Modelling combat strategies in fungal mycelia

Journal of Theoretical Biology 304 (2012) 226–234

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Modelling combat strategies in fungal mycelia Graeme P. Boswell n Department of Computing and Mathematics, Faculty of Advanced Technology, University of Glamorgan, Pontypridd CF37 1DL, United Kingdom

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 October 2011 Received in revised form 12 March 2012 Accepted 27 March 2012 Available online 5 April 2012

Fungal mycelia have a well-established role in nutrient cycling and are widely used as agents in biological control and in the remediation of polluted landscapes. Competition and combat between different fungal communities is common in these contexts and its outcome impacts on local biodiversity and the success of such biotechnological applications. In this investigation a mathematical model representing mycelia as a system of partial differential equations is used to simulate combat between two fungal colonies growing into a nutrient-free domain. The resultant equations are integrated numerically and the model simulates well-established outcomes of combat between fungal communities. The outcome of pairwise combat is shown to depend on numerous factors including the suppression of advancing hyphae in rivals, the degradation of a rival’s established biomass and the utilization and redistribution of available nutrient resources. It is demonstrated how non-transitive hierarchies in fungal communities can be established through switching mechanisms, mirroring observations reported in experimental studies, and how specialized defensive structures can emerge through changes in the redistribution of internal resources. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Mathematical model Fungus Competition Translocation

1. Introduction Fungal mycelia are indeterminate organisms unbounded both spatially and temporally having no predetermined maximum size or age. Indeed, a single Armillaria ostoyae colony is estimated to cover an area of approximately 900 ha and be between 2000 and 8500 years old, placing it amongst the largest and oldest living organisms (Ferguson et al., 2003). In addition to their welldocumented role in nutrient cycling (Wainwright, 1988; Gadd, 1999), certain fungal mycelia form symbiotic connections with plant root systems (Boddy, 1999) and are widely used in numerous biotechnological applications including biological control (Jin et al., 1992; Trujillo, 1992) and remediation (Alexander, 1994; Kohlmeier et al., 2005). Essentially a fungal mycelium is a highly branched and interconnected network of tubes, termed hyphae, that acquire growth materials including carbon, nitrogen, oxygen and trace metals from the local environment and redistribute them through the hyphal network by the process of translocation (Gow and Gadd, 1995; Dix and Webster, 1995). In particular, when internally held material is translocated to the end of a hypha, termed a hyphal tip, it is used to construct new biomass thus extending the hypha essentially by propelling the hyphal tip forward (Gooday, 1995). Hyphal tips themselves are created and lost through the

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processes of branching (the emergence of new hyphal tips, either through the bifurcation of an existing hyphal tip or the emergence from the wall of an existing hypha) and anastomosis (the fusion of hyphal tips with the network) respectively. It is through a combination of tip extension, branching, anastomosis and nutrient translocation that mycelia are able to colonise vast areas and grow through nutritionally deleterious regions by utilizing distal resources. In all of the ecological and biotechnological settings described above, interactions between different species of fungi are common. When there is no mating compatibility a range of responses can be initiated including combat and the formation of specialized biomass structures. One such structure formed on agar concentrations in Petri dishes is an extremely dense and stationary concentration of interlocking hyphae termed a ‘‘barrage’’ (Boddy, 2000). Effectively this is a wall of hyphae constructed by a fungus to defend currently held territory from attack and is impenetrable to a rival mycelia. Pair-wise interactions have typically been investigated experimentally by simultaneously culturing two colonies in a Petri dish (e.g. Boddy, 2000; Evans et al., 2008). In essence, three distinct outcomes of combat have been classified: intermingling, overgrowth and deadlock. Intermingling describes the instance where the hyphae of two non-mutually compatible fungi overlap and there are sustained regions where both fungal colonies coexist. In certain species-specific interactions, particular volatile organic compounds (VOCs), primarily hydrocarbons, are produced by one or other or both of the colonies that degrade hyphae causing the gradual displacement of one fungus by another, a process termed

