the development of fungal networks in complex environments · 2011-10-06 · growth is sparse as is...

Bulletin of Mathematical Biology (2007) 69: 605–634DOI 10.1007/s11538-005-9056-6

ORIGINAL ARTICLE

The Development of Fungal Networks in ComplexEnvironments

Graeme P. Boswella,d,∗, Helen Jacobsb, Karl Ritzc,e,Geoffrey M. Gaddb, Fordyce A. Davidsona

aDepartment of Mathematics, University of Dundee, Dundee, DD1 4HN, UKbDivision of Environmental and Applied Biology, Biological Sciences Institute, Schoolof Life Sciences, University of Dundee, Dundee, DD1 4HN, UK

cSoil-Plant Dynamics Group, Scottish Crop Research Institute, Invergowrie, DD25DA, UK

dPresent address: Division of Mathematics and Statistics, School of Technology,University of Glamorgan, Pontypridd, CF37 1DL, UK

ePresent address: National Soil Resources Institute, Cranfield University, Silsoe, MK454DT, UK

Received: 11 June 2004 / Accepted: 17 May 2005 / Published online: 14 July 2006C© Society for Mathematical Biology 2006

Abstract Fungi are of fundamental importance in terrestrial ecosystems playingimportant roles in decomposition, nutrient cycling, plant symbiosis and pathogen-esis, and have significant potential in several areas of environmental biotechnol-ogy such as biocontrol and bioremediation. In all of these contexts, the fungi aregrowing in environments exhibiting spatio-temporal nutritional and structural het-erogeneities. In this work, a discrete mathematical model is derived that allows de-tailed understanding of how events at the hyphal level are influenced by the natureof various environmental heterogeneities. Mycelial growth and function is simu-lated in a range of environments including homogeneous conditions, nutritionally-heterogeneous conditions and structurally-heterogeneous environments, the latteremulating porous media such as soils. Our results provide further understandingof the crucial processes involved in fungal growth, nutrient translocation and con-comitant functional consequences, e.g. acidification, and have implications for thebiotechnological application of fungi.

Keywords Hybrid cellular automata· Fungal mycelia· Modelling

∗Corresponding author.E-mail address: [email protected] (Graeme P. Boswell).

606 Bulletin of Mathematical Biology (2007) 69: 605–634

1. Introduction

Of the major groups of soil microorganisms, fungi are of fundamental importancewith primary roles in decomposition, plant pathogenesis, symbiosis and the cy-cling of important nutrient elements, e.g. carbon, nitrogen, phosphorus, sulphurand many trace metals (Gadd, 1999). Normally forming at least half the microbialbiomass in many soils, filamentous fungi can often form the dominant component,especially in acidic environments or those receiving high organic inputs. They arefunctionally diverse and, because of their filamentous growth habit, can assist inthe stabilization of soil structure (Paul and Clark, 1989; Tisdall, 1994; Chen andStotzky, 2002). Moreover, as a group, they are also active over a much broaderrange of soil pH values than other major groups of soil organisms (Alexander,1964; Paul and Clark, 1989).

In terrestrial environments, the spatial distribution of nutrient resources is notuniform. This is particularly apparent in mineral soils, where an additional levelof spatial complexity prevails due to the fractal-like nature of the pore networkin the solid phases of the soil. Mycelial fungi are well adapted to growth in suchspatially complex environments, since the filamentous hyphae can grow across sur-faces and also bridge air gaps between such surfaces. This ability is significantlyenhanced by the capacity of many species to translocate materials through hyphaebetween different regions of the mycelium. Thus it has been suggested that hyphaegrowing through nutritionally-impoverished zones of soil, or deleterious regions(e.g. localised deposits of organic pollutants, toxic metals, dry or water-loggedzones etc.) can be supplemented by resources imported from distal regions of themycelium (Morley et al., 1996). This has profound implications for the growth andfunctioning of mycelia, and attendant effects upon the environment in which theylive. The fungal mycelium therefore represents an extremely efficient system forspatial exploration and resource exploitation (Olsson, 1999; Rayner et al., 1999;Ritz and Crawford, 1999; Jacobs et al., 2004).

A common approach to modelling the large-scale spatio-temporal properties offungal mycelia has been to model the fungus and growth-promoting substrates ascontinuous variables so that the model comprises a system of (non-linear) partialdifferential equations (PDEs) (e.g. Edelstein, 1982; Davidson, 1998; Boswell et al.,2003a, and references therein). For example, in Boswell et al. (2003a), five (contin-uous) variables are used to model active hyphal density, inactive hyphal density,hyphal tip density, internal substrate concentration and external substrate concen-tration. This modelling strategy allowed translocation to be explicitly modelled ina variety of habitat configurations as well as certain functional consequences offungal growth, such as acid production. Such an approach is ideal when modellingdense mycelia, for example growth in Petri dishes or on the surfaces of solid sub-strates such as foodstuffs, plant surfaces and building materials. However, whengrowth is sparse as is observed in nutrient-poor conditions or in structurally hetero-geneous environments, such as soils, then a discrete modelling approach is moreappropriate where individual hyphae are identified. Such discrete models havebeen derived previously and usually take the form of computer simulations (e.g.Cohen, 1967; Lindenmayer, 1968a,b; Hutchinson et al., 1980; Bell, 1986; Kotov andReshetnikov, 1990; Ermentrout and Edelstein-Keshet, 1993; Soddell et al., 1995;

Bulletin of Mathematical Biology (2007) 69: 605–634 607

Regalado et al., 1996; Meskauskas et al., 2004a,b) often derived from the statisticalproperties of the experimental system under investigation. Although these mod-els often yield images that are almost indistinguishable from real fungi grown inuniform conditions, they typically use non-mechanistic rules to generate hyphaltip extension and hyphal branching and thus must be reformulated (or at the veryleast completely re-calibrated) to consider the growth of the same species in dif-ferent environments or to consider the growth of a different species. Furthermore,these models typically neglect anastomosis and, because of the overwhelming com-putational difficulties, have as yet neglected translocation—a process crucial in theextension of hyphal tips with particular relevance to modelling colony growth inheterogeneous environments.

