Abstract
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 parameterised 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 counterintutively, on diffusion geometry. We connect our findings to understanding complex behaviour in synthetic and naturally occurring microbial communities.
1 Introduction
Microorganisms display a broad spectrum of interactions that determine the behaviour of microbial communities (Abreu and Taga 2016). Predicting this behaviour is a fundamental challenge in current microbial ecology (Widder et al. 2016). A wealth of experimental data on microbial community structure and dynamics is now available from ‘omics’ approaches (Cooper and Smith 2015, Widder et al. 2016). These, however, need to be complemented by lab-based studies of synthetic consortia and mathematical models to reach a mechanistic understanding of microbial dynamics (Widder et al. 2016, Abreu and Taga 2016). The study of mutualistic interactions between microbial populations is an active area of current resesarch in microbial ecology. Recent experimental studies have investigated synthetic mutualisms between microbes across the kindoms of life. These include strains of enteric bacteria (Harcombe et al. 2014, McCully et al. 2017, LaSarre et al. 2017) and yeast (Allen et al. 2013) engineered to be mutualistic, and synthetic consortia combining wild type microbial species, such as bacterial tricultures (Kim et al. 2008), mixed cultures of algae and fungi (Hom and Murray 2014), and algae and bacteria (Croft et al. 2005, Kazamia et al. 2012, Wang et al. 2014, Segev et al. 2016).
Mutualistic interactions are conventionally modelled using Lotka-Volterra type models, with positive interaction coefficients (Murray 1989). Linear mutualistic Lotka-Volterra models are known to display unrealistic unbounded growth (Murray 1989), but logistic versions have been used to study demographically open mutualistic populations (Thompson et al. 2006), transitions between interspecies interactions (Holland and DeAngelis 2009, 2010), and the steady state dynamics of algalbacterial co-cultures (Grant et al. 2014). Since the pioneering work of May (1973), such models have also been fruitfully employed to describe mutualistic interactions in network models of communities (Okuyama and Holland 2008). In such models the interaction coefficients coupling species together define an interaction or community matrix (for mutualistic interactions the coefficients are positive and symmetric). Significant shortcomings of Lotka-Volterra models have recently been pointed out. For example, when species interact by exchanging metabolites, a metabolite-explicit model does not in general map onto a Lotka-Volterra implicit model (Momeni et al. 2017). Only in special instances does the microbial Lotka-Volterra form provide a good description of the microbial dynamics, e.g. when a fast equilibration approximation holds (Hoek et al. 2016). Resource-explicit models of bacterial mutualisms compare well with experiments in which mutualists are well-mixed (Lee et al. 1976, Wang et al. 2007, LaSarre et al. 2017, McCully et al. 2017). Explicitly modelling resources is critical when studying spatially structured mutualistic systems (not well-mixed) whose interactions are contolled by metabolite dynamics and their spatial transport.
Recent studies have considered spatial aspects of mutualistic and cooperative microbial interactions. Simulations using flux balance analysis (FBA) successfully predict the spatial growth on agar of colonies of synthetically mutualistic enteric bacteria (Harcombe et al. 2014). The FBA approach requires explicit knowledge of every known metabolic biochemical pathway in each mutualistic species, restricting its applicability to mutualisms between metabolically well-characterised organisms. Spatial effects on cheating (Momeni et al. 2013) and genetic drift (Müller et al. 2014) observed in yeast colonies growing on agarose pads have also been modelled explicitly. In these models, coupled cells and nutrients diffusing in two dimensions are simulated to predict how nutrient-mediated interactions control spatial heterogeneity and survival of the populations. In general, interactions have been shown to control the spatial structure of laboratory biofilm communities (Nadell et al. 2016). However, the homogenous environment of nutrient agarose or laboratory biofilm substrates do not possess the intrinsic geometric or topological structure of natural microbial environments, such as the porous matrix of soil or microfluidic analogues (Coyte et al. 2017). Mutualistic microbial dynamics have not thus far been studied in such structured environments, to the best of our knowledge.
Here, we study a model of mutualistic microbial species in a simple geometry representing a minimal unit for a structured environment: populations growing in spatially separated reservoirs, metabolically linked by a channel. As soil can be approximated at the microbial scale as a physical network of growth chambers linked by channels (Pérez-Reche et al. 2012), this model provides the basis for the description of natural complex porous networks or artificial regular networks (see Figure 1) and the microbial interactions therein. The model is generally applicable to auxotrophs cross-feeding remotely. We apply it to make predictions for the dynamics of mutualistic populations of algae and bacteria exchanging vitamin B12 and a carbon source (Kazamia et al. 2012). Our predictions provide new insights into the behaviour of microbial communities residing in structured geometries, both within synthetic consortia in the laboratory and environmental microbial communities.