G.P. Boswell / Journal of Theoretical Biology 304 (2012) 226–234

overgrowth (or displacement). The instance of deadlock arises when the hyphae of both colonies are unable to extend into regions occupied by another and which causes the expansion of the mycelia in the corresponding direction to halt (possibly because of the inhibitory effects of secondary metabolites released by a fungus in the presence of a rival, Evans et al., 2008). Unsurprisingly nutrient availability has a significant role in determining the outcome of pairwise competition with less dominant colonies better able to defend their territory when provided with increased resources (Kennedy, 2010). While different species of fungi can generally be ranked in order of their combat ability, it is apparent that the hierarchal structure is intransitive since certain typically superior competitors can be displaced by generally lesser ones. For example, Phallus impudicus displaces Megacollybia platyphylla, M. platyphylla displaces Psathyrella hydrophilum but P. hydrophilum displaces P. impudicus (Chapela et al., 1988). The only explanation for such intransitive hierarchies is that the response of fungi to combat is species-specific. Indeed, it has been shown that particular VOCs are produced during interactions between Trametes versicolor and Stereum gausapatum that are not produced during self-pairings (Evans et al., 2008) (although it is unknown which fungus is responsible for the production of individual VOCs). Moreover, such species-specific behaviour is not confined to colony displacement; when Lenzites betulina and Vuilleminia comedens are cultured together, the latter undergoes a significant morphological change following contact and forms a stationary ‘‘barrage’’ that is resistant to invasion by the former (Boddy, 2000). While the role of VOCs and secondary metabolites in determining the outcome of fungal combat have been investigated, the role of nutrient translocation has received little attention in both experimental and modelling studies. This is surprising given that there is significant evidence that mycelial cord systems, which comprise a major component of nutrient translocation, change significantly during and following contact with other mycelia (Boddy, 1993, 1999). Moreover changes in fungal morphology often coincide with a sudden shift of the translocation process of internal metabolites (Amir et al., 1995a) and a corresponding increase in turgor pressure within the mycelium (Amir et al., 1995b). Therefore one of the aims of the current investigation is to understand the influence of nutrient translocation in combat between fungi. Despite the numerous experimental studies described above, there are inherent difficulties in attempting to understand how nutrient translocation can impact on combat, principally due to the differences in scale. While fungal colonies are of the order of centimeters (or larger), their constituent component, the hypha, has a diameter of the order of microns. For this reason, numerous mathematical models have been developed that relate the two different scales. However, while there are numerous mathematical models describing the growth and function of a single colony using continuum, discrete and hybrid approaches (see Boswell and Hopkins, 2008 and references therein), there are few such models describing competition and combat between fungal communities. A small number of investigations have focussed on intraspecific competition; for example Schmit (2001) modelled the impact of nutrient status on total biomass. Davidson et al. (1996) adopted a generic approach using activators and substrates to represent the interactions between two fungal communities and demonstrated that spatially heterogeneous structures could be formed using a continuum model but due to the model’s abstract nature, could not be used to make qualitative predictions. An alternative approach was taken by Halley et al. (1994) who developed a cellular automaton model simulating the interactions and succession between four fungal biomasses competing over a

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range of resources. While this model demonstrated the displacement of certain fungal biomasses by their competitors, it essentially relied on the imposition of a transitive hierarchy. Recently Falconer et al. (2011) extended an earlier model by the same authors (Falconer et al., 2005) to account for interactions between different model fungal phenotypes that detected and responded to inhibitors released by each other and demonstrated its role in maintaining greater biodiversity by forming demarcation zones. However because this model does not explicitly consider the underlying mechanisms involved in fungal interactions, its calibration for combat between specific species is problematic. Consequently the predictive power, for example how resource availability and its reallocation by the fungus can influence the outcome of combat, is restricted. As such, there is clearly a need for a simple mathematical model capable of simulating combat between fungal mycelia that is capable of making both qualitative and quantitative predictions including the formation of specialized biomass structures. Furthermore, such a model should explain possible mechanisms for the lack of a transitive hierarchy in the fungal kingdom. In the current investigation a mathematical model, comprising a system of partial differential equations (PDEs) proposed by Boswell et al. (2002) to represent the isolated growth of a fungal mycelium in a single spatial dimension, is adapted to consider pairwise interaction between two non-somatically compatible colonies. The model equations are described in Section 2 and because of their mixed hyperbolic–parabolic type require a suitable numerical solution method, which is also briefly described. Parameter values corresponding to non-combat terms in the model equations are calibrated for a typical fungus. Two distinct parameters relating to combat are varied having initially been selected so that their values are similar in magnitude to corresponding (and calibrated) non-combative processes thereby creating a large number of model phenotypes. The outcome of pairwise combat between these phenotypes is described in Section 3. The simulations demonstrate that different processes are responsible for the initial and then longer-term outcomes of phenotype combat and that certain phenotype triples do not display expected transitive relationships. It is also shown that resource availability and its reallocation to the site of fungal interactions has a significant role in determining the outcome of combat. In particular, sudden increases in nutrient translocation are shown to generate stationary barrages that are resistive to invasion and quantitative predictions are made on the scale of those increases that are necessary to generate such structures.

2. Modelling In the terrestrial environment, combat and competition between rival fungal communities usually occurs in the soil-pore space but is typically investigated in laboratory settings by inoculating either sides of a Petri dish with two species of fungus that grow and subsequently interact with one another. While these interactions are two-dimensional, to model the outcome and understand roles of various combat processes, it is sufficient to consider the interaction fronts along a direct line between the sites of inoculation which can be regarded as a one-dimensional domain. Hence the model constructed below represents the expansion of two fungal biomasses in a one-dimensional setting. 2.1. Model equations A simplification of a previously published mathematical model (Boswell et al., 2002), which itself was an extension of the pioneering model developed by Edelstein (1982), was used to represent an

