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EVOLUTION trait that is opposed by microbe-level selection is equivalent to We assume that host birth rates increase linearly with the den- asking when host-level selection can maintain such a trait. Using sity of helper cells from β/GH in the absence of helper cells to simulations, we show that such a trait can evolve but only when (1 + sb ) · β/GH when a host is fully occupied by helper cells. vertical transmission is stronger than horizontal transmission and The parameter sb thus controls how strongly host birth rates when the host generation time is short enough compared with the depend on their microbiome. Similarly, we assume that host timescale of the evolutionary dynamics at the microbe level. death rates decrease linearly with the density of helper cells, with the parameter sd controlling the strength of this depen- Results dence (Fig. 1). Throughout the main text, we only consider the A Simple Model for Host–Microbe Evolutionary Dynamics. We case where helper cells increase host birth rates without affect- extended a previously developed multilevel selection framework ing host death rates (i.e., we assume that sd = 0). However, we (21) to investigate under what conditions selection can act at the obtained similar results when helper cells decreased host death host level. Throughout, we assumed the simplest possible dynam- rates instead (SI Appendix, Fig. S2). To maintain bounded host ics to reduce the number of model parameters. We consider a population sizes, we assume that host death rates are density microbiome consisting of 2 genotypes: helper cells and neutral dependent (Fig. 1). The generation times of hosts and microbes cells (Fig. 1). Helper and neutral cells are identical except for 1 typically differ strongly, and as a result, we expect evolutionary trait: helper cells increase the reproductive success of their host, change to happen at different timescales for hosts and microbes. but this comes at a cost to their own growth rate. We assume The parameter GH in our model controls how many microbial that microbes have a constant birth rate of β for the neutral cells generations occur within each host generation. and (1 − γ)β for the helper cells, where γ is the cost of helping. We explicitly incorporate the 2 modes of transmission, ver- We expect that, within a given host, microbes can grow to a maxi- tical and horizontal, in our model (Fig. 1). When a host gives mal population size, and thus assume that death rates are density birth, it passes a sample of its microbiome on to its offspring. T = n0 dependent (Fig. 1). Helper cells can transition into neutral cells The strength of vertical transmission vert k is measured and vice versa due to mutations. At each microbial division event, as the size of this sample n0 relative to the microbial carry- there is thus a probability µ that a microbe mutates to the other ing capacity k. We expect that vertical transmission is subject type. Finally, we assume that microbes can only grow within the to sampling variation: the helper frequency in newborn hosts host environment, and we thus only keep track of the microbial will likely differ slightly from that in their parents. We mod- densities within each host. Microbiomes are characterized by the eled sampling variation using a distribution in which we could independently control the degree of variance, allowing us to frequency of helper cells fi (helper frequency in short). In partic- ular, we often show the mean helper frequency in the total host directly test the importance of sampling variation. Specifically, 1 PH we assume that the helper frequency in a newborn host is drawn population hf i = fi . H i from a truncated normal distribution centered on the helper Helper cells increase the reproductive success of their host: frequency of the parent and with a constant variance of σ2. for example, by producing an essential nutrient or by providing Throughout their lives, hosts exchange microbes by horizontal protection against pathogenic bacteria. As a result, helper cells transmission. We assume that microbes migrate between hosts can either increase host birth rates or decrease host death rates. at a constant rate and that migration is random (i.e., no spa- θ tial structure). The strength of horizontal transmission Thoriz = β is measured as the migration rate θ relative to the microbial Microbe level dynamics Host level dynamics growth rate β.

