Fig 1.
Schematic illustration of the model for microbial community with metabolite exchange.
Each cell species α has a chemical reaction network that transforms a single nutrient S transported from the environment for cell growth. The nutrient S is supplied to the environment from the exterior at the rate . Among n chemicals, metabolites (orange circles) are diffusible and exchanged by coexisting species via the environment, while enzymes (green squares) are not.
Table 1.
Parameters and variables in our model.
Fig 2.
Example of leaker-consumer mutualism between two cell species.
(A) A simple example of the mutualism between the leaker (left) and consumer (right) cells. Both have the same network structure that consists of substrate S, enzyme E, ribosome rb, metabolites M1 and M2, and biomass BM, with different rate constants. (B) Schematic illustration of leaker-consumer mutualism. When only a leaker cell is present, the secreted chemical accumulates in the environment and inhibits further secretion (left). The coexistence of other consumer cells is beneficial for both cells as it reduces the concentration of the leaked chemical in the environment (right). (C) Phase diagram of symbiosis depending on and
. Regions M (red), P (green), and NC (black) are delineated by gray lines and represent mutualism, parasitism, and noncoexistence, respectively. The environment volume ratio Venv and the degradation rate in the environment Rdeg are both set at unity. The color denotes the growth rate μ, where a brighter color corresponds to a higher μ. μ with
is just the growth rate of the leaker cells in isolated conditions,
; in region M, it is smaller than μ at the corresponding
value in the panel. (D) Phase diagram of symbiosis depending on the environmental parameters, Rdeg and Venv. The diffusion coefficients of M1 are fixed as
. Red and black diamonds are delineated by gray lines and represent mutualism and noncoexistence, respectively. The rate constants are set as:
, so that the leaker’s growth rate in isolation
with optimal diffusion coefficient
is higher than the consumer’s growth rate in isolation
with
. The other parameters are set as
.
Fig 3.
Example of symbiosis with metabolic exchange via the environment.
(A) An example of randomly generated networks with n = 10. The enzyme labeled on each arrow catalyzes the conversion of the metabolite at the arrowtail to the metabolite or enzyme at the arrowhead. Among n chemicals, chemicals 1 and nenzyme = 2 are enzymes (green squares) and the nutrient chemical 0 and chemicals nenzyme + 1 = 3 to n − 1 = 9 are metabolites (orange circles). The leak-advantage metabolites in isolation conditions (chemicals 3–5, 7, and 9) are highlighted by pink. See also S3(B) Fig for the network of cell species B in panels (C) and (D). (B) Time series of the number of coexisting species through successful invasions by new species and the growth rate of coexisting cell species. (C) Plot of leakage (blue) and uptake (red) fluxes of non-nutrient chemicals from each cell species A-F. (D) Structure of metabolic exchange among six cell species that have different growth rates in isolation. The vertical axis represents the growth rate of each cell species α in isolation, . Cyan and pink arrows indicate the leakage and uptake of each chemical component, respectively. Symbiosis among multiple species increases the growth rate to μsymbiosis (as indicated on the top), which is higher than the growth rate of each cell species in isolation,
. In the numerical simulations in (B)-(D), the parameters were set to
.
Fig 4.
Statistics of symbiosis among randomly generated networks.
(A) Dependence of the frequency of coexisting species on the number of chemical components n. Senv = 0.03, Venv = 3. (B) Dependence of the frequency of coexisting species upon Venv. n = 20, Senv = 0.03. (C) Dependence of the frequency of coexisting species upon Senv. n = 20, Venv = 3. (D) The frequency of coexisting species for random fixed diffusion coefficients (with and without leakage of chemicals that confer leak advantage) without cell-level adaptation. n = 20, Senv = 0.03, Venv = 3. In the “random” case, the diffusion coefficients of chemicals nenzyme + 1 ∼ n − 1 are chosen randomly from a uniform distribution [0.0: 1.0] (see also S6 and S7 Figs). In all the panels, the colored bars show the frequency of symbiosis among two to eight species (with different colors), whereas the black bars show noncoexistence. The frequency for each parameter set was calculated from 50 independent samples of N catalytic networks where the species with the fastest growth in isolation has a leak-advantage chemical in its reaction network. In all the numerical simulations, the other parameters are fixed: .
Fig 5.
Resilience of symbiotic coexistence against the removal of one species.
(A) The upper panel shows the frequency distribution of the number of cell species (0–3) that become extinct when one species is removed. The lower panel shows the ratio of leakage of chemicals from the removed cell species to the total leakage from all cell species to the environment, PLeak,α. Each point corresponds to a sample shown in the upper panel. (B) Average survival ratio (color) against the indices that characterize the diversity of leaking cell species (SCell) and leaked chemical components (SChem). The survival ratio is the number of surviving species after removal of one species, divided by the number of species before the extinction successive to the removal. (If no additional cell species become extinct, this ratio equals one.) The multiple correlation coefficient between survival ratio and (SCell, SChem) is 0.49, while the correlation coefficients between the survival ratio and SCell, between the survival ratio and SChem, and between SCell and SChem are 0.46, 0.43, and 0.65, respectively. For these calculations, we used 55 samples of coexistence of five or six species with n = 20.