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Global dynamics of microbial communities emerge from local interaction rules

['Simon Van Vliet', 'Department Of Zoology', 'University Of British Columbia', 'Vancouver', 'British Columbia', 'Biozentrum', 'University Of Basel', 'Basel', 'Christoph Hauert', 'Department Of Mathematics']

Date: 2022-05

Steady state community properties

Pair approximation allows us to derive the global properties of the community from the local interaction rules between cells. From the neighborhood sizes (r A and r B ) and the maximum growth rates ( and , Fig 2B and 2D), we obtain a system of dynamical equations describing the global, P(A), and local, P(B|A, r A ) and P(A|B, r B ), composition of the system (Fig 2E). By solving the dynamical equation for steady state, we find analytical expressions for the global and local compositions at equilibrium (see S2 Text).

The global composition of the community reaches a steady state in which the frequency of type A is given by: (1)

This equation shows that the equilibrium composition of the community is mostly set by the maximum growth rates of the two types. In general, the type with highest maximum growth rate constitutes the majority (Fig 3A). Also the neighborhood size affects the composition of the community: if the interaction neighborhoods are small, the composition shifts to the type with the faster maximum growth rate (Fig 3A). Increasing the neighborhood size of even a single type moves the community composition closer to the expected composition of a well-mixed system (Fig 3B, colored lines). The neighborhood sizes do not affect the community composition when both cell types have the same maximum growth rate (Fig 3B, black line).

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TIFF original image Download: Fig 3. Community-level properties depend on local interaction rules. (A) The global composition of the community (global frequency of A, given by Eq 1) primarily depends on the ratio of maximum growth rates of the two types. If the neighborhood sizes of both types are large (r A = r B = 10, purple line), the equilibrium frequency approaches that of a well-mixed system (black line, given by Eq 2). If the neighborhood sizes are small (r A = r B = 3, green line), the type with the higher growth rate attains a higher frequency than in a well-mixed system. When either type becomes fixed (i.e., P(A) = 0 or P(A) = 1) cells can no longer reproduce. In natural communities, where populations sizes can change, this would lead to the collapse of the community; however, in our model we assume a constant population size, so in this case the community persists even though there is no longer any turnover of cells (i.e., community productivity drops to 0). (B) The neighborhood size affects the global composition of the community, when the two types have different maximum growth rates. The solid lines show the frequency of type A for spatial systems where r A is held constant at 3, while r B is increased. The equilibrium frequency varies as r B increases: for larger r B , the frequency in a spatial system (solid lines) moves closer to the frequency in a well-mixed system (dotted lines). This result holds when the types have different maximum growth rates (red and green curves) but not when they have equal maximum growth rates (black line). (C) The neighborhood size strongly affects the local composition of the community. Here the two types have the same maximum growth rate and neighborhood size. For both types, the local frequency of the partner cells is much lower than the global frequency, when the neighborhood size is small. (D) The neighborhood size strongly affects community productivity. The community productivity (i.e. the gross production of new biomass) is smaller when cells have smaller interaction neighborhoods, because the local frequency of the partner around each cell is lower. This effect is stronger in communities where the types have different maximum growth (the red and green curves are below the black curve). https://doi.org/10.1371/journal.pcbi.1009877.g003

When the neighborhood size is large (r A , r B ≫ 1), the equilibrium frequency predicted by pair-approximation (Eq 1) simplifies to the expected equilibrium frequency in a well-mixed system (see S2 Text): (2)

Pair-approximation is useful to understand how composition of a community changes when interaction ranges are short and neighborhood sizes small. In this regime, the equilibrium frequency given by pair-approximation (i.e. Eq 1) deviates markedly from that in well-mixed conditions (i.e. Eq 2, see Fig 3). However, when the interaction range is large, the difference in predicted equilibrium frequencies becomes largely unnoticeable (less than 1 percentage point difference when cells have more than 100 neighbors).