2 Model
The model describes two populations of mutualistic microbial species, A and B, interacting at a distance. The mutualistic interactions are predicated on auxotrophy: A requires metabolite V (for “vitamin”), excreted by B; conversely B requires metabolite C (for “carbon”), excreted by A. In formulating the problem we shall first use variables with an overbar to denote dimensional quantities (concentrations, time, space), reserving symbols without typographical modification for appropriately rescaled variables. Populations of A and B, with densities and respectively, reside in two well-mixed reservoirs, of equal volume Γ. These are spatially separated, but connected by a cylindrical channel (length L, cross-sectional area Σ), as in Figure 2. The channel is impervious to cells, but porous to metabolite exchange by diffusion. Population A produces metabolite C with local concentration , which diffuses out of the reservoir and into the channel at (with denoting the position along the channel axis), where it develops a spatial profile and eventually reaches the other reservoir at , where its concentration is . Symmetrically, metabolite V produced by B with concentration , diffuses out at giving , feeding the other reservoir at , generating a concentration .
We first consider dynamics within the channel connecting the reservoirs, within which metabolites obey one-dimensional diffusion equations, with Ds the diffusion coefficients for metabolite S = C or V. The boundary conditions to (2.1) obtained from continuity at the channel-reservoir interface are: . Clearly, one characteristic time scale of the problem is set by diffusive equilibration along the length of the channel, where we anticipate that the diffusion constants of both metabolite species are similar. From Fick’s law, the flux Js (molecules area−1 time−1) of metabolite species S (C or V) entering, say, the left reservoir from the channel is
The rate such molecules enter the reservoir is , and with instantaneous homogenisation there, the rate of change of the reservoir concentration is . The characteristic length will play an important role in the model. If is a typical difference in concentration of S between the two reservoirs, then the typical gradient within the channel is , giving rise, by the arguments above, to an associated rate of change of reservoir concentration scaling as , from which we can identify a characteristic equilibration time
We define the ratio of equilibration and diffusive time scales to be
The regime ζ ≪ 1 is that of fast establishment of the linear concentration profile in the tube relative to changes of concentrations in the reservoirs, while for ζ ≥ 1 the transients within the channel are on comparable time scales to that for changes in the reservoirs. Semi-analytical solutions to the problem of chemical diffusion between two connected reservoirs further demonstrate the existence of these two regimes and the role of the previously identified timescales (see Supporting Information).
We now turn to the population dynamics within the reservoirs, in which we explicitly assume that algae reside in reservoir A and bacteria in B, and that vitamin B12 and carbon are exchanged. The dynamics obey the ordinary differential equations where is the flux of metabolite S = C or V entering the right reservoir. In equations (2.7a) we model cell growth as logistic, with maximum growth rate µi and carrying capacity Ki for species i = A or B. Growth rates are limited by the abundance of the required metabolites. This is modelled using Monod factors (Monod 1949), e.g., for C, , where Kc is the half-saturation constant (and symmetrically for V). Linear death terms, with mortality rates δi for i = A or B, ensure exponential negative growth in the absence of the limiting metabolites. Equations (2.7b) describe the dynamics of metabolite C. This is produced by species A in proportion to its concentration with a rate pc, and diffuses out at 0. In the other reservoir, C is taken up by B. The uptake is assumed proportional to the cell growth rate, the proportionality constant is 1/Yb, where Yb is the yield coefficient (how much metabolite C results in a given concentration of species B). Equations (2.7c) describe the V dynamics, which are completely symmetric to the C dynamics. Although inspired by bacterial-algal symbiosis, it is clear that the structure of these dynamics is quite broadly applicable to mutualistic systems in general.
Identifying the key model parameters
In order to access the general dynamics of remotely cross-feeding monocultures, we nondimensionalise equations (2.7). Because our focus is on the impact of geometry on the biological processes, we choose a scheme accordingly. First, normalize the bacterial and algal concentrations by their respective carrying capacities, the organic carbon and vitamin concentrations by their respective half-saturation concentrations, rescale time by the bacterial growth rate, and rescale space by the length scale of organic carbon diffusion on the time scale of bacterial growth, defining
The ratios of algal and bacterial growth rates and of their diffusion constants, are two additional parameters. With now three characteristic lengths in the problem (L, l, lb) one can form two independent dimensionless ratios. These can be taken to be so that the parameter ζ, defined previously in Eq. 2.6, is ζ = λ/η.
