On the effect of inheritance of microbes in commensal microbiomes

Our current view of nature depicts a world where macroorganisms dwell in a landscape full of microbes. Some of these microbes not only transit but establish themselves in or on hosts. Although hosts might be occupied by microbes for most of their lives, a microbe-free stage during their prenatal development seems to be the rule for many hosts. The questions of who the first colonizers of a newborn host are and to what extent these are obtained from the parents follow naturally. We have developed a mathematical model to study the effect of the transfer of microbes from parents to offspring. Even without selection, we observe that microbial inheritance is particularly effective in modifying the microbiome of hosts with a short lifespan or limited colonization from the environment, for example by favouring the acquisition of rare microbes. By modelling the inheritance of commensal microbes to newborns, our results suggest that, in an eco-evolutionary context, the impact of microbial inheritance is of particular importance for some specific life histories.


Background
Microbial life is ubiquitous in the biosphere [1]. The human body is no exception, as first described by van Leeuwenhoek in the 17th century. We are among the many macroorganisms where diverse microbiomesmicrobial communities living in or on hosts-have been observed [2,3]. As part of their life cycle, members of the microbiome may migrate between hosts and the environment. The migration process has been studied using experimental [4] and theoretical approaches [5,6]. However, some microbes have been found exclusively in hosts [4,7]. How do such microbes persist in the population?
One possibility is the vertical transfer of microbes from parents to offspring [8]. Although there is ample literature about transmission of endosymbionts (e.g. Buchnera and Wolbachia in insects [9]), less is known about extracellular-possibly transient-microbes. Quantifying the low microbial loads in newborns [10] and deciphering the true origin of microbes [11] remains experimentally challenging [12,13]. A few experimental studies have explored the vertical transfer of the microbiome in specific species across the tree of life-including sponges [14], mice [15], cockroach eggs [16], and wheat seedlings [17]. For many others, including humans, there is an ongoing debate on when and how inherited microbes are obtained [11]. Together, these studies suggest there is no universal reliance on microbial inheritance across host species, raising the possibility that even if such associations matter to the host, certain life-history traits may limit their inheritance [13,18]. Relevant traits may include, among others, the extent of environmentally acquired microbes and host lifespan.
Previous theoretical work has studied microbial inheritance in the context of symbiosis-where microbes affect the host fitness. In these models, depending on whether the interaction is positive (mutualism) or negative Page 2 of 13 Zapién-Campos et al. BMC Ecology and Evolution (2022) 22:75 (parasitism) the presence of symbionts is promoted or impeded, respectively. Using multilevel selection arguments, Van Vliet and Doebeli have shown that a symbiosis that is costly for microbes can be sustained only when the host generation time is short and the contribution of inheritance exceeds that of environmental immigration [19]. Following up, in addition to individual inheritance (single contributing parent), Roughgarden analyzed scenarios of collective inheritance (multiple contributing parents) [20]; while Leftwich et al. found a weak influence of the host reproductive mode (sexual or asexual) and mate choice (based on symbiont presence) on the symbiont occurrence [21]. If these host-symbiont interactions persist over evolutionary timescales, they are said to lead to phylosymbiosis-where microbiomes recapitulate the phylogeny of their hosts [22]. Not all co-occurrences between hosts and microbes reflect a fitness impact, however. As suggested by Bruijning et al., the selection on the host-microbiome pair and the microbial inheritance might change with the environment [18]. Moreover, despite taxonomic differences, functional equivalence of microbes in localized host populations could prevail [16]. Microbes might not always influence host fitness [18] nor benefit from influencing it [21]. In this context where there is no active selection of the microbes by the host, the role of microbial inheritance remains largely unexplored [23]. Using a stochastic model, we study the effect of microbial inheritance on the commensal microbiomemicrobes living in hosts but not affecting their fitness. Our model draws parallels to neutral theories in ecology where death, birth, and immigration of individuals are considered [24]. First, we introduce different models of inheritance representative of diverse host species. Then we discuss their effect on microbes present in both hosts and environment, or only present in hosts. We see that inheritance might influence the within-host occurrence and abundance in some cases. However, within the same microbiome, microbial types could be affected differently-while inheritance causes some microbes to increase in frequency, others decrease from it. Moreover, the effects may be transient, rendering life history parameters crucial. Altogether, we highlight the potential and limits of microbial inheritance to modify the composition of commensal microbiomes under different life-history scenarios.

