Physics of Living Systems

Emergence of alternative stable states in microbial communities undergoing horizontal gene transfer

Juken Hong
Wenzhi Xue
Teng Wang author has email address

Key Laboratory of Quantitative Synthetic Biology, Shenzhen Institute of Synthetic Biology, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China

https://doi.org/10.7554/eLife.99593.2

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Figures and data

Horizontal gene transfer (HGT) promotes bistability of communities consisting of two competing microbes.
(a) Schematics of monostable and bistable systems. A monostable system will always rest at the same steady state no matter where the system starts from. In contrast, a bistable system can reach two distinct steady states, depending on the initial ratio between the two competitors. The right panel is a diagram depicting how a microbial community responds to changes on initial species abundances. The red marble stands for the system state, which will always roll “downhill” on the landscape towards stable states at the low points. The x axis represents the community composition and the valleys in the landscape are the stable states (or equilibrium points) when the community composition will stop changing. Depending on parameter values, the landscape can have one valley (called “monostable”) or two valleys (called “bistable”). When the system is bistable, the marble can reach either valley, depending on where it starts.
(b) The phase diagram of the population with (bottom panel, η = 0.2 hr⁻¹) or without HGT (top panel). Different combinations of µ₁ and µ₂ between 0 and 1 hr⁻¹ were tested. For each combination, the initial abundance of each species was randomized 200 times between 0 and 1 following uniform distributions. The system was monostable if all initializations led to the same steady state. Otherwise, the system was bistable. All combinations of µ₁ and µ₂ resulting in bistability were marked in blue in the diagram. Other parameters are γ = 1.1, κ = 0.005 hr⁻¹,D = 0.2 hr⁻¹,. Two states, P (µ₁ = 0.3 hr⁻¹, µ₂ = 0.8 hr⁻¹) and Q (µ₁ = 0.4 hr⁻¹, µ₂ = 0.6 hr⁻¹), were labeled as examples.
(c-d) Population dynamics of P and Q under 100 different initializations. Here, the dynamic changes of species 1’s abundance were shown. Three different HGT rates (0, 0.1, 0.2 hr⁻¹ from top to bottom) were tested. When gene transfer rate increases, the system at Q changed from monostable to bistable, while the system at P remained monostable.

Horizontal gene transfer (HGT) promotes bistability of communities consisting of two competing microbes.
(a) Schematics of monostable and bistable systems. A monostable system will always rest at the same steady state no matter where the system starts from. In contrast, a bistable system can reach two distinct steady states, depending on the initial ratio between the two competitors. The right panel is a diagram depicting how a microbial community responds to changes on initial species abundances. The red marble stands for the system state, which will always roll “downhill” on the landscape towards stable states at the low points. The x axis represents the community composition and the valleys in the landscape are the stable states (or equilibrium points) when the community composition will stop changing. Depending on parameter values, the landscape can have one valley (called “monostable”) or two valleys (called “bistable”). When the system is bistable, the marble can reach either valley, depending on where it starts.
(b) The phase diagram of the population with (bottom panel, η = 0.2 hr⁻¹) or without HGT (top panel). Different combinations of µ₁ and µ₂ between 0 and 1 hr⁻¹ were tested. For each combination, the initial abundance of each species was randomized 200 times between 0 and 1 following uniform distributions. The system was monostable if all initializations led to the same steady state. Otherwise, the system was bistable. All combinations of µ₁ and µ₂ resulting in bistability were marked in blue in the diagram. Other parameters are γ = 1.1, κ = 0.005 hr⁻¹,D = 0.2 hr⁻¹,. Two states, P (µ₁ = 0.3 hr⁻¹, µ₂ = 0.8 hr⁻¹) and Q (µ₁ = 0.4 hr⁻¹, µ₂ = 0.6 hr⁻¹), were labeled as examples.
(c-d) Population dynamics of P and Q under 100 different initializations. Here, the dynamic changes of species 1’s abundance were shown. Three different HGT rates (0, 0.1, 0.2 hr⁻¹ from top to bottom) were tested. When gene transfer rate increases, the system at Q changed from monostable to bistable, while the system at P remained monostable.