Host-Level Selection Can Maintain Helper Cells. Microbes do not receive any direct benefit from helping their host: helper cells grow slower than neutral cells but have the same death rate; con- sequently, microbiomes consisting of only helper cells also reach Vertical transmission at birth Horizontal transmission during life a lower steady-state density than microbiomes consisting of only neutral cells. Helper cells are thus expected to go extinct over evolutionary time in the absence of host-level selection. Indeed, our model shows that helper cells rapidly decrease in frequency when host birth rates are independent of microbiome composi- tion (sb = 0) (Fig. 2B). Helper cells are only maintained at a low frequency (set by mutation–selection balance) due to constant transitions from neutral cells. Fig. 1. We consider a population of H hosts each carrying a microbiome Helper cells could potentially be maintained by host-level consisting of 2 types: helper xi and neutral yi cells. These types are identical except that helper cells pay a cost γ for increasing the host reproductive suc- selection when they increase host reproductive success (or cess. Newborn hosts are colonized by a sample of their parents microbiome, decrease host death rates). The frequency of helper cells which has fixed density n0 and a frequency of helper cells drawn from a decreases within each host due to selection at the microbe level; truncated normal distribution centered on the parent’s helper frequency fp however, the frequency of helper cells within the total host pop- 2 n0 and with variance σ . The strength of this vertical transmission Tvert = k is ulation could increase, because hosts with many helper cells given by the ratio of the size of this sample to the microbial carrying capacity have more offspring and pass on their microbiome (Fig. 2A). β k = δ . There is a constant exchange of microbes between hosts with rate θ. Indeed, our model shows that helper cells can increase to high The strength of horizontal transmission T = θ is given by the ratio of the horiz β frequencies when host birth rates depend on the microbiome migration rate to the microbial birth rate. Microbes have a constant birth composition (sb = 1) (Fig. 2 B and C). rate of (1 − γ)β for helper cell and β for neutral cell, a density-dependent δ · µ death rate ni, and a constant mutation rate between helper and neu- Host-Level Selection Can only Act under Strong Constraints. We per- tral cells. Hosts have a birth rate that increases linearly with the density of helper cells (with slope s ) and a death rate that decreases linearly with den- formed an extensive exploration of the model parameter space to b investigate when selection at the host level can maintain a trait sity of helper cells (with slope sd); moreover, the death rate increases linearly with the number of hosts. GH measures the number of microbe generations that is disfavored by selection at the microbe level (SI Appendix, per host generation. Microbiomes are described by their density ni = xi + yi Fig. S1). We found that 2 conditions have to be met (Fig. 3): and frequency of helper cells fi = xi/ni. 1) vertical transmission Tvert has to be strong compared with

2 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.1909790116 van Vliet and Doebeli Downloaded by guest on October 2, 2021 Downloaded by guest on October 2, 2021 a le n Doebeli and Vliet van and simulations, single of results the θ level T microbe the at dynamics tionary 3. Fig. t frequency cells microbiome helper helper on (mean depend population rates host birth helper host high main- (s when have only composition frequencies are also cells high turn Helper at helper (B) in tained transmission. more offspring vertical with of These hosts because offspring. frequencies selection: more host-level can have to cells cells due helper However, frequency (red). in cells selec- increase neutral microbe-level faster-growing because the frequency, favors in tion decrease (blue) cells helper host, 2. Fig. aaee pc nwihhle el a ec ihfrequen- of high region reach can the cells cells: helper which helper in maintain space parameter can selection for hosts host-level between variation no on. microbe- again act to to is (due selection reproduce there host can helper selection), a host within a level in cells before variation helper extinct if gone any can Likewise, have there longer selection. variation, host-level no this no without be is and compo- there hosts, microbiome between result, same frequencies a the to As verti- converge when sition. quickly transmission, follows: will horizontal hosts as with host). all compared understood single weak be is a transmission can cal within timescale conditions decrease the 2 frequencies (i.e., These helper level microbe which timescale the the over with at compared dynamics enough evolutionary short be to has transmission horizontal and B A horiz ,0.Prmtr ssonin shown as Parameters 5,000. = ooti ifrn ausof values different obtain to mean helper w diinlfcosaeiprati eemnn when determining in important are factors additional Two