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expanding mycelium in a one-dimensional domain. Briefly, a single fungus was modelled by three continuous variables corresponding to hyphal tip density (i.e. the extending end of a hypha), represented by pðx,tÞ, biomass (i.e. hyphal) density, denoted by mðx,tÞ and the concentration of an internally held growth-promoting substrate, represented by sðx,tÞ, which, for purposes of calibration, was taken as a carbon. Hyphal tip extension is the sole mechanism through which mycelia develop and this process is dependent on the supply of internalized material to the hyphal tip to enable such growth and thus is modelled as an increasing function of the substrate s. While under high nutrient concentrations there is likely to be a saturating effect, given the application to nutrient-free domains and the calibration applied (see below), it was reasonably assumed that the rate of tip extension was approximated by a linear function, namely vs (which is similar to the classic model of Prosser and Trinci, 1979, in which the rate of tip extension was assumed proportional to the rate of absorption of tip vesicles at the hyphal tip) and has also been successfully used in previous modelling studies (Boswell et al., 2003a). The trail left behind a tip (corresponding to the tip flux) corresponded to the creation of new hyphae and thus the tip flux increased the biomass density accordingly. Turgor pressure (the build-up of internally located material within the mycelium) and ionic gradients have both been implicated in the formation of new hyphal branches (Prosser and Trinci, 1979; Gow and Gadd, 1995; Watters and Griffiths, 2001) and therefore the creation of hyphal tips through sub-apical branching was modelled at a rate bsm representing the additional dependence of this process on the amount of hyphae from which branching can arise where b was a non-negative constant. The rate of anastomosis was assumed to increase with the density of both hyphae and hyphal tips and so, for simplicity and in the absence of any experimental evidence to the contrary, was represented by the term fmp where f was a non-negative species-specific constant (see also Edelstein, 1982). The degradation of the fungus into the external environment was modelled through biomass decay at a constant per capita rate d. (Autolysis, the recycling of decaying hyphae, was neglected since the model focussed on the early stages of fungal growth where such self-ingestion is uncommon.) Translocation was assumed to comprise both diffusive and advective components (Olsson, 1995) where the latter, consistent with experimental observations (Fisher-Parton et al., 2000; Jacobs et al., 2004) and other modelling investigations (Boswell, 2008; Heaton et al., 2010), was directed towards hyphal tips since they represent the major energy sinks in the model system. The flux of internal substrate was thus modelled by the term Dm@s=@x þ Amð@p=@xÞs where D and A corresponded to the diffusive and advective magnitude of the translocation components respectively and where the dependence of the flux on m encapsulated the essential relationship between the density of the network and its ability to translocate internally located material. To represent the energy costs involved in hyphal tip extension, the internal substrate was locally depleted at a rate proportional to the tip flux while the metabolic cost of advective translocation was assumed to be proportional to the corresponding component of the substrate flux. See Boswell et al. (2002) for full details of the model. Since the current investigation concerned pairwise interactions of fungi, the isolated growth of a second mycelium was modelled through the introduction of a further three model variables corresponding to tip density p0 ðx,tÞ, biomass density m0 ðx,tÞ and internal substrate concentration s0 ðx,tÞ in that fungus. The corresponding model equations thus took a similar form to that described above and it only remains to describe the interaction processes between the two biomasses. When two non-somatically compatible colonies interact they can adopt a range of combative strategies, either aggressive or defensive (Score et al., 1997; Boddy, 2000). Typically a combination of metabolites and VOCs are released by one, or other, or both

fungi that inhibit (or occasionally promote) the growth rate and degrade existing hyphae in the opposite colony (Schoeman et al., 1996; Evans et al., 2008) and where the type and amount of VOC produced is species specific and dependent on the degree of contact between colonies (Hynes et al., 2007). For simplicity, neither the metabolites nor VOCs were modelled explicitly but instead the functional consequence of the interaction between the two colonies was considered. To represent the inhibition of hyphal tip expansion through the detection of secreted metabolites, it was reasonably assumed that hyphal tips corresponding to the first colony were degraded at a rate proportional to the local biomass density of the second colony (and vice versa). While this did not strictly alter the speed of hyphal tip extension, it nonetheless provided a simple mechanism that essentially reduced the effective expansion rate of each biomass in the presence of the other. (There are alternative approaches to inhibiting the tip flux—see discussion in Section 4.) Hyphal degradation in response to VOC production was modelled as being proportional to the respective biomass densities since they corresponded to the source of VOCs and its resultant impact. By applying the above assumptions, the mathematical model was thus represented by the following system of six PDEs: tip movement

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ branching anastomosis tip degradation through rival zffl}|ffl{ z}|{ zfflfflffl}|fflfflffl{ @p @ ¼  ðvspÞ þ bsm  fmp  a0 m0 p, @t @x @m ¼ @t

new hyphae

z}|{ vsp

hyphal death



z}|{ md

biomass degradation through rival

zffl}|ffl{ c0 m0 ,



translocation: diffusion & directed

@s ¼ @t

ð1Þ

ð2Þ

directed translocation cost

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ growth cost  zffl}|ffl{ @ @s @p Dm Ams  gvsp  @x @x @x

zfflfflfflfflfflfflffl}|fflfflfflfflfflffl  ffl{ @p kAms , @x

@p0 @ 0 0 ¼  ðv0 s0 p0 Þ þb s0 m0 f m0 p0 amp0 , @x @t

ð3Þ

ð4Þ

@m0 0 ¼ v0 s0 p0 m0 ðd þcmÞ, @t  0   @p  @s0 @ @s0 @p0 0 D0 m 0 ¼ A0 m0 s0 g 0 v0 s0 p0 k A0 m0 s0  , @x @t @x @x @x