The number of alternative stable states in multi-species communities increases with HGT rate.
(a), A schematic of the stability landscape of a population composed of multiple competing species with or without HGT. Each alternative stable state is characterized by a different dominant species (shown in filled circles).
(b), The emergence of alternative stable states in a five-species community with gene transfer. We simulated the dynamics of 500 parallel communities, each with the same kinetic parameters but different initial compositions. The species compositions of these communities at three different timepoints were shown by principal component analysis from left to right. Without HGT (η = 0), all populations converged into a single stable state. In contrast, with HGT (η = 0.2 hr^-”), four different attractors appear. Here, populations reaching different stable states were marked with different colors. Other parameters are , µ₁ = 0.52 hr⁻¹, µ₂ = 0.48 hr⁻¹, µ₃ = 0.44 hr⁻¹, µ₄ = 0.60 hr⁻¹, µ₅ = 0.31 hr⁻¹, γ = 1.05, κ = 0.005 hr⁻¹, D = 0.2 hr⁻¹.
(c), The number of stable states (top panel) and the multistability coefficient χ increase with HGT rate in communities consisting of 8 competing species. We calculated the number of stable states by randomly initializing the species abundances 500 times between 0 and 1 following uniform distributions. Then we simulated the steady states and clustered them into different attractors by applying a threshold of 0.05 on their Euclidean distances. The data were presented as mean ± standard deviation of 10 replicates. Each replicate corresponded to a different combination of randomized species growth rates.

The number of alternative stable states in multi-species communities increases with HGT rate.
(a), A schematic of the stability landscape of a population composed of multiple competing species with or without HGT. Each alternative stable state is characterized by a different dominant species (shown in filled circles).
(b), The emergence of alternative stable states in a five-species community with gene transfer. We simulated the dynamics of 500 parallel communities, each with the same kinetic parameters but different initial compositions. The species compositions of these communities at three different timepoints were shown by principal component analysis from left to right. Without HGT (η = 0), all populations converged into a single stable state. In contrast, with HGT (η = 0.2 hr^-”), four different attractors appear. Here, populations reaching different stable states were marked with different colors. Other parameters are , µ₁ = 0.52 hr⁻¹, µ₂ = 0.48 hr⁻¹, µ₃ = 0.44 hr⁻¹, µ₄ = 0.60 hr⁻¹, µ₅ = 0.31 hr⁻¹, γ = 1.05, κ = 0.005 hr⁻¹, D = 0.2 hr⁻¹.
(c), The number of stable states (top panel) and the multistability coefficient χ increase with HGT rate in communities consisting of 8 competing species. We calculated the number of stable states by randomly initializing the species abundances 500 times between 0 and 1 following uniform distributions. Then we simulated the steady states and clustered them into different attractors by applying a threshold of 0.05 on their Euclidean distances. The data were presented as mean ± standard deviation of 10 replicates. Each replicate corresponded to a different combination of randomized species growth rates.

