Evolution of Vertical and Oblique Transmission Under Fluctuating

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EVOLUTION PNAS PLUS oblique transmission, at a rate dependent on the trait frequency Proof: If we rewrite Eq. 4 as x 0 = x · f (x), it can be seen that in the parental generation, occurs in addition to classical ver- f (1) = 1, and for ρ > 0 and 0 < x < 1, tical transmission. We then ask how fluctuations in selection interact with the rate of oblique transmission to affect evolu- f (x) > 1 when wA > wB , [5] tionary dynamics and how the rate of oblique transmission itself f (x) < 1 when wA < wB . might evolve. ∗ In our formulation, both the parental phenotype and the dis- Hence, as wA > 0 and wB > 0, both fixations in A or in B (x = 1 tribution of phenotypes in the whole population contribute to an for fixation in A and x ∗ = 0 for fixation in B) are equilibrium offspring’s phenotype. Using conventional modifier theory (27), points of Eq. 4. Moreover, if xt is the value of x at the tth gen- we show that, in a symmetric cyclic selection regime with cycles eration (t = 0, 1, 2, ... ), from Eqs. 4 and 5, we have, for any of periods 1 or 2, an allele reducing the rate of vertical trans- 0 < x0 < 1 and all t = 0, 1, 2, ..., mission is expected to increase in frequency when rare and in so doing, to increase the mean fitness of the population. However, xt+1 > xt when wA > wB , [6] for cycles of greater length or period asymmetry, interesting non- xt+1 < xt when wA < wB , monotonicities emerge both in the uninvadable rate of vertical transmission and in the rate that maximizes the geometric time and since x ∗ = 1 or x ∗ = 0 is the only equilibrium point, we have average of the population mean fitness, which we will refer to as the “geometric mean fitness.” We develop the models in very limt→∞xt = 1, for all 0 < x0 ≤ 1, when wA > wB , [7] large populations with cyclic selection and with random fitness limt→∞xt = 0, for all 0 ≤ x0 < 1, when wA < wB . and also in the case where drift occurs via sampling from generation to generation in a finite population. Therefore, fixation of the favored phenotype is globally stable.

Model Periodically Changing Environment. Suppose the environment Consider an infinite population whose members are character- changes periodically, such that the favored phenotype changes ized by their phenotype φ, which can be of two types, φ = A or after a fixed number of generations. Simple examples are φ = B, with associated frequencies x and (1 − x), respectively. A1B1 = ABABAB... , in which the favored phenotype switches We follow the evolution of x over discrete nonoverlapping gen- every generation, or A2B1 = AABAABAAB... , where every erations. In each generation, individuals are subject to selection, two generations, in which selection favors A, are followed by a where the fitnesses of A and B are wA and wB , respectively. single generation, in which selection favors B. In general, AkBl An offspring inherits its phenotype from its parent via vertical denotes a selection regime, in which the period is of (k + l) gen- transmission with probability ρ and from a random individual in erations, with k generations favoring phenotype A followed by l the parental population via oblique transmission with probabil- generations favoring B. ity (1 − ρ). Therefore, given that the parent phenotype is φ and Let W be the fitness of the favored phenotype and w be that assuming uniparental inheritance (28), the conditional probabil- of the other phenotype, where 0 < w < W . Rewrite Eq. 4 as 0 0 0 ity that the phenotype φ of the offspring is A is x = FA(x) = xfA(x) when A is favored and x = FB (x) = xfB (x) when B is favored. Then, 0 (1 − ρ)x + ρ if φ = A P(φ = A|φ) = , [1] (1 − ρ)x if φ = B x(1 − ρ)(W − w) + ρW + (1 − ρ)w f (x) = A x(W − w) + w where x = P(φ = A) in the parent’s generation before selection. 0 W − w Therefore, the frequency x of phenotype A after one genera- = 1 + ρ(1 − x) , tion is given by the recursion equation Wx + w(1 − x) [8] x(1 − ρ)(w − W ) + ρw + (1 − ρ)W 0 wA f (x) = x = ρ x + (1 − ρ)x B x(w − W ) + W w [2] wA wB w − W = x (1 − ρ)x + ρ + (1 − x) (1 − ρ)x , = 1 + ρ(1 − x) . w w wx + W (1 − x) where w is the mean fitness, namely If xt denotes the frequency of the phenotype A at generation t w = wAx + wB (1 − x). [3] starting with x0 initially, then as we are interested in the values of xt for t = n(k + l) with n = 0, 1, ... at the end of complete Eq. 2 can be rewritten as periods, we can write h w − w i x 0 = x 1 + ρ(1 − x) A B x = F (x ), n = 0, 1, 2, ..., [9] w (n+1)(k+l) n(k+l) [4] x(1 − ρ)(w − w ) + ρw + (1 − ρ)w = x · A B A B . where F is the composed function x(wA − wB ) + wB F = FB ◦ FB ◦ · · · ◦ FB ◦ FA ◦ FA ◦ · · · ◦ FA . [10] In what follows, we explore the evolution of the recursion Eq. | {z } | {z } 4, namely the equilibria and their stability properties, in the cases l times k times of constant environments and changing environments. Clearly, since FA(0) = FB (0) = 0 and FA(1) = FB (1) = 1, both Constant Environment. When the environment is constant, the fit- fixations in A or in B are equilibrium points. An interesting question is when these fixations are locally stable. We concentrate on ness parameters wA and wB do not change between generations, ∗ 0 x = 0, the fixation of the phenotype B. As x = FA(x) = xfA(x) and we have the following result. 0 for k generations and x = FB (x) = xfB (x) for l generations, the

Result 1. If 0 < ρ ≤ 1 and both wA and wB are positive with linear approximation of F (x) “near” x = 0 is wA6 =wB , then ﬁxation in the phenotype A (B) is globally stable k l when wA > wB (wA < wB ). F (x) ' fA(0) fB (0) x. [11]

