Microbial mutualism at a distance: The role of geometry in diffusive exchanges

The exchange of diffusive metabolites is known to control the spatial patterns formed by microbial populations, as revealed by recent studies in the laboratory. However, the matrices used, such as agarose pads, lack the structured geometry of many natural microbial habitats, including in the soil or on the surfaces of plants or animals. Here we address the important question of how such geometry may control diffusive exchanges and microbial interaction. We model mathematically mutualistic interactions within a minimal unit of structure: two growing reservoirs linked by a diffusive channel through which metabolites are exchanged. The model is applied to study a synthetic mutualism, experimentally parametrized on a model algal-bacterial co-culture. Analytical and numerical solutions of the model predict conditions for the successful establishment of remote mutualisms, and how this depends, often counterintuitively, on diffusion geometry. We connect our findings to understanding complex behavior in synthetic and naturally occurring microbial communities.

Figure 1

Diffusive cross feeding at a distance. Auxotrophic microbial populations $A$ and $B$ (concentrations $\overline{a}$ and $\overline{b}$) reside in well-mixed reservoirs of equal volume $\mathrm{\Gamma }$ separated by a channel of length $L$ and cross section $\mathrm{\Sigma }$. Microbe $A$ produces a carbon source $C$, of homogeneous concentration ${\overline{c}}_a$, in its reservoir. This diffuses through the channel, forming a profile $\overline{c}(\overline{x},\overline{t})$, a function of position along the channel $\overline{x}$ and time $\overline{t}$. On reaching the reservoir where microbe $B$ resides the concentration is homogenized to ${\overline{c}}_b$. Symmetrically, the vitamin $V$ produced by microbe $B$ in its reservoir at concentration ${\overline{v}}_b$ diffuses to reservoir $A$ creating a profile $\overline{v}(\overline{x},\overline{t})$, homogenized to ${\overline{v}}_a$ in the reservoir. Here, this general model is applied to an algal-bacterial partnership.

Figure 2

Transient dynamics of a bacterial population fed through a channel that allows metabolite diffusion from a remote carbon source. The diffusive exchange geometry controls the dynamics through the nondimensional channel length $\lambda$ and reservoir equilibration length $\eta$. Model solutions predict that (a) for fixed $\lambda =3$, increasing $\eta$ delays the time of peak bacterial growth and curtails growth due to a limited carbon-source flux; (b) for fixed $\eta =10$, increasing $\lambda$ significantly delays peak growth, with an impact on the maximum bacterial concentration attained. The delay as measured by ${\tau }_{\max }$, the time of maximal growth rate, is proportional to ${\lambda }^2$ (inset). For all simulations, initial nondimensional bacterial and carbon concentration are $b_0=5\times {10}^{-4}$ and $c_a(t=0)=10$; other parameters are from Table 1.

Figure 3

Coexistence diagram illustrating the long-time fate of mutualistic populations in terms of initial concentrations. (a) At a fixed equilibration length $\eta =3$, increasing channel length $\lambda$ causes the coexistence region to shrink progressively. (b) On the other hand, the response to an increase in $\eta$ for fixed channel length $\lambda =3$ is nonmonotonic. The coexistence initially contracts, then expands, and finally contracts again. The grey lines in both plots corresponds to the membrane limit for which $\lambda \to 0$ and equilibration of metabolite concentrations between the two flasks is instantaneous. This provides the maximum possible concentration parameter space for mutualistic coexistence. The coexistence boundaries were determined by solving Eqs. (II.14) and (II.15) numerically using the parameters in Table 1 (see the Appendix). (c) Along the transect (dotted red line) in (a) corresponding to a conserved ratio of initial concentrations $b_0/a_0=20.0$, the critical initial algal concentration $a_0^c$ above which coexistence occurs is an increasing monotonic function of the length of tube $\lambda$. (d) Using the same transect in (b), the nonmonotonic behavior of the critical algal concentration $a_0^c$ with $\eta$ is clearly revealed.

Figure 4

The time taken for the populations to relax to equilibrium (crash or coexistence) depends on the geometric parameters $\lambda$ and $\eta$. Here, we plot these times for fixed initial microbial concentrations $(a_0,b_0)$ (assuming, as before, no initial metabolites within the diffusion geometry). Times for populations reaching coexistence are shown in white up-pointing triangles, and those for populations that will crash in black down-pointing triangles. (a) For fixed $\eta =3$, the relaxation time increases with $\lambda$ up to the critical value at the coexistence boundary (where it diverges). On the other side of this critical value it decreases. (b) For fixed $\lambda =3$, the dependence of the time as a function of $\eta$ shows a similar divergence when approaching a transition between extinction and survival. For the initial concentrations $(a_0,b_0)$ here chosen, two of these transitions are possible, with extinction for low and high values of $\eta$ and coexistence for intermediate values. Both panels correspond to $a_0=2\times {10}^{-2}$ and $b_0=3\times {10}^{-4}$.

Figure 5

Schematic of a diffusively coupled microbial network representing (a) a structurally and microbially heterogeneous network as a realistic representation of soil [31]; (b) a crystalline network that can be engineered in the laboratory. The nodes of this physically structured network represent reservoirs of different volumes filled with different growing microbial species diffusively exchanging metabolites via porous channels, as described in the model formulated in this work. Diffusive exchanges are parameterised by sets of geometric parameters, as such as the lengths ${\lambda }_{ij}$ of the channels connecting nodes.

Figure 6

Evolution of the concentration $c_L$, in a reservoir initially devoid of chemical, diffusively coupled to a reservoir filled with initial concentration $c_0\left(0\right)=1$. The concentration was evaluated numerically from the inverse Laplace transform of $f_L$, given in Eq. (A6). Each red curve in panels (a)–(c) corresponds to a numerical evaluation for value of the parameter $\zeta$ equal to $1,0.1$, and 0.01 respectively. Dash-dotted lines are the corresponding nondimensional versions of the approximation of $c_L$ as a saturating exponential as given in Eq. (A2), while dashed lines correspond to the linear approximation $c_L=\zeta t$. Note the change of scale of the time axis for different values of $\zeta$, where time itself has been rescaled by $L^2/D$.

Figure 7

Experimental results and theoretical fits on growth of co-cultures. Rows (a)–(f) display results from six independent growth experiments for M. loti and L. rostrata co-cultures, for different starting values and ratios of the two species. For each experiment, from left to right, the panels show the algal concentration $\overline{a}$, the bacterial concentration $\overline{b}$, and, when data are available, the vitamin concentration $\overline{v}$. Continuous thick lines show the average value over a set of replicates, with the interval of $+/-$ one standard deviation shown as a shaded area. The fits from the model with parameters from Table 1 are shown with dashed black lines. Number of replicates per experiment from (a) to (f) is $n=6$, 3, 5, 5, 4, and 4. Large downward shaded areas represent on this logarithmic scale time points for which standard deviation is comparable to the mean.

Figure 8

Chambers for proof-of-principle experiments. (a) Sketch of the platform holding the modified flasks during assembly. (b) Sketch of the diffusive plug filled with polyacrylamide (PAM) gel, used to connect the two flasks in experiments of mutualism at a distance.

Figure 9

Example of oscillations of (a) concentrations of cells and (b) concentration of metabolites during the time evolution of a co-culture at a distance system before convergence. Initial parameters are close to the boundary between survival and extinction ($\lambda =2,\eta =3,a_0=2\times {10}^{-2},b_0=3\times {10}^{-4}$, and no initial nutrients).