The effects of HGT on population multistability when MGEs promote or reduce the strength of interspecies competition.
(a-b), The schematics of HGT promoting or reducing competition.
(c), For populations of two species, when MGEs promote competition, increasing HGT rate enlarges the area of bistability region in the phase diagram. Here, δ describes the effect of mobile genes on the competition strength. Positive δ represents HGT promoting competition. In numerical simulations, we tested three different δ values (marked in different colors). When calculating the area of bistability region, we randomized µ₁ and µ₂ 500 times between 0 and 1 hr^-1 following uniform distributions while keeping and constants. For each pair of growth rates, we randomized the initial abundance of each species 200 times between 0 and 1 following uniform distributions. The system was monostable if all initializations led to the same steady state. Otherwise, the system was bistable. Then we calculated the fraction of growth rate combinations that generated bistability out of the 500 random combinations. Other parameters are γ₁ = γ₂ = 1.1, κ = 0.005 h⁻¹, D = 0.2 h⁻¹.
(d), When MGEs reduce competition, the area of bistability region decreases with HGT rate. Three negative values δ were tested and shown as examples here.
(e), For populations of 5 species, when MGEs promote competition, the number of stable states increases with HGT rate. We calculated the number of stable states by randomly initializing the species abundances 500 times. Then we simulated the steady states and clustered them into different attractors by applying a threshold of 0.05 on their Euclidean distances. The data were presented as mean ± standard deviation of 10 replicates. Each replicate corresponds to a different combination of randomized species growth rates. Other parameters are γ_ij = 1.1, κ = 0.005 h⁻¹, D = 0.2 h⁻¹, δ = 0.5.
(f), When MGEs reduce competition, the number of stable states decrease with HGT rate. The data were presented as mean ± standard deviation of 10 replicates. δ = −0.5 was used in the simulation.
(g), For populations of two species without HGT, reducing the magnitude of λ (promoting ecological neutrality) enlarges the area of bistability. When gene transfer rate equals zero, changing δ does not influence bistability. When calculating bistability probability, we randomized λ₁ and λ₂ 500 times between −α and α following uniform distributions. α represents the magnitude of λ variations. µ₁ and µ₂ were calculated as and , respectively. The overall competition strength was calculated as and . We tested different combinations of α (the magnitude of λ variations) and δ values. We then calculated the fraction (f₀) of growth rate combinations that generated bistability out of the 500 random combinations. Other parameters are γ₁ = γ₂ = 1.1, κ = 0.005 h⁻¹, D = 0.2 h⁻¹.
(h), In populations of two species with HGT, reducing the magnitude of λ (promoting ecological neutrality) or promoting δ value enlarges the area of bistability (f₁). η being 0.4 was tested as an example here.
(i), The effect of HGT on community bistability depends on how mobile genes modify growth rates and competition. Compared with the population without HGT, gene transfer promotes bistability when δ is zero or positive, while reduces bistability when δ is largely negative.

The effects of HGT on population multistability when MGEs promote or reduce the strength of interspecies competition.
(a-b), The schematics of HGT promoting or reducing competition.
(c), For populations of two species, when MGEs promote competition, increasing HGT rate enlarges the area of bistability region in the phase diagram. Here, δ describes the effect of mobile genes on the competition strength. Positive δ represents HGT promoting competition. In numerical simulations, we tested three different δ values (marked in different colors). When calculating the area of bistability region, we randomized µ₁ and µ₂ 500 times between 0 and 1 hr^-1 following uniform distributions while keeping and constants. For each pair of growth rates, we randomized the initial abundance of each species 200 times between 0 and 1 following uniform distributions. The system was monostable if all initializations led to the same steady state. Otherwise, the system was bistable. Then we calculated the fraction of growth rate combinations that generated bistability out of the 500 random combinations. Other parameters are γ₁ = γ₂ = 1.1, κ = 0.005 h⁻¹, D = 0.2 h⁻¹.
(d), When MGEs reduce competition, the area of bistability region decreases with HGT rate. Three negative values δ were tested and shown as examples here.
(e), For populations of 5 species, when MGEs promote competition, the number of stable states increases with HGT rate. We calculated the number of stable states by randomly initializing the species abundances 500 times. Then we simulated the steady states and clustered them into different attractors by applying a threshold of 0.05 on their Euclidean distances. The data were presented as mean ± standard deviation of 10 replicates. Each replicate corresponds to a different combination of randomized species growth rates. Other parameters are γ_ij = 1.1, κ = 0.005 h⁻¹, D = 0.2 h⁻¹, δ = 0.5.
(f), When MGEs reduce competition, the number of stable states decrease with HGT rate. The data were presented as mean ± standard deviation of 10 replicates. δ = −0.5 was used in the simulation.
(g), For populations of two species without HGT, reducing the magnitude of λ (promoting ecological neutrality) enlarges the area of bistability. When gene transfer rate equals zero, changing δ does not influence bistability. When calculating bistability probability, we randomized λ₁ and λ₂ 500 times between −α and α following uniform distributions. α represents the magnitude of λ variations. µ₁ and µ₂ were calculated as and , respectively. The overall competition strength was calculated as and . We tested different combinations of α (the magnitude of λ variations) and δ values. We then calculated the fraction (f₀) of growth rate combinations that generated bistability out of the 500 random combinations. Other parameters are γ₁ = γ₂ = 1.1, κ = 0.005 h⁻¹, D = 0.2 h⁻¹.
(h), In populations of two species with HGT, reducing the magnitude of λ (promoting ecological neutrality) or promoting δ value enlarges the area of bistability (f₁). η being 0.4 was tested as an example here.
(i), The effect of HGT on community bistability depends on how mobile genes modify growth rates and competition. Compared with the population without HGT, gene transfer promotes bistability when δ is zero or positive, while reduces bistability when δ is largely negative.