Growth in structured environments is the main focus of our work, and hencein the following, the mycelium will be modelled as a discrete structure. Moreover,anastomosis and translocation will be explicitly included and to facilitate the lat-ter, the substrates will be, as is most appropriate, treated as continuous variables.Therefore, the model is of hybrid cellular automaton type.

Cellular automata (CA) are a popular means of simulating a variety of biolog-ical and physical phenomena (e.g. Caswell and Etter, 1993; Durrett and Levin,1994; Halley et al., 1994; Boswell et al., 1998). The essence of CA is that the spatialdomain is divided into an array of “cells” whose states change according to pre-scribed transition rules. These state transition rules typically contain a stochasticcomponent and depend on the status of neighbouring cells (typically only nearestneighbour, but see examples in Durrett and Levin, 1994, for more complex in-teractions between cells). In this way, simple, local rules can produce seeminglycomplex, global, behaviour. However, the global behaviour of CA can be sensitiveto the choice of the local transition rules and the corresponding parameter val-ues. Indeed, the selection of suitable transition rules and the corresponding modelcalibration, coupled with the choice of the cell state space, are the main intrinsicdifficulties in developing meaningful CA. To overcome these inherent difficulties,in this paper we present a cellular automaton in which the state transition rules aredirectly related to corresponding processes in a previously calibrated and testedcontinuum model (Boswell et al., 2003a). The parameters (and their values) usedhere are precisely those detailed in Boswell et al. (2003a), which are in turn derivedfrom experimental observations (Boswell et al., 2002).

To the authors’ knowledge, a novel property of the model system derived here isthe simultaneous use of a combination of “cell” models (which are used for mod-elling internal/external substrate and hyphal tips) and “bond” models (which areused for modelling active/inactive hyphae). “Cell” models are ideal for modellingthe movement of individual particles since each cell (or site) can take a value cor-responding to the current state of that cell (for example, the presence or absenceof a tip). However, this cellular approach is not suitable for modelling the devel-opment of a network since adjacent cells need not be connected. For example,consider two adjacent fungal hyphae that run parallel to one another. If a purelycell-based approach was used the two hyphae would automatically be connectedto one another and thus would allow the transfer of internally-located material. Inreality, this is unlikely to be the case—such connections will only occur at anasto-mosis points. However, a bond-based approach explicitly models any connections


between adjacent hyphae and thus can accurately model the formation of and in-ternal substrate redistribution inside parallel hyphae showing various degrees ofanastomosis.

The model is described in Section 2 and is derived using a discretization methodthat offers a simple and meaningful link between classical continuum models andthe often more appropriate discrete formulation (see, e.g. Anderson et al., 1997;Anderson and Chaplain, 1998). This method is similar in spirit to the derivationof CA via a consideration of biased random walks: both techniques result in thederivation of a “master equation” in which the rate of change of the value of asystem variable at a given point is related, via “transition probability rates” (move-ment probability rates), to the values of this variable at a number of neighbouringpoints (see, e.g. Othmer and Stevens (1997); Hillen and Othmer (2000) for the re-lated underlying theory of biased random walks and Plank and Sleeman (2004)for a recent review and comparison of different techniques for generating masterequations. See also Zheng et al. (2005); Sun et al. (2005) for recent adaptationsand applications of the Anderson discretization method.).

In the context of the continuum fungal model of Boswell et al. (2003a), thediscretization procedure allows certain key processes, including hyphal inactiva-tion and reactivation, anastomosis and branching to be treated in a more detailedmanner. These processes represent interactions of elements within the automaton(representing tips with hyphae, for example). Therefore, as discussed in Ander-son (2003), the discretization technique employed here generates a CA that willnot reduce to the original continuum equations in the limit as the space and timesteps are reduced to zero. This is not the intention of this technique. Rather, it is toprovide a framework on which a far more detailed automaton can be constructed.

The model is calibrated for the fungus Rhizoctonia solani, a ubiquitous soil-borne fungus that shows strong saprotrophic growth, can be a pernicious plantpathogen and also has potential use in areas of biotechnology (Ogoshi, 1987;Cartwright and Spurr, 1998; Thornton and Gilligan, 1999; Bailey et al., 2000). InSection 3, the model is applied to what represents an initially homogeneous en-vironment and various quantities including the fractal dimension of the modelbiomass structure and the production of acidity are compared to data obtainedexperimentally. The processes responsible for substrate translocation are reconsid-ered in Section 4 to simulate growth in nutritionally heterogeneous environmentsand the modelled quantities are compared to experimentally-obtained data con-cerning a variety of growth domains. In Section 5, the effect of temporal nutritionalheterogeneity is considered culminating in Section 6 in which fungal growth is sim-ulated in an environment emulating porous media that exhibits spatio-temporalnutritional and structural heterogeneity. Finally, in Section 7, the implications ofthe modelling results are discussed in terms of both the natural functions of fungiand their potential for applications in biotechnology.

2. A model for fungal growth

Fungi are well adapted for growth in soils due to their branching architectureand filamentous growth habit. In general, fungal hyphae elongate strictly by the


apical deposition of wall skeletal polysaccharides (Gooday, 1995). Underlying thisgrowth process is the forward movement of a variety of types of vesicles that pro-vide new cell membrane material and thus generate the extension of hyphal tips.As a result of this hyphal tip movement, hyphae are able to penetrate solid sub-strata such as chitin and, by the excretion of lytic enzymes, are able to uptakevarious nutrients in the form of solutes.

Translocation is an important process that allows the redistribution of internalmetabolites throughout the fungal mycelium. The ability of fungi to translocatehas long been acknowledged (Littlefield et al., 1965) while more recent studieshave focussed on the role of translocation in growth and function in heteroge-neous environments. In particular, Olsson (1995) demonstrated that several fungi,including R. solani, exhibit at least two translocation mechanisms: simple diffusionand the active movement of intracellular metabolites from regions of local excessto regions of local scarcity.

