On the effect of inheritance of microbes in commensal microbiomes

TLDR: In this article, a mathematical model was developed to study the effect of the transfer of microbes from parents to offspring and found 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 favoring the acquisition of rare microbes.

Abstract: Background. 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. Results. 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. Conclusion. 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.

Figures

Figure 5: Persistence of lineage taxa in hosts. A microbial taxon is initially present in hosts x1(0) > 0, but not in the pool of colonizers, p1 = 0. (A) The frequency within a host decreases through time. For some conditions, Eq. (8), there is a period of increase. If the taxon is transmitted to offspring before the gain is lost, this might persist in the host population (although extinction within the parent occurs sooner). (C) Low immigration (m → 0) and fast occupation of available space (α0 → 0) allow increase and prolong the time before extinction, Eq. (8). Large initial available space (xi + oi → 0) and lineage taxon fractions (x1/(x1+o1)→ 1) also prolong this time. (B) After the increase stops (x∗1), the average time to extinction is shorter for large immigration (m → 1) and a smaller fraction of the taxon. (D) At the host population level, lines indicate the death probability after which most hosts lose the lineage taxon (τ0.5), Eq. (9). The early increase shown in (A) only occurs within the darkened area. The distribution of microbes inherited, Fig. 1B and Eq. (4), affects the initial load and fraction of lineage taxa in offspring. Asymmetric inheritance in low microbial loads might preserve lineage taxa as well as symmetric inheritance in high loads. We set N = 105. Each point corresponds to 104 simulated hosts.

Figure 4: Effect of asymmetric inheritance on the average frequency of a taxon in hosts. Cases without inheritance and inheritance are compared. Inheritance is symmetric if offspring receive 9% of their parent’s microbiome on average (ai = 0 and bi = 9). Inheritance is asymmetric if offspring receive 9% of taxon 1 and 1% of other taxa (ai = 0 and b1 = 9, bi 6=1 = 99 in Eq. (4)). Available space within hosts is occupied more easily for α0 → 0. Single parameters are modified from the condition p1 = 10−2, m = 10−2, τ = 10−4, and N = 105. (A-B) The average frequency increases for larger abundances in the pool of colonizers (p1), immigration (m), and α0 → 0. An asymmetric inheritance has no effect, as hosts are not fully occupied within their lifetime (Fig. Sup. 2 and Fig. Sup. 3). (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 104 hosts.

Figure 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 [24] or D. melanogaster [25]). Others might provide minimal (eg. humans [11]) or large fractions of their microbes (eg. fragmentation of some sponges, corals, fungi and plants [26, 27]), 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 ai and bi in Eq. (4).

Figure 3: Average microbial load in hosts under microbial inheritance. (A) Starting from a condition where all hosts are initially empty, the average frequency of microbes in hosts increases through time before reaching an equilibrium. In this particular case, inheritance makes such equilibrium abundance larger only when hosts are occupied rapidly, α0 → 0. This increase results from a host distribution where higher microbial loads are more common (Fig. 2B). The cases shown in (A), with parameters m = 10−2, τ = 10−4, and N = 105, are indicated by the triangles in (B-D). A single parameter is varying in (B-D). (B) Changes of migration from the pool of colonizers, m, have minimal effect (notice the scale). As m→ 1, more microbes colonize the hosts. Still the average microbial load only increases if the loss of microbes to inheritance is less than the gain from proliferation. (C) The effect of changes to host death probability, τ , are much larger and maximal at intermediate τ . A faster occupation of hosts makes the effect of inheritance larger for shorter living hosts, τ → 1. (D) In contrast to the occurrence (Fig. 2E), changes in the carrying capacity for microbes, N , have a larger intermediate effect. Faster occupation of hosts makes the effect peak for larger N . Points and bars in (B-D) indicate the average and standard deviation of 6 simulation pairs, with vs. without inheritance, with 104 hosts each. Offspring receive 9% of their parent’s microbiome on average, ai = 0 and bi = 9 in Eq. (4). The whole distributions are shown in Fig. Sup. 2.