2 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1719171115 Ram et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 dclycagn niomns ohfiain a eunstable, be may polymorphism. fixations protected a both producing environments, changing odically eut3. Result reuvlnl,if equivalently, or if in general For ii) product the by mined unstable. a tal. et Ram entails sumption Conclusions. Since that Observe 2. Result case the with start We one. than larger is Eq. it From if unstable and one than of uct stability local the Hence, xtosusal iigtefloigresult. following the giving unstable fixations B choosing by n ti mosbeta ohfiain r tbe Furthermore, stable. are Eq. fixations by both that since impossible is it and hrfr,following Therefore, n htof that and Since i) f A a Proof: Proof: + 1 B A (29). polymorphism protected a is there and lost, be in fixations in two fixation the and between symmetry total is there btntbt)cnb tbe nadto,w a aeboth have can we addition, In stable. be can both) not (but l (0) b f susal if unstable is A k is 0 ρ (0) > k < ρ < f W k A With If k hrfr,bt xtosaeusal if unstable are fixations both Therefore, 1. Let h oa tblt of stability local The f 0 + 1 = (0) log ehave we 8, B w k = − k (0) B f k l w = a 1 12, B 0 a is n h bv eutas od when holds also result above the and and + (0) k and l < ρ < k + 1 = , a and f l = a A l A log h odto o oa tblt ffiainin fixation of stability local for condition the , + 1 k > a l ρ l (0) 0 sas ntbe hs ete hntp can phenotype neither Thus, unstable. also is f b ; k A W prpitl,fiainon fixation appropriately, 1 x < l eut2, Result 0 b 1 ρ b (0) f f k > ∗ ρ + 1 l A A > and < w W w and > − > = 0 = w (0) (0) n iial,fiainin fixation similarly, and 1, 1 w w −w < 0 k w 1 l − W ρ . +l and 0 < 0 W W slclysal fti rdc sless is product this if stable locally is k l , W x < < and and W and f ∗ x f f xto on fixation , w B B − B b ∗ w 0 0 = (0) (0) (0) < 0 = < with w + 1 = < f b B olwn Eq. Following 1. l f a sdtrie yteprod- the by determined is W B 0 + 1 = (0) + 1 = k h xto of fixation the , k log l l +l (0) b < < 0 + 1 k ntecs of case the in ρ(1 < ρ < a 1. 1, > > < + 1, ρ 1 ρ B − 1 w k w −W sunstable. is W when ρ) xto of fixation 1, ρ log − W A w (W W n u as- our and , b − W rfiainon fixation or A wW > 0 0 W B − fixation 11, sunstable is AkBl < < 0. sdeter- is , A k w . w W . ) and 2 < > < peri- [17] [16] [15] [19] [18] [14] [13] [12] B W B 1. w is , , , o h nqaiiso Eq. of inequalities the Now rqec fe n yl of cycle one after frequency h w xtosaeusal,adteeeit nqestable F unique a exists there and unstable, are polymorphism. fixations two the 4. Result following. the have we ment, of of values polymorphism gives polymorphic that more or one have equilibria. to expect we and polymorphism, in fixation ther fact the on of only B cycle depend a fixations in two that, phenotypes the of which properties in of stability cycle order a the within favored on depend not (i.e., and lo as Also, As is equation equilibrium The F Also, positive one H and negative one 0 roots, real two has in equation B polynomial degree fourth a equation x F A B < 0 B 1 = 0 (x Proof: of case simple the For h ierapoiainof approximation linear The if only and if consistent are inequalities These and sfavored is (x = ( 0 x x F = ) = ) < ρ < x ab ) ∗ h te qiiracrepn osltoso quadratic a of solutions to correspond equilibria other the , A 2 α < A F F (1 (x > F x 1 B A h atrdtrie nqeplmrhs.Let polymorphism. unique a determines latter The 1. r qiiracrepnigt h solutions the to corresponding equilibria are 2 − s Q α ) B Let = ntecase the In ,wihi reb Eq. by true is which 1), (x (y (1 i.S1 Fig. 2 = 1 ρ)(w (x hr,b Eq. by where, , F (2 1 = − = ) = ) and l W H = ) and A x ρ)(W ie n o hi re ntecce hnnei- When cycle. the in order their not and times − (x 0 0, = (0) A − x y − W eteiiilfeunyof frequency initial the be and 0 ρ α k ) y x W )(W o htin that nor lutae h eainhpbetween relationship the illustrates (k Q Q w < 2 (1 (1 /b log(1 Then, . − + x ) . 1 = (1) = (0) w 2 + 2 log − − α w w 2xW + + < 0 A1B − ) l ρ)(w ρ)(W 2 ) < a α k W [x H w 2 + generations,A A1B 1 (2 (2 8, h udai equation quadratic the 0, + ) (w ) y x 19 (x x 1 ehave we , [x A xw < (w (1 − − − − l x + (W B ( ∗ − − odi n nyif only and if hold with eeain.Teeoe h local the Therefore, generations. W 1 l = − = ) (1 and 1, W ρ)(W ρ)(W F α − W < w r tbe hr saprotected a is there stable, are −w eidclycagn environ- changing periodically ρ)(w A1B 0 F W − 14. (x − − α + ) + ) W + ) k 0 = x B ρ)(W w 0 w ) B ∗ 0 /b log(1 + ) = + ) , near F W + ) − rfiain o different for fixation, or , − − ρw ρW < ρ < 1 with , log A W (2 w w ] x NSEryEdition Early PNAS w (x 2 H W − eeto.Then, selection. sfavored is w (1 + ic h xtosin fixations the Since . ) ) ] a + ) − (1 + 2 x w 1 1. = (1) ) > < 1 α ∗ ) + ) ρ)(W hc eue to reduces which , W . 2 0 = 0. 0 −w − A and − 1 = /b log(1 w [ρw ρ [ρW and ρ)w )W A (Eq. x − and (1 + k 0 ∗ and satisfying , w < (1 + . , ie and times x k x Q ) does 11) ) , − w 0 0 = | . < l (x ρ)W B eits be − < and f10 of 3 log 0 = ) x [20] [23] [22] [21] [24] ρ)w and [26] W are [25] 0 ] = a ρ . , ]