The influence of epistasis on the role of HGT in mediating population bistability.
(a-b) With magnitude epistasis, increasing HGT rate still promotes bistability in populations of two competing species.
(c-d) With sign epistasis, increasing HGT rate reduces the area of bistability region. For each type of epistasis, we considered two scenarios. In a and c, host genetic background influences the growth rate effect of only one MGE, while in b and d, host genetic background influences both MGEs. ξ₁ and ξ₂ are defined as λ₂₁/λ₁₁ and λ₁₂/λ₂₂, respectively. ξ₁ >0 and ξ₂ >0 represent magnitude epistasis, while ξ₁ < 0 or ξ₂ < 0 represents the sign epistasis. Other parameters are γ₁ = γ₂ = 1.1, κ = 0.005 h⁻¹, D = 0.2 h⁻¹.

HGT creates chances of multistability for communities under strong environmental selection.
(a-b), The schematics of communities under strong selection. Without HGT, only the donor species carrying the MGE can survive. HGT allows the other species to acquire MGE from the donor, creating opportunities for the other species to survive under strong selections.
(c), The phase diagram of two-species populations transferring an MGE. Different combinations of µ₁ and µ₂ between 0 and 1 hr⁻¹ were tested. All combinations of µ₁ and µ₂ that led to bistability were marked in blue in the diagram. Other parameters are γ = 1.1, κ = 0.005 hr⁻¹, D = 0.2 hr⁻¹, . Here, we assumed that the ARG was initially carried only by species 1. Bistability was more feasible when µ₂ >µ₁.
(d), Increasing HGT rate enlarges the area of bistability region in the phase diagram of two competing species transferring an MGE. Three different competition strengths were tested (marked in different colors).
(e-f), For communities of 5 or 8 species under strong selection, increasing HGT rate promotes the emergence of alternative stable states. We calculated the number of stable states by randomly initializing the species abundances 500 times. Then we simulated the steady states and clustered them into different attractors. The carriage of the mobile gene changed the species growth rate from 0 to a positive value µ_i. When calculating the number of stable states in multi-species communities, we randomly drew the µ_i values from a uniform distribution between 0.3 and 0.7 hr⁻¹. The data were presented as mean ± standard deviation of 10 replicates. Each replicate corresponds to a different combination of randomized species growth rates. Other parameters are γ_i,j = 1.5, κ = 0.005 h⁻¹, D = 0.2 h⁻¹.

HGT creates chances of multistability for communities under strong environmental selection.
(a-b), The schematics of communities under strong selection. Without HGT, only the donor species carrying the MGE can survive. HGT allows the other species to acquire MGE from the donor, creating opportunities for the other species to survive under strong selections.
(c), The phase diagram of two-species populations transferring an MGE. Different combinations of µ₁ and µ₂ between 0 and 1 hr⁻¹ were tested. All combinations of µ₁ and µ₂ that led to bistability were marked in blue in the diagram. Other parameters are γ = 1.1, κ = 0.005 hr⁻¹, D = 0.2 hr⁻¹, . Here, we assumed that the ARG was initially carried only by species 1. Bistability was more feasible when µ₂ >µ₁.
(d), Increasing HGT rate enlarges the area of bistability region in the phase diagram of two competing species transferring an MGE. Three different competition strengths were tested (marked in different colors).
(e-f), For communities of 5 or 8 species under strong selection, increasing HGT rate promotes the emergence of alternative stable states. We calculated the number of stable states by randomly initializing the species abundances 500 times. Then we simulated the steady states and clustered them into different attractors. The carriage of the mobile gene changed the species growth rate from 0 to a positive value µ_i. When calculating the number of stable states in multi-species communities, we randomly drew the µ_i values from a uniform distribution between 0.3 and 0.7 hr⁻¹. The data were presented as mean ± standard deviation of 10 replicates. Each replicate corresponds to a different combination of randomized species growth rates. Other parameters are γ_i,j = 1.5, κ = 0.005 h⁻¹, D = 0.2 h⁻¹.

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