857):105–110.621Slatkin, M. (1981). Fixation probabilities and fixation times in a subdivided population. Evolution, 35(3):477–488.622Stathos, A. and Fishman, L. (2014). Chromosomal rearrangements directly cause underdominant F1 pollen sterility in Mimulus623lewisii − Mimulus cardinalis hybrids. Evolution, 68:3109–3119.624Stephan, W. (1996). The rate of compensatory evolution. Genetics, 144(1):419–426.625Szendro, I. G., Schenk, M. F., Franke, J., Krug, J., and de Visser, J. A. G. M. (2013). Quantitative analyses of empirical fitness626landscapes. Journal of Statistical Mechanics: Theory and Experiment, P01005.627Van Valen, L. (1973). A new evolutionary law. Evolutionary Theory, 1:1–30.628Wade, M. and Goodnight, C. (1998). Perspective: the theories of Fisher and Wright in the context of metapopulations: when629nature does many small experiments. Evolution, 52(6):1537–1553.630Walsh, J. (1982). Rate of accumulation of reproductive isolation by chromosome rearrangements. American Naturalist,631120(4):510–532.632Weinreich, D. and Chao, L. (2005). Rapid evolutionary escape by large populations from local fitness peaks is likely in nature.633Evolution, 59(6):1175–1182.634Weinreich, D. M., Watson, R. A., and Chao, L. (2005). Perspective: sign epistasis and genetic constraint on evolutionary635trajectories. Evolution, 59:1165–1174.636Weissman, D. B., Desai, M. M., Fisher, D. S., and Feldman, M. (2009). The rate at which asexual populations cross fitness637valleys. Theoretical Population Biology, 75(4):286–300.638Weissman, D. B., Feldman, M., and Fisher, D. S. (2010). The rate of fitness-valley crossing in sexual populations. Genetics,639186(4):1389–1410.640White, M. J. D. (1978). Modes of speciation. W.H. Freeman and Company, San Francisco, CA.641Whitlock, M. and Phillips, P. (2000). The equisite corpse: a shifting view of the shifting balance. Trends in Ecology and642Evolution, 5347:347–348.643Wright, S. (1931). Evolution in Mendelian populations. Genetics, 16(2):97–159.644Wright, S. (1932). The roles of mutation, inbreeding, crossbreeding and selection in evolution. Proceedings of The Sixth645International Congress of Genetics, 1:356–366.646A. Dynamic single mutants647Here we calculate the waiting time for a successful double mutant to arise, starting from a populationthat is composed entirely of residents. While single mutants are far from mutation-selection balance, |sijt| <1 ∀ i 6= j, we can write x22 as a function of t by replacing xij in Equation (4) with the appropriate xij in21. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; Equation (3). With rare mutation, rare single mutants, and weak selection on single mutants, the expectednumber of generations until a successful double mutant appears, T , solves1u22N= µ1111(22)+ T[µ1111(22)[1− µ1111(21)∗ − µ1111(12)∗] + µ1111(21)∗µ2111(22) + µ1111(12)∗µ1211(22)+13µ1111(21)∗s21µ2111(22) +13µ1111(12)∗s12µ1211(22) +13µ1111(21)∗µ1111(12)∗r1221(22)]+ T 2[µ1111(21)∗µ2111(22) + µ1111(12)∗µ1211(22) + µ1111(21)∗µ1111(12)∗r1221(22)− µ1111(22)[µ1111(21)∗ + µ1111(12)∗]]+ T 3[23µ1111(21)∗µ1111(12)∗r1221(22) +13µ1111(21)∗s21µ2111(22) +13µ1111(12)∗s12µ1211(22)]+O(5). (A1)The O(5) terms disappear and the equation is exact when single mutants are neutral, sij = 0. Otherwise,648with selection against single mutants, the higher order terms can only be ignored as long as the crossing649time, T , is much smaller than the inverse of the selection coefficients, s21 and s12. Because Equation (A1)650is a cubic in T , its solution is cumbersome (see supplementary Mathematica file). Here we examine two651scenarios which give more interpretable approximations for T .652Without selection on single mutants (s21 = s12 = 0) and without recombination from single mutants to653double mutants [r1221(22) = 0] the T3 term in Equation (A1) vanishes. In addition, if the crossing time T is654long, the dominant term is the one proportional to T 2. Solving for T from this term alone gives655T ≈[u22N[µ1111(21)∗µ2111(22) + µ1111(12)∗µ1211(22)]]−1/2, (A2)where we have ignored double mutants arising instantaneously [µ1111(22) = 0]. Equation (A2) shows that the656crossing time without selection on or recombination among single mutants is roughly proportional to N−1/2657generations. The crossing time decreases with N because increasing N increases the per generation input of658mutations. Holding mutation input θklij (mn) = Nµklij (mn) constant, the crossing time becomes proportional659to N1/2. When the single-mutation transmission probabilities are equal [µ1111(21)∗ = µ1111(12)∗ = 2µ2111(22) =6602µ1211(22) = µ] and we calculate the first appearance of any double mutant (successful or not; u22 = 1),661the expected time until the first double mutant appears simplifies to the neutral genetic case without662recombination, T ≈ 1/√µ2N (equation 8 in Christiansen et al., 1998). Equation (A2) clarifies the role of66322. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; the various, potentially different, mutation probabilities µij11(kl) on the time until the first double mutant,664while also allowing us to ignore double mutants that are lost.665When there is recombination between single mutants to produce double mutants [r1221(22) > 0] and the666crossing time, T , is smaller than the inverse of the selection coefficients, s12 and s21, the dominant term667in Equation (A1) is proportional to T 3. This term is positive when recombination is frequent relative to668selection against single mutants. Again, if the time T is long we can use this term alone to approximate T ,669which gives670T ≈ 31/3[u22N[µ1111(21)∗µ1111(12)∗r1221(22) + s21µ1111(21)∗µ2111(22) + s12µ1111(12)∗µ1211(22)]]1/3 , (A3)where we have once again ignored the instantaneous production of double mutants. Notice that, for a671given mutation input θklij (mn), when there is recombination between single mutants, the crossing time is672roughly proportional to N1/3 generations (rather than N1/2 generations without recombination), implying673that recombination between single mutants tends to shorten the expected time until the first (successful or674unsuccessful) double mutant arises. However, because recombination can also occur between residents and675double mutants (reducing u22) Equation (A3) shows that the waiting time until the first successful double676mutant is minimized at intermediate levels of recombination.677Equation (A3) reduces to T ≈ 1/ 3√Nrµ2/3 (equation 9 in Christiansen et al., 1998) when we ignore678the weak selection against single mutants (sij = 0), there is equal mutation probability at each locus679[µ1111(21)∗ = µ1111(12)∗ = µ], and we wait until the first double mutant appears, successful or not (u22 = 1).680Once again our analysis clarifies the role of the various, potentially different, mutation probabilities µklij (mn)681on the waiting time until the first successful double mutant. Equation (A3) also allows (weak) selection on682single mutants and incorporates transmission bias, which we explore more fully in the main text.683Figure A1 compares the approximations derived here (Equations A2 and A3) with that derived in the684text assuming mutation-selection balance is reached before crossing (Equation 8). The approximations given685by Equations (A2) and (A3) break down as the depth of the valley (δ = −s21 = s12) increases such that the686crossing time becomes long, T > 1/δ.687B. Diffusion approximation688Here we take the large population limit (N → ∞), scale time such that one unit of time in the scaled689diffusion process (τ ∈ Z≥0) is Nα generations in the unscaled Markov process (4t = τNα) and define new69023. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; frequency parameters Y (τ) = iτ/Nβ and Z(τ) = jτ/Nβ , with 0 < α, β < 1.691We are concerned with three quantities for each variable 4Y and 4Z. The first is the infinitesimal mean692µY (y) = limN→∞E[4Y |Y (τ) = y = i/Nβ ] = limN→∞NαNβE[4i|it = i]. (B1)The second quantity is the infinitesimal variance693σ2Y (y) = limN→∞E[(4Y )2|Y (τ) = y = i/Nβ ] = limN→∞NαN2βE[(4i)2|it = i]. (B2)And the third quantity of interest is a higher (n > 2) infinitesimal moment694limN→∞E[(4Y )n|Y (τ) = y = i/Nβ ] = limN→∞NαNnβE[(4i)n|it = i]. (B3)We can similarly calculate µZ(z), σ2Z(z), and a higher moment in 4Z.695The final quantity of interest is the scaled “killing rate”696κ(y, z) = limN→∞NαP˜Hij ≈ limN→∞NαNx′22u22, (B4)where the approximation assumes x′22u22 << 1.697For the Markov chain to converge to a diffusion process as N →∞ we require: 1) µY (y) and µZ(z) to be698finite; 2) σ2Y (y), σ2Z(z), and κ(y, z) to be positive and finite; and 3) some higher moment (in both 4Y and6994Z) to be equal to zero (Karlin and Taylor, 1981). We first take a hint from the genetic case (Christiansen700et al., 1998) and scale transmission probabilities as701bklij (mn) =Bklij (mn) : m ∈ {i, k}, n ∈ {j, l}Bklij (mn)N2 +O(1/N3) : m 6∈ {i, k}, n 6∈ {j, l}Bklij (mn)N +O(1/N2) : otherwise(B5)In the genetic case this can be interpreted as making the probability of mutation proportional to the inverse702of population size µ = B/N . Then, as N →∞ mutation probability decreases (µ→ 0), such that mutation703input B = Nµ is constant. This prevents the process from taking large jumps in frequency space, which704violate the diffusion process (Karlin and Taylor, 1981).705In order for the transmission parameters to satisfy the logical constraint∑2m,n=1 bklij (mn) = 1 the diffusion70624. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; also requires, as N →∞, that707Bijij (ij) = 1 +O(1/Nβ) (B6)and708Bklij (ij) +Bklij (kl) = 1 +O(1/Nβ) (B7)when either {i 6= k, j = l} or {i = k, j 6= l}. In words, the sum total mutation probability for parents AiBj709and AkBl must be relatively small, at most on the order of 1/Nβ .710Finally, our approximation requires weak selection, relative to w11 = 1. In particular, total selection on711single mutants must be weak, on the order of 1/Nβ ,712wij [bij11(11) + b11ij (ij)] = 1 + Sij/Nβ +O(1/N2β) (B8)for i 6= j, where Sij is the scaled selection strength. And selection on double mutants must also be weak,713such that714s22 = S22/N +O(1/N2). (B9)With the above assumptions (Equations B5-B9) the Markov chain converges to a diffusion process as715N →∞ when716α = β = 1/2 : r2112(22) ≤ O(1/N1/2)1/3 : otherwise(B10)This scaling implies that if recombination between single mutants to make double mutants r2112(22) is less717likely that mutation (which is on the order of N−1/2; Equation B6), then the time until the process is killed718scales with N1/2. Meanwhile, if recombination is more likely than mutation the killing time scales with719N1/3. These results align with our semi-deterministic analysis (Equations A2 and A3).720When α = β the infinitesimal variances are σ2Y (y) = y and σ2Z(z) = z. The infinitesimal means and the721killing term depend on the probability of recombination. When recombination is rare the single mutants are722expected to reach higher frequencies and therefore have a greater influence on the dynamics. To simplify,723when recombination is rare [r1221(22) ≤ O(1/N1/2)] we assume weak transmission bias for residents mating724with single mutants [bij11(11) + b11ij (11) = 1 + O(1/Nβ)] and for single mutants mating with each other725[bklij (ij) + bijkl(ij) = 1 + O(1/Nβ)]. We further assume weak viability selection on single mutants, wij =72625. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; 1−O(1/Nβ), regardless of recombination. The infinitesimal mean is then always727µY (y) = B1111(21)− y S21 (B11)and similarly for µZ(z). The first term, B1111(21) ≈ b1111(21)N , describes mutation to single mutants in728resident-resident matings and the second term, with S21 ≈ s21Nβ , describes the removal of single mutants729by selection (both through transmission bias when mating with the resident and survival).730The killing terms are731κ(y, z) =u22w22[y[B2111(22) +B1121(22)] + z[B1211(22) +B1112(22)] + y z R1221(22)]: r1221(22) ≤ O(1/N1/2)u22 y z r1221(22)∗ : otherwise(B12)where R1221(22) ≈ r1221(22)N1/2 describes a (low) probability of recombination. The first line shows that732the process can be killed by mutations in single mutants that mate with residents [Bij11(22) + B11ij (22) ≈733 (bij11(22) + b11ij (22))N ] or by rare recombination between single mutants to produce double mutants R1221(22).734When recombination is more likely than N−1/2 the process is essentially always killed by recombination735r1221(22) (second line in Equation B12).736C. Stochastic crossing times737Neutral single mutants without recombination. With no chance of recombination from single mutants to738double mutants [r1221(22) = 0] we have scaling parameter β = 1/2. Then, without selection on single mutants739(s21 = s12 = 0) and with some mutational symmetry between the two loci [b¯2111(22) = b¯1211(22) = b¯m11(22)], the740single mutants are equivalent and we can concern ourselves with only their sum, ξ = y + z. Letting m be741either single mutant type (m = 21 or 12), Equation (12) reduces to74212ξd2T˜ (ξ)dξ2+[B1111(21) +B1111(12)]dT˜ (ξ)dξ− u22ξ[Bm11(22) +B11m (22)]w22 T˜ (ξ) = −1 (C1)where Bklij (mn) = bklij (mn)N .743When there are an infinite number of single mutants a successful double mutant is produced immedi-744ately, giving one boundary condition limξ→∞ T˜ (ξ) = 0. The second boundary condition is dT˜ (0)/dξ =745−[B1111(21) +B1111(12)]−1, which can be derived directly from Equation (C1) by setting ξ = 0 (see appendix74626. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; A in Christiansen et al., 1998, for a more complete derivation).747The solution to the boundary value problem, evaluated at ξ = 0, corresponding to the expected number748of generations until a successful double mutant arises when beginning with only residents, T = N1/2T˜ (0),749is then750T =N1/2Γ[1/2]Γ[B1111(21) +B1111(12)]Γ[1 +B1111(21) +B1111(12)]√u222[Bm11(22) +B11m (22)]w22, (C2)where Γ[·] is the gamma function. Setting mutation probabilities equal [B1111(21) = B1111(12) = 2Bm11(22) =7512B11m (22) = θ = Nµ] reduces Equation (C2) to the neutral genetic case (equation 27 in Christiansen752et al., 1998) divided by√u22w22 because we census after selection and consider double mutant fixation.753By separating the various mutational terms our analysis clarifies that, while the crossing time is inversely754proportional to the mutation probability from residents to single mutants, it is inversely proportional to the755square root of mutation probabilities from single mutants to double mutants. The crossing time is therefore756increased much more by a reduction in mutations from residents to single mutants than it is by a reduction757in mutations from single mutants to double mutants.758When mutations from residents to single mutants [b1111(21) and b1111(12)] are rare, an approximation for759the crossing time, in terms of our unscaled parameters, is760T ≈ 1N [b1111(21) + b1111(12)]√u224µm11(22)∗ . (C3)Increasing the mutational supply of single mutants [N(b1111(21) + b1111(12))] or the probability of mutation761from single mutants to successful double mutants [u22µm11(m)∗] decreases the amount of time we expect to762wait before a successful double mutant arises. Holding mutation input, θ, constant, Equation (C3) shows763that the crossing time without recombination is roughly proportional to N1/2 generations, aligning with the764semi-deterministic analysis (Equation A2) and indicating that, for a given mutational input, genetic drift765increases the speed at which fitness valleys are crossed.766Neutral single mutants with recombination. With recombination the scaling parameter is β = 1/3. We767can reduce and solve Equation (12) with recombination when the frequencies of single mutants remain768proportional to one another, such that we need follow only cµ y = z = ξ, where cµ is a constant. This769requires mutation input [Nb1111(21), Nb1111(12)] to be large enough to make the dynamics of y and z relatively770deterministic. We further assume no selection on single mutants (s21 = s12 = 0). We then have cµ y = z771for all time, t, when the ratio of mutation probabilities is cµ [i.e., cµb1111(21) = b1111(12)] and we begin with77227. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; cµy(0) = z(0). Equation (12) then collapses to773ξ2(1 + cµ)d2T˜ (ξ)dξ2+B1111(21)dT˜ (ξ)dξ− u22cµr1221(22)∗ ξ2 T˜ (ξ) = −1. (C4)The boundary conditions are limξ→∞ T˜ (ξ) = 0 and dT˜ (0)/dξ = −B1111(21)−1. The solution to the774boundary-value problem, evaluated at ξ = 0, in units of generations, T = N1/3T˜ (0), is775T =25/3pi311/6Γ[2/3] N1/3(1 + cµ)Γ[2(1 + cµ)B1111(21)/3]Γ[2[1 + (1 + cµ)B1111(21)]/3]3√u22cµ(1 + cµ)r1221(22)∗. (C5)Letting cµ = 1, B1111(21) = θ = Nµ, and r1221(22) = r/2 reduces Equation (C5) to the neutral genetic776case (equation 30 in Christiansen et al., 1998) divided by 3√u22w22 because we census after selection and777consider double mutant fixation. Our result extends the insight of Christiansen et al. (1998) by allowing778the frequencies of single mutants to differ, cµ 6= 1. Holding average mutation input [(1 + cµ)Nb1111(21)/2]779constant, Equation (C5) shows that the crossing time is minimized when there are equal numbers of the two780single mutants (cµ = 1) and increases as the asymmetry grows. This occurs because recombination is most781effective in creating double mutants when the single mutants are equally frequent.782Converting the full solution back in terms of our unscaled parameters and letting the mutation probability783b1111(21) be small, we have the approximation784T ≈ 22/3pi35/6Γ[2/3]2 1N2/3b1111(21) 3√u22cµ(1 + cµ)r1221(22)∗ . (C6)Holding mutation input [Nb1111(21)] constant, Equation (C6) shows that the crossing time is roughly propor-785tional to N1/3 generations, aligning with the semi-deterministic analysis (Equation A3).786D. Stochastic simulations787We performed stochastic simulations to verify our analytical and numerical results. Simulation code is788supplied in the supplementary Mathematica file. Briefly, we performed random multinomial sampling of789genotypes with frequency parameters given by Equation (1) and transition probabilities defined in Tables7902-4. Crossing time simulations ended on double mutant fixation and the generation in which this occurred791was recorded as the crossing time. Crossing time was averaged over all trials (103 trials in Figure 1, 102792trials in Figure 3 and 5). Crossing probability simulations ended on resident or double mutant fixation and79328. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; the genotype which fixed in each trial was recorded. The crossing probability was calculated as the fraction794of trials in which the double mutant fixed (103 trials in Figure 2 and 4, 105 trials in Figure 6).79529. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; x x xxxxx x xxxxx xxxx010 00020 00030 00040 000Tx x xxxxx x xxxxx x xxxxk = - 110000k = 0k = 11000010-6 10-5 10-4 0.001 0.01 0.1r0200040006000800010 00012 000Figure 1: Expected number of generations until a double mutant begins to fix, T , as a function of the probability ofrecombination, r. The dots show the full semi-deterministic solution (numerical solution to Equation A1, including higher orderterms, allowing both recombination and mutation to generate double mutants). The solid curves show the semi-deterministicresults when (top) mutation-selection balance is first reached (Equation 8) and (bottom) mutation-selection balance is notreached and crossing can occur by recombination (Equation A3). The dashed line gives the crossing time when crossingoccurs by mutation only, before mutation selection balance is reached, and single mutants are selectively neutral (EquationA2). The X’s are mean simulation results (Appendix D). The grayscale corresponds to (dark) distortion favouring singlemutants, k = 10−4; (medium) the Mendelian case, k = 0; and (light) distortion favouring wild-type, k = −10−4. Parameters:µ = 5× 10−7, N = 106, w22 = 1.05, and (top) w21 = w12 = 1− 10−3 and (bottom) w21 = w12 = 1− 10−5.30. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; xxxxxxxxxxk = 110000k = 0k = - 11000010-8 10-7 10-6 10-5Μ0.010.020.050.100.200.501.00Pxxxxxxxxxxxxxx10-5 10-4 0.001 0.01 0.1r0.0050.0100.0500.1000.5001.000Figure 2: The probability, P = 1−u, of crossing the valley given an initial stock of single mutants (with no further mutationsfrom resident-resident matings) as a function of the rate at which single mutants produce double mutants (top) withoutrecombination (Equation 15) and (bottom) by recombination only (Equation 17). The X’s are simulation results (AppendixD). Parameters and grayscale as in Figure 1 (bottom) with i0 = j0 = 1000.31. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; xxxxxxxxx10-11 10-9 10-7 10-5 0.001 0.1Ν1101001000104105TxxxxxMutation-selection balance HMSBLBefore MSBStochastic10-4 0.01 1 100 104cΜ101001000104105TFigure 3: Expected number of generations until a double mutant begins to fix, T , as a function of (top) the mutationprobability in locus B, ν, and (bottom) the relative mutability of the two loci, cµ = ν/µ. The top panel holds the mutationprobability in locus A (µ = 5 × 10−7) constant while the bottom panel holds the average mutation probability [(µ + ν)/2 =µ(1 + cµ)/2 = 5 × 10−7] constant. The solid curves show the stochastic crossing time by recombination with neutral singlemutants (Equation C5). The dashed curves show the semi-deterministic results when crossing occurs (black) before (EquationA3) and (gray) after (Equation 8) mutation-selection balance is first reached. The dots show the full semi-deterministic solution(numerical solution to Equation A1, including higher order terms, allowing both mutation and recombination to generate doublemutants). The X’s are mean simulation results (Appendix D). Parameters as in Figure 1 (bottom), except w22 = 2.01, whichensures s22 ≥ 0 ∀ cµ.32. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; xxx x xxx x xnuclear mutation & i0 constanttotal mutation & i0+ j0 constant1 10 100 1000 104cn0.00.20.40.60.81.0PFigure 4: The probability, P = 1−u, of crossing the valley given an initial stock of single mutants (with no further mutationsfrom resident-resident matings) as a function of the ratio of the initial numbers of single mutants and mutation probabilities,cµ = ν/µ = cn = j0/i0 (Equation 17). The gray curve holds the initial number of A2B1 (i0 = 100) and the mutationprobability in the A locus (µ = 5 × 10−7) constant and varies the initial number of A1B2 (j0) and the mutation probabilityin the B locus (ν). The black curve holds the initial number of single mutants (n0 = i0 + j0 = 200) and average mutationprobability [(µ+ ν)/2 = 5× 10−7] constant. The X’s are simulation results (Appendix D). Other parameters as in Figure 3.33. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; x x xxxxxxxxWeak A2B2 transmission advantageStrong A2B2 transmission advantage0.0 0.1 0.2 0.3 0.4 0.5 q1101001000104105TFigure 5: Expected number of generations until the new combination of cultural traits (A2B2) begins to fix, T , as a functionof the probability of inheritance, q, of the new traits singly (A2B1, A1B2) over the previous combination (A1B1). The curvesshow the estimate given mutation-selection balance is first reached (which assumes A2B1 and A1B2 are disfavoured, q < 0.5;Equation 8). The dots show the full semi-deterministic solution (numerical solution to Equation A1, including higher orderterms, allowing both recombination and mutation to generate double mutants). The X’s are mean simulation results (AppendixD). The transmission advantage for the new combination of cultural traits is either weak (thin curves, small dots: p = 0.51) orstrong (thick curves, large dots: p = 0.6). Parameters: N = 103, µ = 10−3, r = 0.01, and w21 = w12 = w22 = 1.34. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; x xxxxxx xxxxxNo recombination È r = 0Recombination È r = 0.010.45 0.46 0.47 0.48 0.49 0.50 q10-510-40.0010.010.11PFigure 6: The probability, P = 1 − u, that the new combination of cultural traits (A2B2) fixes given an initial number ofA2B1 and A1B2 (with no further mutations from resident-resident matings) as a function of the probability of inheritance,q, of the new traits singly (A2B1, A1B2) over the previous combination (A1B1). The grey curves show the probability ofcrossing in the absence of recombination (r = 0; Equation 15), with a strong (thick curves: p = 0.6) or weak (thin curves:p = 0.51) transmission advantage for the new combination of cultural traits. The black curves show the probability ofcrossing by recombination only (r = 0.01; Equation 17). The X’s are simulation results (Appendix D). With such a largemutation probability, crossing can be more likely without recombination, which has the added effect of breaking apart the newcombination. Parameters as in Figure 5 with i0 + j0 = 20 and c = 1.35. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; 020 00040 00060 00080 000100 000T∆ =1100000010 00020 00030 00040 00050 00060 000∆ =11000010-6 10-5 10-4 0.001 0.01 0.1r0100 000200 000300 000400 000∆ =11000Figure A1: Expected number of generations until a double mutant begins to fix, T , as a function of recombination, r, given(gray) mutation-selection balance is first reached (Equation 8) or mutation-selection balance is not reached and (black, solid)crossing can occur by recombination (Equation A3) or (black, dashed) crossing occurs by mutation only and −s21 = −s12 =δ = 0 (Equation A2). The dots show the full semi-deterministic solution (numerical solution to Equation A1, including higherorder terms, allowing both recombination and mutation to generate double mutants). The mutation-selection balance estimate(gray) performs better than the dynamic estimates (black) when δ T > 1, and vice-versa. Parameters: symmetrical, Mendelianinheritance with N = 105, s22 = 0.05, and µ = 5× 10−7 (see supplementary Mathematica file).36. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; Table 1: Parameters used throughout textSymbol Descriptionxij frequency of AiBj in the current generationx′ij expected frequency of AiBj in the next generationwij viability of AiBj relative to viability of A1B1V normalizing factorN number of individuals in the populationt time, in units of generationsbklij (mn) probability AiBj mother and AkBl father produce AmBn offspringb¯klij (mn)∗ average probability of surviving AmBn offspring from AiBj x AkBl mating,12wmn[bklij (mn) + bijkl(mn)]µklij (mn)∗ probability of surviving mutant offspring, b¯klij (mn)∗, m 6∈ {i, k}, n 6∈ {j, l}rklij (mn)∗ probability of surviving recombinant offspring, b¯klij (mn)∗, m ∈ {i, k}, n ∈ {j, l},mn 6∈ {ij, kl}b¯klij (mn)average transmission probability before selection, b¯klij (mn)∗/wmn[similarly for µklij (mn) and rklij (mn)]sij selection on AiBj in a resident population, 2b¯ij11(ij)∗ − 1T generations until first successful double mutant arisesu22 probability that a double mutant begins a lineage that will fixit number of A2B1 individuals in generation t (similarly for A1B2, jt)X(t) numbers of single mutants in generation t assuming no double mutants, (it, jt)4i change in number of A2B1 individuals, it+1 − it (similarly for A1B2, 4j = jt+1 − jt)α, β scaling parameters in diffusion processτ scaled unit of time, t/NαY (τ) scaled frequency of A2B1, iτ/Nβ (similarly for A1B2, Z(τ) = jτ/Nβ)µY (y) first moment of 4Y = Y (τ + 1)− Y (τ) given Y (τ) = y = i/Nβ (similarly for Z)σ2Y (y) second moment of 4Y given Y (τ) = y (similarly for Z)κ(y, z) rate diffusion killed by successful double mutants given Y (0) = y, Z(0) = zBklij (mn) scaled transmission probability, bklij (mn)NβR1221(22) scaled (rare) recombination from single mutants to double mutants, r1221(22)N1/2Sij scaled selection on AiBj in population of residents, sijNβT˜ (y, z) scaled time until first successful double mutant given Y (0) = y, Z(0) = z, TNαm index for single mutant types when equivalent (e.g., sm = s21 = s12)cµ mutation rate at locus B relative to locus A, ν/µcn initial number of A1B2 individuals, relative to A2B1, j0/i0ξ scaled frequency of single mutants (y + z or ci y = z, depending on assumptions)u(y, z) probability no successful double mutant appears given Y (0) = y, Z(0) = zn0 initial number of single mutants, i0 + j037. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; Table2:Transmissionprobabilities,bklij(mn),withsegregationdistortion(autosomalkilling).Recombinationoccurswithprobabilityr,followedbyautosomalkillingofstrength0≤k≤1,andmutationwithprobabilityµ.Whenk>0thekillingallelesarethemutantalleles(A2andB2)andwhenk<0thekillingallelesaretheresidentalleles(A1andB1).ParentsOffspringMotherFatherA1 B1A2 B1A1 B2A2 B2A1 B1A1 B1(1−µ)2µ(1−µ)µ(1−µ)µ2A1 B1A2 B11−k2(1−µ)21+k2(1−µ)1−k2µ(1−µ)1+k2µ+1−k2µ2A1 B1A1 B21−k2(1−µ)21+k2(1−µ)1−k2µ(1−µ)1+k2µ+1−k2µ2A1 B1A2 B2(1−r)1−k2(1−µ)2(1−r)1−k2µ(1−µ)+r2(1−r)1−k2µ(1−µ)+r2(1−r)1+k2+(1−r)1−k2µ2A2 B1A1 B11−k2(1−µ)21+k2(1−µ)1−k2µ(1−µ)1+k2µ+1−k2µ2A2 B1A2 B101−µ0µA2 B1A1 B2r1−k21−r2(1−µ)1−r2(1−µ)r1+k2A2 B1A2 B201−µ201+µ2A1 B2A1 B11−k2(1−µ)21+k2(1−µ)1−k2µ(1−µ)1+k2µ+1−k2µ2A1 B2A2 B1r1−k21−r2(1−µ)1−r2(1−µ)r1+k2A1 B2A1 B2001−µµA1 B2A2 B2001−µ21+µ2A2 B2A1 B1(1−r)1−k2(1−µ)2(1−r)1−k2µ(1−µ)+r2(1−r)1−k2µ(1−µ)+r2(1−r)1+k2+(1−r)1−k2µ2A2 B2A2 B101−µ201+µ2A2 B2A1 B2001−µ21+µ2A2 B2A2 B2100038. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; Table 3: Transmission probabilities, bklij (mn), with cytonuclear inheritance. The A locus is biparentally inherited with µ themutation probability from A1 to A2. The B locus is uniparentally inherited with ν the mutation probability from B1 to B2.Parents OffspringMother Father A1B1 A2B1 A1B2 A2B2A1B1 A1B1 (1− µ)(1− ν) µ(1− ν) (1− µ)ν µνA1B1 A2B11−µ2 (1− ν) 1+µ2 (1− ν) 1−µ2 ν 1+µ2 νA1B1 A1B2 (1− µ)(1− ν) µ(1− ν) (1− µ)ν µνA1B1 A2B21−µ2 (1− ν) 1+µ2 (1− ν) 1−µ2 ν 1+µ2 νA2B1 A1B11−µ2 (1− ν) 1+µ2 (1− ν) 1−µ2 ν 1+µ2 νA2B1 A2B1 0 1− ν 0 νA2B1 A1B21−µ2 (1− ν) 1+µ2 (1− ν) 1−µ2 ν 1+µ2 νA2B1 A2B2 0 1− ν 0 νA1B2 A1B1 0 0 1− µ µA1B2 A2B1 0 01−µ21+µ2A1B2 A1B2 0 0 1− µ µA1B2 A2B2 0 01−µ21+µ2A2B2 A1B1 0 01−µ21+µ2A2B2 A2B1 0 0 0 1A2B2 A1B2 0 01−µ21+µ2A2B2 A2B2 0 0 0 139. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; Table4:Transmissionprobabilities,bklij(mn),withculturalinheritance.Parentaltraitcombinationsarebrokenupwithprobabilityr,followedbybiasedtransmission(A2B1andA1B2arepasseddownoverA1B1withprobabilityq,A2B2ispasseddownoverA1B1withprobabilityp),andmutationwithprobabilityµ.ParentsOffspringMotherFatherA1 B1A2 B1A1 B2A2 B2A1 B1A1 B1(1−µ)2µ(1−µ)µ(1−µ)µ2A1 B1A2 B1(1−q)(1−µ)2(1−q)µ(1−µ)+q(1−µ)(1−q)µ(1−µ)(1−q)µ2+qµA1 B1A1 B2(1−q)(1−µ)2(1−q)µ(1−µ)(1−q)µ(1−µ)+q(1−µ)(1−q)µ2+qµA1 B1A2 B2(1−r)(1−p)(1−µ)2(1−r)(1−p)µ(1−µ)+r2(1−µ)(1−r)(1−p)µ(1−µ)+r2(1−µ)(1−r)[(1−p)µ2+p]+rµA2 B1A1 B1(1−q)(1−µ)2(1−q)µ(1−µ)+q(1−µ)(1−q)µ(1−µ)(1−q)µ2+qµA2 B1A2 B101−µ0µA2 B1A1 B2r(1−p)(1−µ)21−r2(1−µ)+r(1−p)µ(1−µ)1−r2(1−µ)+r(1−p)µ(1−µ)(1−r)µ+r[p+(1−p)µ2]A2 B1A2 B20(12 −p+q)(1−µ)0(12 −p+q)µ+12 −p+qA1 B2A1 B1(1−q)(1−µ)2(1−q)µ(1−µ)(1−q)µ(1−µ)+q(1−µ)(1−q)µ2+qµA1 B2A2 B1r(1−p)(1−µ)21−r2(1−µ)+r(1−p)µ(1−µ)1−r2(1−µ)+r(1−p)µ(1−µ)(1−r)µ+r[p+(1−p)µ2]A1 B2A1 B2001−µµA1 B2A2 B200(12 −p+q)(1−µ)(12 −p+q)µ+12 −p+qA2 B2A1 B1(1−r)(1−p)(1−µ)2(1−r)(1−p)µ(1−µ)+r2(1−µ)(1−r)(1−p)µ(1−µ)+r2(1−µ)(1−r)[(1−p)µ2+p]+rµA2 B2A2 B10(12 −p+q)(1−µ)0(12 −p+q)µ+12 −p+qA2 B2A1 B200(12 −p+q)(1−µ)(12 −p+q)µ+12 −p+qA2 B2A2 B2000140. CC-BY-NC-ND 4.0 International licensenot peer-reviewed) is the author/funder. It is made available under aThe copyright holder for this preprint (which was. http://dx.doi.org/10.1101/025502doi: bioRxiv preprint first posted online Aug. 26, 2015; 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.52383.1-0042086/manifest

Comment

Related Items