Abstract
Quantifying the strength, sign, and origin of species interactions, along with their dependence on environmental context, is at the heart of prediction and understanding in ecological communities. Pairwise interaction models like Lotka-Volterra provide an important and flexible foundation, but notably absent is an explicit mechanism mediating interactions. Consumer-resource models incorporate mechanism and the resource landscape dependency, but describing competitive and mutualistic interactions is more ambiguous. Here, we seek to bridge this gap by modeling a coarse-grained version of a species’ true, cellular metabolism to describe resource consumption via uptake and conversion into biomass, energy, and byproducts. This approach does not require detailed chemical reaction information, but it provides a more explicit description of underlying mechanisms then pairwise interaction or consumer-resource models. Using a model system, we find that when metabolic reactions require two distinct resources we recover Liebig’s Law and multiplicative co-limitation in particular limits. In between these limits, we derive a more general phenomenological form for consumer growth rate, and we find corresponding rates of secondary metabolite production, allowing us to model competitive and non-competitive interactions (e.g., mutualism). Using the more general form, we show how secondary metabolite production can support coexistence even when two species compete for a shared resource, and we show how differences in metabolic rates change species equilibria. Building on these findings, we make the case for incorporating metabolism to update the phenomenology we use to model species interactions.
A central goal in ecology is to understand and predict the dynamics in communities of interacting species (Holt, 1977; Loreau, 2010; Vellend, 2010, 2016). Mathematical models allow us to generate and test theoretical predictions, and the development of such models leads to a hierarchy of challenges. The first challenge is determining an appropriate functional form describing species dynamics. A range of functional forms with increasing complexity has been used and each has strengths and weaknesses which are often context dependent (Holling, 1959; Abrams, 1982; DeAngelis et al., 1989; Murdoch et al., 2003; Mougi & Kondoh, 2012). Second, we need to parametrize these equations, for example by quantifying the strength and sign of species interactions in a given environmental context. While some attempts have been made to parametrize real-life systems, accurately fitting interaction strengths remains challenging in both empirical and theoretical work (Schoener, 1983; Tilman, 1987; Ives et al., 2003; Carrara et al., 2015; Terry et al., 2017; Barner et al., 2018). Finally, we may wish to determine how a change in the environmental context will modify species interactions, dynamics, and even coexistence. Integrating each of these goals will lead to the development of robust models which can predict the dynamics of complex communities, even when the environmental landscape changes within and across ecosystems.
The Lotka-Volterra equations provide an example of mathematical models which has been widely used for close to a century (Lotka, 1932; Volterra, 1926). These equations characterize species interactions in terms of the net, direct effect of one population on another’s growth rate, so that in the case of two species with abundances N1 and N2:
Here, r1 and r2 are per capita growth rates when species are rare, and the parameters aij (collectively called a community matrix) represent intra- and interspecific interactions. Empirically, it has been extremely difficult to reliably estimate these parameters (Schoener, 1983; Tilman, 1987). Even where it has been possible to infer or approximate pairwise interactions (Stein et al., 2013; Marino et al., 2013; Fisher & Mehta, 2014; Bucci et al., 2016), it may be difficult to translate the inferred interactions in different environmental contexts. These models lack an explicit description of the mechanisms mediating interactions (Abrams, 1983; Grilli et al., 2017). For example, if two species compete, it is often because consume common resources (Gause & Witt, 1935; MacArthur, 1970; Schoener, 1983). However, these models assume that resource dynamics can be safely ignored because resource dynamics are faster than consumer dynamics (MacArthur, 1970). This exposes an important context-dependence of Lotka-Volterra type equations: the strength and even the sign of a pairwise interaction may depend on what resources are present (Xiao et al., 2017). As such, landscape variation can influence species composition due to differences in competitive ability and the context dependence of species interactions (Cadotte & Tucker, 2017).