EVOLUTION PNAS PLUS 0 From our assumptions on ρ, w, and W , we have FA(x) > 0 for 0 A 0 ≤ x ≤ 1. Observe that the numerator of FB (x) is linear in ρ; its value when ρ = 1 is wW > 0, and when ρ = 0, it is

x 2(w − W )2 + 2xW (w − W ) + W 2 = [x(w − W ) + W ]2 > 0. [27]

0 0 0 Hence, FB (x) > 0 for all 0 ≤ x ≤ 1, and H (x) = FB FA(x) 0 FA(x) is positive when 0 ≤ x ≤ 1. Thus, H (x) is monotone increasing for 0 ≤ x ≤ 1; H (x) > x is monotone increasing for 0 < x < x ∗, and H (x) < x is monotone increasing for x ∗ < x < 1. ∗ Starting from any initial value 0 < x0 < 1, we have xt → x as t → ∞. Fig. S2 A, C, and E illustrates how the frequency of A changes over time in the A1B1 regime of cycling selection. For more general cyclic fitness regimes, the polynomial that gives the equilibria is of higher order, and it is conceivable that more than one stable polymorphism could exist for given values B of ρ, W , and w. We have been able to show that, when neither fixation in A nor fixation in B are stable, in the AkBk case, this cannot occur. In fact, we have the following.

Result 5. In the AkBk selection regimes, if the fixations in A and B are locally unstable, a single stable polymorphic equilibrium exists. The proof of Result 5 is in SI Text. Fig. S3A shows the stable equilibrium frequencies x ∗ as a function of ρ, W , and w in the A1B1 regime. For AkBk selection regimes from k = 1 to k = 40, Fig. S4 illustrates the convergence to a single stable polymorphism. We have not been able to prove that, for selection regimes AkBl with l 6 = k, there is a single stable polymorphic equilibrium when the two fixations are unstable. However, the numerical examples in Fig. S1 for AkBl and in Fig. 1 and Fig. S5 for the special case A1B2 all exhibit a single stable polymorphic equilibrium when fixations in A and B are unstable. These numerical Fig. 1. Stable frequency of phenotype A and geometric mean fitness in results suggest that, for W > w > 0 and 0 < ρ < 1, the high-order selection regime A1B2 as a function of the vertical transmission rate ρ and equilibrium polynomial has only a single root corresponding to a the fitness of the disfavored phenotype w.(A) Stable frequency of phe- globally stable polymorphism. Fig. S6 shows that this is the case notype A at the end of each three-generation cycle. (B) Geometric aver- for the A3B10 regime. age of the stable population mean fitness over the three-generation cycle: (w* · w** · w***)1/3. Gray contour lines join ρ and w combinations that Randomly Changing Environment. We now consider the case result in the same stable value. In all cases, W = 1. where the environment changes according to a stochastic process. Without loss of generality, assume that the fitness param- In our case, there are two constant equilibria x ∗ = 0 and x ∗ = 1 eters at generation t (t = 0, 1, 2, ... ) are 1 + st for phenotype corresponding to fixation in B and A, respectively. We can char- A and 1 for phenotype B, where the random variables st for acterize the stochastic local stability of these fixations with the t = 0, 1, 2, ... are independent and identically distributed. Also following results, and proofs are in SI Text. assume that there are positive constants C and D, such that P(−1+ C < st < D) = 1. ∗ Result 6. Suppose E [log(1 + ρst )]> 0. Then, x = 0, the fixa- Corresponding to Eq. 4, with wA = 1 + st and wB = 1, the tion of phenotype B, is not stochastically locally stable. In fact, recursion equation is P(limt→∞xt = 0) = 0.

1 + ρst + xt (1 − ρ)st ∗ xt+1 = xt t = 0, 1, 2, . . .. [28] Result 7. Suppose E[log(1 + ρst )] < 0. Then, x = 0, the fixation 1 + xt st of phenotype B, is stochastically locally stable. In particular, if ∗ E(st ) ≤ 0, x = 0 is stochastically locally stable. As {xt } for t = 0, 1, 2, ... is a sequence of random variables, the Using the general notation for the fitness parameters wA and notion of stability of the two fixation states needs clarification. wB , the stochastic local stability of fixation in B is determined Following Karlin and Lieberman (30) and Karlin and Liberman h i by the sign of E log 1 − ρ + ρ wA , and that of fixation in A is (31), we make the following definition. wB h i Definition: “Stochastic local stability” is defined as follows. A wB ∗ determined by the sign of E log 1 − ρ + ρ . For example, constant equilibrium state x is said to be stochastically locally wA ∗ stable if, for any ε > 0, there exists a δ > 0, such that |x0 − x | < δ if the sign of the first is negative, fixation in B is stochastically implies locally stable, and when it is positive, with probability of one, convergence to fixation in B does not occur. It is also true that, if ∗ E(wA/wB ) ≤ 1, then fixation of B is stochastically locally stable. P lim xt = x ≥ 1 − ε. [29] t→∞ Following Eq. 14, for all realizations of wA and wB ,

∗ Thus, x is stochastically locally stable if for any initial x0 sufﬁ- ∗ ∗ ciently near x the process xt converges to x with high proba- w w log 1 − ρ + ρ A + log 1 − ρ + ρ B > 0. [30] bility. wB wA