Following the previously successful approach of Boswell et al. (2003a), the fun-gal mycelium is here modelled as a distribution comprising three components: ac-tive hyphae, inactive hyphae and hyphal tips. Active hyphae refer to those hyphaeinvolved in nutrient uptake, branching and translocation while hyphal tips denotethe ends of these hyphae. Inactive hyphae denote hyphae no longer directly in-volved in translocation, branching or uptake and represent, for example, moribundhyphae.

We distinguish between nutrients located within the fungus (internal) and thosefree in the outside environment (external). Internally-located material is used toextend hyphal tips (and thus a hypha may be regarded as a “trail” left behind ahyphal tip as it moves), initiate branching, and drives nutrient uptake. In all en-vironments, a combination of elements are required for growth; of particular im-portance are carbon, hydrogen, nitrogen, sulphur, phosphorus, oxygen, and otherelements including metals. However in this study, for simplicity, it is assumed thata single generic element is drives growth. We assume that this substrate is carbonbecause of its central role in fungal energy-yielding metabolism, biosynthesis andacid excretion (Gadd, 1988; Boswell et al., 2002).

We will restrict our attention to modelling growth in the plane. This is appro-priate for comparison with growth experiments in Petri dishes and for studies ofthin soil sections. Of course, growth in soils will be fully three-dimensional, but weview the restriction to the plane as being a relevant and necessary first step.

2.1. Model construction

The internal and external substrate distributions are modelled using a hexag-onal array and the substrate concentrations are assumed to be homogeneouswithin each hexagonal cell. However, this standard “cellular” approach is notideal for modelling the hyphal network since it would force all adjacent “cells”occupied by hyphae to be connected to each other. Therefore, to model thehyphal network we apply a “bond”-type model (cf. site and bond percolationmodels, e.g. Stauffer, 1985): the mycelium is modelled using the triangular lat-tice embedded in the hexagonal lattice used to model the substrate distributions.Specifically, the vertices of the triangular lattice (located at the centres of the


hexagonal cells) correspond to the possible locations of hyphal tips, while theedges connecting adjacent vertices denote the possible locations of hyphae. Noticethat this particular lattice configuration has the further desirable property that thebranching angles are modelled as 60◦, which is a much better approximation to thehyphal branching characteristics in many mycelial fungi, including R. solani, thanthe orthogonal branching angles (and subsequent orthogonal growth) that wouldbe obtained through the use of a square lattice. We define �x to be the distancebetween the centres of adjacent hexagonal cells. Thus a physical interpretation of�x is that it can be no smaller than the mean diameter of a hypha (which is, forexample, approximately 10 µm for R. solani—see below).

The model consists of five variables, each of which is defined on a hexagonalarray: m(k, t) and m′(k, t) respectively denote the status of active and inactive hy-phae in cell k at time t , p(k, t) denotes the presence or absence of a hyphal tip,and si (k, t) and se(k, t) respectively denote the level of internal and external sub-strate. The state space of the cells containing substrate, si (k, t) and se(k, t), arenon-negative continuous variables while p(k, t) takes either the value unity or zerorespectively denoting the presence or absence of a hyphal tip in that cell. The vari-ables m(k, t) and m′(k, t) hold information on the presence or absence of hyphaeconnecting cell k to its six neighbours. Thus the state space of both m(k, t) andm′(k, t) is discrete and takes one of 26 possible values, corresponding to the pres-ence or absence of hyphae in each one of the six directions.

2.1.1. Tip movement and hyphal creationThe migrating hyphal tips of many fungi (including R. solani) do not in generalreact to gradients of external nutrients (Gooday, 1975) although they may ex-hibit chemotropic behaviour in response to gradients of toxic metals (see, for ex-ample, Fomina et al., 2000). Indeed, hyphal tips in most mycelial fungi, includ-ing R. solani, grow predominantly in a straight line but exhibit small directionalchanges at seemingly random moments in time (see, for example, Dix and Web-ster, 1995). This directed growth habit is essentially caused by the structure of thehyphal walls and the manner in which new wall material is incorporated at thehyphal tip.

We assume that two processes are involved in tip movement: a dominating de-terministic (i.e. convective) component, accounting for the directed nature of tipmovement, and a stochastic (i.e. diffusive) element accounting for the small vari-ations in the direction of hyphal growth. The directed nature of tip movementcan be modelled by assuming a hyphal tip avoids its own hyphal trail whereasthe random element of movement can be modelled by diffusion (Boswell et al.,2003a). Thus tip movement is essentially modelled as a biased random walk onthe vertices of the embedded triangular lattice and is assumed to depend upon thestatus of the hexagonal cell currently occupied by the tip. Note that each hexag-onal cell containing a tip must also contain a single hypha. The orientation ofthis hypha determines from which cell the tip had originated; this information isimportant for modelling the straight-line growth habit of hyphal tips. Note alsothe caveat detailed above concerning the interaction of these tips with hyphaeand with each other, which prevents an exact comparison with a biased randomwalk.


Fig. 1 The movement probabilities of a hyphal tip (solid circle) during the time step �t areρ = 0, µ = Dpsi

�t�x2 , λ = Dpsi

�t�x2 + vsi

�t�x , η = Dpsi

�t�x2 , and ξ = 0. The probability of remaining

stationary is defined to be 1 − (µ + λ + η). The thick solid line denotes an existing hypha andthe dashed lines denote the possible hyphal tip extension routes. See the text for details on thederivation of these probabilities.

