Evolution of transmission mode in conditional mutualisms with spatial variation in symbiont quality

While some symbioses are always mutualistic or parasitic, others have costs and benefits that depend on environmental factors. The environmental context may itself vary in space, in some cases causing a symbiont to be a mutualist in one location and a parasite in another. Such spatially conditional mutualisms pose a dilemma for hosts, who might evolve (higher or lower) horizontal or vertical transmission to increase their chances of being infected only where the symbiont is beneficial. To determine how transmission in hosts might evolve, we modeled transmission evolution where the symbiont had a spatially conditional effect on either host lifespan or fecundity. We found that over ecological time, symbionts that affected lifespan but not fecundity led to high frequencies of infected hosts in areas where the symbiont was beneficial and low frequencies elsewhere. In response, hosts evolved increased horizontal transmission only when the symbiont affected lifespan. We also modeled transmission evolution in symbionts, which evolved high horizontal and vertical transmission, indicating a possible host-symbiont conflict over transmission mode. Our results suggest an eco-evolutionary feedback where the component of host fitness that a conditionally mutualistic symbiont influences affects its distribution in the population, and, through this, the transmission mode that evolves.

. We call these interactions conditional mutualisms. Symbiont effects may vary based 31 on abiotic factors (e.g. nutrient availability (Cheplick et al., 1989) or temperature (Baker 32 et al., 2013)) or biotic factors (e.g. the presence of a third species which parasitizes the host 33 (Smith, 1968)). The abiotic or biotic context may in turn vary in space. In some cases, the 34 symbiont may change from a mutualist to a parasite depending on the location. For exam- 35 ple,the endophytic fungus Epichloë coenophiala increases the biomass of tall fescue (Festuca 36 arundinacea) seedlings in nutrient-rich soil, while decreasing host biomass in nutrient-poor 37 soils (Cheplick et al., 1989). Different temperatures produce a similar pattern in the nutri-38 ents provided by the Symbiodinium endosymbionts of corals. Clade D members of Symbio-39 dinium provide less nitrogen than Clade C symbionts except at high temperatures, where 40 they provide equivalent nitrogen and more carbon (this pattern is thought to explain the 41 geographic distribution of Clade C and Clade D symbioses) (Baker et al., 2013). 42 Such spatially conditional mutualisms pose a dilemma for hosts in deciding how to ac-43 quire their symbionts. In general, assuming no correlation between horizontal and vertical 44 transmission, hosts are predicted to evolve reduced vertical (parent-to-offspring) trans-45 mission of parasites and increased vertical transmission of mutualists (Yamamura, 1993). 46 Hosts may also evolve decreased susceptibility to horizontal transmission of parasites, including when resistance comes at a cost to fecundity, if reproduction is local (Best et al.,48 juvenile survival (as in the interaction between jellyfish and the juvenile scads they protect 84 from predators) affect lifespan without having any influence on the reproductive output of 85 hosts who survive to adulthood (Bonaldo et al., 2004). ( When the symbiont affects host fecundity, we assume infected hosts have higher fecun-149 dity than uninfected hosts in M-patches, and that the reverse is true in P-patches. When 150 the symbiont affects host lifespan, we assume all hosts have equal fecundity. 151 If the parent host is infected, its offspring has a chance to acquire the symbiont via vertical transmission. For a vertical transmission probability v, the probability that a host patch q gives birth to an uninfected or infected offspring is After birth, newborns disperse to a new patch with probability d or stay in their natal 152 patch with probability 1 − d. We assume that newborns must mature somewhat before 153 they become susceptible to horizontal infection, such that there is a window of time after 154 dispersal and before establishment where newborns may acquire the symbiont horizon-155 tally, as is the case for many horizontally transmitted symbioses (Bright and Bulgheresi,156 2010). For simplicity, we assume that when newborns arrive in the patch, they make con-157 tact with a single neighbor, who, if infected, may infect the newborn with probability h. where N is the population size, and m q is the average mortality in patch q.
where t is the time in units of one host birth per time step (see Appendix A for deriva-177 tion).  outcome. They also suggest that a conditional mutualism that has more costs than bene-408 fits may actually be better for hosts than more "mutualistic" conditional mutualisms, by 409 increasing hosts' chances of evolving transmission modes that contain the symbiont to 410 locations where it is beneficial. 411 The simulations largely confirm that our results hold for finite populations. However, 412 they suggest an alternative way that hosts in small populations may respond to a con-413 ditional mutualism when dispersal rate is low. If dispersal rate is small enough relative Using Equations 1 and 2, and taking into account the fact that newborn hosts may enter a patch via dispersal, the rate of change in the fraction of infected hosts is where q represents patch M or P , and q ′ is the other patch. Note that the rate of change is It is possible that some of the equilibrium fractions of infected hosts may not be stable.
To find stable equilibria, we select those solutions of equation 4 for which the eigenvalues of the Jacobian are negative. The Jacobian is defined as We find the eigenvalues of the Jacobian at each equilibrium numerically using Mathemat-503 ica (supplement) and select those equilibria that are stable for invasion analysis. 505 We can now investigate transmission mode evolution when transmission is a host trait.