4 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1719171115 Ram et al. 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Fig. a tal. et Ram Eq. to frequencies leading pheno-genotype rationale the Following by denoted are generation, given and n h oie ou sslcieynurl ehv h follow- the have we neutral, selectively table: ing is phenotypes locus two modifier the the by and determined are fitnesses the As genotypes: A ewe hs ie,fiaino ihrpeoyei o stochas- not is phenotype either of fixation lines, these between 6, Result and = ihpoaiiy1 probability with etcltasiso rate transmission vertical M 1 ti mosbeta ohfiain r simultaneously are fixations both that impossible is it , + A tcatclclsaiiy h gr hw h rqec fphe- of frequency the shows figure The stability. local Stochastic M be s pheno-genotype fe 10 after t E ρ e h etcltasiso ae eemndby determined rates transmission vertical the Let . [log(1 and = ρ mA frequency 0.1. fitness and w , − B 6 mB = ρ eeain navr ag ouaineovn na in evolving population large very a in generations P + ,rsetvl,where respectively, 1, − epciey hs hr r orpheno- four are there Thus, respectively. , , A ρ h rylnsmr obntosof combinations mark lines gray The p. and MA, w w A A B nacs where case a in .Tefinse fphenotypes of fitnesses The 28). = )] n npriua,tesohsi local stochastic the particular, in and A is and 0 x x 1 0 0 , 2 1 Am AMB MA mB mA MB = w x x n t aineicessas increases variance its and , ρ ρ 2 x 1 0 A 0 0,adtevria transmis- vertical the and 000, 1/10, B E , 1 [log(1 x , i.3sostedynamics the shows 3 Fig. . ihfeunista,a a at that, frequencies with , s 3 0 x t and , 2 w s x nEq. in , t 2 B − x w is h etgeneration next the 2, 3 A ρ and , s x + ihprobability with 4 and 0 w x P P ρ are 28 3 A w w A x B w 4 = )] fettefre- the affect respectively. , B w x .According 0. 4 B r identi- are A A and . p and p and B [31] and are m B ρ s n eeain fe w eeain,w have we generations, two after generation, ond hr h olna transformation nonlinear the where w general the on depends the and of exists case simple the particular, invasion A1B of in to and independent environments 32) type, changing (27, favored stability the external of its allele check by we present, is variants. genetic haploid as viewed if introduced were be might which recombination, to the assumptions, these A/B under that, Note with A 3. Fig. that seen be can it and equilibrium rate associated x T with loicess h tesso phenotypes of fitnesses The increases. also with distribution polymorphic a reaches rate transmission vertical phe- the of frequency the of Dynamics environment. notype changing randomly a in with equation quadratic the of root tive r ohepnnilrno aibe ihepce auso two. of values expected with variables random exponential both are 00 B 2 wx wx wx wx and trigwt tbeeulbim hr nythe only where equilibrium, stable a with Starting o the For x r h rqec vectors. frequency the are = 0 4 3 2 1 w α 1 0 0 0 0 w hntpcdcooyd o neg ntiganalogous anything undergo not do dichotomy phenotypic sgvnb Eq. by given is B w = = = = 2 A yln niomn,weeauiu tbepolymorphism stable unique a where environment, cycling h enfins,gvnby given fitness, mean the , A feto etcltasiso rate transmission vertical of Effect r dnial n neednl itiue admvrals As variables. random distributed independently and identically are x , vrtm trigat starting time over ntefis eeainand generation first the in 1 w w w w α = ∗ M 1 A A A A = W A1B , x x x x AkBl osatevrnetawy ed ofixation to leads always environment constant A . α 3 3 1 1 x 2 1 , 0 (1 (1 ∗ (1 (1 − w pcfidi Eq. in specified (x = 1 w − − ρ, B − − W ae hnol the only when case, ae.Seicly rmEq. from Specifically, case). = P ρ)(x = 0 P ρ)(x 1 ∗ )(x w + 32 < ρ < w √ 1 , )(x A ρ 2 w W Ww 2 (x 1 n h olna transformation nonlinear the and with − + IText (SI 1 x + − + 1 ρ 00 + − x x x 2 + and 1, x 0 nrae rm001t .,tefrequency the 0.5, to 0.001 from increases q 1 x = 4 ∗ − x 4 = w · 3 + ) ,0) 0, , w 3 x + ) (2 + ) (1 T w + ) 10 3 A Solving 22. + ) 2 − w − = −5 (T a opttoa nlssof analysis computational a has < w ρ B P Q E ρ)(W B ρ) w 0 x w hntefinse fphenotypes of fitnesses the when 1 w (x x exists. + A x 1 x 2 (x < , ∗ x 2 B A t 4 ρ + (W ) ), w and e hrfr,assume therefore, We, ρ. 0 < w (x → (1 w = ) = npeoyepolymorphism phenotype on = (1 m B w B NSEryEdition Early PNAS 2 2 1 − < w atr u h variance the but faster, 0.5 B − T = x B, − + − . α leei rsn with present is allele 2 x Q x , M 1 w W ρ)(x w (1 W 4 1 2 w x ∗ x P w (x (1 ) x A 4 /m ) B )(x nqestable unique a , − sgvnb Eq. by given is 2 ). 2 steol posi- only the is Here, . and 32 A/B 0 = ) − 2 = 4 + + ρ 2 + )(x P with ou n the and locus W α w + Ww x )(x 1 B phenotypes gives 4 1 respectively, , x x ntesec- the in + ) 4 + x 1 + w + ) m , , + A | x x α ρ 3 0 = x 0 f10 of 5 allele P ) and , x 3 [33] [36] [34] [32] [35] 0 = 00 ) W , 32 = , ,

EVOLUTION PNAS PLUS AB C

Fig. 4. Consecutive fixation of modifiers that reduce the vertical transmission rate in selection regime A1B1. The figure shows results of numerical simulations of evolution with two modifier alleles (Eq. 32). When a modifier allele fixes (frequency > 99.9%), a new modifier allele is introduced with a vertical transmission rate one order of magnitude lower (vertical dashed lines). (A) The frequency of phenotype A in the population over time. (B) The frequency of the invading modifier allele over time. (C) The population geometric mean fitness over time; Inset zooms in to show that the mean fitness increases slightly with each invasion. Invading alleles are introduced at frequency 0.01%; whenever their frequency drops below 0.01%, they are reintroduced. Parameters: vertical transmission rate of the initial resident modifier allele, ρ0 = 0.1; fitness values: W = 1 and w = 0.5. The x axis is on a log scale, as each sequential invasion takes an order of magnitude longer to complete. Fig. S12 shows w = 0.1 and 0.9.