ð5Þ

ð6Þ

where a and c (and a0 and c0 ) are parameters representing the hyphal tip and biomass degradation processes respectively. 2.2. Initial data and boundary conditions To best represent typical experimental protocol (e.g. Evans et al., 2008), the model equations were considered on the interval (0,1) where the biomasses, the model tips and internal substrates were initially positioned at either end of the domain and then expanded towards one another. Thus the initial data was taken as ( ( m0 if x o0:25, m00 if x 4 0:75, mðx,0Þ ¼ m0 ðx,0Þ ¼ 0 otherwise, 0 otherwise, ( pðx,0Þ ¼ ( sðx,0Þ ¼

p0

if x o0:25,

0

otherwise,

s0

if x o 0:25,

0

otherwise,

( p0 ðx,0Þ ¼ ( s0 ðx,0Þ ¼

p00

if x 4 0:75,

0

otherwise,

s00

if x 4 0:75,

0

otherwise:

This approach is consistent with initial data used in previous modelling investigations representing a ‘‘plug’’ of mycelium expanding into an uncolonized spatial region (Boswell et al., 2002).

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Table 1 Default parameter values and initial data calibrated for Rhizoctonia solani from Boswell et al. (2002) where substrate was regarded as a carbon and were rescaled for the purpose of numerical simulation, see Boswell et al. (2003b) for full details. n Model phenotype a ¼ 10, c ¼ 0:1 had this value changed to A ¼10  4 cm3 day  1 per tip and 10  3 cm3 day  1 per tip on selected simulations of combat with phenotype a ¼ 1000, c ¼ 10 following the collision of their leading edges (see text for details). nn This value was varied in one set of simulations to investigate the effect of resource variations on combat. Parameter/variable

Description

Calibrated value

v b f d D A g k

Tip speed Branching rate Anastomosis rate Hyphal decay rate Internal substrate diffusion Internal substrate advection Growth cost Active translocation cost

105 cm (mol glucose)  1 day  1 107 branches cm  1 hyphae day  1 101 fusions in cm hyphae day  1 10  1 day  1 10  2 cm3 day  1 10  5 cm3 day  1 per tipn 10  7 mol glucose cm  1 per tip 10  8 cm  1

p0 m0 s0

Initial tip density Initial biomass density Initial internal substrate/boundary condition

100 tips cm  2 100 cm  1 4  10  5 mol glucose cm  2nn

The biomasses were assumed to have access to their own continually replenished resource away from the interaction region and to best represent this situation fixed boundary conditions on the internal substrate concentrations were implemented on the appropriate boundaries. Specifically sð0,tÞ ¼ s0 ,

s0 ð1,tÞ ¼ s00 ,

while zero flux boundary conditions were imposed, where needed, on the remaining model variables. 2.3. Numerical solution The model equations (1a)–(1f) are of mixed hyperbolic–parabolic type and because the hyperbolic terms dominate (see calibration below), it was crucial for application that the numerical scheme employed to solve the equations both conserved mass and preserved the positivity of the model variables. To this end, the model equations were solved numerically using the method described in Gerisch et al. (2001) and further developed in Boswell et al. (2003b) by using carefully selected finite difference approximations. The non-linear diffusion terms were treated using mid-point central difference approximations of second-order accuracy while the hyperbolic terms were treated using the method of lines coupled with flux-limiters (specifically van Leer’s limiter function, Sweby, 1984) to preserve the positive nature of the solutions and which was also second order accurate. The resultant system of ordinary differential equations was numerically integrated using a three-stage second order accurate Runge–Kutta scheme where elements of the corresponding Butcher tableaux were carefully selected to preserve the positivity of the differential equations (see Boswell et al., 2003b for full details). An adaptive time step method, based on Richardson extrapolation, was used to perform the integration and while the time step size was not specifically selected to preserve positivity, negative solutions were not observed in any of the numerical simulations (because of the good positivity properties elsewhere). Finally, to ease the implementation of the adaptive time step routine (which used tolerances based on absolute and relative errors), the calibrated variables and parameters (see below) were rescaled so that the model variables all had a similar order of magnitude (the unscaled model variables were easily recovered by multiplying by a suitable constant, see Boswell et al., 2003b for full details). The results presented in the current study use the original and unscaled variables and parameters. 2.4. Calibration Boswell et al. (2002) calibrated a subset of Eqs. (1) for the fungus Rhizoctonia solani grown on an agar mixture where the substrate was

taken to be carbon. Hence the calibrated parameter values and initial data obtained in that study were used in the current investigation as typical values for a generic fungus (Table 1). To investigate the qualitative impact of the interaction parameters on combative outcomes, 25 model phenotypes were formed by independently selecting values for a (the tip inhibition rate) and c (the biomass degradation rate) where a A f0; 10,50; 100,1000g,

c A f0,0:1,0:5,1; 10g:

