On the application of network theory to arbuscular mycorrhizal fungi–plant interactions: the importance of basic assumptions
Ecologists working at the interface between fungal and plant ecology have been puzzling over the qualitative and quantitative nature of the arbuscular mycorrhizal (AM) association for a long time (Fitter, 2006; Kiers et al., 2011). An important advance has been the use of network theory, which has been proposed in two independent articles in New Phytologist very recently (Chagnon et al., 2012; Montesinos-Navarro et al., 2012). Both Chagnon et al. (2012) and Montesinos-Navarro et al. (2012) argued that the community matrices obtained by fungal ecologists to address the structure of AM fungal assemblages in plant roots can actually be viewed as bipartite networks. In this kind of network, plant and fungal taxa represent the nodes, and patterns of fungal occurrences are the links connecting the two sets (fungi vs plants or vice versa) of nodes. In practice, the presence–absence matrix coding for fungal occurrences across plants can be viewed as the interaction matrix that describes the topology of the dynamical network linking plants and AM fungi. Following the assumption that the AM fungi–plant species co-occurrence matrix is an interaction matrix, Chagnon et al. (2012) and Montesinos-Navarro et al. (2012) applied network analysis to this matrix and calculated key metrics such as nestedness and modularity. The former quantifies the extent to which smaller assemblages (e.g. fungi associated with a particular plant species) are a subset of larger ones (e.g. fungi associated with the plant species of a forest) and the latter describes clustering patterns in which certain nodes connect more densely with each other than with other nodes in the network (Guimera & Nunes Amaral, 2005; Ulrich et al., 2009). If the co-occurrence matrix used to calculate these metrics is an interaction matrix, then nonrandom patterns in these structural elements of the network imply specific functional consequences. For example, as clearly explained by Chagnon et al. (2012) and the references they cited, in the case of AM fungi–plant interactions, strong nestedness means strong specialization, whereas strong modularity could imply that species can coexist thanks to mechanisms such as complementarity of functional traits.
These arguments and the analyses employed by both Chagnon et al. (2012) and Montesinos-Navarro et al. (2012) are well grounded in network theory and, as such, represent a valuable contribution. However, a fundamental point remains to be explored: is the co-occurrence matrix analyzed by these authors really an interaction matrix? In this Letter, we address this essential point, as we believe that the state of the art in AM fungi–plant research makes it challenging to assume that ‘co-occurrence = interaction’.
First, we recall some basic theory to show what the interaction matrix is supposed to summarize. Second, we point out some basic facts that imply that co-occurrence ≠ interaction. Finally, we offer some constructive suggestions to contribute to network theory as applied to the AM fungi–plant interactions.
Co-occurrence and interaction matrices: what’s the difference?
Bipartite networks are a particular type of network, and networks are objects that can be described mathematically by graph theory (Chartrand, 1985; Cartozo et al., 2006; Pascual & Dunne, 2006). In ecology, this theory has been classically applied to food webs (Pimm, 2002) and, only more recently, to mutualistic systems (Jordano et al., 2003; Jacquemyn et al., 2011). The idea behind ecological networks is that the main players of the network, which in terms of graphs are represented by the vertices or nodes, are dynamically linked. Thus, the network represents just the topology of the system, more or less in the same way that the human skeleton is just a coarse, although structurally essential, representation of the human body. Crucially, the interaction matrix has a strongly dynamical nature for at least two different but mutually reinforcing reasons. In the short and medium term, it determines and (given a quantitative model) compactly describes the dynamics of an ecological system (typically, but not only, population dynamics). Over a longer timescale, it must also be interpreted as the outcome of the coevolution of species in the ecosystem considered, that is, as the result of many events involving speciation, extinction, and immigration of species. In other words, the interaction matrix determines, but is also (on a longer timescale) determined by, ecosystem dynamics (Caldarelli et al., 1998; Drossel et al., 2001). The first implication is usually made explicit in dynamic models such as a system of generalized Lotka–Volterra equations, whereby one can represent a classical food web model (Case, 2000):
where V is the state variable describing each and every of the S vertices (in our case species) in the graph or food web, ri is a constant that relates to the intrinsic dynamics of each vertex and the coefficients aij are the vertices–species interaction terms. This being a system of linear equations, we can write it in matrix form so that all possible interaction terms can be described by what some ecologists call the community matrix A and others call the interaction matrix (May, 1973a; Case, 2000):
Several other choices for the dynamics are possible, involving more complicated functional responses (Caldarelli et al., 1998; Drossel et al., 2001). In any case, the interaction matrix will strongly affect the outcomes of population dynamics (May, 1973a; Allesina & Tang, 2012). If the framework is expanded in order to consider long-term speciation and extinction events, the interaction matrix will be a key element involved in the determination of such events, but its structure will also change and adapt in response to the evolutionary dynamics itself (this is the second implication). Indeed, it has been shown that many properties of real food webs have a consistent interpretation as signatures of this continuous feedback between dynamics and topology (Caldarelli et al., 1998; Drossel et al., 2001). In all these cases, the matrix A is, in general, a weighted matrix. This matrix, when binarized, represents the purely topological information; that is to say, it describes ‘who eats whom’ or, more generally, ‘who interacts with whom’. If quantitative pieces of information are known, then the network dynamics can be described in full, for example in terms of the efficiency of resource distribution (Pascual & Dunne, 2006). However, topology in its own right already contains valuable pieces of information in terms of the dynamical stability of the system (May, 1973b) or allometric scaling related to the delivery of resources and maintenance of the network (Banavar et al., 1999; Garlaschelli et al., 2003). In this sense, network analysis can be applied to binary matrices such as those analyzed by Chagnon et al. (2012) and Montesinos-Navarro et al. (2012). However, a fundamental premise is that each and every link represented in matrix A must have been demonstrated to exist in nature, otherwise the dynamical system of Eqns 1, 2 would not be a proper model of the network under investigation. In this sense co-occurrence by no means implies interaction.