K frequency = H = ihneach Within (A) cells. helper maintain can level host the at Selection eeto a nyata h otlvludrsrnetcniin.Tema eprfeunyi hw safnto ftetmsaeo h evolu- the of timescale the of function a as shown is frequency helper mean The conditions. stringent under level host the at act only can Selection β θ w ifrn eeso apigvrac r hw:teS ftesmln distribution sampling the of SD the shown: are variance sampling of levels different Two . 5,000. Microbe level f selection i b aiswdl ewe ot;tedsrbto ssonfor shown is distribution the hosts; between widely varies = ) h vrg rqec fhle el vrtetotal the over cells helper of frequency average the 1); ie[.. helperfrequencyinhost( time [a.u.] T horiz τ AB M frequency

/τ more vertical and , xetfor except S1, Table Appendix, SI helper mean

H transmission A, and τ C Lower M h otgnrto time generation host the 2)

fraction of host Host level selection = T hf vert -2 1 βγ relative tomicrobialdynamics 0 2 4 0 sson (C shown. is i) 1 4 22 -2 host generationtimeshort /T and 00 =0.10 =0.02 2 eaiet h otgnrto time generation host the to relative horiz 0 B, -2 epciey l te aaeesaea hw in shown as are parameters other all respectively; , Lower -2 0 hwtesm aabtaeae ihndsrt is evre h cost the varied We bins. discrete within averaged but data same the show rqec of Frequency ) τ M 0 G fthe of H f i ) = τ 10 H 2 lasdces nfeunydet irb-ee eeto.We selection. microbe-level to due frequency in decrease always Cells. Helper of Maintenance ogtm eid htaepootoa oteivreo the of inverse the to over 1 maintained section proportional (τ be are rate can that heritability migration periods result, time a As long over. cells take acquired horizontally can before density microbiome steady-state density, the its (τ dominates, reaches steady-state transmission lost vertical its rapidly when contrast, reaches is microbiome heritability and the environment the before of (T that long dominant resembles composition is those den- microbiome transmission than the steady-state horizontal the its extent reached When change larger has sity. birth microbiome much horizon- the after a when microbes: directly arriving to the arrive composition that of microbiome cells dynamics acquired growth ver- tally nonlinear and horizontal the in of depends by 4 strength it (Fig. relative that transmission the found tical on We way 1). nonlinear section approx- a analytical Appendix, an (SI derived we imation calculate and to model numerically, our timescale used by this We host environment. the the microbes from by obtained entered dominated become that has when composition words, cells microbiome other the In of 50%. frequency reaches transmission the vertical which at birth ie,teohrhssi h ouain.I te od,the words, other decreases In heritable is population). composition time. the shifts over community in which environment thus to hosts the host degree from other to a obtained the parent microbes its of (i.e., from by microbiome obtained dominated microbes The being by dominated 4B). being decreases (Fig. from Conse- cells 4A). time transmitted cells (Fig. vertically over acquires rate of by proportion constant it colonized the a life, quently, is at its host transmission during horizontal a com- but by birth, cells, microbiome At transmitted the heritable. vertically be cells, to helper needs maintain position to selection Time. level with Decays Composition Microbiome of Heritability detail. more micro- sections, in the 2 requirements next these on the explore strongly In and will S1 ). we 3) more Fig. (Fig. depends Appendix, (SI higher rate composition is biome birth variance host sampling the the when when larger is cies edfietehrtblt timescale heritability the define We τ -2 1 0 2 4 H 0 22 -2 4 = 2 ). G β 0 H -2 n h ai fvertical of ratio the and -2 σ 0 her s00 in 0.02 is ∝ . S1 Table Appendix , SI 1/θ 0 B Fg 4 (Fig. ) A ihna sltdhs,hle cells helper host, isolated an Within and n . in 0.1 and 2 1 0 .Ti olnaiyi caused is nonlinearity This C). T vert mean helper her B < = frequency B; τ and Fg 4 (Fig. 1) n k A, her 0 NSLts Articles Latest PNAS ohrzna transmission horizontal to Upper stetm fe host after time the as C γ and and n irto rate migration and B IAppendix, SI B, horiz and Upper o host- For > t | .In C). τ > T f7 of 3 show vert her ), ,

