A holobiont is a composite organism consisting of a host together with its microbiome, such as a coral with its zooxanthellae. Hosts and their microbiomes are often intimately integrated in behavior, physiology, anatomy, and development. To explain this integration, some investigators contend that selection operates on holobionts as a unit and view the microbiomes genes as extending the hosts nuclear genome to jointly comprise a hologenome. Holobiont selection then operates on holobionts with their hologenomes by analogy to how ordinary natural selection operates on individuals with their genes. Other investigators point out that vertical transmission of microbiomes is uncommon. They contend that holobiont selection cannot be effective because a holobionts microbiome is an acquired condition rather than an inherited trait. The disagreement between these positions invites a simple mathematical model to see how holobiont selection might operate and to assess its plausibility as an evolutionary force.\n\nHere I present two variants of such a model. In one variant, juvenile hosts obtain microbiomes from their parents (vertical transmission). In the other variant, microbiomes of juvenile hosts are assembled from source pools containing the combined microbiomes of all parents (horizontal transmission). According to both variants, holobiont selection indeed causes evolutionary change in holobiont traits. Therefore, holobiont selection is plausibly an effective evolutionary force with either mode of microbiome transmission.\n\nFurthermore, the modeling employs two distinct concepts of inheritance, depending on the mode of microbiome transmission: collective inheritance whereby juveniles inherit a sample of the collected genomes from all parents as contrasted with lineal inheritance whereby juveniles inherit the genomes from only their own parents. Collective inheritance may also apply to the evolution of soil and other microbes that feature large amounts of horizontal gene transfer and may underlie cultural evolution wherein each generation inherits a sample from the collected knowledge of the preceding generation. A distinction between collective and lineal inheritance also features in theories of multilevel selection.