Time is modelled in discrete steps of size �t . Therefore after each time step �t ,there is a probability distribution of tip movement into neighbouring cells. Thevalue of �t is initially set (and moreover checked at each subsequent time step) tobe sufficiently small to ensure that these movement probabilities are all boundedbetween zero and unity. If any movement probability falls outside this range, thenthe simulation is aborted and restarted with a reduced value of �t . That we allowthe time step to vary sets this method apart from that developed by Andersonet al. Also, as indicated in the legend of Fig. 1, the probability of no movementin our CA is defined to be P(no movement) = 1 − P (movement). This is againdifferent to the corresponding movement probability derived by Anderson et al.In particular, our movement probabilities always sum to unity, provided the timestep �t is chosen sufficiently small. We view this to be a more natural scaling inthis context as it allows a fixed spatial distance (the grid spacing—�x in our case)to be traversed in varying amounts of time, i.e. for the direct modelling of a varyingtip velocity (rather than prescribing the distance travelled in a fixed time step to bezero or some fixed value). We now motivate and define these state space transitionrules.

The structure of the hyphal tip and the associated hypha is such that the hyphaltip cannot grow back on itself since that space is already occupied by the (immo-bile) existing hypha. Thus we set the probability of a tip performing a completereversal of direction to be zero. Moreover, “sharp” bending in hyphae is not gen-erally observed in unobstructed domains. Consequently the corresponding proba-bilities (ρ and ξ in Fig. 1) are also set to zero. Therefore, after each time step �t ,a hyphal tip is assumed to either remain stationary or move forward into one ofthe three cells in front of the hyphal tip. We assume that tip diffusion is distributeduniformly in these three possible directions, while tip convection occurs only inthe single direction in line with the tip’s trailing hypha. To encapsulate the centralrole of internal substrate in tip movement, both components of tip movement are


assumed to depend (linearly) on internal substrate. To aid the construction andcalibration of these movement probabilities, we relate the discrete process of tipmovement to the continuous version of tip flux considered in Boswell et al. (2003a).Specifically, we choose the diffusion and convection movement probabilities suchthat they relate to the coefficients in a finite-difference discretization of the contin-uous PDE processes. Hence, we assume that the probability of tip movement dueto diffusion between adjacent cells during the current time step (t, t + �t) is

P (movement due to diffusion) = Dpsi (k, t)�t�x2

, (1)

while the probability of convective movement is

P (movement due to convection) = vsi (k, t)�t�x

, (2)

where Dp and v are non-negative parameters denoting the magnitude of the dif-fusive and convective component of hyphal tip movement. The movement prob-ability in (1) is similar to the corresponding coefficient in a central difference ap-proximation used in the numerical treatment of parabolic PDEs. Moreover, be-cause of how the hyphal distribution is modelled, the state of a given (hexagonal)cell contains information on the state of neighbouring (hexagonal) cells and hencegradients of hyphal density across neighbouring cells are implicitly included in themovement probability (2). In this way, the movement probability in (2) is similar tothe coefficient obtained in a finite difference approximation of a hyperbolic PDE.These similarities allow us to use the corresponding calibrated parameter valuesfrom the continuum model.

If a model tip moves, for example, from cell k at time t to a neighbouring cellq after the time step, the location of the resultant hypha (the new connection be-tween the centres of cells k and q, which recall is of length �x) is recorded inthe entries m(k, t + �t) and m(q, t + �t). Moreover, the direction of growth isalso recorded. The cost of hyphal tip extension and resultant hypha creation isaccounted for by a change in the internal substrate in the originating cell. We rea-sonably assume (see also Boswell et al., 2003a) that the cost of producing a unitlength of hypha is a constant, c2, and thus when a tip moves between neighbouringcells the internal substrate in the originating cell is reduced by an amount c2�x.

Pulsatile growth, i.e. regular slow-fast tip growth was first observed some timeago, (see Lopez-Franco et al., 1994). Recently, experimental evidence has beenpresented that suggests the variation in tip growth rate is random (see Sampsonet al., 2003). Certainly, the tip growth mechanism included in our model producesa random “move/do not move” event at each time step and as such produces arandom variation in hyphal tip extension rate.

2.1.2. Branching and anastomosisNew hyphal tips are created by branching. Both turgor pressure and the build-up of tip vesicles have been implicated (Webster, 1980; Gow and Gadd, 1995)in the commencement of the branching process. As in Boswell et al. (2003a), we


assume that internal substrate is responsible for branching and, in the absenceof any experimental evidence to the contrary, we assume that the branching rateis linear with respect to the amount of internal substrate. Thus with probabilitybsi (k, t)�t , where b is a non-negative constant, a branching event may occur incell k during the time interval (t, t + �t). The possible branching directions froma hypha are determined by the orientation of that hypha. Specifically, consistentwith the physiology of R. solani, a new branch is assumed to only arise at acuteangles to the direction of hyphal growth. The location of the new branch is uni-formly selected from all possible branching routes and if none are available (e.g. ifall branching directions are occupied by hyphae) then no branch is created.

Anastomosis, the fusion of a hyphal tip into another hyphae, is modelled as oc-curring when a model tip moves into a hexagonal cell that already contains biomass(either an active or inactive hypha). The model tip is removed from the corre-sponding array and a new connection is established in the network.

2.1.3. Maintenance and inactivation of active hyphaeActive hyphae in mycelial networks require maintenance. If not appropriatelymaintained, for example, by the reinforcement of hyphal walls through the con-tinual deposition of skeletal polysaccharides (Gooday, 1995) hyphae appear to be-come inactive and cease to play a major role in growth and translocation.

The maintenance of active hyphae is modelled by assuming that a constantamount of internal substrate is required to maintain a unit length of hyphae perunit time. Specifically, for each active model hypha originating in cell k, the main-tenance of that model hypha for a time step �t corresponds to a subtraction ofc5�x�t from si (k, t). If the subtraction results in the internal substrate being be-low some quantity ω > 0 it is assumed that insufficient energy is available to main-tain the model hypha and therefore it becomes inactive. In such circumstances, theamount of internal substrate in cell k is set to be ω (assumed to be bound in thewall materials or stored within the vacuole).

