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Animals are associated with a microbiome that can affect their reproductive success. It is therefore important to understand how a host and its microbiome coevolve. According to the hologenome concept, hosts and their microbiome form an integrated evolutionary entity, a holobiont, on which selection can potentially act directly. However, this view is controversial and there is an active debate on whether the association between hosts and their microbiomes is strong enough to allow for selection at the holobiont level. Much of this debate is based on verbal arguments, but a quantitative framework is needed to investigate the conditions under which selection can act at the holobiont level. Here we use multilevel selection theory to develop such a framework. We found that selection at the holobiont level can in principle favor a trait that is costly to the microbes but that provides a benefit to the host. However, such scenarios require rather stringent conditions. The degree to which microbiome composition is heritable decays with time, and selection can only act at the holobiont level when this decay is slow enough, which occurs when vertical transmission is stronger than horizontal transmission. Moreover, the host generation time has to be short enough compared to the timescale of the evolutionary dynamics at the microbe level. Our framework thus allows us to quantitatively predict for what kind of systems selection could act at the holobiont level.
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