An alternate approach is to model competitive interactions as the explicit result of shared resource consumption (Grover, 1990; Tilman, 1980; Tilman et al., 1982; Litchman, 2003; Abrams, 2009). For example, in the case of two species with abundances N1 and N2 competing for a single shared resource, R, the prototypical consumer-resource model is: where ρ describes the environmental input rate of an abiotic resource, ai describes the resource uptake rates, ϵi describes the resource use efficiency, and μi are the species mortality rates. These models produce species interactions as an emergent property dependent on shared resource consumption, and so the issue of inferring species interactions is no longer quite the right question—though there is now a challenge in determining consumer feeding preferences. Assuming we can infer or otherwise estimate those preferences, one critical aspect of the environmental context is now explicitly characterized, via resource input rates like ρ. As such, species dynamics across resource landscapes can be understood better than in the case of Lotka-Volterra, where the effect of the environment is implicit (Tilman, 1977; Grover, 1990, 2011).
The more explicit mechanism gives an advantage, but there is a cost. Lotka-Volterra equations are extremely flexible and can straightforwardly incorporate a mixture of antagonistic and mutualistic interactions simply by altering the signs of entries in the community matrix, aij (Mougi & Kondoh, 2012; Allesina & Tang, 2012). But for consumer-resource models we have to be careful about how those mechanisms are formulated. Consumption can take a variety of forms depending, for example, on whether resources are substitutable or essential (Tilman, 1980), and mutualistic interactions can occur via resource production and exchange (Loreau, 2001; Freilich et al., 2011; Zelezniak et al., 2015). The latter in particular is under explored relative to consumption (Butler & O’Dwyer, 2018), and an open question is to what extent the precise form of exchange might affect community dynamics and species coexistence.
So how do we retain the advantages of consumer-resource models, but also incorporate the flexibility of models of direct species interactions? Current consumer-resource models are largely agnostic to what happens inside cells or organisms. For many systems, this approach may be valid especially when the resources (i.e., prey) belong to higher trophic levels, self-regulate, and/or closely match the stoichiometric requirements of the consumer (Sterner & Elser, 2002; Cherif & Loreau, 2007; Hall, 2009). However, most microorganisms consume abiotic resources which do not self-regulate. Second, any single resource consumed generally does not satisfy the full stoichiometric requirements. For example, heterotrophic microorganisms require organic carbon, but they still require nitrogen, phosphorus, and other resources to grow and reproduce. Because growth depends on multiple resources, dynamics may depend on a single limiting resource (e.g., Liebig’s Law of the Minimum von Liebig & Gregory, 1842; Odum, 1959) or an interaction between resources (e.g., multiplicative co-limitation Harpole et al., 2011). Likewise, the consumption and transformation of resources depends on how cells produce biomass, energy, and metabolic byproducts, and species interactions may therefore depend on metabolic rates and byproduct production. Furthermore, all of these processes may depend on environmental conditions like temperature, but it is not clear how to incorporate this information. Here, we propose that going one level deeper into cellular metabolism will allow us to generalize consumer-resource models in a meaningful way and give them the same flexibility to describe multiple interaction types as models of direct species interactions.
Results
The Metabolically Informed Consumer-Resource Model
We know a substantial amount about the internal physiology of cells, and there have been large advancements in the development of flux balance models which use biochemistry and genomics to describe (to some level of approximation) every reaction that occurs within a cell (Kauffman et al., 2003; Orth et al., 2010). More recently, these models have been applied in the context of entire microbial communities and their interactions (Embree et al., 2015; Zomorrodi & Segrè, 2016; Pacheco et al., 2018). However, we propose that including a full description of a metabolic network may not be required to develop a useful ecological model. Here, we model the internal dynamics via simplified metabolic networks, which require less knowledge of the particular species’ idiosyncrasies but still capture the major metabolic events transforming resources. Our simplified metabolic model is based on a basic fermentation reaction, homolactic fermentation, which uses glucose and phosphate and produces lactate (Fig. 1). If used strictly for energy production, one glucose and two phosphate molecules yields two molecules of lactate and generates chemical energy in the form of two ATP (Gottschalk, 1986). However, to produce biomass, some of the available glucose and phosphate must be used for anabolic metabolism—forming new biomass and maintaining cellular stoichiometry. In this reaction, the glucose is used to form new biomass and to produce chemical energy via fermentation. The energetic component results in the production of the byproduct, lactate, which is exported back into the environment (i.e., excretion); therefore, efficiency is emergent property determined by the balance between the biomass and energy production.