Inactive hyphae can be degraded in the external environment and the possibilityof this event is modelled by a Poisson process of constant rate di . Model hyphaethat have degraded are removed from the appropriate array cells.

2.1.4. Substrate uptakeIt has been observed that internal substrate is generally absorbed autocatalytically,that is internal substrate is used to acquire external substrate by active transportacross the plasma membrane (facilitated by the production and exudation of pro-tons and other metabolites—see later). The acquisition rate must therefore dependon the amount of internal substrate available to perform the active transport, theamount of external substrate available for absorption, and the hyphal surface areaover which the absorption occurs.

As in Boswell et al. (2003a), we assume that the uptake rate depends onboth hyphal surface area and the amount of internal substrate. Moreover, inthe experimental protocol used to calibrate the model (and in most terrestrialgrowth habitats) the external substrate levels are such that saturating behaviour is


unlikely to be observed. Thus, following Boswell et al. (2003a), the change in in-ternal substrate in cell k through uptake after the time step �t is

�uptakesi (k) = c1si (k, t)m(k, t)se(k, t)�t, (3)

where c1 is a non-negative constant and m(k, t) denotes the total length of hyphaein cell k at time t (taking one of the values 0,�x/2,�x, . . . , 3�x).

The depletion of external substrate is modelled in a similar fashion, namely

�uptakese(k) = −c3si (k, t)m(k, t)se(k, t)�t, (4)

where c3 is a non-negative constant. The cost of uptake is accounted for by settingc1 < c3 so that there is an imperfect conversion of external substrate into internalsubstrate.

2.1.5. TranslocationIn line with experimental evidence (see e.g. Olsson, 1995; Jacobs et al., 2002), weassume that two different translocation mechanisms are responsible for nutrient(in particular, carbon) reallocation in R. solani: simple diffusion and the activemovement of intracellular metabolites from regions of local excess to regions oflocal scarcity.

In Jacobs et al. (2004), a number of experiments are described that measurethe translocation rates of 13C in growing and established colonies of R. solani be-tween two domains of different nutritional status. The experiments suggest thatonly newly-formed hyphae (and associated hyphal tips) use active translocation,while older, established hyphae use diffusion as the major means of internal nu-trient reallocation. Therefore, we model active translocation as occurring in thevicinity of hyphal tips. Furthermore, it is reasonable (and consistent with exper-imental observations, see, for example, Fisher-Parton et al., 2000; Jacobs et al.,2004) to assume that the active translocation of material is directed towards hy-phal tips.

Internal substrate is modelled as a continuous variable defined in the cells of thehexagonal lattice and consequently, translocation is most appropriately modelledusing a deterministic process. Clearly, translocation can only act between cells thatare connected by active hyphae. Standard conservation laws show that the changein internal substrate in cell k due to translocation is

�transsi (k) = Mk,k1 + Mk,k2 + · · · + Mk,k6 ,

where Mk,kj denotes the amount of material that has passed across the six bound-aries of cell k during the time period �t and where k1, . . . , k6 denote the six neigh-bours of cell k. As outlined above, there are two processes governing the move-ment of material between neighbouring cells k and q, say: a diffusive component(Mpas

k,q ), and a convective component (Mactk,q). We first consider the contribution of

substrate diffusion.


Simple diffusion between cell k and cell q is characterized by (see Anderson andChaplain, 1998, for further details on the discretization method)

Mpask,q =

⎧⎨

⎩

Di�t si (q,t)−si (k,t)�x2 ,

if cells k and q are connectedby an active hypha,

0, otherwise,(5)

where Di ≥ 0 is a species-dependent constant describing the diffusion of internalmaterial. Implicit in (5) is the reasonable assumption that all hyphae exhibit similardiffusion characteristics.

The active translocation of internal substrate is modelled by a convective fluxdirected towards hyphal tips. Specifically, internal substrate is moved to a cell con-taining a hyphal tip from a neighbouring cell only if there exists an (active) hyphaconnecting them. In Boswell et al. (2003b), a numerical method is described thatis used to treat a continuous version of a related flux by exploiting the positivityproperties of flux limiters (see also LeVeque, 1992). An analogous version of thatnumerical method yields the approximation

Mactk,q =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

Dap(k,t)−p(q,t)

�x si (q, t) �t�x ,

if p(k, t) − p(q, t) > 0 and cells k andq are connected by an active hypha,

Dap(k,t)−p(q,t)

�x si (k, t) �t�x ,

if p(k, t) − p(q, t) < 0 and cells k andq are connected by an active hypha,

0, otherwise,

(6)

where Da is a species-dependent non-negative constant describing the strength ofactive translocation. To account for the metabolic cost of active translocation, aquantity c4 Mact

k,q is subtracted from cell si (k, t) or cell si (q, t) depending on whetherp(k, t) < p(q, t) or p(k, t) > p(q, t) respectively and where c4 is a positive con-stant.

2.2. Calibration and initial data

The model was calibrated using the parameter values obtained from specificgrowth experiments where a plug of R. solani Kuhn anastomosis group 4 (R3)(IMI 385768) was cultured on mineral salts media (MSM) containing 2% glucose(w/v) at 30 ◦C. The approximate tip velocity and branching rate were estimatedby image analysis techniques on enlarged images of mycelia grown over a 15 htime period while other parameters, namely the diffusion of internal and externalsubstrate and the uptake rate of the substrate, were taken from the literature (e.g.Jennings et al., 1974; Olsson, 1994, 1995; Olsson and Gray, 1998). See Boswell et al.(2002) for full details of the calibration procedure. The time scale used in all figuresrepresent days. Again by comparison with radial growth experiments, an approx-imate length scale was determined in which the diameter of each cell represents0.1 mm. Therefore, the simulations illustrated in the figures below, correspond togrowth domains of approximately 1 cm2.