The external stability of x ∗ to the introduction of the modifier Geometric Mean Fitness and Rate of Vertical Transmission allele M with rate P is determined by the linear approximation Under fluctuating selection, the geometric mean fitness of geno- matrix L. We prove the following result in SI Text. types has been shown to determine their evolutionary dynamics (8, 30, 33). For the evolution of mutation rates that are controlled Result 8. L has two positive eigenvalues, and by genetic modifiers, the stable mutation rate and the mutation rate that maximizes the geometric mean fitness of the population i) when P > ρ, the two eigenvalues are less than one; seem to be the same when the period of environmental fluctu- ii) when P < ρ, the largest eigenvalue is larger than one; and ation is low enough (24). We can ask the same question here: iii) when P = ρ, the largest eigenvalue is one. is the stable rate ρ∗ the same as the rate ρˆ that maximizes the

We conclude that, in the A1B1 selection regime, an allele m producing vertical transmission rate ρ is stable to the introduction of a modifier allele M with associated rate P if P > ρ, A and it is unstable if P < ρ. Thus, in this case, evolution tends to reduce vertical transmission and hence, increase the rate of oblique transmission, and there is a reduction principle for the rate of vertical transmission (27, 32). The evolutionary dynamics of the reduction in ρ under the A1B1 cycling regime are shown in Fig. 4, which also illustrates the change in phenotype frequencies over time. In the case of identically distributed random fitnesses wA and wB , Fig. 5 shows an example of the success of modi- B fiers that reduce ρ. We have not, however, been able to prove that there is a reduction principle for this class of fluctuating fitnesses. Values of the evolutionarily stable vertical transmission rate, ρ∗, for some AkBl examples (SI Text and Fig. S10 have analyt- ical details) are recorded in Table 1 for different values of w relative to W = 1. Interestingly, with w = 0.1, the evolutionarily stable value of ρ is zero for the A1B2 regime but not for the A3B10 and A5B30 regimes, in which the only stable values are those that lead to fixation of phenotype B (e.g., ρ > 0.4625 and ρ > 0.1489, respectively); these are, therefore, neu- Fig. 5. Consecutive fixation of modifiers that reduce the vertical transmis- trally stable (Fig. S10). AkBk results are plotted in Fig. 6B. In the sion rate ρ under symmetric randomly changing selection. The figure shows ∗ A2B2 regime, ρ = 0, and there is reduction of vertical transmis- results of numerical simulations of evolution with two modifier alleles (Eq. sion for all selection values tested. However, for AkBk regimes 32). When a modifier allele fixes (frequency > 99.9%), a new modifier allele with k > 2, we find ρ∗ 6 = 0, and depending on w, ρ∗ can be as is introduced with a vertical transmission rate one order of magnitude lower high as 0.95. In Table 1, blank values for ρ∗ indicate that our (vertical dashed lines). (A) The frequency of phenotype A in the population method was numerically unstable and that a precise value for over time. (B) The frequency of the invading modifier allele over time. Invad- ρ∗ ρ∗ ing alleles are introduced at frequency 0.01%; whenever their frequency could not be obtained. This is why, in Fig. 6B, no points drops below 0.01%, they are reintroduced. Parameters: vertical transmis- are shown for AkBk with k > 19. In Table 1, the word “fixa- sion rate of the initial resident modifier allele is ρ0 = 0.1, and the ratio of tion” indicates that fixation of B occurs, at which point there fitness values is w /w = 10 with probability 0.5 and w /w = 0.1 also with ∗ A B A B can be no effect of modification of ρ; ρ cannot be calculated in probability 0.5. The x axis is on a log scale, as each sequential invasion takes such cases. an order of magnitude longer to complete.