Pair-approximation is especially relevant if we want to investigate the local effects of spatial structure. The same global community composition can correspond to many different local spatial arrangements of the two cell types. For example, the two cell types could be highly mixed in space or completely segregated. The spatial arrangement matters because cells interact only locally: the growth of a cell depends on the local frequency of the partner type, P(B|A, r A ) and P(A|B, r B ), which can be different from the global frequency, P(B) and P(A), when the cell types are clustered in space. A relevant quantity to describe a spatial system is thus the ratio between the local and global frequency of the partner type, and pair approximation shows that at steady state: (3)

This equation shows that the frequency of the partner type in the interaction neighborhood is lower than one would expect from the global composition of the community (Fig 3C). This happens because cells place offspring close to themselves when they divide. As a consequence, the local frequency of the partner type is reduced by as much as 50% when the neighborhood size is small (e.g., when cells interact with three neighbors).

Both cell types are affected by this reduction in the local frequency of the partner type. This can be understood as follows: in a spatial system, cells of both types place their offspring close by in space and as a result they form patches. Within each patch, the frequency of the partner type is (much) lower than the global frequency. When interaction ranges are short, cells on average interact mostly within their own patch, and thus mostly interact with their own type (see Fig C in S3 Text). The average cell thus interacts with fewer cells of the other cell type than it would in a well-mixed system. In other words: in spatial systems, the local frequency of the partner type is always lower than its global frequency (as is shown by Eq 3). The difference will grow smaller as the interaction range becomes large compared to the patch size (Fig C in S3 Text). Pair-approximation shows indeed that the local frequency of the partner type approaches the global frequency when the neighborhood sizes grow large (i.e. less than 1% reduction in frequency of partner type when cells have more than 100 neighbors).

The dimensionality of the system can have a strong effect on the community properties. Given the same interaction range (R), cells growing in two-dimensional sheets have fewer neighbors (∝ R2) compared to cells growing in three-dimensional structures (∝ R3). The interaction range (i.e. the distance over which molecules are exchanged) is similar in two and three dimensions because it mostly depends on the ratio of uptake and diffusion rate (see S1 Text). However, the number of cells within the interaction range varies in two and three dimensions: cells in two-dimensional systems have fewer neighbors than cells in three-dimensional systems, which results in a stronger reduction of the local frequency of the partner type.

The reduction in the local frequency of the partner type has an important consequence for the community: cells have less access to the resources they need for growth and as a result birth rates go down. To quantify this effect, we calculate the average birth rate of cells in the community; this quantity corresponds to the gross rate at which new biomass is produced and we refer to it as the community productivity. It is important to note that the net change in biomass (i.e., the change in population size) also depends on the rate of cell death/loss. In our model we assume a constant population size, so the net rate of biomass production is always zero. Biologically, this corresponds to biofilms where cell growth and loss balance each other, e.g., because cells at the edge of the biofilm are flushed away. Even in such scenarios where population sizes are constant, community productivity is still an important quantity as it measures the turn-over rate of cells, which in turn affects many relevant quantities such as the overall metabolic rate in the community. We can derive an analytical expression for the community productivity and compare it to the productivity of an equivalent community growing in well-mixed condition (see S2 Text).

For cross-feeding communities the productivity in spatial systems is always lower than that in well-mixed systems and this difference increases as interaction neighborhoods become smaller (Fig 3D). This reduction in productivity is more pronounced when the two cell types have different maximum growth rates. We call such communities asymmetric. Our model shows that the more asymmetric a cross-feeding community is, the more the community’s productivity is hindered by small neighborhood sizes, to the point that the community collapses when neighborhood sizes are very small (Fig 4A). Increasing the neighborhood size of either one (Fig 4B), or both cell types (Fig 4A), can prevent the collapse of the community. This finding shows that there is a limit to the stability of cross-feeding interactions in spatially structured communities: cross-feeding cell types with different maximum growth rates that can stably coexist in well-mixed environments might not be able to survive in spatially structured environments. For cross-feeding communities, spatial structure can thus have a detrimental effect.

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