A.2 Transmission Mode Evolution -Host Control
We want to find the invasion fitness of a mutant host with slightly different horizontal and vertical transmission rates than the resident. To do this, we can think of the growth of the mutant when rare as a multitype branching process (Lehmann et al., 2016). We write a matrix (X t ) that gives the expected number of mutants produced by an uninfected or infected mutant in each patch every time step (measuring time in units of host births, t).
Rows of X t correspond to the location and infection status of mutants produced. The first two rows correspond to uninfected and infected mutants produced in patch M, and the third and fourth rows are the same for patch P. Columns of X t correspond to the type of mutant producing a new mutant (or "producing" itself by surviving to the next time step).
Columns are in the same order as rows. To find X t , let A be a matrix that gives the probability a mutant gives birth to an un-506 infected or infected offspring that successfully establish in each patch (rows and columns 507 in same order as in X t ). Let B be a matrix that gives the probability that an uninfected or 508 infected mutant in each patch dies. Then where I is the identity matrix and indicates that besides giving birth and dying, mutants 510 may simply persist in the population from one time step to the next. 511 We can get the probabilities in A from the product of Equations 1 and 2. The probabilities we need for A are the following: Pr(Uninfected mutant produces infected offspring) Pr(Infected mutant produces uninfected offspring) =        Pr(Infected mutant produces infected offspring) if offspring stays in q Using the above probabilities of mutant reproduction, we can write A as Because all cases in Equations 5 -8 have a 1 N term, we can re-write A as To find B, we start from the fact that, if a newborn establishes in patch q, an adult host in the patch has a 2 N · m mq chance of dying (since there are N 2 hosts in each of patch M and P). Because the population is comprised almost entirely of residents, the probability that a newborn establishes can be approximated using the probability that a newborn resident establishes. For patch q, where the other patch is q ′ , a host (mutant or resident) with mortality m has a probability of dying of Pr(A given host in patch q dies) = 2m N m q Pr(A newborn resident establishes in q) where Pr(A newborn resident establishes in q) = The 1 2 in the probability a resident establishes is due to the fact that each patch represents 513 only half of the population and thus has its probability of reproducing normalized by 1 2f . 514 We separate it out from the rest of the expression (b q ) to make it easier to deal with A -B 515 later. This gives 516 Pr(A given host in patch We can then write B as All the nonzero entries of B have a 1 N term. We can re-write B in terms of 1 N and B ′ , a matrix that does not depend on N .
Then we can write X t as One problem with X t is that as N → ∞, X t → I. To fix this, we rescale time in units of τ = tN . Then the expected number of mutants produced per mutant of each patch and infection status can be written as As the population size goes to infinity, we get the following formula for X τ 517 lim N →∞ The mutant should invade if the leading eigenvalue of X τ > 1 when the resident is 518 at equilibrium. Assuming mutations in transmission mode are small, we can trace the 519 evolutionary trajectory of a population by seeing which mutant with similar transmission 520 rates can invade, and then looking to see what transmission rates allow invasion of that 521 mutant when it is the resident. Practically, this means finding the derivative of the leading 522 eigenvalue of X τ at a range of resident transmission rates (a positive derivative means 523 a mutant with a slightly higher transmission rate can invade, and a negative derivative 524 means one with a lower transmission rate can invade). We then use these derivatives to 525 trace the path of transmission mode evolution.

527
When transmission is a symbiont trait, we again investigate the invasion fitness of a mu-528 tant with slightly different transmission rates than the resident. We will follow the same 529 general procedure as for host control. However, since a mutant symbiont should spread 530 in the population if it can infected more hosts than the resident symbiont, we will track 531 the number of mutants in units of hosts infected. 532 Let X t be the expected number of hosts infected with mutant symbionts in patches M and P by a mutant symbiont in each patch. The first and second rows of X t will give the infections produced in patches M and P, respectively. The columns of X t will likewise correspond to the location of the symbiont that produces the new infection. The probability that a mutant symbiont dies depends on the rate of newborn hosts establishing in its patch. This is given by Equation 10, which will be the diagonal entries