Model and methods
Consider the host-microbiome system depicted in Fig. 1A. A population of hosts is colonized by a set of microbes, and each microbial taxon i has a constant frequency p i in the environment. The total number of microbes a host can contain is finite and given by N, leading a to competition for space. Each newborn empty host B A Fig. 1 Host-microbiome dynamics and microbial inheritance in our model. A Dark blobs indicate hosts, coloured-and empty-circles indicate microbes and empty-space, respectively. Within the hosts, microbes go through a death and immigration-birth process, with new residents migrating from the pool of colonizing microbes with probability m or replicating within a host with probability 1 − m . For microbes, each host is an identical habitat. The host population is at a dynamic equilibrium, every timestep there is a probability τ that a host death occurs, immediately followed by the birth of a new one. The newborn obtains a sample of its parent microbiome according to a probability distribution. B The probability distribution of the fraction of the parental microbiome inherited vary across host taxa-among others, influenced by development, reproduction and delivery mode. Certain hosts might not transfer microbes (eg. C. elegans [25] or D. melanogaster [26]). Others might provide minimal (eg. humans [11]) or large fractions of their microbes (eg. fragmentation of some sponges, corals, fungi and plants [27,28]), while others might be centred around a fixed value (eg. seeds of plants [17]). In our model, we control this probability distribution through the parameters a i and b i in Eq. (4) inherits a set of microbes from its parent, chosen at random within the host population. The inherited sample, taken off the parental microbiome, is drawn according to a probability distribution (Fig. 1B). After this initial seeding, only the death, immigration and replication of microbes can modify the host microbiome. Through these processes, the microbial populations within the host can decrease or increase by one individual each time step. After one microbe is selected to die, migration from the pool of colonizers occurs with probability m, while duplication of a resident microbe, or non-replacement, occurs with probability 1 − m . This process ends with the host death, which occurs with probability τ at each time step. We assume that the number of hosts does not change, so that a host death is followed by the birth of a new empty host, for which the process described above is repeated.

Transition probabilities
Our aim is to describe the dynamics of the microbiome load and composition, focusing in particular on how a certain microbial taxon experiences it. Within a specific host, the frequency of the i-th taxon is denoted by x i (for i ≥ 1 ), and of the remaining other microbes by o i = j� =i x j . The frequency of available space is then given by x 0 = 1 − x i − o i , the difference between the available space N and the space occupied by microbes normalized by N. The transition probabilities from state {x i , o i } that are due to the microbial dynamics are given by the product of the probability of host survival ( 1 − τ ) and the probability of death of a certain microbial type followed by an immigration or birth event. These events produce changes in the frequencies of magnitude 1 N . First, microbial taxa can replace each other when a microbe dies and is replaced by another one, In Eq. (1a), one microbe of type i dies and is replaced by a microbe of another type, either by immigration from the environmental pool or by replication within the same host. Similarly, in Eq. (1b), a microbe of another type dies and is replaced by a microbe of type i. In these equations, α 0 controls the establishment of microbes in hosts-the ability to occupy available space-going from fast for α 0 = 0 , to slow if α 0 is positive. For α 0 > 1 and without migration, microbes cannot be maintained in hosts.
Microbes may also occupy previously available space, such that the microbial abundance increases, (1a) Finally, microbes may decrease in abundance when a microbe selected for death is not replaced, , thus Eq. (1e) and (1f ) decrease when the microbial load is higher.
The transition probabilities due to the hosts dynamics are given by the product of the probability of host death and birth of an empty host ( τ ), by the probability to inherit certain microbes, where 1/(H − 1) is the probability of choosing a parent p in the population of H − 1 potential parents, and i ] are the probabilities of transfer of x i and o i microbes from the parent to the offspring, respectively. The probability to transfer more microbes than the parent can provide is zero, i.e.

Distribution of inherited microbes
In our model, parents can seed the microbiome of their offspring. A sample of the parental microbiome is vertically transmitted according to a probability distribution function, Eq. (2). In addition to the case without microbial inheritance, observed in some egg-laying and polyp producing organisms, and analyzed elsewhere [29], at least three qualitatively distinct cases may be defined (Fig. 1B), depending on host development, reproduction, and mode of delivery.