The values selected for a (in units cm day  1) ensure that the effective inhibition of a rival phenotype’s hyphal tips is either equivalent to or greater than the natural loss of tips due to anastomsosis (or none at all if a ¼0, see Table 1). The values of c (also in units cm day  1) correspond to the degradation of hyphae at a rate equivalent to or greater than the natural hyphal decay rate (or none at all if c ¼0, see Table 1). Pairwise combat was considered between each of these 25 model phenotypes (generating 325 such pairings, including all possible self-pairings) where each phenotype had access to equal amounts of continually replenished resource (i.e. s0 ¼ s0 0 ) and the outcomes were investigated by setting the relevant calibrated initial data at either end of the domain (and reversing the sign of v0 in Eq. (1d) to ensure the model tips were directed towards the centre of the domain). The model equations were solved as described above over a period of time representative of 3 days. The position of the leading edge of the biomasses were recorded throughout the simulations (where the edge was defined to be the first position where its density exceeded 10  1 cm  1, a value that was successfully used in previous investigations to define the edge of a growing fungal colony, Boswell et al., 2002). To investigate the role of a species-specific switch in the rate of nutrient translocation (in particular to test the hypothesis that stationary ‘‘barrages’’ can be formed through a sudden increase in nutrient translocation), a pair of the model phenotypes a ¼ 1000, c ¼ 10 and a0 ¼ 10, c0 ¼ 0:1 was selected where the latter phenotype was displaced by the former. The numerical simulations were repeated for this pairing where the advective translocation component in the initially displaced biomass was increased by a factor of 10 and then 100 following contact of the leading biomass edges to simulate the effect of increased nutrient translocation in fungal combat. The effect of resource level on combative outcomes for a different pair of the model phenotypes was also considered by systematically reducing s0 from its calibrated value but maintaining s0 0 , which also indirectly allowed the investigation of the effect of biomass asymmetries on the outcome of combat.

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Fig. 1. The position of the leading edge of the model biomass (defined as the first location where the biomass exceeds 10  1, see text for details) as obtained from solving the model equations (1) for different interaction parameters is plotted as a function of time. The biomasses first collided at t¼0.8 days and then changed depending on the parameters a, c (for the biomass starting on the left hand boundary, represented by the dotted curve) and a0 , c0 (for the biomass starting on the right hand boundary, represented by the solid curve) where (a) a ¼ 10, c ¼ 0:1, a0 ¼ 10, c0 ¼ 0:1; (b) a ¼ 100, c ¼ 0:1, a0 ¼ 100, c0 ¼ 0:1; (c) a ¼ 10, c ¼ 0:1, a0 ¼ 100, c0 ¼ 0:1; (d) a ¼ 10, c ¼ 0:1, a0 ¼ 10, c0 ¼ 10; (e) a ¼ 10, c ¼ 0:1, a0 ¼ 50, c0 ¼ 0; (f) a ¼ 1000, c ¼ 10, a0 ¼ 10, c0 ¼ 0:1; (g) a ¼ 1000, c ¼ 10, a0 ¼ 10, c0 ¼ 0; (h) a ¼ 1000, c ¼ 10, a0 ¼ 0, c0 ¼ 0:1.

3. Results 3.1. Tip inhibition and biomass degradation Under equal resource availability the initial development of the biomasses, prior to their collisions, were symmetric and identical in all cases. The distribution of model tips became highly peaked at the expanding biomass edge and progressed as a travelling wave towards the centre of the domain. The biomass density increased behind the wavefront of model tips over time while the internal substrate concentration was generally a decreasing function from the edge of the domain but exhibited a small increase at the biomass edge as a result of translocation in response to model tip density. (See Boswell et al., 2002 for typical behaviours of the solution of the model equations prior to biomass collisions.) After a time representative of approximately 0.6 days the biomass edges were separated by a distance corresponding to 1 mm and by a time corresponding to 0.8 days the biomass edges had collided and began to overlap. Following the collisions of the biomasses, the initial overlapping region, and thus the position of the leading edges of the biomasses, changed depending on the relative interaction parameter values (i.e. depending on a,a0 ,c and c0 ). Generally those phenotypes with larger values of a (or a0 ) were initially less overlapped by their rivals than those with smaller values of a. For example, in the self-pairing between the model phenotype with a ¼ 10, c ¼ 0:1, there was a much larger overlap than in the self-pairing between the model phenotype with a ¼ 100, c ¼ 0:1 (Fig. 1(a) and (b)). This overlap region then changed over time in a manner mostly determined by the relative values of the parameter c (and c0 ); those phenotypes with larger values of c typically displaced those with smaller values of c, either continuously (when the difference between the c-parameters in the phenotypes was large), or until a state of deadlock had been reached (when there was a smaller difference between the value of the parameters c). For example, the model phenotype a ¼ 10, c ¼ 0:1 obtained a state of deadlock when simulated against the phenotype a ¼ 100, c ¼ 0:1 (Fig. 1(c)) but was continuously displaced by the phenotype a ¼ 10, c ¼ 10 (Fig. 1(d)). In certain phenotype pairings, typically where a 4 a0 but c oc0 , initial gains

Fig. 2. The position of the leading edge of model biomasses in pairwise combat at a time representing t ¼ 2.4 days was obtained from the numerical solution of Eqs. (1) . Each of the 25 blocks corresponds to combat involving one of the 25 model phenotypes investigated with the parameter values of a and c indicated on the axes. Each block comprises 5  5 cells whose shading represents the position of the leading biomass edge of that phenotype with respect to the centre of the domain when combat is simulated against each of the 25 rival model phenotypes arranged in the same configuration as the overall block structure (i.e. a increasing horizontally and c increasing vertically within each block and taking the same values as indicated on the axes). Red indicates that the phenotype has made progress into its rival’s territory, green indicates that the biomass edge has remained stationary following collision while blue indicates it has been displaced by its rival. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)

made by the first biomass were often eroded to a state of deadlock (Fig. 1(e)). Furthermore the speed of any displacement was typically dependent on the relative differences between the parameters a of the model phenotypes. For example, the model phenotype a ¼ 1000, c ¼ 10 displaced the phenotypes a ¼ 10, c ¼ 0:1 and a ¼ 10, c ¼ 0 at approximately the same rate (Fig. 1(f) and (g)) whereas its displacement of the phenotype a ¼ 0, c ¼ 0:1 was