Figure 2: Microbial occurrence in hosts under microbial inheritance. (A) Starting from a condition where all hosts are initially empty, the microbial occurrence increases through time. At first sight, this increase is largely independent of α0 and the inheritance of microbes. A closer look at equilibrium abundance reveals that inheritance increases the occurrence, in this case, regardless of how rapidly hosts are occupied (α0). (B) The increase results from a distribution of microbial load across the host population where the microbe-free state is less common. A microbial load of 10−5 corresponds to 1 microbe per host. In (C-E), single parameters are modified from the case shown in (A-B) (with parameters m = 10−2, τ = 10−4, and N = 105, indicated by the triangles in (C-E)). (C) A large migration from the pool of colonizers, m → 1, hinders any effect of inheritance on occurrence as hosts are readily colonized. The change peaks and decreases for smaller m, as for m → 0 hosts are less likely to be colonized. The change can even be negative for slowly occupied hosts where the few colonizing microbes are lost to stochasticity. (D) The gain from inheritance is maximal for intermediate values of host death probability, τ . Long living hosts, τ → 0, are colonized even without inheritance. Short living hosts, τ → 1, are less likely to be colonized and thus transmit microbes through inheritance. (E) The carrying capacity for microbes of a host, N , and α0 do not alter the gain from inheritance. Points and bars in (C-E) indicate the average and standard deviation of 6 simulation pairs, with vs. without inheritance, with 104 hosts each. Offspring receive 9% of their parent’s microbiome on average, ai = 0 and bi = 9 in Eq. (4). The whole distributions are shown in Fig. Sup. 2.

Full text

On the effect of inheritance of microbes1
in commensal microbiomes2
Rom´an Zapi´en-Campos
1
, Florence Bansept
1
, Michael Sieber
1
, and Arne Traulsen
1,*
3
1
Max Planck Institute for Evolutionary Biology, Pl¨on, Germany4
1
zapien@evolbio.mpg.de, bansept@evolbio.mpg.de, sieber@evolbio.mpg.de5
*
Corresponding author: traulsen@evolbio.mpg.de6
Background. Our current view of nature depicts a world where macroorganisms7
dwell in a landscape full of microbes. Some of these microbes not only transit but8
establish themselves in or on hosts. Although hosts might be occupied by microbes for9
most of their lives, a microbe-free stage during their prenatal development seems to be10
the rule for many hosts. The questions of who the first colonizers of a newborn host11
are and to what extent these are obtained from the parents follow naturally.12
Results. We have developed a mathematical model to study the effect of the13
transfer of microbes from parents to offspring. Even without selection, we observe that14
microbial inheritance is particularly effective in modifying the microbiome of hosts with15
a short lifespan or limited colonization from the environment, for example by favouring16
the acquisition of rare microbes.17
Conclusion. By modelling the inheritance of commensal microbes to newborns, our18
results suggest that, in an eco-evolutionary context, the impact of microbial inheritance19
is of particular importance for some specific life histories.20
Keywords: microbiome, host, colonization, microbial inheritance, mathematical model.21
1
.CC-BY 4.0 International licenseperpetuity. It is made available under a
preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in
The copyright holder for thisthis version posted September 24, 2021. ; https://doi.org/10.1101/2021.09.21.461237doi: bioRxiv preprint