(1c) Firstly, microbial inheritance could be low. For example in mammals, newborns get microbes attached to epithelia or fluids during delivery [8,11]. These represent a small fraction of the parental microbiome, leading to distributions centred at frequency zero decaying towards one. Secondly, certain hosts, including some sponges, corals, fungi and plants [27,28], are able to reproduce by fragmentation, where a breaking body part generates a new individual. Such fragments could carry a faithful microbiome composition, leading to distributions centered at frequency one decaying towards zero. Finally, hosts that produce embryos that can disperse, eg. seeds, might transfer a microbiome sample contained within these physical structures [17].
We modelled such diverse parental microbiome samplings ( x i ) using the beta distribution for the probability i ] to inherit x i microbes from parent p. This probability distribution can take arguments in the range from zero to the current frequency of a microbe in the parent p, x where B is the beta function [30], 1/B a normalization constant, and a i and b i are shape parameters. The expected value of our beta distribution is a i +1 a i +b i +2 . The special case of a i , b i = 0 leads to a uniform distribution, where the parental microbes are distributed randomly between parent and offspring. Other combinations of a i , b i ≥ 0 produce different unimodal distributions (Fig. 1B). The case of a i > b i skews the distribution towards full inheritance of the parental microbes, i -all the i-th microbes from the parent could be transferred to the offspring. The case of a i < b i skews the distribution towards non-inheritance of microbes of type i to offspring, x i = 0 . Finally for a i = b i , the distribution is symmetric and the parental microbes are likely to be equally distributed between parent and offspring. In most of our analyses a i and b i are the same for all microbial taxa. Only for non-neutral, asymmetric inheritance, we will set different a i and b i for the focal taxon ( x i ) and the set of others ( o i ). To illustrate the effect of a i and b i , on average, an offspring inherits ≈ 9% of the parental microbes of taxon 1 for a 1 = 0 and b 1 = 9 , while only ≈ 1% is inherited for a 1 = 0 and b 1 = 99.
Throughout the results, we focus on distributions with a maximum at microbial frequency zero decaying towards x (p) i , which we call 'low inheritance' (Fig. 1B). In our model, the low inheritance and the 'full inheritance' scenarios (distributions with maximum at frequency x (p) i decaying towards zero) are equivalent. The number of microbes is conserved, so that inheritance happens through the splitting of the parental microbiome between the parent and the offspring. Because the transition rates only depend on the current state, no age can be assigned to parent or offspring-hosts are indistinguishable from each other and the labels "parent" and "offspring" are interchangeable. As a consequence, at the host population level, the splitting of microbes in the low inheritance scenario-where the parent transfers a small fraction of microbes-and the full inheritance scenario-where most microbes are transferred-are equivalent. Finally, we address under which circumstance a 'seed-like inheritance' leads to different results.

Stochastic simulations
In order to simulate the microbiome dynamics of individual hosts we formulated the model as a stochastic differential equation. We solved this equation numerically using the Euler-Maruyama method [31]. Starting from is the covariance matrix of the change of x . Further, W is a vector of uncorrelated random variables sampled from a normal distribution with mean 0 and variance t , the stochastic contribution. That W is normally distributed arises from the time independence and identical distribution of the noise. A detailed description connecting Eqs. (1) and (5) is provided in Additional file 1: Material A.1. For most of their life, hosts are independent of each other, only newborns are influenced by others when they acquire their initial microbiome. A given host lives for a duration sampled from an exponential distribution τ e −τ t , with mean 1/τ . We solve Eq. (5) for that interval. Immediately after a host dies, the microbiome of a newborn is assembled according to Eq. (2). We repeat these steps for all hosts until the total simulation time is reached.
As a result of stochasticity, each host trajectory is different. We look into the statistical description of the microbiome composition across the host population.

