Fig. 1

Microbiome parental inheritance and environmental acquisition under Moran process. (a) The diagram shows a population of hosts evolving under a standard Moran process, where the green circle indicates a dying host and the red circle indicates a reproducing host. The reproduced offspring will replace the dead one. N (where N is the population size) time steps of the abovementioned events are equivalent to one host generation. (b) The yellow arrows signify the flow of microbes from hosts to environment (pooled environment is a collection of microbes available in the current host population) and from environment to hosts (both fixed environment and pooled environment contribute to host microbiomes at next time step and the ratio of y% and 1-y% is determined by the environment assembly mode ME _y ). Parental inheritance of microbes only exists for newly reproduced offspring: M (the total number of microbes per host) microbes are inherited from parents at birth. All the hosts acquire t microbes from local environments at each time step: \( t=\frac{aM}{N} \), where M is the total number of microbes per host, N is the host population size, a satisfies \( x=\frac{1-{e}^{-a}}{a} \), and x is the expected percentage of parental microbes over one host generation determined by MA _x (see derivations in Methods)