1. Background22
Microbial life is ubiquitous in the biosphere [1]. The human body is no exception, as first described23
by van Leeuwenhoek in the 17th century. We are among the many macroorganisms where diverse24
microbiomes – microbial communities living in or on hosts – have been observed [2, 3]. As part25
of their life cycle, members of the microbiome may migrate between hosts and the environment.26
The migration process has been studied using experimental [4] and theoretical approaches [5, 6].27
However, some microbes have been found exclusively in hosts [4, 7]. How do such microbes persist28
in the population?29
One possibility is the vertical transfer of microbes from parents to offspring [8]. Although there is30
ample literature about transmission of endosymbionts (e.g. Buchnera and Wolbachia in insects [9]),31
less is known about extracellular – possibly transient – microbes. Quantifying the low microbial loads32
in newborns [10] and deciphering the true origin of microbes [11] remains experimentally challenging33
[12, 13]. A few experimental studies have explored the vertical transfer of the microbiome in specific34
species across the tree of life – including sponges [14], mice [15], cockroach eggs [16], and wheat35
seedlings [17]. For many others, including humans, there is an ongoing debate on when and how36
inherited microbes are obtained [11]. Together, these studies suggest there is no universal reliance37
on microbial inheritance across host species, raising the possibility that even if such associations38
matter to the host, certain life-history traits may limit their inheritance [13, 18]. Relevant traits39
may include, among others, the extent of environmentally acquired microbes and host lifespan.40
Previous theoretical work has studied microbial inheritance in the context of symbiosis – where41
microbes affect the host fitness. In these models, depending on whether the interaction is positive42
(mutualism) or negative (parasitism) the presence of symbionts is promoted or impeded, respec-43
tively. Using multilevel selection arguments, Van Vliet and Doebeli have shown that a symbiosis44
that is costly for microbes can be sustained only when the host generation time is short and the45
contribution of inheritance exceeds that of environmental immigration [19]. Following up, in addition46
to individual inheritance (single contributing parent), Roughgarden analyzed scenarios of collective47
inheritance (multiple contributing parents) [20]; while Leftwich et al. found a weak influence of the48
host reproductive mode (sexual or asexual) and mate choice (based on symbiont presence) on the49
2
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preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in
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symbiont occurrence [21]. If these host-symbiont interactions persist over evolutionary timescales,50
they are said to lead to phylosymbiosis – where microbiomes recapitulate the phylogeny of their51
hosts [22].52
Not all co-occurrences between hosts and microbes reflect a fitness impact, however. As suggested53
by Bruijning et al., the selection on the host-microbiome pair and the microbial inheritance might54
change with the environment [18]. Moreover, despite taxonomic differences, functional equivalence55
of microbes in localized host populations could prevail [16]. Microbes might not always influence56
host fitness [18] nor benefit from influencing it [21]. In this context where there is no active selection57
of the microbes by the host, the role of microbial inheritance remains largely unexplored [23].58
Using a stochastic model, we study the effect of microbial inheritance on the commensal micro-59
biome – microbes living in hosts but not affecting their fitness. First, we introduce different models60
of inheritance representative of diverse host species. Then we discuss their effect on microbes present61
in both hosts and environment, or only present in hosts. We see that inheritance might influence62
the within-host occurrence and abundance in some cases. However, within the same microbiome,63
microbial types could be affected differently – while inheritance causes some microbes to increase64
in frequency, others decrease from it. Moreover, the effects may be transient, rendering life history65
parameters crucial. Altogether, we highlight the potential and limits of microbial inheritance to66
modify the composition of commensal microbiomes under different life-history scenarios.67
2. Model and methods68
Consider the host-microbiome system depicted in Fig. 1A. A population of hosts is colonized by69
a set of microbes, and each microbial taxon i has a constant frequency p
i
in the environment.70
The total number of microbes a host can contain is finite and given by N. Each newborn empty71
host inherits a set of microbes from its parent, chosen at random within the host population. The72
inherited sample, taken off the parental microbiome, is drawn according to a probability distribution73
(Fig. 1B). After this initial seeding, only the death, immigration and replication of microbes can74
modify the host microbiome. Through these processes, the microbial populations within the host75
3
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preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in
The copyright holder for thisthis version posted September 24, 2021. ; https://doi.org/10.1101/2021.09.21.461237doi: bioRxiv preprint

BA
Figure 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
[24] or D. melanogaster [25]). Others might provide minimal (eg. humans [11]) or large fractions of their
microbes (eg. fragmentation of some sponges, corals, fungi and plants [26, 27]), 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).
can decrease or increase by one individual each time step. After one microbe is selected to die,76
migration from the pool of colonizers occurs with probability m, while duplication of a resident77
microbe, or non-replacement, occurs with probability 1 − m. This process ends with the host death,78
which occurs with probability τ at each time step. We assume that the number of hosts does not79
change, so that a host death is followed by the birth of a new empty host, for which the process80
described above is repeated.81
2.1. Transition probabilities82
Our aim is to describe the dynamics of the microbiome load and composition, focusing in particular83
on how a certain microbial taxon experiences it. Within a specific host, the frequency of the i-th84
taxon is denoted by x
i
(for i ≥ 1), and of the remaining other microbes by o
i
=
P
j6=i
x
j
. The85
frequency of available space is then given by x
0
= 1−x
i
−o
i
. The transition probabilities from state86
{x
i
, o
i
} that are due to the microbial dynamics are given by the product of the probability of host87
4
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preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in
The copyright holder for thisthis version posted September 24, 2021. ; https://doi.org/10.1101/2021.09.21.461237doi: bioRxiv preprint