Inheritance can increase the occurrence of microbes in hosts with low microbial loads
Without microbial inheritance, which will be our reference case throughout, any microbe occurring inside a host has to have migrated from the environment during the host lifespan. As a result, a low environmental migration or short host lifespan can be limiting [29]. The transfer of microbes from parents to offspring during birth could increase the probability of observing any microbes in hosts, P[x i + o i > 0] inh. . We quantified the change in the probability of occurrence relative to its microbe-free birth condition P[ Using this observable, we investigated the role of life history in modulating the effect that inheritance has on the microbiome. We quantified this for a single microbial taxon, x i , as well. Fig. 2 shows a condition where, in the absence of inheritance, hosts are not fully occupied by microbes. This results from a short host lifespan ( τ ) and low microbial immigration from the pool of colonizers (m). We tested the effect of the 'low inheritance' mode ( Fig. 1B) for increasing rates of establishment of microbes ( α 0 → 0 ) and other life-history parameters.
Inheritance impacts the occurrence of microbes by increasing the number of hosts with at least one colonizing microbe (Fig. 2B). The effect is most prominent in scenarios where without inheritance, most of the hosts are microbe-free. However, the maximum increase occurs at intermediate immigration and host lifespans (Fig. 2C, D). For high immigration, m → 1 , hosts are readily occupied by microbes, so inheritance brings no change. This is similar for a long host lifespan, τ → 0 . On  the other hand, if immigration is limited, m → 0 , or host lifespan short, τ → 1 , microbes never occur in hosts, so parents cannot transmit microbes to their offspring. Inheritance might decrease the occurrence of microbes if the transfer-which splits the parental microbiome between parent and offspring-makes microbes more susceptible to stochastic fluctuations. This occurs if the microbial frequency of the parent is already low-for example when migration is limiting and microbes proliferate slowly (Fig. 2C). This phenomenon might be pronounced for individual taxa. Our analyses from the perspective of a single taxon (Additional file 1: Fig.  S1) found multiple instances where inheritance might decrease the occurrence (Additional file 1: Fig. S1C-F), but also have a larger effect in situations where the occurrence increases. Additionally, the effect on single taxa depends strongly on the carrying capacity for microbes, N (Additional file 1: Fig. S1F compared to Fig. 2E). Competition for space favours taxa according to their frequency in the pool of colonizers, p i (Additional file 1: Fig. S1C). Abundant taxa outcompete rare ones as space is more limited, but only until a point, after which there is no benefit-they readily occur without inheritance. In other words, in microbiomes composed by many taxa, the taxon-level effect of inheritance in terms of occurrence is relative to their environmental abundance.

Inheritance can increase the microbial abundance in hosts, but mostly of those abundant in the environment
Modifying the presence of microbes is not the only effect-inheritance also alters the microbial load considerably. Using the distribution of microbial frequencies in hosts, we quantified the change in the average microbial load of microbes as compared to its microbe-free birth condition, Similarly to Eq. (6), we quantified this observable for a single microbial taxon, x i , as well.
When looking at the distribution of microbial load and microbial frequencies in hosts, the effect of the 'low inheritance' mode ( Fig. 1B) is two fold-while hosts with small microbial frequencies might experience the largest increase in microbes, hosts with large frequencies can see the largest decrease of microbes ( Fig. 2B and Additional file 1: Fig. S2). Thus, both at the level of the microbial load and at the level of a single microbial taxon, hosts with small and large loads become less frequent. Inheritance makes hosts resemble each other to a greater extend (see the reduced spread of the distributions in Additional file 1: Fig. S2 and Additional file 1: Fig. S3). This is equivalent to the effect of increased immigration, which also tends to make microbiomes similar to one another, but increased inheritance does not favour the preservation of the microbial diversity from the pool of colonizers-in contrast to immigration.
An increase in the average microbial load is observed for some conditions (Additional file 1: Fig. S2). Analogously to the occurrence, such increase peaks at intermediate host death probabilities τ ; but also at intermediate carrying capacities N (Fig. 3C, D). The limited time for host colonization impedes any microbial inheritance ( τ → 1 ), while for τ → 0 or small N, hosts are fully occupied even without it. The relative effect of inheritance is less for large carrying capacities. A faster occupation of available space ( α 0 → 0 ) displaces the effect to larger host death probabilities and capacities for microbes. Finally, because the main limitation is the short host lifespan ( τ ), the influence of immigration (m) is minimal (see the scale in Fig. 3B and Additional file 1: Fig. S4C).