EVOLUTION A 1 total C vertical -1 -6 horizontal -3 -3 density 0 0 B 1 10 -6 3 2.0 6 0.5 0.1 -9 9 horizontal transmission -9 -6 -3 -1 vertically frequency

transmitted 0 -3 log10 time [a.u.] vertical transmission

Fig. 4. Microbiome composition is only heritable when vertical transmission dominates horizontal transmission. (A) Horizontally acquired cells eventually dominate over vertically acquired cells. The densities of the total population and of vertically and horizontally acquired cells were calculated analytically

(SI Appendix, Eqs. S7 and S8) and are shown for the case Tvert = 2Thoriz.(B) Frequency of vertically transmitted cells decreases over time in a nonlinear way. The frequency of vertically inherited cells was calculated analytically (SI Appendix, Eq. S8) and is shown for 4 different ratios of vertical to horizontal transmission. Microbiome composition is heritable (i.e., dominated by vertically transmitted cells) for long periods when vertical transmission dominates

(dark blue) and for short periods when horizontal transmission dominates (light blue). (C) Heritability timescale τher is short when vertical transmission Tvert is weak compared with horizontal transmission Thoriz. The heritability timescale τher is defined as the time after birth at which the frequency of vertically acquired cells reaches 0.5 and was calculated numerically (SI Appendix, Eq. S8).

can analytically show that this decrease follows a sigmoidal curve decreases during a parent’s lifetime, and the higher the sampling with a timescale that is inversely proportional to the cost of variation has to be in order to maintain helper cells (Fig. 5B). helping: τM = 1/(βγ) (Fig. 5A and SI Appendix, section 2). For When there is strong sampling variance, helper cells can be helper cells to be maintained by host-level selection, hosts have maintained at intermediate frequencies even in the absence of to give birth before all helper cells are lost. The host generation host-level selection as long as the cost of helping is low (SI time τH = GH /β thus has to be short enough compared with the Appendix, Fig. S3). timescale τM of the evolutionary dynamics at the microbe level High sampling variation allows host-level selection to maintain (Fig. 5B). The bigger the difference in generation time between helper cells in a larger region of parameter space, but it low- hosts and microbes, the lower the cost of helping has to be in ers the maximally achieved helper frequency (Fig. 5B). This is order to maintain helper cells (Fig. 5B). because the helper frequency is bounded between 0 and 1. When Sampling variation is an essential component in our model, a parent already has a high helper frequency, the frequency in its as it is the only process that can increase the frequency of offspring can increase by only a small amount but can decrease by helper cells over evolutionary time. Within an isolated host (no a large amount. As a result, the distribution of helper frequencies migration), helper frequencies always decrease over time due in newborn hosts becomes skewed toward lower helper frequen- to microbe-level selection. As a result, newborn hosts inherit a cies, and this decreases its average value. In regions of parameter helper frequency that is lower than the helper frequency with space where the average helper frequency is high, increasing which their parents were born. Over many host generations, sampling variance increases the skew in the distribution of helper helper cells would thus disappear. Migration between hosts can- frequencies in newborn hosts. As a result, the average helper not stop this process; migration can redistribute helper cells frequency in the total population has a lower steady-state value between hosts but cannot increase their total number. However, in this region (Fig. 3B and SI Appendix, Fig. S1). For a similar sampling variation can prevent helper cells from being lost; it reason, increasing sampling variance will increase the average can create some offspring that have helper frequencies that are helper frequency in regions of parameter space where the aver- higher than those of their parents. To maintain helper cells over age helper frequency is low (Fig. 3B and SI Appendix, Fig. S1). evolutionary time, sampling variation has to be strong enough Although sampling variation at host birth is essential in our to offset the decrease in helper frequency that occurred during model, other mechanisms can in principle be envisaged to main- a parent’s lifetime in a sufficiently large number of offspring. tain helper cells. The essential requirement for such mechanisms The higher the cost of helping, the more the helper frequency is that they can increase the frequency of helper cells in newborn

lower cost and/or ABcost = 0.0010.01 0.1 faster host dynamics i 1 1 0.02 0.03 0.04 0.10 0.20 mean helper frequency 0.50 helper frequency f 0 0 -20 2 time [a.u.]