Here, we assume that cellular metabolism relies on the interaction of sugar and phosphate, producing new biomass and a byproduct (lactate). Given that we consider one phosphate molecule, this already simplifies the resource requirements relative to the true process. But we will see general principles emerge.
R1 here is the sugar, R2 represents a source of phosphorus, and R3 is the metabolic byproduct (lactate). We model this intracellular reaction as occurring at rate β. Next, we assume that our organism grows in a resource replete environment, such as a chemostat, and therefore resource inputs are constant. In addition, we assume that metabolic rates are not limited by any other factors and that resources do not inhibit growth. Therefore, we can balance fluxes for the internal cell densities of the three resources (labeled M1, M2 and M3). This leads to: for uptake rates k1 and k2, which can depend in an arbitrary way on external resource concentrations R1, R2 and R3, outflow (i.e., export) rates ν1, ν2 and ν3, and biomass production νbio. We make a natural assumption that export of each resource is determined by passive excretion: i.e. that να = λαMα for each metabolite, where lambdaα is a species and resource-specific constant.
We can now solve this system of polynomial equations for internal resource concentrations (see Methods). Using these equations, we built a set of generalized, co-limited consumer-resource equations using uptake rates and a simplified flux balance analysis as the only building blocks. For simplicity, we assume that resource uptake and export are passive processes. First, we focus on one species and model its uptake rate kα of resource α as Cα1Rα, where Rα is the external (environmental) concentration of this resource. Applying this assumption, we derive a general set of equations for the consumer and three resources: where ρα and ηα are inflow and outflow rates for each of the three resources, and μ1 is the mortality rate of the consumer.
We now note two limits (see Methods). First, when β is large relative to the other rates the consumer-resource equations become: and therefore we recover Liebig’s law (i.e. the net growth rate of the consumer is min(R1C11, R2C21)). Using these equations in numerical simulations, we observe that the consumer abundance saturates (Fig. 2A), and the final abundance depends on the inflow rate of the more limiting resource (Fig. 2B). As such, the model in this limit behaves similar to classical consumer-resource models. In addition, we generate novel terms for the byproduct production rate, which in this fast reaction rate limit is ≃ N1 min(R1C11, R2C21), and we find corresponding equations for the uptake and export of glucose and phosphate.
On the other hand, when β is small relative to the other rates the consumer-resource equations become:
Hence, we recover multiplicative co-limitation by the two resources (i.e., the net growth rate of the consumer is ). Using numerical simulations, we observe that consumer abundance, N1 saturates as expected but now includes a growth lag-phase (Fig. 2C). However, the final abundance depends on the inflow rate of both resources (Fig. 2D). An increase in the flux rate of either resource will yield a higher final population abundance.
In summary, from this coarse-grained representation, we recover two classic outcomes of consumer-resource theory by taking limits of the internal reaction rate β. We can also generalize these classic limits for intermediate β, in a way that is not currently used in consumer-resource models and falls neither into the category of Liebig’s Law nor multiplicative co-limitation. Finally, we find functional forms for the production rate of lactate (and excretion of other resources) in each of these limits. Our model therefore demonstrates how we can generalize the functional form of consumer-resource models by considering realistic, simplified intracellular processes.
Two Species Model
We now modify the model above to incorporate a second species. Here, both species use R2 (phosphate), but the second species, N2, uses a combination of R2 and R3 (lactate) to generate new biomass. While we are using this as a model with both competition (e.g., shared resources) and facilitation (e.g., metabolic cross feeding) interactions, it also represents the natural cross-feeding interaction between lactate producing and lactate consuming bacteria found in human and animal digestive systems (Duncan et al., 2004). These metabolic cross-feeding interactions are common in microbial systems (Mee et al., 2014; Tasoff et al., 2015; Zelezniak et al., 2015) and have industrial applications (Jiao et al., 2012). As such, our model demonstrates how competition and facilitation mediate species dynamics and coexistence conditions and can be used to understand natural and engineered microbial systems.