Although higher microbial loads might be reached with inheritance if space is limited (Additional file 1: Fig. S2C), abundant microbial taxa might increase at the expense of rare ones (Additional file 1: Figs. S3D and S4D-E). Such reduction is exacerbated by the fast occupation of available space α 0 → 0 . Interestingly, this might happen as a result of longer host lifespans as well, if hosts are rapidly occupied by inherited microbes. Such a condition favours abundant microbes in the pool of colonizers. Instead, if the occupation is slower, rare microbes increase in frequency, derived from the added benefits of inheritance and a more influential immigration (m).
A particularly relevant question is whether the frequency of a microbial taxon in a specific host ( x i ) can be larger than in the pool of colonizers ( p i )-i.e. a benefit is obtained from the host association. We observe this even in the absence of inheritance (Additional file 1: Fig.  S3), where stochastic colonization results in some host containing microbial frequencies larger than in the pool ( p i ). The average frequency across hosts, however, can be larger only when space limitation increases the competition. In this context, inheritance may, in fact, decrease the chances of such outcome, by relating the hosts to each other (Additional file 1: Fig. S3C, D).

Preferential inheritance can temporally lead to specific microbial taxa overrepresentation
A potential mechanism to increase the average frequency of taxa beyond their frequency in the pool of colonizers ( p i ), is preferential inheritance. The asymmetry in inheritance could stem from differences in microbial properties, but also a host's direct or indirect influence. We studied such possibility by manipulating the distribution of the sample inherited, Eq. (4). Focusing on a 'low inheritance' mode, we decreased the inheritance of other microbial taxa relative to taxon i, from equal if offspring receive 9% of every taxa on average, to preferential if they receive 9% of taxon i but 1% of others. For the same parameters as before (Fig. 4), we observe no effect if the host lifespan is limiting. In this case, regardless of the frequency in the pool of colonizers ( p i ), preferential inheritance does not alter the average frequency of the i-th microbial taxon in hosts (Fig. 4A), similarly for the probability of immigration m (Fig. 4B). This holds even for fast occupation of available space, α 0 → 0 . Only for longer host lifespan, τ → 0 , preferential inheritance leads to an increase (Fig. 4C). Besides the almost exclusive occupation of hosts by the i-th taxon ( x i → 1 ), the maximum effect is constrained to intermediate τ . This is because the effect of preferential inheritance is transitory for longer living hosts, after which they continue approaching their long term equilibrium, x i → p i . For faster occupation of available space the gain spans a wider range and shorter host lifespans ( τ → 1 ). Varying the immigration ( p i and m) when the host lifespan is not limiting increases the influence of asymmetric inheritance (Additional file 1: Fig. S5); especially if the effect of inheritance is not overwhelmed by the long term equilibrium originating from immigration (where x i → p i ). For hosts with short lifespan and limited immigration (in our example τ = 10 −4 and m = 10 −2 ), the gain from preferential inheritance is largest for decreasing carrying capacity for microbes, N (Fig. 4D). As shown in Fig. 4D, inheritance itself might not benefit all microbial taxa. For some taxa, only preferential inheritance can lead to larger frequencies than without inheritance.

Persistence of lineage microbial taxa in hosts
An extreme case of reliance on microbial inheritance are microbes present in hosts but absent from the environment ( p i = 0 ) [1,7]. We refer to these microbes as "lineage taxa". We investigated the conditions allowing their persistence under different life-history scenarios (Fig. 5).