Fig. 5. Host generation time has to be short enough compared with the timescale of the evolutionary dynamics at the microbe level. (A) Within a given host, the frequency of helper cells decreases following a sigmoidal curve with timescale τM = 1/(γβ). The frequency of helper cells was calculated analytically (SI Appendix, Eq. S23) for an isolated host (no migration) and is shown for 3 different costs γ starting from an initial frequency of 0.9 (solid lines) or 0.3 (dashed lines). (B) Host-level selection can only maintain helper cells when the host generation time τH = GH/β is short enough compared with the timescale of the evolutionary dynamics at the microbe level τM. This ratio τM/τH = 1/(γGH) depends on the cost of cooperation γ and the number of microbe generations 2 per host generation GH. Higher sampling variance σ allows for host-level selection over a larger region of parameter space but lowers the maximal value of the helper frequency. We varied the cost γ to obtain different values of τM/τH, and all other parameters are as shown in SI Appendix, Table S1.

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EVOLUTION neutral cells and vice versa. The death rate is δ · ni(t) for both helper and 1) Calculate birth (Bi) and death (Di) rates for each host. neutral cells, where ni(t) = xi(t) + yi(t) is the total microbial density in host 2) Iterate with time step δt; in each δt time step, at maximum a single i. With these assumptions, the microbial dynamics are given by host-level event (birth or death) can occur. If there are H hosts in the populations, there are 2H + 1 possible events that can happen in this dxi time step: any of the H hosts can reproduce, and host i reproduces with = (1 − µ)(1 − γ)β · xi + µβ · yi − δni · xi [1] dt probability P(bi) = Bi · δt; any of the H hosts can die, and host i dies with probability P(di) = Di · δt; or no host event happens, with probability PH dyi P = 1 − i P(bi) + P(di). We randomly select 1 of these possible events = (1 − µ)β · yi + µ(1 − γ)β · xi − δni · yi. [2] dt based on their relative probabilities. If a host gives birth, we add a new host to the population with a microbiome of density n0 and helper fre- quency f given by Eq. 5. Newborn hosts cannot give birth or die until the Host Dynamics. Hosts dynamics are modeled using an individual-based 0 next ∆t time step. If a host dies, we remove that host and its microbiome approach. The birth rate Bi of host i increases linearly with the density from the population.  xi  of helper cells: Bi = b 1 + sb · ; the helper density is normalized by k 3) Update microbiome state using Euler method: xt+1 = xt + ∆t · gx (xt , yt ) the microbial carrying capacity k to ensure that host birth rates remain and yt+1 = yt + ∆t · gy (xt , yt ), where gx and gy are given by the right- bounded. The death rate Di increases linearly with the number of hosts hand site of Eqs. 6 and 7, respectively. and decreases linearly with the density of helper cells in the host: Di =  x  The time step δt was dynamically adjusted to the number of hosts d 1 − s · i · H(t). The constants s and s control how strongly host d k b d in the population such that the probability of having more than 1 host reproductive success depends on microbiome composition and thus, control event per time step was less than 0.01. We ran the simulations until the the strength of selection at the host level. Throughout the main text, we helper frequency reached its steady-state level; this condition was evalu- use sd = 0. It is convenient to rewrite these rates as ated automatically by requiring that the variation in helper frequency over a moving time window (104 time units) was below a threshold value. All β  xi  Bi = 1 + sb · [3] simulation results were averaged over a moving time window of 1,000 GH k time units.