G.P. Boswell / Journal of Theoretical Biology 304 (2012) 226–234

it none-the-less clearly demonstrates a counter-intuitive and nonlinear relationship between phenotype and combat outcome.

considerably faster (Fig. 1(h)). Within the time frame examined, the phenotypes with greater values of a and c generally performed better than those phenotypes with smaller values of a and c while the only model phenotypes that experienced complete displacement corresponded to those with a¼0 and where the displacement arose only in combat with phenotypes having large values of both a and c (Fig. 2).

3.3. Phenotypic-specific changes in translocation It was shown above, through the tracking of the position of the leading biomass edges, that the phenotype a ¼ 1000, c ¼ 10 continually displaced the phenotype a ¼ 10, c ¼ 0:1 but the rate of displacement decreased over time (Fig. 1(f)). This decrease in the rate of displacement arose since as the invading phenotype progressed, it encountered increased densities of the biomass being displaced and because in Eqs. (1) the decay of a rival biomass and its hyphal tips was assumed proportional to the density of the opposing biomass, the phenotype being displaced was therefore better able to defend its territory than where its biomass density was small (Fig. 4(a)). This observation suggests the potential existence of a critical biomass density that prevents the displacement of one phenotype by a rival, similar to the stationary ‘‘barrages’’ displayed by fungal colonies described above. Thus to examine the hypothesis that a change in translocation rates may be responsible for the formation of such barrages, the model equations were solved again for the same phenotype pairing except the displaced phenotype ða ¼ 10, c ¼ 0:1Þ experienced an increase in its advective translocation rate after the leading biomass edges had collided. An order of magnitude increase in the active translocation rate following the collision of the biomass edges did not significantly alter either the position of the leading biomass edges (Fig. 5(a)) or the biomass densities

3.2. Transitivity in phenotypes Every possible phenotype triple that could be formed from the 25 model phenotypes described above was examined for transitivity in the replacement of one biomass by another (resulting in the analysis of 25C3 ¼2300 such triples). Each phenotype triple displayed the transitive property; that is in the phenotype triple ðP 1 ,P2 ,P 3 Þ if P1 displaced P2 in pairwise combat and P2 displaced P3 in pairwise combat, P1 always displaced P3. However, while the majority of triples were strongly transitivity (i.e. if P1 displaced P2 and P2 displaced P3, then P1 displaced P3 more than it displaced P2 and more than P2 displaced P3), certain triples were only weakly transitive (e.g. the displacement by P1 of P3 was less than its displacement of P2). For example, the phenotype a ¼ 1000, c ¼ 1 marginally displaced the phenotype a ¼ 100, c ¼ 10 before entering a state of deadlock (Fig. 3(a)), the phenotype a ¼ 100, c ¼ 10 significantly displaced the phenotype a ¼ 10, c ¼ 0:1 (Fig. 3(b)) and while the phenotype a ¼ 1000, c ¼ 1 displaced the phenotype a ¼ 10, c ¼ 0:1, its displacement was less than that by the phenotype a ¼ 100, c ¼ 10 (Fig. 3(c)). While this is not the same as the phenotype-triple being intransitive,

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Fig. 3. The position of the leading biomass edge is plotted over time during pairwise combat between three model phenotypes. Solid line denotes a ¼ 1000, c ¼ 1; dashed line denotes a ¼ 100, c ¼ 10; dotted line denotes a ¼ 10, c ¼ 0:1. (a) Phenotype a ¼ 1000,c ¼ 1 (starting at the x¼ 0 boundary) marginally displaces the phenotype a ¼ 100, c ¼ 10 (starting at the x ¼1 boundary) before entering a state of deadlock. (b) Phenotype a ¼ 100, c ¼ 10 (starting at the x¼ 1 boundary) significantly displaces phenotype a ¼ 10, c ¼ 0:1 (starting at the x ¼0 boundary). (c) While phenotype a ¼ 1000, c ¼ 1 (starting at the x¼ 1 boundary) displaces phenotype a ¼ 10, c ¼ 0:1 (starting at the x¼ 0 boundary) the displacement is less than that observed in (b) despite the ability of phenotype a ¼ 1000, c ¼ 1 to displace the superior combatant in (b).

Fig. 4. The biomass densities obtained by solving model equation (1) for combat between phenotypes a ¼ 1000, c ¼ 10 (starting on left) and a0 ¼ 10, c0 ¼ 0:1 (starting on right) is shown at times corresponding to 0.6 days (solid line), 1.2 days, (dashed line), 1.8 days (dot–dash line) and 2.4 days (dotted lines). Parameter values are those in Table 1 except the active translocation rate for the phenotype starting on the right hand side is increased following collision of the leading biomass edges. (a) A0 ¼ 105 throughout the simulation, (b) A0 ¼ 105 for t o 0:8, A0 ¼ 104 for t Z 0:8, and (c) A0 ¼ 105 for t o 0:8, A0 ¼ 103 for t Z 0:8. All other parameters are given in Table 1.