6 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1719171115 Ram et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 and is equilibrium stable the at fitness mean where a tal. et Ram Eq. From w of the in equilibrium ble 9. Result result. fluctuat- lowing under the fitness For mean selection? geometric ing the of value equilibrium the of equilibrium stable † the at fitness AkBl mean geometric the maximizes that k of Values 1. Table ltsteproof. the pletes Thus, because Now, the of middle the at fitness mean the Eq. 050 30 50 20 20 20 30 12 20 12 20 12 20 12 12 12 050 50 30 50 30 50 30 30 oethat Note 1 1 1 1 1 2 2 1 30 10 10 5 10 3 3 3 2 30 30 5 5 ∗ Proof: rate. transmission vertical the of value uninvadable the is ρ* ρ. = 35, B cycle. w hc losu ordc h rbe opoete of properties to problem the reduce to us allows which , w ∗∗ dx ∗ Z d If W In sdcesn in decreasing is ρ 36, w = u otesmer ewe h w phenotypes two the between symmetry the to due ∗ W = ∗ = = 1 1 2 2 1 2 2 2 2 q l A1B 0 = 1. W > − (1 x < 2 x ∗ (1 w ∗ x − > − h tbefeunyo phenotype of frequency stable the 1, x − W ∗ (stable ρ* ∗ and − ρ w ρ) 2(2 ≤ 2 1 = A1B , (1 + ρ)(W 2 − w 2 1 A1B {w (W 2 1 0 − 0.1 0.1 0.9 0.5 0.1 0.9 0.5 0.1 0.9 0.5 0.9 0.5 0.5 0.1 0.9 0.9 0.5 0.1 0.9 0.5 0.1 0.5 0.9 0.5 0.1 0.9 0.1 n therefore, and , ∗ (1 − ≤ ρ } ρ)Z sa nraiglna ucinof function linear increasing an is 1 − † − if ρ − 2(2 1 − 2(2 niomn sadcesn function decreasing a is environment x and ρ) ≤ dx w eeto eie ehv h fol- the have we regime, selection ρ)(W ∗ w W )w ) hntema tesa h sta- the at fitness mean the then 1, − − ∗ ) 2 /d 4 + ρ)(W − + ρ = )Z ρ >0.4265 >0.1489 ρ w − ˆ 2(2 x Ww Fixation Fixation Fixation Fixation Fixation Fixation 0000.000000 0.000000 0.000000 0.000000 0000.000000 0.000000 0000.00065 0.000000 0000.000000 0.000000 0.000000 0.000000 0000.000000 0.000000 0.95686 0.99581 0.9768 0.99136 0.96331 0.98304 0.94643 0.91209 0.84924 sngtv.UigEq. Using negative. is (optimal ∗ − w A1B W (W dx ρ* − − ) Z w . > ρ)(W + w ∗ ∗ − /d 1 h geometric The 0. ) · w , w w yl;i hscase this in cycle; < ρ ρ) − ∗∗ + ) − Z where , hc com- which 0, w w ) . ρ ˆ 0.00031 0.00027 0.000000 0.22107 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.24347 0.223925 0.15419 0.193280 sterate the is Fixation Fixation Fixation Fixation Fixation Fixation A w ρ ˆ s by is, ∗∗ [37] [39] [38] 37, x w ∗ A ∗ : is . hnslcini togadtedrto ewe eeto fluctuations selection cases, rate. between stable all duration In the for long. and is strong calculated shows is line be selection dashed when not The (k longer. could rate short are (blue) transmission cycles are selection cycles vertical when 1 selection lower one when or roughly zero to higher from either increases rate rapidly with stable modifiers evolutionary S5 by The (Figs. (B) decrease sition). slow favor- between a quickly by (com- cycles zero selection phenotype is fitness when ing mean transmission) population oblique the of plete average geometric the mized in 6. Fig. w with that, see w we the 6A, in with Fig. positive while in to of zero Also values from the 6. changes from Fig. different in substantially shown zero, also was all At of values the in small fitness mean metric and A1B decreasing with fitness ie emti enfins,aegvni al 1, Table in given are fitness, mean geometric mizes between match to,nml (Eq. namely ation, phenotype that pose population Wright–Fisher finite A the to use due we Let model, drift model. deterministic random above the of in effect size the include To Size Population Finite and S8, B A − 0 = 0 = i.4ilsrtsteices vrtm ftegoercmean geometric the of time over increase the illustrates 4 Fig. napplto ffie size fixed of population a in AkBk AkBk w k A2B −1 1 1 .5, .1, 0 = hc t h ausfor values the fits which , ins otml n vltoaysal etcltasiso rate transmission vertical stable evolutionary and “optimal” Fitness eie h ausof values The regime. h etcltasiso rate transmission vertical The (A) regime. selection IText. SI ρ ρ ˆ 2 ˆ . al hw that shows 1 Table 1, eie htw etdwith tested we that regimes X xed . for 0.2 exceeds eie,nml ohaezr.Fg 1B Fig. zero. are both namely regimes, sbten01 n .4for 0.24 and 0.15 between is w A X t t = 0 = A nteifiiepplto oe ntenx gener- next the in model population infinite the in eoetenme fidvdaswt phenotype with individuals of number the denote Nx and ρ w h hneocr at occurs change the .5, ∗ W hc antb nae,and invaded, be cannot which , hsma tesicessas increases fitness mean this , lo let Also, . 2) B = n hnarpl rniin to transitions abruptly then and 1. ρ IText SI 31 taplmrhceulbimi the in equilibrium polymorphic a at and A1B ≤ x ρ N ˆ a eal nhww acltdthe calculated we how on details has 0 S6 w k k and ersn h rqec fthe of frequency the represent 2 ρ ≤ = ˆ tthe at > aedtiso h buttran- abrupt the on details have eie n eseta,for that, see we and regime, 4) h ausfor values The (41). 0.5 and oedtiso h mis- the on details More 50. 9det ueia instability numerical to due 19 AkBk ,wihcno einvaded be cannot which ρ*, ρ ∗ t ρ w NSEryEdition Early PNAS r h aein same the are 12 hgnrto,adsup- and generation, th ∗ k 0 = r ogl eo In zero. roughly are ≤ eie with regimes nFg ,with 6, Fig. In 31. = k h au of value the .9, ≤ hw h geo- the shows hc maxi- which ρ, hl with while 50, ˆ ≈ ρ is S7 Figs. followed 0.2 ρ ˆ w decreases. htmaxi- that = 0 = | k w r2) or 1 A1B ρ f10 of 7 ≥ ∗ . = 1, and as , 12, 0.1 P ρ ρ 1 ˆ ˆ ,

EVOLUTION PNAS PLUS w x 0 = ρ A x + (1 − ρ)x. [40] The proof of Result 11, based on induction on n, is given in SI Text. w Using the moments in Eq. 45, the fixation probability u(x) and the expected time T (x) to fixation from an initial frequency of x Then, according to the Wright–Fisher model (34), Xt+1, the number of individuals of phenotype A at generation (t + 1), is can be computed, where s is replaced by sn /n. We find determined by the probability −2ρ Sn x 1 − e n u(x) = , [46] Sn N j N −j −2ρ PX = j | X = Nx = x 0 1 − x 0 [41] 1 − e n t+1 t j and T (x) can be computed similarly. for j = 0, 1, 2, ..., N . Thus, the fluctuations in the numbers of In the case of the AkBl cycling environment, we write n = k + phenotypes A and B in the population of size N are generated l, and if wA = W , wB = w for k generations and wA = w, wB = by the Wright–Fisher Markov chain, where, given that Xt = Nx, W for l generations, we have 0 Xt+1 has a binomial distribution with parameters (N , x ). S = S = N (k − l)(W − w). [47] This Markov chain has two absorbing states, Xt = N and Xt = n k+l 0, corresponding to the two fixations in A and B, respectively, and we are interested in the fixation probabilities and the time Fig. 7 shows an example of how (k − l), ρ, and (W − w) in Eq. 46 to fixation of these two absorbing states as functions of the initial for u(x) interact to affect fixation probabilities. More examples are illustrated in Fig. S11. frequency x and also of ρ, wA, and wB . To these ends, we use a {X } diffusion approximation to the process t , which allows us to Discussion compute u(x), the probability that phenotype A goes to fixation when its initial frequency is x, namely Nonchromosomal modes of phenotypic transmission are receiv- ing increasing attention (36–38), especially with respect to their 1 − e−2ρsx potential role in adaptation and maintenance of diversity (39). Here, we have focused on a dichotomous phenotype transmit- u(x) = −2ρs . [42] 1 − e ted through a combination of parental and nonparental transmission. In addition to the roles that these transmission modes The expected time to fixation in A starting from an initial fre- play in the dynamics of phenotypic diversity in large and small quency of x is given by populations, we have also investigated a genetic model for the evolution of the transmission mode itself. 1 − u(x) Z x e2ρsξ − 1 u(x) Z 1 1 − e−2ρs(1−ξ) T (x) = dξ + dξ, Our model differs markedly from that of Xue and Leibler (26), ρs 0 ξ(1 − ξ) ρs x ξ(1 − ξ) who took the individual phenotypic distribution (i.e., the proba- [43] bility that an individual develops one of a set of phenotypes) to be the inherited trait. In our model, the transmitted trait is the where u(x) is given in Eq. 42, and in generations, T (x) is mul- phenotype itself. Thus, with two phenotypic states A and B, we tiplied by N (the derivation is in SI Text). Unfortunately, the track the frequency x of A, whereas Xue and Leibler (26) focus integrals in Eq. 43 cannot be done in closed form unless ρs = 0, on the dynamics of the per-individual probability πA of learning in which case u(x) = x and T (x) = −2x ln x − 2(1 − x) ln(1 − x) the phenotype A. One interpretation of our model is as a mean (ref. 34, p. 160), and only numerical computation of T (x) is possible for specified values of x, ρ, and s. For the fixation probability u(x), we have the following result.