survival, 1 − τ, by the probability of death of a certain microbial type followed by an immigration88
or birth event. These events produce changes in the frequencies of magnitude
1
N
. First, microbial89
taxa can replace each other when a microbe dies and is replaced by another one,90
T
o
i
+
x
i
−
= (1 − τ) x
i

m(1 − p
i
) + (1 − m)
o
i
α
0
x
0
+ x
i
+ o
i

(1a)
T
o
i
−
x
i
+
= (1 − τ) o
i

mp
i
+ (1 − m)
x
i
α
0
x
0
+ x
i
+ o
i

. (1b)
In Eq. (1a), a microbe of type i dies and is replaced by another microbe, either by immigration from91
the environmental pool or by replication within the same host. Similarly, in Eq. (1b), a microbe of92
another type dies and is replaced by a microbe of type i.93
Alternatively, microbes may occupy previously available space, such that the microbial abundance94
increases,95
T
o
i
+
x
i
= (1 − τ) x
0

m(1 − p
i
) + (1 − m)
o
i
α
0
x
0
+ x
i
+ o
i

(1c)
T
o
i
x
i
+
= (1 − τ) x
0

mp
i
+ (1 − m)
x
i
α
0
x
0
+ x
i
+ o
i

. (1d)
Finally, microbes may decrease in abundance, when a microbe selected for death is not replaced,96
T
o
i
x
i
−
= (1 − τ) x
i

(1 − m)
α
0
x
0
α
0
x
0
+ x
i
+ o
i

(1e)
T
o
i
−
x
i
= (1 − τ) o
i

(1 − m)
α
0
x
0
α
0
x
0
+ x
i
+ o
i

. (1f)
In these equations, α
0
controls the establishment of microbes in hosts – the ability to occupy97
available space – going from fast for α
0
= 0, to slow if α
0
is positive. For α
0
> 1 and without98
migration, microbes cannot be maintained in hosts.99
The transition probabilities due to the hosts dynamics are given by the product of the probability100
of host death and birth of an empty host (τ), by the probability to inherit certain microbes,101
T
∆o
i
∆x
i
= τ
X
p
1
H − 1
ω
i
[∆x
i
, x
(p)
i
]ω
i
[∆o
i
, o
(p)
i
], (2)
5
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preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in
The copyright holder for thisthis version posted September 24, 2021. ; https://doi.org/10.1101/2021.09.21.461237doi: bioRxiv preprint

Frequently asked questions

Q1. What are the contributions mentioned in the paper "On the effect of inheritance of microbes in commensal microbiomes" ?

The questions of who the first colonizers of a newborn host 11 are and to what extent these are obtained from the parents follow naturally. The authors have developed a mathematical model to study the effect of the 13 transfer of microbes from parents to offspring. By modelling the inheritance of commensal microbes to newborns, their 18 results suggest that, in an eco-evolutionary context, the impact of microbial inheritance 19 is of particular importance for some specific life histories. 

Q2. What is the effect of inheritance on the frequency of the host?

The reduced variability of the early microbiome, makes hosts with initially large frequencies of the microbial taxon less likely. 

Q3. What is the probability of occurrence of microbes in hosts?

(A) Starting from a condition where all hosts are initially empty, the average frequency of microbes in hosts increases through time before reaching an equilibrium. 

Q4. How many hosts have a low inheritance?

For both, low and seed-like inheritance, offspring receive 9% of their parent’s microbiome on average (ai = 0 and bi = 9 for low inheritance, and ai = 9 and bi = 99 for seed-like inheritance in Eq. (4)). 

Q5. What is the microbial occurrence in the host?

In this particular case, inheritance increases the occurrence if hosts are colonized rapidly, α0 → 0. (B) The hosts now contain the taxon in small frequencies. 

Q6. What is the probability of occurrence and frequency within hosts?

(A) The probability of occurrence and frequency within hosts increase for higher abundances in the pool of colonizers, p1 → 1, and (B) larger migration from the environment, m → 1.