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3.4. Resource dependence

themselves (Fig. 4(b)). However, an increase by a factor of 100 in the active translocation rate resulted in a wave of internal substrate directed towards the expanding end of its biomass and generated a dense wave of model tips that progressed similarly. The model tips increased the biomass density significantly at the edge of the existing biomass and formed a barrage against the superior competitor resulting in deadlock (Figs. 5(b) and 4(c)). While the increased active translocation resulted in the formation of a dense wall of biomass, it did so at the expense of biomass behind this defensive structure. Indeed, after the representation of 2.4 days growth, the density of the inferior phenotype biomass at a representative distance of 2 mm behind its leading edge was approximately an order of magnitude less when the 100-fold increase in active translocation was implemented compared to when it was not (Fig. 4).

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To investigate how resources influence combative outcomes, Eqs. (1) were solved as described above to represent the instance where phenotypes a ¼ 1000, c ¼ 10 and a ¼ 10, c ¼ 0:1 engaged in combat but the resource available to the former was varied, i.e. the value of the substrate s0 was reduced from its calibrated value, while the resource available to the latter remained fixed, i.e. s0 0 retained its calibrated value. It was shown above that under equal resource availability, the former phenotype continually displaced the latter (Fig. 4(a), see also Fig. 6(a)). However, in simulations with reduced values of s0, the biomass expansion of the corresponding phenotype was less (because of the reduction in internal substrate) and following the collision of the biomasses, the rate of biomass displacement similarly slowed (Fig. 6(a)–(c)). When the internal substrate s0 was reduced to one quarter of its calibrated value, a state of deadlock was immediately obtained following the collisions of the biomass (Fig. 6(d)) and when s0 was reduced to one-tenth of its initial calibrated value biomass corresponding to the once ‘‘dominant’’ phenotype was in fact slightly displaced before entering deadlock (Fig. 6(e)). Under extremely small values of s0, there was no expansion of the corresponding biomass but it was not displaced by the rival biomass following their collision (Fig. 6(f)).

4. Discussion

3

Understanding and predicting the outcome of competition and combat between fungal communities is crucial to ensure their successful biotechnological application. Because of the inherent difficulties in adopting a purely experimental approach, for example accounting for the vast range of volatile organic compounds (VOCs) produced during interactions (Boddy, 2000), this mathematical modelling exercise has focussed on key features

Fig. 5. The position of the leading biomass edge is plotted over time during pairwise combat between the model phenotypes a ¼ 1000, c ¼ 10 (dashed line) and a0 ¼ 10, c0 ¼ 0:1 (solid line) where other parameter values are those given in Table 1 except for the active translocation rate for the phenotype a0 ¼ 10, c0 ¼ 0:1 starting on the right hand side which was increased following collision of the leading biomass edges. (a) A0 ¼ 105 for t o 0:8, A0 ¼ 104 for t Z 0:8, and (b) A0 ¼ 105 for t o 0:8, A0 ¼ 103 for t Z 0:8. All other parameters are given in Table 1.

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Fig. 6. The position of the leading biomass edge obtained by solving the model equation (1) for combat between phenotypes a ¼ 1000, c ¼ 10 (dashed line) and a0 ¼ 10, c0 ¼ 0:1 (solid line) corresponding to different resource regimes. The substrate available to the phenotype a ¼ 1000, c ¼ 10 starting on the x¼ 0 boundary was varied: (a) s0 ¼ 4  105 , (b) s0 ¼ 3  105 , (c) s0 ¼ 2  105 , (d) s0 ¼ 1  105 , (e) s0 ¼ 4  106 , (f) s0 ¼ 106 (all units mol glucose cm  2).

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involved in fungal combat. Essentially the suppression of a rival’s growth coupled with the degradation of its biomass is sufficient to generate commonly observed outcomes of fungal combat. It is well established that apical extension rates in fungal colonies can change when in the presence of other fungi and, in certain circumstances, can be halted altogether (Evans et al., 2008). In order to represent the decrease of tip movement in one fungal phenotype in the presence of a rival, the current modelling investigation assumed that the tips in a growing fungus were destroyed by a rival. An alternative approach that reduced tip flux could have represented tip movement in one phenotype as a decreasing function of biomass from a rival phenotype. However, while this alternative approach initially appears preferable, it has a major failing. In the adopted approach, hyphal tips were removed from the simulation and could no longer contribute to biomass growth while in the latter approach they would continue to exist but would not move. Hence during colony displacement those hyphal tips would remain in position while the biomass to which they were originally attached was degraded. This is not only undesirable from a physical perspective, but in any subsequent biomass expansions, these hyphal tips would immediately become activated again creating unrealistic growth. Further experiments are needed to more fully understand which, if either, of these instances are found in competing fungal colonies; for example by removing one fungal mycelium from a pair engaged in a state of deadlock and observing the growth of the remaining mycelium may provide such insights. An important component of the modelling exercise has been the inclusion of a growth promoting substrate since this has been shown to play a crucial role in combative outcomes between fungal mycelia (Kennedy, 2010). Indeed, the ability of a fungus to translocate resources towards the expanding edge of a mycelium is fundamental to its existence both when developing in isolation and also when in combat with other fungi. While localized resources have been shown to influence the outcome of interspecific competition (White et al., 1998), nutrient reallocation from other regions of the mycelia is a decisive factor in combat outcome (Figs. 5 and 6). Variations in the amount and reallocation of this substrate is sufficient to explain much behaviour displayed by fungal colonies engaging in combat. Upon increasing the advective component of translocation by a sufficient amount (in the simulations by a factor of 100), an otherwise inferior phenotype was able to defend itself against attack by an otherwise superior combatant through the construction of a stationary ‘‘barrage’’, representing a dense wall of hyphae. Such structures are widely seen in fungi, for example where sclerotia (a resting state comprising a collection of knotted hyphae containing resources) are formed in many mycelia when stressed, such as in periods of antagonism by others or when local resources are scarce (Boddy, 2000). Indeed, while it has been previously shown that the formation of sclerotia coincides with a change in the translocation rate of internal metabolites (Amir et al., 1995a), the above modelling demonstrates firstly that a sudden increase in nutrient translocation is sufficient to produce similar formations, and secondly quantitative predictions are made on the scale of such increases. In all the simulations performed in this investigation, the expansion of a biomass declined when it encountered increasing densities of a rival biomass (provided a and c were not both simultaneously zero in the opposing phenotypes). This simple observation highlights the defensive role that hyphal structures have in fungal combat. Indeed, for a given phenotype pairing, the outcome of combat was partly influenced by the initial biomass densities at the point of contact (e.g. through nutrient availability, Fig. 6) indicating that not only are the interaction processes crucial in determining the outcome of combat, but that biomass