Result 10. When s > 0, so that the phenotype A is favored, the fixation probability u(x) is monotone increasing in ρ. The proof of Result 10 is in SI Text. Fig. S9 compares the fixation probability and time to fixation derived numerically from simulating the Wright–Fisher Markov chain with the diffusion- derived values of u(x) and T (x). The fit is seen to be very good. Note that, when N is large, the Wright–Fisher model exhibits persistent fluctuation around the deterministic expectation, as shown by the orange traces in Fig. S2. We can also develop a diffusion approximation for the case of a cycling environment. Suppose that selection changes in cycles of length n, such that, within the cycle, the fitness parameters are t t wA, wB for t = 1, 2, ..., n. Also, let

t 1 t t X s ' w − w , S = s , t = 1, 2, ..., n. [44] N t A B t i i=1

Following Karlin and Levikson (35), we have the following result. Fig. 7. Fixation in a finite population with different ratios of selection 2 k Result 11. The mean µ(x) and variance σ (x) of the change in the periods l . Fixation probability of phenotype A when starting with a sin- k−l frequency of A in one generation for the diffusion approximation in gle copy in a population of size N: u(1/N) = (1 − exp(−2ρ k+l (W − w))/(1 − the case of a cycling environment AkBl, where k + l = n, are k−l exp(−2Nρ k+l (W − w)) (Eqs. 46 and 47). k and l are the numbers of generations in which phenotypes A and B, respectively, are favored by selection. µ(x) = ρSn x(1 − x) Here, fitness of the favored phenotype is W = 1, fitness of the disfavored 2 [45] phenotype is w = 0.5, and the population size is N = 10, 000. Fig. S11 shows σ (x) = nx(1 − x). additional examples.

8 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1719171115 Ram et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 a tal. et Ram neo hsdsrbto sgetrfrlre ausof values larger involve conditions for be stability greater local may stochastic is both the E vari- because distribution case, The is this this result. This may of in distribution cyclic stable polymorphic ance of a stochastically case and be unstable, the phenotypes in both in fixations stability is not While local stability variation. to local fitness stochastic analog cyclically, appropriate varying the than rather ables such produce conditions. to initial seem the not alternate on does dependence between model differences haploid fitness Our homozygote generations. frequencies the allele be as initial all well on which could as depending under evolution equilibrium the conditions polymorphic with and stable, found fixations and allelic (33) both Jayakar and fit- (compare Haldane cycle homozygote two-generation a equal as with strength with in However, alternating ness, polymorphism. geno- of of mean loss) geometric Eqs. the (compare of Jayakar fitness relevance and typic the Haldane showed regimes, first fitness (33) cycling with models netic without transmission allows phe- it disfavored because the reproduction. selection, protect to from said notype be can transmission oblique pro- 2 which in consider rate (40), death coexistence: effect species species, storage maintains two unstable. the selection are to from fixations similar tection both is if rate result polymorphism the This stable and unique differences a fitness is the both Eq. the of on ρ hence, inequalities depend the and which by fixations given 20, both are polymorphism of of instability protection the the In determine stable. that globally is morphism hr sasnl tbeplmrhceulbim(ihpheno- (with equilibrium polymorphic stable single types a is there If (the stable. values be fitness symmetric cycling particu- AkBl In deterministically complicated. with more much lar, general, in are, frequencies retarded. is be fixation trans- oblique to to more approach with polymorphism mission, but a environment, constant allow a not in achieved does (Eq. model selection undergone phenotypic have simple who parents the from ing offspring an that chance becomes added the represents rate, transmission type: rate probabilities. phenotype individual of bution where (26), Leibler and Xue x of model the to approximation value .Rgr R(98 osbooycntanculture? constrain biology Does (1988) AR Rogers (2012) LA 4. Dugatkin CT, Bergstrom 3. .By ,RcesnP 19)Wyde utr nraehmnadaptability? human increase culture does Why (1995) PJ Richerson R, Boyd effects 6. The learning: social of model evolutionary An (1988) PJ Richerson R, Boyd 5. ‘tradition.’ Acheulean the surrounding questions On (2008) JA Gowlett SJ, Lycett 2. (1981) MW Feldman LL, Cavalli-Sforza 1. h tt normdl steaeaeo h ouaindistri- population the of average the is model, our in state the , ecie h hnei rqec fspecies of frequency in change the describes fvria rnmsin ecnetr ht with that, conjecture We transmission. vertical of hntefitnesses the When ge- population diploid two-allele one-locus, deterministic In ihflcutn niomns h yaiso h phenotype the of dynamics the environments, fluctuating With nacntn niomn,tehge h etcltransmission vertical the higher the environment, constant a In Sociobiol pp NJ), Hillsdale, Associates, Erlbaum (Lawrence Jr BG 29–48. Galef TR, Zentall eds variation. spectives, temporal and spatial of 831. Archaeol World Approach titative log[1 h atrteapoc ofiaino h aoe pheno- favored the of fixation to approach the faster the ρ, A1B A A oe) ti mosbefrfiainin fixation for impossible is it model), and if − A 16:125–143. ,Kri n iemn(1 xeddterslsof results the extended (31) Liberman and Karlin 1), w ρ n ifrn rwhrates growth different and ρ, ylann rmteprn’ ouainatrlearn- after population parent’s the from learning by A + B A k > 40:295–315. ρ(w = .I the In 5). (Result present) PictnUi rs,Princeton). Press, Univ (Princeton and w l B o xml,nihrfiaini tbe and stable, is fixation neither example, for , A /w B or ihoelpiggnrtos nequal an generations, overlapping with , w B B A )]