To be consistent with typical experimental protocol, in the following simulationsthe initial data is chosen to represent a plug of active mycelium being placed onto afresh solid substrate. Specifically, the central hexagonal cell is set to contain active


hyphae radiating out in all six possible directions and the six surrounding cells areset to hold hyphal tips that are connected by active hyphae to the central cell. Thus,the initial biomass distribution is ‘star-shaped’.

3. Initially homogeneous environments

Homogeneous conditions, for example, uniform agar distributions on Petri dishes,are the simplest environments in which the growth of mycelial fungi can be exam-ined. There is a significant body of qualitative and quantitative data on biomassdensities, structural properties (e.g. the fractal dimension of the mycelium) andmetabolic activities, e.g. acid production, in such conditions (see, for example, Lit-tlefield et al., 1965; Ritz and Crawford, 1990; Carroll and Wicklow, 1992; Gow andGadd, 1995; Sayer et al., 1995; Olsson and Gray, 1998; Boswell et al., 2003a, andreferences therein).

3.1. Distribution and diffusion of external substrate

To replicate an initially homogeneous environment, the initial external substratelevel was set to be uniform throughout the growth domain. Consistent with theinitial data in the calibration experiment where the growth medium had 3 × 10−5

mol glucose cm−2, we set se(k, 0) = 3√

32 × 10−5�x2 since each hexagonal cell has

area√

3�x2

2 .The change in external substrate in cell k after time �t caused through substrate

diffusion is

�difse(k) = Mdifk,k1

+ · · · + Mdifk,k6

, (7)

where Mdifk,q denotes the flux of external substrate between cells k and q during �t

and k1, . . . , k6 denote the six neighbouring cells of cell k. The external substrate isassumed to diffuse in a standard Fickian manner and therefore we set

Mdifk,q = De�t

se(q, t) − se(k, t)�x2

, (8)

where De is a non-negative constant.To replicate the conditions of mycelia growing in Petri dishes, zero-flux bound-

ary conditions are applied on the edge of the hexagonal lattice. These boundaryconditions are implemented by defining Mdif

k,kj= 0 where appropriate to prevent

substrate diffusing outside the growth domain. It has been determined experimen-tally that the diffusion coefficient of glucose in agar is 4 × 10−6cm2s−1 (or 0.3456cm2 day−1, see e.g. Olsson, 1995) and hence in all that follows, De is set to be thisvalue.

The zero-flux boundary conditions described above impose further restrictionson the movement of hyphal tips. To prevent tip movement outside the growth do-main, the tip movement probabilities are appropriately rescaled to be consistentwith experimentally observed behaviour: the movement probabilities that would


Fig. 2 Development of the model mycelium in an initially homogeneous environment. Parametervalues are v = 105, Dp = 104, b = 106, Di = De = 0.3456, c1 = 10, c2 = 10−7, c3 = 102, c4 = 10−8,c5 = 10−9, and ω = 10−13. The initial amount of external substrate in each cell was se0 = 10−11

while a similar amount was used for the initial internal substrate (confined to the cells initiallycontaining biomass). The model network is shown at times (a) t = 0, (b) t = 0.33, (c) t = 0.66,and (d) t = 1.

take the tip outside the growth domain are set to be zero and the “excess” prob-ability (i.e. that originally taking the tip outside the domain) is uniformly redis-tributed between the remaining movement possibilities. If no transitions are avail-able, the hyphal tip is assumed to “stagnate” and is effectively “removed” from thesimulation.

3.2. Biomass development in homogeneous conditions

The model system was calibrated and initial data was applied to what representedan active plug of mycelium being placed onto an initially homogeneous growthhabitat configuration as described above. A network of biomass expanded from


the centre of the growth domain resembling the typical branched interconnectedstructure of a mycelium (Fig. 2). The biomass network was most dense at the centreand the density decreased towards the biomass periphery (cf. biomass densitiesmeasured experimentally and those obtained by a continuum model, Boswell et al.,2002, 2003a, respectively).

It has long been recognized that a mycelial network displays certain fractal prop-erties. This observation was quantified by Ritz and Crawford (1990) who approx-imated the “fractal dimension” of mycelia at various stages of growth using thebox-counting technique (see, for example, Barnsley, 1988, for a description of themethod). It was observed that the fractal dimension of the mycelial networks ex-amined increased from 1.4 to approximately 2.0 as the colony expanded from aninoculum.

Similar to real mycelia, the model biomass distribution also resembles a fractal-like structure (Fig. 2 and cf. Regalado et al., 1996). However, because of the regu-lar geometry underlying the biomass network, the resultant structure is not strictlyfractal. It is worth noting that real mycelia are not strictly fractal either and that the“fractal dimension” determined as described above is purely an effective methodof quantifying the structure of the mycelial distribution. However, provided the“fractal dimension” of real mycelia and the model biomass are approximated ina consistent manner, a comparison between the two structures is possible. Hence,the fractal dimension of the model biomass was determined using the box-countingmethod with boxes of sides between 5�x and �x/5. (Smaller box sizes than thiswould fail to capture the fractal-like properties of the biomass network and in-stead treat the model structure as a series of straight lines; a similar minimum boxsize was used by Ritz and Crawford, 1990.) The box-counting method is applied bysuper-imposing a square lattice over the triangular lattice and counting the num-ber of squares containing any amount of model biomass. The fractal dimension isthen approximated in the standard fashion by determining of the gradient of theregression line of log N on log r , where N and r denote the number of boxes andthe side of the boxes respectively.

The fractal dimension of the model biomass was calculated for four differentgrowth conditions; the first representing the conditions of the calibration exper-iment, and the remaining representing progressively lower concentrations of nu-trients (i.e. modelled with progressively lower values of se in each cell). As in theexperimental case, the fractal dimension of the biomass network increased overtime (Fig. 3). The initial rate of increase in the fractal dimension appears to beindependent of the level of the external substrate suggesting that the initial levelof internal substrate (the same in each simulation) determines the structure dur-ing this phase (see also Fig. 3 (d)). However, once this initial expansion phase hadpassed, the external substrate levels did affect the fractal dimension of the biomassstructure. Indeed, with larger levels of external substrate, the fractal dimension ofthe biomass structure increased faster and to a greater final value. Notice that inthe simulation representing the calibration experiment conditions, the fractal di-mension of the biomass structure increased over time from approximately unity toa maximum of 1.8 (Fig. 3), which compared favourably with the experimental dataof Ritz and Crawford (1990).