That co-occurrence does not imply interaction is a well known fact in community ecology. Indeed, nestedness analysis is nothing new in community ecology, the concept dating back to the late 1930s, as recently reviewed in Ulrich et al. (2009). Many mechanisms could underlie nestedness, which does not strictly apply to networks. For example, one of the very first explanations of this pattern in island biogeography was in terms of immigration and extinction dynamics (see reviews in Ulrich et al., 2009), which are usually very weakly related to the dynamics assumed to take place in the establishment and maintenance of ecological networks. In the case of island biogeography or, more generally, community ecology, null models based on the co-occurrence matrix are a perfect tool to test for the assembly rules that might have caused the observed matrix to diverge from randomly generated matrices. Nestedness certainly is one of the properties of the co-occurrence matrix that might be analyzed in this framework. Nevertheless, the property analyzed being the same, if one makes the further assumption that the co-occurrence matrix is an interaction matrix then one can also infer functional/interaction properties from structural ones, as in fact done in Chagnon et al. (2012) and Montesinos-Navarro et al. (2012). Thus our point is that these inferences can actually be unwarranted if the main assumption ‘co-occurrence = interaction’ is not carefully verified and demonstrated for each element of matrix A.
Is an AM fungi–plant co-occurrence matrix an interaction matrix?
Chagnon et al. (2012), in the last paragraph of their paper, point out some of the limitations of their analysis, making it clear that there is no quantitative evidence that the frequency at which the interactions shown in their matrix A take place is a proxy for the strength of the interaction. Here we ask whether it is even correct to say ‘the frequency at which interactions take place’. For example, in systems such as pollinators (Vázquez et al., 2005), the frequency of an interaction is the frequency of a biological relationship that is certainly taking place, because pollinators are objectively seen to convey pollen. Instead, in the case of the matrices analyzed in Chagnon et al. (2012) and Montesinos-Navarro et al. (2012), the only certain fact is that a certain AM fungal taxon occurs within the root of a certain plant species. As AM fungi are obligate biotrophs of plants and given the data quality associated with next-generation sequencing techniques, the authors made the conceptual step that occurrence of fungi in plant roots must signify an interaction. Here we want to show the caveats of this assumption.
First and foremost, there is a fundamental problem with the definition of the vertices of the network from the fungal side. In fact, molecular data provide fungal ecologists with operational taxonomic units (OTUs) and it is not clear to what extent OTUs are a good proxy for biological interacting units (species or population in this case); nor is there a general consensus on the phylogenetic scale (i.e. cutoff % molecular dissimilarity used to define molecular OTUs; Powell et al., 2011) at which OTUs should be defined, as also clearly pointed out by Montesinos-Navarro et al. (2012; see their Table 2). This state of the art implies that the real nature of the links in matrix A is actually unknown, as even the definition of the vertices making up the network is fuzzy. Interestingly, this fuzziness might be a structural element of fungal biology and brings the main point back to the problem of properly defining individuals and populations in this group of organisms (Mikkelsen et al., 2008; Rosendahl, 2008; Caruso et al., 2012): the same taxonomic unit, be it an OTU or a classical taxon such as Glomus, might be present in two different plant species, but this by no means implies that the two plant species are interconnected: simply, it is not known whether in the real physical network the fungal units must be viewed as either very pervasive clones or a myriad unconnected entities that originate from different propagules. In addition to that, it is known that the amount of AM fungal colonization observed in roots does not correlate with the amount of AM fungal DNA retrieved in real-time PCR (Gamper et al., 2008), whereas root colonization is a reasonably good proxy for establishing whether interactions are taking place or not. Overall, these arguments imply two key points: first, an OTU might be present but could be defined at a phylogenetic scale that causes a confounded or even incorrect definition of the interactions depicted in matrix A (e.g. the OTU definition led to a ‘lumping’ of separate entities that might have different functional roles); second, an OTU might be detected but its interaction with the plant could actually be negligible or occurring on such an occasional basis that it should be considered biologically insignificant, more or less to the same extent to which some predators might happen to feed on an item that is not usually a fundamental part of their diet. In the AM fungi–plant relationships, interactions take place between the fungal mycelium and the plant root, and it is at this scale that each and every co-occurrence should be mechanistically demonstrated also to be an actual interaction (Kiers et al., 2011). Only in this way would one then be allowed to place that specific interaction into Eqns 1 and 2 and thus into matrix A.