Within a host, lineage taxa go through the stages sketched in Fig. 5A. Depending on the context, after host birth, their frequency might either decrease or increase. If decrease occurs, in a neutral context this trend will not change during the host life. In fact, events of microbiome inheritance will further decrease the frequency in the parent. We found that on average, lineage taxa increase while the inequality holds ( Fig. 5C and Additional file 1: Material A). Therefore, lineage taxa increase before reaching carrying capacity, favoured by their fast proliferation ( α 0 → 1 ), but restricted by migration (m). Because the microbial load increases through time ( x i + o i → 1 ), alongside the initial state, Eq. (8) limits the time of increase. Note that on average, the maximum frequency of lineage taxa (8) 4)). Available space within hosts is occupied more easily for α 0 → 0 . Single parameters are modified from the condition p 1 = 10 −2 for the frequency of taxa 1 in the pool of colonizers, m = 10 −2 for immigration, τ = 10 −4 for host death, and N = 10 5 for carrying capacity. A, B The average frequency increases for larger abundances in the pool of colonizers ( p 1 ), immigration (m), and α 0 → 0 . An asymmetric inheritance has no effect, as hosts are not fully occupied within their lifetime (Additional file 1: Figs. S2 and S3). C Longer host lifespans, τ → 0 , increase the average frequency and effect of asymmetric inheritance. The gain is maximal at intermediate τ . Inheritance has more influence before hosts are fully occupied. After this, hosts resemble the colonizers pool. D The average frequency increases with competition for space (smaller N). While the symmetry of inheritance decreases the average frequency as a result of the reduced initial microbiome variability, asymmetry increases it. Each simulation included 10 4 hosts . From this point on, a decrease driven by the immigration of environmentally present microbes (m) and stochasticity follows. For sufficiently long time, such decrease may lead to their extinction (Fig. 5B).
There is a trade-off between the duration of the increase and the maximum frequency of lineage taxa. While small initial microbial loads lead to long durations but small frequencies (as a result of immigration, Eq. 8), the opposite is true for high initial loads abundant in lineage taxa. Once increase stops, the time to extinction is proportional to the lineage taxa frequency, Fig. 5B.
Putting these two times together, the extra time from the increase is behind the subtle effect of the initial microbial load on the total extinction time, Fig. 5D. A reduced migration ( m → 0 ) and fast occupation of available space ( α 0 → 0 ) simultaneously increase the frequency and time before extinction.
Looking at the population level, a condition for lineage taxa persistence emerges-namely, an increase of frequency in each host followed by transfer to offspring of a frequency at least equal to that received at birth. This is possible only while the frequency in the parent is larger than initially, Fig. 5A. The largest frequencies are expected at intermediate time.
In this context, host lifespan, and thereafter the probability of host death ( τ ) become fundamental. From the distribution of host death events, τ e −τ t , we see most hosts die early on, potentially while lineage taxa are still abundant; τ → 0 results in longer living hosts-those more likely to lose lineage taxa. We estimated the probability of host death at which a fraction z of hosts loses the taxa, where t z , the time at which lineage taxa remain present in a fraction z of the host population, is obtained from the distribution of extinction times. Based on the former observations (Fig. 5D), our model predicts that regardless of the distribution of inherited microbes (Fig. 1B), preferential inheritance of lineage taxa in small microbial loads might favour their persistence as effectively as large but non-preferential microbial loads.

When the distribution of inherited microbes matters
We proposed that a finite set of shapes captures most of the possible microbial inheritance distributions (Fig. 1B)-low, high, and seed-like inheritance-all characterized by the most likely microbiome fraction transferred to the offspring. So far, we have focused on the impact of low inheritance on the microbiome (Figs. 2, 3, 4, 5). As mentioned before, because we enforce the conservation of microbes in our model -i.e. the microbes are transferred from the parent host to the offspringthe outcome of low and high inheritance is equivalent. In other words, although the parental microbiome is distributed differently, the outcome is indistinguishable at the host population level, because hosts are indistinguishable. When referring to certain life-histories, other distribution shapes may alter the impact of microbial inheritance. To find out differences between the effect of seed-like inheritance and our former results (where we assumed low inheritance) we compared the occurrence and average microbial loads.