Fig. 3 The fractal dimension of the model biomass increases as the initial distribution expands inwhat represents an initially homogeneous environment. The fractal dimension is measured usingthe box-counting method but with a suitable restriction on the minimum box size to account forthe regularity of the underlying model structure. (a) shows the fractal dimension of the modelbiomass expanding in what represents the calibration conditions, se = se0 , (b) denotes the fractaldimension where se = se0 /10, (c) where se = se0 /100 and (d) where se = 0.

3.3. Acidification of the environment as a consequence of fungal growth

Acidification of the environment is a frequently observed consequence of fun-gal growth and arises from, for example, proton efflux and organic acid excre-tion. The production of acidity allows the fungus to solubilize otherwise insolublecompounds (Gadd and Sayer, 2000; Gadd, 2001; Jacobs et al., 2002) but also hasa wider affect on soil geochemistry (Gadd, 1999). It has been shown that R. solanionly produces acidity in the presence of a utilizable carbon source (Jacobs et al.,2002). This is represented in the model by internal substrate and the productionof acidity can be related to this quantity. As in Boswell et al. (2003a) and in theabsence of any experimental evidence to the contrary, we assume the productionof acidity is (linearly) proportional to the amount of internal substrate, and onceexuded, is assumed to diffuse throughout the growth domain. Thus, in terms of apartial differential equation where a(x, t) denotes the acid concentration at posi-tion x at time t , the acidity is modelled by

∂ a∂t

= Dacid∇2a + c6si , a(x, 0) = B(x), (9)


where B(x) denotes the underlying acidity in the growth medium prior to inocula-tion, Dacid is the diffusion coefficient of the acid, and c6 is a non-negative constant.Consistent with the standard definition of pH, the model pH is then taken to be− log10(a(x, t)). The cost associated with acid production is taken account of in thegeneral costs for uptake discussed above.

The PDE (9) was discretized using the techniques described above and accord-ingly a new variable a was introduced, where a(k, t) denotes the amount of acidityin cell k at time t . The parameters Dacid and c6 were determined from experimentsconducted in homogeneous media (see Boswell et al., 2003a, for details).

The model system was calibrated as described above and applied to what rep-resented an initially homogeneous growth environment. As the biomass expandedfrom the centre of the growth domain (as in Fig. 2), acidity was produced through-out the biomass network (Fig. 4). The diffusion of acidity created a zone of acidi-fication, which extended beyond the periphery of the biomass, and was similar tothat observed in experimental systems (see, e.g. Jacobs et al., 2002).

4. Heterogeneous nutrient distributions

The use of filamentous fungi in biotechnological applications is increasing andconsequently the study of fungal growth in patchy environments has received in-creased attention (see, for example, Olsson, 1995; Ritz, 1995; Boddy, 1999; Pers-son et al., 2000; Jacobs et al., 2002). These and other studies have shown howfungi use translocation to reallocate nutrients and assist growth and function inenvironments where nutrients can be spatially and temporally localized. In suchenvironments, two distinct growth phases have been identified; exploration andexploitation. The exploration phase is adopted in low-nutrient environments andfeatures fast moving hyphal tips coupled with minimal branching, resulting in asparse mycelial network. The exploitation phase is adopted in high-nutrient con-ditions and features slower moving hyphal tips and increased branching and anas-tomosis, resulting in a dense mycelial network. However, the precise mechanismsthat determine which growth strategy is adopted are unknown. Moreover, it is alsounknown whether the phases are a combination of independently-acting processesor the consequence of a single process. We propose that the exploration and ex-ploitation phases are controlled predominantly by the translocation process andthat changes in the active translocation process alone account for the switch be-tween exploration and exploitation phases.

4.1. Revised translocation mechanism

A slight extension of the basic model is required to test the above hypothesis. First,we distinguish between two types of hyphae; those in substrate-rich conditions(where the amount of external substrate exceeds a certain level, φ, say) and thosein substrate-poor conditions (where the amount of external substrate is less thanor equal to φ). The only difference between these two classes of hyphae is in therate at which internal substrate is actively translocated: the active translocationrate in substrate-poor conditions, denoted by Da,p, is assumed to be an order of


Fig. 4 The model acidity, plotted using a universal pH colour indicator, increases from pH 7(green) to pH 3 (red) as the model mycelium expands from its initial distribution. Parametervalues are as in Fig. 2 with Dacid = 0.3456 and c6 = 10−10. The acidity is shown at times (a) t = 0,(b) t = 0.33, (c) t = 0.66, and (d) t = 1.

magnitude greater than the active translocation rate in substrate-rich conditions,denoted by Da,r , as calibrated previously. Note that an increased translocation ratewill have an increased cost associated with it. However, we assume the uptake,maintenance, branching rates and growth costs to be identical for both distincthyphal types (see discussions in Section 7).

The growth environments considered in the model system (see below) reflectthose considered in experimental systems where the external substrate was con-fined to a number of regions within the growth domain. Thus, there is a clear parti-tion between those hyphae in substrate-rich conditions and those in substrate-poorconditions and so we henceforth set φ, defined above, to be zero.


Fig. 5 The internal substrate distribution (blue) and model biomass network (red) is plotted foroutgrowth into a substrate-free region at times (a) t = 0, (b) t = 0.1, (c) t = 0.2, and (d) t = 0.3.The only external substrate in the system is confined to a central square region of the domainwhose boundary is shown in (a). The dark blue and light blue cells respectively denote high andlow levels of internal substrate. For purposes of improved visualisation, the internal substrate sta-tus within each model hypha is represented by the colour of the hypha’s corresponding hexagonalcell.