There are three pairs of parameters remaining which capture the relative strength of cellular death, uptake and production in bacteria and algae respectively. They are: the ratios of death rate to maximum growth rate of bacteria and algae, which define mortality parameters which must be less than 1 for any population increase to occur; and finally, for both carbon and vitamin, the ratios of the typical uptake rate to the typical rate of change define the uptake parameters for both carbon and vitamin, the ratios of the typical production rate to the typical rate of change define the production strengths With these rescalings, the dimensionless evolution equations are where now the dimensionless fluxes are . These equations are to be solved together with the diffusion equations for c and v on the interval x ∈ [0, λ], ensuring continuity of fluxes and concentrations at the ends of the tube. Equations (2.14) were solved numerically to explore the role of diffusive geometry on mutualistic coexistence. We used the nondimensional parameters shown in Table 1, corresponding to the mutualistic association between Lobomonas rostrata, a B12-requiring green alga, and Mesorhizobium loti, a B12-producing soil bacterium (Kazamia et al. 2012). Parameter values were obtained by fitting growth and vitamin B12 assay data from co-culture experiments with this synthetic mutualism (see Supporting Information and Figure A.2).
Before discussing the results from numerical solutions of the dynamical system of our model, we note that it exhibits a set of fixed points for which the cell concentrations vanish (a = b = 0) at any combination of residual metabolite concentrations. More interestingly, a single positive fixed point exists given by equations (A.7) below under the conditions of equations (A.8) (see Supporting Information). This positive fixed point corresponds to an established co-culture at a distance.
Feeding on a distant passive source
Before considering the fully coupled system dynamics, we consider the case of a single auxotrophic species B, concentration b, residing in a reservoir initially free of a growth-limiting metabolite coupled by the channel (also initially nutrient-free) to a strong source of the metabolite with initial concentration . This source consists of a reservoir filled with limiting metabolite. The long time steady-state for the model is always extinction of B once it has exhausted the remote resource. However, separation of the microbial population from the source modifies the transient population dynamics. Recalling the nondimensional channel length and equilibration length , we can define the nondimensional timescales tdiff = λ2 and teq = λη as the ratios between the typical times of diffusion and of equilibration between reservoirs, and the biological growth timescale τb = 1/µb. These ratios gauge the relative rates of diffusion/equilibration and growth. We require teq and tdiff ~ 1 for diffusion to transport metabolites to species B, stimulating its growth.
We have solved the remotely-fed single microbe limit of the model (see Supporting Information) to predict the dynamics of the rhizobial bacterium Mesorhizobium loti fed from a remote glycerol carbon source. Figure 3 shows the transient growth dynamics in the regime for which both geometric parameters λ and η impact the dynamics. We first consider the effect of diffusive reservoir equilibration, quantified by η for a fixed channel length λ. For large η, teq is large: diffusive equilibration in the reservoir is much slower than growth. Thus, the instantaneous flux from the carbon source reservoir to the bacterial reservoir is below what the bacteria need to grow to carrying capacity. As a result, increasing η decreases the value of the peak bacterial concentration (preceding the inevitable decay), as well as delaying the onset of growth (Figure 3a). Next we fix η and vary λ. Since the diffusive timescale scales like λ2, increasing λ progressively delays the onset of bacterial growth (Figure 3b inset). Large λ values also correspond to weaker carbon source gradients across the tube, and thus a ‘slow-release’ nutrient flux. Consequently, a less concentrated population can be sustained for longer by the remote source (Figure 3b). The passive source case we have just considered demonstrates the critical role played by both geometric parameters λ and η in setting the timescale of transients, but also the peak microbial numbers achievable on a finite resource.