We found most changes are minimal, however, differences appear for extreme parameters. A seed-like inheritance might better guarantee the occurrence of microbes in extremely adverse life-histories-e.g. rare environmental migration ( m → 0 ) and short host lifespan ( τ → 1 ) simultaneously (vertical axis on Additional file 1: Fig.  S6A, B). Exceptions could arise for a slower occupation of available space ( α 0 ). For individual microbial taxa, changes are greater in occurrence as well (Additional file 1: Fig. S7); however, derived from the competition for limited space (N), the effect of a seed-like inheritance (9) is case-specific. Moreover, both maximum increase and maximum decrease occur at intermediate m (Additional file 1: Fig. S7B) and τ (Additional file 1: Fig. S7C). In microbiomes composed of taxa with different environmental frequencies ( p i ), while some taxa gain, others lose from inheritance (Additional file 1: Fig. S7A). Under less adverse conditions, seed-like inheritance might allow larger microbial loads. That is the case when either host lifespan (horizontal axis on Additional file 1: Fig. S6A) or migration (Additional file 1: Fig. S6B) is limiting. The gain from a seed-like inheritance can be large, especially for a small carrying capacity for microbes N (Additional file 1: Fig. S6C). The consistent microbial transfer and reduced variation are beneficial. Nonetheless, at the single taxon level, gains are minimal (Additional file 1: Fig. S7). At this level, a limiting carrying capacity for microbes, where competition increases, might even lead to a decrease (Additional file 1: Fig. S7D). In this case, the variation provided by the low inheritance mode is more beneficial.
In summary, regardless of the distribution of microbes inherited (Fig. 1B), life-history seems intrinsically linked to the effect of microbial inheritance on the microbiome composition.

Discussion
The impact of microbial inheritance on host-associated microbial communities is largely unknown. In this work, we explored its potential effects under diverse life-history scenarios, including multiple distributions of microbes inherited (Fig. 1). Using a model free of selection-i.e. without microbial fitness differences or effect on host fitness-we shed light on the conditions where microbial inheritance may influence the microbiome composition, showing its impact but also its limits.
Our work emphasizes the role of life-history over hostmicrobe associations (Figs. 2, 3). Even without symbiotic benefits, the inheritance process itself might alter the microbiome composition [21]. Using a discrete generation model, Zeng et al. considered microbial inheritance in neutral associations over evolutionary timescalesspecifically, its effect on the microbial diversity and the distribution of frequencies [23] . Our results, however, highlight the relevance of within-generation probabilistic events-environmental colonization, host lifespan, or carrying capacity for microbes-as ecological drivers to constrain inheritance.
A crucial constraint is the host lifespan. Similarly to Van Vliet and Doebeli, but without any impact on the host fitness, we observe that the environmental acquisition of microbes makes the effects of inheritance transient (Figs. 2D, 3C and 4C) [19]. Short-living hosts (relative to the microbial timescale) could influence their commensal microbiome over their whole lives, without immunity or other means, long-living hosts could do it only during the first stages of development. The rapid proliferation of inherited microbes or isolation from the environment might prolong the period of influence. This is in contrast to Van Vliet and Doebeli, where selection within isolated hosts acts against costly symbiosis, reducing the period of mutualists presence.
We observed that the effect of inheritance may differ between taxa. Microbiomes assembled entirely from the environment are prone to variation when migration between hosts is rare [18,29]. Inheritance might increase the presence of certain microbes, but in contrast to environmental migration, inheritance reduces the variation between hosts and potentially their microbial diversity. This reduction, which especially affects rare taxa, is more pronounced if the carrying capacity is limited (Additional file 1: Fig. S1 and Additional file 1: Fig. S4), where competition is larger. Bruijning et al. have shown that under selection, such decreased variation and diversity could be detrimental for adaptation to changing environments [18].
Initially, we assumed no distinction between microbial taxa-only their frequency determined the population dynamics (Eq. 1). This could be modified in at least two ways. First, fitness differences could influence the birth and death rates of microbes. Although this is certainly relevant, it diverts from our focus on inheritance. Instead, we addressed a possibility crucial for inheritance-the asymmetric transfer of microbes (Fig. 4). Such asymmetry could emerge from differences in microbial capabilities at play during the transfer process, including oxygen tolerance [15] (obligate anaerobes tend to be transmitted vertically) and sporulation [32] (spores might allow the transfer of oxygen-sensitive bacteria). Alternatively, hosts could selectively transfer certain microbes to their offspring [9]. Interestingly, we observe that inheritance alone is not always beneficial; some taxa might only benefit when transferred asymmetrically (Fig. 4).
We have emphasized the importance of looking at rare taxa. Such is the case of lineage taxa (Fig. 5), microbes absent from the environment that only propagate by inheritance. Our results indicate the importance of modelling the stochasticity and conservation of microbesonly in this way did we appreciate that inheritance can lead to stochastic loss (Figs. 2, 3) and that persistence of lineage taxa may be prolonged by asymmetric inheritance (Fig. 5D). Because microbial frequencies are often small, the omission of stochastic effects from models could lead to misestimate the impact of inheritance.