β  xi  H(t) Di = · 1 − sd · , [4] GH k KH Continuous Investment Model. Next, we consider a model where the degree to which a microbe helps its host is a continuous trait: the cooperative where GH = β/b is the number of microbial generations per host generation investment 0 < w < 1. The investment level was discretized into N = 100 and KH = b/d is the host carrying capacity. equally sized bins; the investment level in bin j is wj = (2j − 1)/(2N), j = (1, 2, ... , N). We tracked xj(t), the density of microbes with investment level Vertical Transmission. When a new host is born, it is seeded with a micro- i x j in host i, over time. Each host is characterized by the total microbial den- 0 N j 1 2 N biome of fixed total density n0 and a helper frequency f0 = n , which is P 0 sity ni = j xi and investment distribution (xi , xi , ... , xi ). Microbes with 2 drawn from a truncated normal distribution with constant variance σ and investment level wj have a birth rate of (1 − wj · γ)β. With probability a mean value equal to the helper frequency in its parent fp: (1 − µ), their offspring inherits the same investment level, and with prob- ability µ/2, their offspring mutates to an investment level of wj ± . We 2 f0 ∼ N (fp, σ ) with constrained 0 < f0 < 1. [5] set  = 1/N so that mutations only happen between adjacent investment bins. There is a density-dependent death rate δ · ni and constant migration between hosts with rate θ. The dynamics for xj(t) are then given by: Horizontal Transmission. Microbes leave a host at a fixed rate θ and are i distributed evenly among all other hosts in the population. The inflow of j PH θ dxi j j µ j−1 j−1 helper cells into host i is given by xj, where θ · xj is the number j6=i H−1 = (1 − µ)(1 − w γ)βxi + 2 (1 − w γ)βxi of helper cells leaving hosts j and 1/(H − 1) is the fraction of those cells dt ending up in host i. As horizontal transmission acts continuously, we mod- H θ j µ j+1 j+1 j j X xk + (1 − w γ)βx − δnix − θx + , [8] ify the dynamical equations for the microbiome dynamics (Eqs. 1 and 2) by 2 i i i H − 1 including the terms for the inflow and outflow of microbes: k6=i

0 N+1 H where xi = xi = 0. The host birth and death rates are given by dxi X θxj = (1 − µ)(1 − γ)βxi + µβyi − δnixi − θxi + [6] dt H − 1 β j6=i Bi = (1 + sb · W) [9] GH

β H(t) H Di = · , [10] dyi X θyj GH KH = (1 − µ)βyi + µ(1 − γ)βxi − δniyi − θyi + . [7] j dt H − 1 x ·wj j6=i PN i respectively, where W = j k is the cumulative investment of all microbes in host i (the normalization by the microbial carrying capacity k Model Parameters. We can reduce the number of independent parameters ensures that 0 < W < 1). Vertical transmission is implemented by randomly in our model by measuring time in units of the inverse microbial birth rate drawing N0 discrete samples from the parent investment distribution with 1/β and by measuring microbial densities in units of their carrying capacity j j probabilities P(w = w ) = xp/np. The resulting offspring investment distri- k = β/δ. In all simulations, we thus arbitrarily set β = δ = k = 1. bution is normalized to a total density of n0. The model is implemented identically to the 2-type model described above but now using Eq. 8 to Model Implementation. The model was solved numerically using code update microbiome state and Eqs. 9 and 10 to calculate host rates. implemented in Python [code available on Zenodo (25)]. We updated the microbial densities and host birth and death rates using a coarse time step ACKNOWLEDGMENTS. We thank Alma Dal Co, Gil Henriques, and the ∆t and used a fine time step δt ≤ ∆t to implement host birth and death other members of the laboratory of M.D. for comments and discussion. events assuming constant rates. We used this procedure to keep computa- S.v.V. was funded by Schweizerischer Nationalfonds zur Forderung¨ der tion times reasonably short and confirmed that we obtain identical results Wissenschaftlichen Forschung Postdoc.Mobility Fellowship 175123, and if we update microbial dynamics and hosts rates every δt time step. At each M.D. was supported by Natural Sciences and Engineering Research Council time step ∆t, we performed the following steps. of Canada Discovery Grant 219930.

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