Remotely cross-feeding populations
Next, we consider auxotrophic populations in separate reservoirs, exchanging limiting metabolites through a connecting channel. As mentioned earlier, we apply the model to an algal-bacterial system, obtaining our parameters from experiments where the phototrophic alga L. rostrata, auxotrophic for vitamin B12, is grown in co-culture with the heterotrophic bacterium M. loti. The algal and bacterial populations in their reservoirs have initial concentrations, a0 and b0, respectively. Neither carbon source nor vitamin (the limiting metabolites) are initially present in the reservoirs and channel. The coexistence diagrams in Figure 4a,b show what values in the initial concentration parameter space give rise to long-term mutualistic coexistence or a population crash due to metabolite deprivation. These fates are the possible fixed points of our model, which we shall also refer to as model equilibria. Figure 4a displays the boundary between these two regions for different values of the channel length λ for a fixed value of the equilibration length η. In Figure 4b crash-coexistence boundaries are instead shown for different equilibration lengths η at fixed λ. Also shown on both diagrams is the membrane limit (bottom-left grey line). In this limit the distance between reservoirs vanishes (λ → 0) and they are simply separated by a membrane impervious to cells, as has been demonstrated experimentally in co-culturing/metabolomic experiments (Paul et al. 2013). We assume instantaneous equilibration of metabolite concentrations across the membrane in this limit. It is thus an ideal case in which exchanges are not limited by diffusion dynamics along the tube nor by the geometry of the problem, and as such represents an interesting common reference case to understand the impact of both the channel length λ and the equilibration length η.
We see that increasing the channel length has the effect of pushing the crash-coexistence boundary toward higher initial microbial concentrations (Figure 4a,c). Coexistence is achieved in the membrane limit for initial concentrations lower than those for finite λ. The boundary between crash and coexistence regions shifts quantitatively with λ, but does not change significantly qualitatively. Its shape is revealing: if the initial concentration of bacteria b0 is not too large, coexistence depends weakly on b0, and very strongly on the initial algal concentration a0. For low enough bacterial concentrations, the smallest critical initial algal concentration for which coexistence will occur increases with λ. These features are reasonable considering that there is a diffusive delay in the metabolite exchange between reservoirs: if the delay is too long, auxotrophs will difficultly recover in the absence of a limiting nutrient. However, we note that the model does not predict any critical length above which recovery is impossible: longer separations will simply restrict the establishment of co-existence to cases with very high initial populations.
The effect of the reservoir equilibration length η on the coexistence diagrams is more subtle. Recall η is the nondimensional ratio of growing volume to metabolite exchange area, which controls diffusive equilibration in the reservoirs. For small η, the crash-coexistence boundary sits above the membrane limit boundary toward higher initial concentrations. This boundary is then pushed toward lower initial concentrations for intermediate values of η while still sitting above the membrane limit (as expected given that the membrane limit corresponds to the ideal case of instantaneous equilibration for no separation length), before raising to higher initial values for high values of η (Figure 4b,d). The general shape of the boundary is preserved for all η. To understand the nonmonotonic dependence of the boundary shift with η, we note η/λ is the reservoir/channel volume ratio. Thus, with λ fixed, changing η takes the populations through three regimes: i) the reservoir volume is small compared to that of the channel, η/λ ≪ 1; ii) the volumes are the same size, η/λ ~ 1; iii) the channel volume is smaller than that of the reservoir, η/λ ≫ 1. In regime i), the equilibration time teq = λη is small, but a large channel volume relative to the reservoirs dilutes any metabolite produced, making metabolites inaccessible to the microbial partner and preventing co-existence. In regime iii), the relative channel volume is small, but co-existence is impeded due to the long equilibration time teq ≫ 1, which slows down significant metabolite exchanges between reservoirs. Finally, in regime ii), where reservoirs and channel have similar volume and teq ~ 1, mutualistic coexistence is favoured.
Aside from the co-existence or crash fixed points just discussed, we can use the model to analyse the transient dynamics leading to these equilibria. In particular, it is illuminating to evaluate the relaxation time taken for remote populations to reach the fixed points for a given initial microbial concentration in reservoirs assumed initially devoid of metabolites, as previously. Numerical solutions of the model equations show that this time varies as λ is increased across the co-existence/crash boundary for given η, as shown in Figure 5a. It is clear that the time to relax to the equilibrium rises sharply on either side of the critical λ at the boundary. This slow relaxation for λ values close to the bifurcation between extinction or co-existence is accompanied by oscillatory transients (see Figure A.4). Similar considerations apply to the dependence of this time on the equilibration length η for a given λ, within that case there is the possibility of two boundaries between extinction and survival, see figure 5b. We thus predict a complex behavior of the time needed to reach steady-state in such connected mutualistic systems, with the potential for slow relaxation if geometrical parameters are close to critical values between extinction and co-existence.