if and n h feto h aineof variance the of effect the and , Evolution oilLann:PyhlgcladBooia Per- Biological and Psychological Learning: Social 16 w B utrlTasiso n vlto:AQuan- A Evolution: and Transmission Cultural and w > B w Nro,NwYork). New (Norton, r rae srno vari- random as treated are o h aneac (or maintenance the for 17) A AkBl Here, . w A1B A ae onson bounds case, and mAnthropol Am 1 A normodel, our In A. 1 − and ae hspoly- this case, A h oblique the ρ, w B and k hn Eq. then, ; B 6 = .This 2). oboth to l B there , 90:819– can- Ethol l /k ρ. 3 eebr ,TtmE 14)Gn eobnto nEceihacoli. Escherichia in recombination Gene (1946) EL Tatum salmonella. J, in spa- Lederberg exchange Genetic and (1952) 13. temporally J Lederberg in ND, strategies Zinder learning 12. of Evolution (2014) MW Feldman K, intensifi- Aoki and emergence 11. the for model strategy mixed A (2006) K Aoki JY, Wakano 10. fslcin(htis, (that selection of and mutation, selection. recombination, secondary or of induced affect modifiers migration neutral be while must tion, it of modifier although a modifiers, that genetic noted for scenario unusual an is fact, In 1). in explored values value uninvadable of the reduction regime, While straightforward. not of than fitnesses the for when However, erations. case selection is random this and the that a seems of in it control true the iteration, also under numerical is From it modifier. when genetic decrease to tends transmission fixation. to time probability expected the the reducing increasing selec- and makes effective, transmission more vertical of tion level greater a case, ulation small stable evolutionarily the as same the value is fitness mean geometric of k values larger for true in especially is of this instability; dynamics numerical the of frequen- of modifier phenotype dependence the the small, When exceedingly fixation. become to cies system the of proach tnodCne o opttoa,Eouinr n ua Genomics. Human and Evolutionary Computational, for Center Stanford ACKNOWLEDGMENTS. would approaches two mod- the worthwhile. the of be between of classes exploration discrepancies additional two and that overlaps suggest the and from interesting are findings population els coevo- different from gene–culture theory The of modifier tradition genetics. with the together in theory squarely lutionary is treated model paper The ways. this different in qualitatively transmis- in so nonparental do and they parental sion, incorporate both here lyzed of value intermediate an for mum with decreasing AGR the show curves They our of similarity a is for there lineage, parental its from learns parameter whose large (26), very Leibler for 0.99 6A reach Fig. can ing and 0.8 above of remains values larger For 0.9 of AkBk value the (i.e., between selection of to strength tive the on strongly with regimes (w .Ak ,Wkn Y eda W(05 h mrec fsca erigi tem- a in learning social of emergence The (2005) MW Feldman JY, Wakano K, mathematical A Aoki learning: social 9. of Evolution (2004) MW Evolutionary Feldman K, learning: Aoki social JY, versus Wakano Individual 8. (1996) J Kumm K, Aoki MW, Feldman 7. ≥ i.6A Fig. lhuhtemdl fXeadLilr(6 n htana- that and (26) Leibler and Xue of models the Although h eedneo h oie yaiso h strength the on dynamics modifier the of dependence The the in that, shown have We ilyvral niomns eiwo theory. of review A environments: variable tially environment. natural changing periodically Biol a in learning social of cation model. theoretical A environment: changing porally analysis. environment. fluctuating a in analysis A AkBk w /w 19). B A1B 70:486–497. 0 = k ρ eie and regimes B r dnial itiue n needn ewe gen- between independent and distributed identically are W u agrvle of values larger but , ∗ ) ho ou Biol Popul Theor ρ ˆ nthe in .1 eie when regimes ilceryices as increase clearly will vlto famdfiro etcltasiso is transmission vertical of modifier a of evolution 1, 1 = Tbe1 hw httevalue the that shows 1) (Table ρ and ihteaypoi rwhrt AR fXeand Xue of (AGR) rate growth asymptotic the with ∗ and k nrae hrl as sharply increases .For ). > ρ A1B ρ ∗ 0.5 u ueia nlsssosthat shows analysis numerical our 2, sse vnwt the with even seen is eoe xrml ifiutt eetdeto due detect to difficult extremely becomes AkBk w ihtercrei the in curve their with hsrsac a upre,i at ythe by part, in supported, was research This AkBl 1 w 66:249–258. 0 = AkBl and when w ρ k eie with regimes efind we .9, sntnurl tafcspiayselec- primary affects it neutral; not is , eie with regimes ssal(i.6B (Fig. small is ρ A2B ˆ eeto eie oecomplicated more regimes selection k sbten01 n .5 while 0.25, and 0.15 between is W nhoo Sci Anthropol η ρ nalta h G a maxi- a has AGR the that entail ∗ A1B stert twiha individual an which at rate the is 2 1 = sntzr o l ftefitness the of all for zero not is ρ k η eeto eie.For regimes. selection . nrae beyond increases nrae.I h nt pop- finite the In increases. scmlctdb h ap- the by complicated is ) 1 ρ ˆ rcNt cdSiUSA Sci Acad Natl Proc η ae h aeo vertical of rate the case, k 0 = A1B nthe in NSEryEdition Early PNAS 104:209–231. urAnthropol Curr w > ρ while , ρ ˆ 0 = 2 cusi the in occurs AkBk 2 n al )(with 1) Table and htmxmzsthe maximizes that Fg 6B (Fig. niomn.For environment. . AkBk h difference the 1, Bacteriol J u (x ρ environment. ∗ ) 46:334–340. k Nature k eiefor regime ffixation of scoeto close is Compar- . ρ n Table and ˆ 91:3–19. 2 = ho Popul Theor 64:679–699. depends | w A2B 158:558. AkBk This . f10 of 9 rela- ρ A k 2 ∗

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