Gut mutualists can persist in host populations despite low fidelity of vertical transmission

Abstract

Humans harbour diverse microbial communities, and this interaction has fitness consequences for hosts and symbionts. Understanding the mechanisms that preserve host-symbiont association is an important step in studying co-evolution between humans and their mutualist microbial partners. This association is promoted by vertical transmission, which is known to be imperfect. It is unclear whether host-microbial associations can generally be maintained despite 'leaky' vertical transmission. Cultural practices of the host are expected to be important in bacterial transmission as they influence the host's interaction with other individuals and with the environment. There is a need to understand whether and how cultural practices affect host-microbial associations. Here, we develop a mathematical model to identify the conditions under which the mutualist can persist in a population where vertical transmission is imperfect. We show with this model that several factors compensate for imperfect vertical transmission, namely, a selective advantage to the host conferred by the mutualist, horizontal transmission of the mutualist through an environmental reservoir and transmission of a cultural practice that promotes microbial transmission. By making the host-microbe association more likely to persist in the face of leaky vertical transmission, these factors strengthen the association which in turn enables host-mutualist co-evolution.

Keywords: Microbiota; cultural transmission; horizontal transmission; host–microbe association; mathematical modelling.

Graphical abstract

Figure 1.

A schematic showing the transmission or change of microbiota types across one generation (indicated by the arrows) in the basic model. M is the proportion of hosts that carry the mutualist M⁺ and N is the proportion of hosts that lack the mutualist M⁻. The primes (′) indicate variables in the next generation. The mutualist is shed into the environment with probability γ and acquired from the environment with probability βE_m. We refer to the combination of these processes as horizontal transmission. The mutualist can fail to transmit to the next generation owing to leaky vertical transmission with probability λ.

Figure 2.

Schematic of the model with cultural factors. The parameters are defined in Table 1. The arrows indicate the transmission or loss of microbes or the cultural practice across one generation. The primes (′) indicate variables in the next generation.

Figure 3.

Dynamics of equilibria and threshold values in a culturally homogeneous population (the basic model) obtained by solving Equations (1)–(4). (a) The proportion of microbiota types at equilibria against λ (the leakiness of microbial vertical transmission), from numerical solutions. An estimate of the equilibrium is obtained when the difference between two consecutive iterations is smaller than an error of 1 × 10⁻⁵. The leakiness decreases the proportion of the mutualist in the population M⁺ and the environment E_m. The black dotted lines represent the lower and upper thresholds (T₁ and T₂), as labelled. (b) and (c) The thresholds of λ shift the equilibria of the basic model as a function of β and s (Table 2). Unless indicated otherwise, the parameters are set at γ = 0.15, β = 0.1, s = 0.1 and c = 0.1.

Figure 4.

The proportion of microbiota types at equilibrium against λ (the leakiness of microbial vertical transmission) for the extended model with cultural practice. The curves show equilibria obtained numerically by solving Equations (7)–(12) using γ = 0.15, λ = 0.1, β = 0.1, s = 0.1, c = 0.1, α = k = 0.1 and δ = 0.02. An estimate of the equilibrium is obtained when the difference between two consecutive iterations is smaller than an error of 1 × 10⁻⁵. Increasing leakiness of vertical transmission reduces the proportion of the mutualist in the population and the environment. The black dotted line represents the threshold .

Figure 5.

Heatmap of threshold leakiness as a function of the equilibrium frequency of hosts with the cultural practice, and benefit to the host s for three values of the strength of horizontal transmission βγ/c. The threshold leakiness is the value of λ above which the mutualist will go extinct. The green lines are contours of the threshold at the values given in the labels. In all three heatmaps α = 0.1.