Interestingly, the algal and bacterial concentration fixed points, a*, b* respectively, are independent of λ and η, as already mentioned (see equations (A.7)). Larger separation (increasing λ) or weaker diffusive coupling to the reservoirs (increasing η) increases delays in chemical exchanges and reduces the extent of the mutualistic co-existence region. However, these geometric changes do not alter the microbial concentration fixed points, which have the same values as in the membrane limit: high densities of mutualistic microbes can be achieved even with weak or slow diffusive coupling. This equilibration is possible thanks to supply of metabolites (whose concentrations are also geometry-independent, see equations (A.7)) from the partner reservoir. A sufficiently large metabolite gradient across the channel is required to support the equilibrium metabolite and cell concentrations. Indeed, the model predicts an increase in the metabolite concentration at the production reservoir. For example, if the equilibrium concentration of vitamin B12 in the algal reservoir is , then at the bacterial reservoir we predict , where the function f can be obtained by comparison with equation (A.7). The same applies for carbon. This metabolite enrichment is an interesting prediction of the model. The concentration excess at the production reservoir is linear in both separation λ and equilibration length η: two parameters with which enrichment could be experimentally controlled. As an example, for the L. rostrata and M. loti mutualism using λ = 1.25 and η = 2 (all other parameters as before) our model predicts a sevenfold enrichment of vitamin B12 in the bacterial reservoir compared to the algal side.
3 Discussion
Microbial populations often interact by diffusive exchange of metabolites in structured environments, such as the porous matrix of soil. Metabolite diffusion is known to play an important role in determining microbial dynamics in unstructured environments (Allen et al. 2013, Momeni et al. 2013, Harcombe et al. 2014, Hom and Murray 2014, Nadell et al. 2016). Current models of microbial interactions, however, do not explicitly model diffusive transport in geometrically confining habitats. A recent theoretical study has investigated microbial invasion in soil networks (Pérez-Reche et al. 2012), but interactions were modelled stochastically, without considering diffusive exchanges. How the geometry of diffusive exchanges constrains microbial interactions remains an important open question. We have addressed this here by modelling a minimal geometrical unit of microbial interaction: two mutualistc populations in finite volume reservoirs linked by a diffusive channel. The model was solved to predict the diffusively mediated interactions of mutualitistic algae and bacteria, whose dynamics in co-culture have been experimentally characterised (Kazamia et al. 2012). Two key geometrical parameters control the diffusive exchange of metabolites between the populations: the separation λ (the nondimensional channel length) and the equilibration length η (the nondimensional ratio of growing volume to metabolite exchange area). Model solutions allow prediction of whether initial concentrations of algae and bacteria will result in mutualistic coexistence or population crash (the model equilibria) for given values of the geometrical parameters λ and η. In particular, we can draw the boundary between regions exhibiting these two equilibria for given initial microbial concentration, and predict how this boundary shifts when the values of the geometrical parameters are changed.
The model makes several interesting predictions. For instance, coexistence between mutualistic partners can be achieved only if the numbers of one or both partners are abundant; low initial numbers will lead to a crash. This feature is qualitatively independent of diffusive geometry (λ or η), like the shape of the coexistence boundary itself (approximately flat for a broad range of bacterial concentrations, falling very rapidly thereafter, see figure 4). It has an intuitive explanation: an initially high concentration of one of the two species will produce a large initial amount of metabolite, which allows the partner species to grow and recover, even from initially very low numbers. A more surprising result is that mutualistic populations at a distance can achieve as high a steady concentration as in a mixed environment. The effect of the diffusive geometry is only to modify the transient dynamics and raise the initial cell concentration values required to avoid a crash. The fact that, given enough time, separated cross-feeding mutualists might reach as high numbers as populations in proximity is a counterintuitive of great potential significance for microbial ecology. This contrasts with the case of a population feeding from a distant passive resource (figure 3), for which maximum achievable concentrations do depend strongly on geometric coupling.
A final prediction of the model to highlight is the nonmonotonic dependence of the boundary position as the equilibration length η is varied. As one might expect, increasing the channel length λ (at fixed equilibration length η and bacterial concentration b0) increases the critical concentration of algae that will support co-existence with bacteria. On the other hand (for fixed λ and b0) the critical algal concentration varies nonmonotonically, falling and then rising again with increasing η. The dependence on λ is intuitive: separating the partners further increases a diffusive delay, which we recall scales like λ2, so that more algae are required to support coexistence at a distance. The nonmonotonic behaviour with η is less obvious. It results from a dilution of metabolites in the volume of the channel for low values of η, requiring higher initial densities for successful coexistence, and from weak fluxes of metabolites into the homogenisation volume when η is large. With respect to these two extremes, coexistence is more easily achieved at intermediate values of η. This is another counterintuitive prediction, which highlights the value of explicitly accounting for diffusive transport in modelling mutualistic interactions.