First, we define two distinct internal metabolic processes, one for each consumer species:
Consumer N2 may also produce a metabolic byproduct, but we have not included such a process here because we are focusing on competition for R2 and facilitation through the production of R3 by species N1. Importantly, we now have two internal reaction rates, β1 and β2. Here we focus on how these rates, both relative to each other and also to the other rates in the model, affect species coexistence. This approach demonstrates the power of modeling coarse-grained metabolic processes as the mechanism underlying species interactions. Furthermore, it allows us to explore the potential for changes in species coexistence due to metabolic (i.e., reaction rates) and landscape (i.e., inflow rates) factors.
Generalizing the approach in the previous section, we define the fluxes for an individual belonging to species N1 as: for internal concentrations M1 and M2, and uptake rates ki1. While for an individual of species N2 we have:
We also assume that all resources can be secreted from both consumers, but to simplify the model slightly we will assume that the specific export rates are equal, λ, broadly consistent with passive diffusion across sufficiently similar cell wall types. With this assumption, we can solve for internal equilibrium in both cell types (see Methods). Finally, we can put all of this together to generate a set of equations for both species and all three resources.
To focus on the effects of internal reaction rates and the resource landscape, we will further simplify our model by making a few assumptions. We will assume that the outflow rates for each resource are the same, so that ηi = η, and that the per capita mortality rates for each consumer are equal, so that μi = μ. We will also again assume that the per capita uptake rate of resource i by species j can be written as Cij Ri. We will then determine the effects of internal reaction rate by independently changing the value of β for each species. In addition, we will change the landscape conditions by exploring the inflow rates for each resource ρi. Given these assumptions, our two species consumer-resource model is:
Using numerical simulations, we model the consumer dynamics to determine species dynamics and equilibrium conditions (Fig. 4). First we consider when the internal metabolic rates, βi, are the same. When internal metabolic rates are both high, species coexist at a density determined by the shared resource inflow rate (i.e., ρ2) until the inflow rate of the unshared resource, R1, exceeds the inflow rate of the shared resource, R2. When ρ1 is greater than ρ2 (i.e., ρ1/ρ2 > 1), species N1 will outcompete species N2 for R2, and N2 will become rare (Fig. 4A). These findings expand the expectations of Liebig’s Law to two cross-feeding species and demonstrate both species dynamics and equilibrium abundances across various resource landscapes. In contrast, when internal metabolic rates are low, species relative abundances are determined by their respective required resources and both species have growth lag-phases. These findings expand the expectations of multiplicative co-limitation to two cross-feeding species. However, since species N1 will not be resource limited when ρ1 is greater than ρ2, then species N2 will maintain a higher relative abundance across wider resource landscape (Fig. 4B, see Eq. 7). Together, we find that if internal rates are the same but either high or low compared to the other rates in our model, then our results expand the expectations of Liebig’s Law and multiplicative co-limitation to the two-species system with a metabolic dependency. In addition, we find that coexistence depends on both the internal metabolic rates and the resource inflow rates even when uptake rates are the same.
Finally, we consider the dynamics and equilibria when the internal metabolic rates, βi, are the different. We find that, when the rates differ the outcome depends on which species has the higher metabolic rate. When the byproduct producer, N1, has the higher rate, then the results are similar to when both species have high internal metabolic rates (Fig. 4C). We do note, however, two important differences: 1) species N2 exhibits a growth lag-phase, and 2) both R1 and R2 are depleted as the species reach an equilibrium. However, when species N2, has the higher rate, then the coexistence conditions and high relative abundances for both species are greatly expanded. In fact, we find coexistence with moderate abundances along all inflow rates tested and the final abundances of both species are determined only by the shared resource, R2 (Fig. 4D). In addition, we find that both species exhibit growth lag-phases and that R2 and R3 are now the depleted resources. These findings highlight how variation in both the internal metabolic reaction rates and in the environmental conditions can influence species interactions and change expectations for coexistence.