Vertical transfer of microbes might occur in the most diverse host species [12,18], with only a few exceptions [3]. A great diversity of reproduction and delivery modes might, in turn, determine the distribution of their inheritance-namely the number of microbes transferred and its probability. A comparison of two qualitatively distinct distributions (low and seed-like inheritance in Fig. 1B), indicates they might influence the presence and frequency of microbes differently (Additional file 1: Fig. S6). A consistent cargo in seeds might guarantee the presence of certain microbes in plants [17], who might sometimes benefit from being the first colonizers [29]. In contrast, greater variation might be expected for mammals, where changing amounts of microbes are obtained from epithelia during delivery [11,12]. Overall, these intrinsic differences might affect the ecological and evolutionary dynamics of hosts and microbes.
We found that microbial inheritance is effective only for some life-histories. While it has been shown that symbiosis [19] and fidelity of inheritance [18] can evolve driven by selection, our results suggest the evolution of life-history traits itself, independent of symbiosis, can impact the relevance of microbial inheritance. Interestingly, the emergence of symbiosis could lead to selection acting on the more evolvable and impactful traits-not only the fidelity of inheritance [18]. On the other hand, because lineage taxa are constantly facing extinction some level of selection might be needed to facilitate their preservation-e.g. by the alignment of reproductive interests of host and microbes or mutual benefits.
Investigating microbial inheritance experimentally poses technical challenges [11]. However, developments using diverse host species [14][15][16][17], suggest that our predictions could be tested experimentally. Firstly, that inheritance is more influential at intermediate host lifespan, environmental migration, or carrying capacity (Figs. 2, 3). Related host species with diverse life histories could be compared [33]; alternatively, control could be increased using model organisms amenable to manipulate such traits [34]. Secondly, that the maximum lineage taxa frequency changes with life-history (Eq. 8), could be tested using germ-free or gnotobiotic hosts [17]. Finally, the effect of distinct distributions of microbes inherited (Fig. 1) could be surveyed.
Our approach simplifies the complexity of natural microbiomes. Although this setup allowed us to isolate the effect of microbial inheritance on the microbiome, a natural step forward would be considering other ecological and evolutionary features of microbes and hosts, e.g. phylogeny and niche structure. These could interact with inheritance to preserve or out-compete certain microbes or even lead to priority effects-where the establishment of new taxa in a community depends on their order and time of arrival [35]. In addition, commensals only represent a fraction of the microbiome, the presence of other symbionts could be studied. This requires a deeper empirical knowledge about microbemicrobe and host-microbe interactions. Secondly, a flux of microbes from hosts to the environment could ease the persistence of some microbes. Thirdly, the host population structure could be included [36]. In such a scenario, subpopulations characterized by different microbiomes could emerge [21]. Moreover, critical connectivity might be needed for inheritance to be effective. Finally, we did not account for specific reproductive ages (or development). This might be particularly relevant because, as we have shown, the effect of inheritance erodes through time.

Conclusion
Microbial inheritance can influence the occurrence and abundance of microbes within the host-associated commensal microbiome. Even the persistence of microbes absent from the environment could be facilitated in some cases. These findings extend to diverse scenarios of inheritance representative of different host species. However, inheritance is not a silver bullet, instead life-history in terms of environmental immigration, early microbial proliferation, and host lifespan limit its magnitude and temporal extent. Only certain naturally occurring host-microbiome pairs might meet such conditions to exploit its benefits.
Additional file 1: Fig. S1. Occurrence of a microbial taxon in hosts under microbial inheritance. Fig. S2. Microbial load distribution across a host population, with or without microbial inheritance. Fig. S3. Frequency of a microbial taxon distribution across the host population, with or without inheritance. Fig. S4. Average frequency of a microbial taxon in hosts under microbial inheritance. Fig. S5. Effect of asymmetric inheritance on hosts with longer lifespans. Fig. S6. Difference in microbial load between 'low' and 'seed-like' inheritance. Fig. S7. Difference in the frequency of a microbial taxon between 'low' and 'seed-like' inheritance.