Our findings have implications for the microbial ecology of synthetic consortia. This is an active area of investigation, with several recent studies on microbial mutualisms (Kim et al. 2008, Kazamia et al. 2012, Allen et al. 2013, Harcombe et al. 2014, Hom and Murray 2014, Wang et al. 2014, Segev et al. 2016, McCully et al. 2017, LaSarre et al. 2017). None thus far have addressed the role of diffusive geometry on these interactions, which could test the predictions of our model. A preliminary experiment in which batch cultures of algae and bacteria grow linked by a channel allowing metabolite diffusion (filled with a hydrogel to prevent cross-contamination) demonstrates the possibility of establishing remote mutualisms, see Supporting Information. Further, it provides preliminary confirmation that vitamins accumulate in the B12 producer (bacteria) flask, as predicted by our model (equation (A.7)). The experiment provides a ‘proof of concept’ and a blueprint for further experiments using our connected flask set-up. These should explore how the population behaviour varies with the geometrical parameters, and if the stark predictions of the model, such as the nonmonotonicity of the crash-coexitence boundary with η, are borne out experimentally. Alternatively, experiments using diffusively coupled microfluidic chambers [e.g. as in the work of Kim et al. (2008)], could be used, noting that modifications would be necessary to account for stochastic effects associated with the small cell numbers in such systems (Khatri et al. 2012). As well as being tested, the model could be used to describe other synthetic consortia in which populations also interact diffusively across porous hydrogels (Kazamia et al. 2012, Harcombe et al. 2014) or microfluidic structures (Kim et al. 2008). It is straightforward to extend the model to account for two or three-dimensional diffusive exchanges appropriate to these systems.
The present model may also provide the foundation for a physical description of microbial networks, e.g. consortia for cooperative biosynthesis (Hom et al. 2015, Cavaliere et al. 2017) or microbial communities in soil, or spatially coupled biofilms (Liu et al. 2017). Indeed, as mentioned earlier, at the microbial scale, soil can be approximated as a physical network of growth chambers linked by channels (Pérez-Reche et al. 2012). In establishing the key geometric parameters that govern the most elementary unit in a network, namely two diffusively linked nodes (reservoirs), the present work provides a basis for describing population dynamics in a two- or three-dimensional network of coupled nodes (Figure 1). It is left to future work to take up the significant challenge of studying such networks, particularly when there is inhomogeneity in the diffusive couplings and stochasticity in the populations themselves. This view of microbial networks centering on the physics of diffusion could also help refine interaction matrix models of microbial communities and extend them beyond contact interactions (Mathiesen et al. 2011). An interesting possibility is that interaction networks could be simplified by constraints deriving from diffusion geometry.
Aside from the microbial networks mentioned above, the model may also be a relevant interpretative tool to understand the behaviour of structured environmental communities with diffusive exchanges, such as river biofilms (Battin et al. 2016) or sediment layers (Pagaling et al. 2014). Moreover, knowledge of the mechanisms for metabolite exchange between spatially separated organisms is important to gain insight into how such communities initiate in the natural environment, and the drivers and constraints on the evolution of mutualisms within them (Kazamia et al. 2016).
Acknowledgments
We thank J. Kotar and R. Bowman for discussions. We thank the Cavendish and G. K. Batchelor Laboratory workshops for assistance, in particular D. Page-Croft. F.J. Peaudecerf gratefully acknowledges support from Mines ParisTech and from a Raymond and Beverly Sackler Scholarship. O.A. Croze, M.A. Bees and A.G. Smith gratefully acknowledge support from the Engineering and Physical Sciences Research Council (EP/J004847/1). O.A. Croze also acknowledges support from a Royal Society Research Grant and the Winton Programme for the Physics of Sustainability. R.E. Goldstein acknowledges support from an EPSRC Established Career Fellowship (EP/M017982/1) and the Schlumberger Chair Fund. V. Bhardwaj was in receipt of a studentship from the Gates Cambridge Trust. F. Bunbury is in receipt of a studentship from the UK Biotechnology and Biological Sciences Research Council (BBSRC) Doctoral Training Partnership.