Discussion
Classic formulations for pairwise interactions and consumer resource dynamics have each led to insights regarding species coexistence, community stability, population self-regulation (Schoener, 1983; Barabás et al., 2017; Allesina & Tang, 2012; Leibold & McPeek, 2006). Here, we identify trade-offs with each approach. Lotka-Volterra (and related) equations provide a flexible approach to modeling a range of interactions between species but are unable to generalize across environmental variation because they do not provide an unambiguous way to include the resource landscape. While it may be possible to modify per capita growth rates to be a function of environmental conditions (e.g., temperature) and the resource landscape, it is not clear how information would be included the species interaction terms. However, these models allow positive and negative species interactions to be explored straighforwardly. Consumer-resource models explicitly include the interaction between the resource landscape and consumers, but at the expense of introducing more explicit mechanism, and therefore more choices in the way interactions are implemented. Including positive interactions through the production of resources has led to new predictions regarding the stability of communities (Butler & O’Dwyer, 2018), indicating that incorporating resource exchange may be important for understanding the dynamics of real communities. However, we don’t know how sensitive these results may be to the precise way consumption and exchange are formulated. Here, we argue that incorporating metabolism more explicitly into consumer-resource dynamics will allow us to explain a broader range of community dynamics and natural phenomena, with less ambiguity in the functional form of interactions, and these metabolic rates may also reveal how environmental conditions like temperature which can change metabolic rates contribute to species dynamics.
In including these processes, we reformulated the classical consumer-resource model to independently include resource uptake, internal metabolic rates, and byproduct export. As such, our model is more complex than the Lotka-Volterra equations but not as complex as a full multi-species flux balance analysis. We found that when internal metabolic dynamics are included in addition to uptake two common models of resource limitation (Liebig’s Law and multiplicative co-limitation) appear in particular limits of the internal reaction rates. In this model, we also make predictions for the functional form of the production of metabolic byproducts, and we balance the requirements for growth, energy, storage, and export. We further expand our model to include a second species which uses the metabolic byproduct of the first species. In this metabolically-informed approach to consumer-resource models both species interactions (competitive and mutualistic) and resource use efficiency are emergent properties of the system. In addition, we show how internal metabolic reaction rates and the resource landscape determine species dynamics and equilibria. We find that the metabolic rates can change when resources are metabolically limiting, and therefore our model shows how how metabolic rates and the resource landscape change the interactions between cross-feeding species. We further show that, if interacting species have different metabolic rates, species and resource equilibria can change while maintaining the competitive and facilitative interactions among species. We propose that when extended more broadly, this approach will lead to mechanistic predictions for the role of positive interactions along stress gradients (Callaway & Walker, 1997; Brooker & Callaghan, 1998), the ability of species interactions to stabilize or de-stabilize communities (Butler & O’Dwyer, 2018; Allesina & Tang, 2012), and the mechanisms underlying biodiversity ecosystem function relationships (Duffy et al., 2007; Flynn et al., 2011; Cardinale et al., 2012). In short, we propose that metabolically-informed consumer-resource dynamics will provide a platform to explore the consequences of cooperative and competitive interactions across environmental contexts.
Methods
Solution for Polynomial Equations
We can solve the single-species polynomial equations (Eq. 4) for internal resource concentrations to obtain: while biomass production is given by where depends on uptake, export, and reaction rates.
Limits of the general set of equations
We now solve Eq. 5 for two limits. The first limit is where β is large relative to the other rates,
Note that we have to keep the O(1/β) terms for M1 and M2, because for at least one of the two (depending on whether k1 > k2 or k1 < k2) the O(1) term vanishes in this limit of fast reaction rate β.
The second limit is where β is small relative to the other rates,
0.1 Internal Equilibrium in Two-Species Model
For the two-species model (Eqs. 9 & 10), we assume that all resources can be secreted from both consumers, but to simplify the model slightly we will assume that the specific export rates are equal, λ, broadly consistent with passive diffusion across sufficiently similar cell wall types. Based on these assumptions, we can solve for internal equilibrium in both cell types to obtain: where, similarly to the one species case, the function .