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asymmetries also play a decisive role. This feature arose because the processes of tip and biomass degradation were assumed to increase with the total biomass of the rival phenotype and as an invading phenotype made progress into territory occupied by a rival, it encountered increased densities of biomass (e.g. Fig. 4(a)) that slowed its growth (e.g. Fig. 1(f)), sometimes resulting in deadlock (e.g. Fig. 1(c)). As a consequence the modelling thus predicts that deadlock in fungal combat is a result of the combined processes of hyphal tip and biomass degradation. When the sole difference between two combatants was their ability to attack their rival (i.e. the sole difference was in the values of the parameters a and c) the phenotype that initially gained the most territory following the collision of their biomasses corresponded to that with the greatest value of a, i.e. the phenotype that was better able to suppress the growth of its rival. However, after this initial transient phase, the phenotype that eventually dominated corresponded to that with the greatest degradation rate of the biomass of its rivals (i.e. the phenotype with the greatest c). In this way, a transitive hierarchy of the model phenotypes could be formed according to the ability of a fungus to degrade biomass in its rivals; thus if a fungus X displaces a fungus Y that in turn displaces a fungus Z, it followed that X will displace Z. While this hierarchical structure existed in the model phenotypes, it was observed that the transitivity was often only weak; that is if fungus X displaces Y that in turn displaces Z, X can displace Z but not necessarily as quickly or as much as it displaces Y or as quickly or as much as Y displaces Z. Experimental investigations have shown the existence of intransitive hierarchies (Boddy, 2000); i.e. fungus X can displace Y that in turn can displace Z but which in turn can displace X. One previously proposed explanation for this intransitive hierarchy is that the VOCs produced during competition are specific to the fungal pairings involved. However, in such pairwise interactions it is currently unknown as to which VOC is produced by which fungus and its role, if any, in the combat process. In the model phenotypes examined in the current investigation, this situation is analogous to a switch in the value taken by either the parameter a or c (or both) in one or both phenotypes following contact with a particular rival. For example, if X denotes the model phenotype a ¼ 100, c ¼ 1, Y the phenotype a ¼ 50, c ¼ 0:5 and Z the phenotype that typically takes a ¼ 10, c ¼ 0:1 but when in contact with X becomes a ¼ 1000, c ¼ 10, then ðX,Y,ZÞ would form an intransitive triple with the only change arising in the typically inferior phenotype. Despite extensive numerical investigation, it was not possible to form a triple of model phenotypes that could display this intransitivity without enforcing a phenotype-specific change in the interaction parameters. While these simulations appear to add further support to the claim that species-specific changes in combat processes are responsible for intransitivity between fungal communities, as discussed above (Figs. 4 and 5) a similar result could also be achieved through a suitable increase in the translocation rate of internalized resources (and, indeed, it is well established in experimental investigation that nutrient reallocation patterns do change in periods of combat in a species-specific manner, Boddy, 2000). Despite the inability to exclude one of these alternative hypotheses, the modelling has demonstrated that species-specific changes in metabolic processes are involved during fungal combat to generate the intransitive hierarchies observed in fungal communities and that they only need arise in the typically inferior combatant. Fungi are among the most diverse and versatile groups of organisms and form crucial components in many ecosystems through their roles in nutrient cycling and their ability to form symbiotic connections with plant root systems. Furthermore, many recent biotechnological advances (e.g. Gadd, 1992) have

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involved the application of specific species of fungi to environments containing other fungal communities. In such settings, competition and combat for resources is common and therefore improving the understanding and predicting the outcome of these interactions can further improve the efficiency and potential of such applications. This investigation has focussed attention on a small number of physiological traits that appear to be crucial in determining the outcome of such interactions. Extending these ideas, utilizing both mathematical modelling and experimental data, will further aid in the understanding of fungal community dynamics.

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