Another limit of network analysis as applied by Chagnon et al. (2012) and Montesinos-Navarro et al. (2012) is that the network is bipartite. This is, of course, correct in the framework of analyses that address truly mutualistic interactions between two groups of organisms. However, this approach implies that the vertices of one set of organisms, say the plants, are not directly interacting, at least in the formulation used by these authors. This might actually be a fairly incomplete picture of the AM fungi–plant interactions, at least in terms of accounting for competitive dynamics and niches in both plants and fungi (Bever et al., 1997, 2010; Dumbrell et al., 2010). In practice this means that many equations are missing in the system of our Eqn 1.
Network theory and the AM fungi–plant system: a call for defining the interaction matrix that describes this system
All in all, we think that the pioneering contributions by Chagnon et al. (2012) and Montesinos-Navarro et al. (2012) have the potential to spur fundamental research on the nature of the AM fungi–plant interaction. In fact, we definitely agree on the basic fact that this interaction is worth modeling by means of network theory, although we also pointed out that the state of the art of research on this mutualistic system is not mature enough to shift from patterns of community structure (e.g. nestedness) to functions.
As we have discussed, the key step will be to define biologically robust interaction matrices. There are examples in the literature of how to progress from pure co-occurrence matrices to co-occurrence matrices that might represent a proxy for interaction matrices. Recently, network analysis has been applied to another fungi–plant system, namely the Orchis–fungi (Tulasnellaceae and Ceratobasidiaceae) association (Jacquemyn et al., 2011). Although in this case, too, network analysis was based on a co-occurrence matrix where fungi were indentified molecularly, several pieces of information were available that suggest that the analyzed matrix could be, at least partially, a valid proxy for an interaction matrix. Fungi were amplified from roots that were checked for mycorrhizal colonization, and at the same time some of the fungi were experimentally shown to be truly mycorrhizal in the field. Jacquemyn et al. (2011) also provide the reader with arguments about the employed OTUs and their taxonomic and phylogenetic relatedness to groups of fungi that are known to be essential for the life cycle of Orchis species. In fact, in this case, it is usually the plant that very much depends on the fungus to complete the life cycle. Last but not least, results from the network analysis were integrated with results from a phylogenetic approach (Blomberg et al., 2003; Ives & Godfray, 2006) that showed a clear phylogenetic signal in the association between the targeted plant genus (Orchis) and the most abundant fungal taxon (Tulasnellaceae). Given this result and the theoretical link between interaction matrices and evolutionary processes (Caldarelli et al., 1998; Drossel et al., 2001), it is reasonable to assume that the latter have shaped the interaction between plants and the analyzed fungal taxon. Evolutionary processes causing phylogenetic signals in ecological and life traits of fungi have also been detected in the AM fungi–plant association (Powell et al., 2009). Thus, in the very near future it will likely be possible to integrate the network approach proposed by Chagnon et al. (2012) and Montesinos-Navarro et al. (2012) with a phylogenetic approach (Jacquemyn et al., 2011) and we suggest that this integration may allow one to focus on those pairs of co-occurring AM fungi and plants that are more likely to interact. Nevertheless, even in the case analyzed by Jacquemyn et al. (2011) there might still be fungi–plant pairs that are just co-occurring. Thus, with regard to AM fungi and plants, the main point is that one should exploit all available information that allows the refining of AM fungi–plant co-occurrence matrices in terms of pairs of units that are very likely to interact in space and time. A possible outcome of this refinement could be a reanalysis of a subset of the matrices already analyzed by Chagnon et al. (2012) and Montesinos-Navarro et al. (2012). With regard to this subset and new datasets, an essential part of the quest for robust interaction matrices will be to mine the existing literature and to pursue experimental research that shows not only effective AM fungi colonization but also taxa-specific fungal effects on the plants (or vice versa). Stable isotope probing is a very promising approach to achieve this task; using this tool we now know that not all the fungi present in the root take up carbon from the plant within a certain temporal window (Vandenkoornhuyse et al., 2007), which highlights the challenges of disentangling co-occurrence and interaction.
Thus, we conclude that the main elements to define biologically robust interaction matrices are, first, a better understanding of AM biology in terms of defining the biological units that really interact with plants, be these at the mycelium, the population, and/or the species levels or above; and second, an exhaustive sampling of these interactions in terms of the spatiotemporal extent to which they take place. In fact, as also argued by Chagnon et al. (2012) and Montesinos-Navarro et al. (2012), the more exhaustive the sampling, the more statistically and phylogenetically robust will be matrix A and, as a consequence, the more well grounded will be the inferences of functional effects in structural patterns.
T.C. was supported by the Alexander von Humboldt Foundation. We are grateful to two anonymous reviewers for very constructive comments that improved an earlier version of the manuscript.