4.2. Distribution of internal substrate

The cells corresponding to a central square were set to contain external substratelevels used in the calibration experiment, while the level was set to zero in theremainder. In order to examine the consequence of a wholly localised nutrient re-source (for example, a wood block), in the model the diffusion of external substratewas assumed to occur between cells within the central square but not across the


boundary of this square. The central cell was “inoculated” with biomass as beforeand the model system was allowed to develop in the manner described above.

The biomass expanded over the central square and model hyphae extendedinto the zero-substrate region. The increased rate of translocation induced moreinternal substrate to be carried at tips extending across zero-substrate regionsthan compared to tips extending across substrate-rich conditions (Fig. 5). Suchan internal substrate distribution is consistent with experimental observations ofthe distribution of internal metabolites in the apical compartments of fungal hy-phal (Fisher-Parton et al., 2000).

The increase in active translocation caused hyphal tips to move faster. However,as a consequence of the increased translocation, the branching rate along the trail-ing model hyphae was reduced since less internal substrate was held in the hyphae.

4.3. New resources

An additional consequence of carrying increased amounts of internal substrate atthe tip is that the uptake rate of external substrate immediately behind the tip isincreased (see Eq. (3)). Thus active translocation allows more rapid exploitationof any newly-encountered resources (see also Boswell et al., 2002).

Following a set up very similar to that used in, for example, Boddy (1999) inwhich two wood blocks are placed in a large soil assay, we initially inoculate a cen-tral block of substrate with model biomass as described above. It was observed thatthe model hyphae colonised this central block and then explored the surroundingzero-substrate cells displaying the sparse distribution typical of developing mycelia(Fig. 6(a, b)). However, on encountering a second block of substrate, the modelgrowth dynamics changed and the second block was colonised by slow-moving,branching hyphae (Fig. 6(c, d)).

5. Temporal nutritional heterogeneity

It is generally observed that inactive hyphae can be reactivated under certain cir-cumstances. For example, old, inactive mycelia of R. solani can be reactivated,generating new hyphal growth, branching, uptake, etc. simply by the topical addi-tion of suitable amounts of glucose. This reactivation process is clearly an impor-tant strategy used by fungi to survive in terrestrial systems where, perhaps, longperiods of time can elapse between the supply of fresh resources, for example leaflitter. Thus, whereas nutrient translocation is crucial for fungal growth in envi-ronments exhibiting spatial heterogeneity, hyphal reactivation would appear to beimportant for fungal growth in environments exhibiting temporal heterogeneity insuch resources.

5.1. Reactivation process

Consistent with experimental observations, we assume the reactivation processdepends only on the status of externally-located nutrients. Therefore, we as-sume that inactive hyphae originating from cell k are reactivated if the external


Fig. 6 The model biomass (red) expands from what represents a localised food source (blue) intoa domain that contains a distant resource but is otherwise empty of nutrients. The model biomassis shown at times (a) t = 0, (b) t = 0.5, (c) t = 1 and (d) t = 1.5.

substrate level in cell k, se(k, t), exceeds a certain level, ψ , say. In such circum-stances, the variables m(k, t), m′(k, t), m(kj , t) and m′(kj , t), j = 1, . . . , 6, are ap-propriately updated, where k1, . . . , k6 denote the neighbouring cells of cell k.

Our own observations revealed that inactive hyphae of R. solani were reacti-vated by the addition of small amounts of glucose (approximately 10% of thatinitially in the growth medium MSM) and hence we calibrate the parameter ψ

accordingly.

5.2. Reactivated model biomass

The model system was solved in what represented a zero-nutrient environmentwith initial data as described above. The biomass expanded from the initial dis-tribution using the initially-supplied internal substrate. This resource was rapidly


depleted and hyphal inactivation ensued throughout the biomass network exceptat the (distant) tips, which, because of the active translocation process, had suffi-ciently high local amounts of internal substrate to prevent inactivation (Fig. 7(a,b)). When all the hyphae at the biomass centre were inactive, a block of sub-strate was introduced to the centre of the growth domain (Fig. 7(c)) by settingthe corresponding cells se(k, t) to be the calibrated value determined earlier forMSM. In order to isolate the consequences of introducing a localised resource, theparameter De was set to be zero. The inactive hyphae in contact with this substrateblock were immediately reactivated and substrate uptake automatically followed.As the internal substrate levels increased, branching occurred and more hyphaewere created capable of further uptake producing a second “wave” of biomass.The external substrate was then depleted through uptake by this second “wave”of biomass growth (Fig. 7(d)–(f)).

6. Nutritional and structural heterogeneity

Soils, the terrestrial growth habitat of R. solani, exhibit spatio-temporal, nutri-tional and structural heterogeneity. The structural heterogeneity in soils is deter-mined by the relative location of soil particles and the resulting pore space. Thenutritional heterogeneity is strongly modulated by the ground-water distribution,which itself depends on the architecture of the pore space. In non-saturated soils,water films prevail around pore walls and larger pores are air-filled. Nutrients arein general confined to such water films.

Experimental studies of mycelial growth in soils has typically consisted of exam-ining thin slices of soils (see, for example, Otten et al., 2001; Harris et al., 2002,2003). These soil slices are in essence a two-dimensional object and hence variousproperties, such as the fractal dimension of the growth habitat and the location andabundance of biomass within the growth habitat, can be easily compared betweenthe model and experimental systems.

6.1. Modelling soil-like structures

We constructed artificial structures that emulate heterogeneous porous media suchas soils by randomly “removing” (possibly overlapping) hexagonal blocks of cellsfrom the growth domain. These “removed” cells exhibit the same characteristicsas cells on the edge of the growth domain; that is the cells have zero-flux bound-ary conditions (for the purpose of external substrate diffusion) and also cannot bepenetrated by model hyphal tips. A vector R is defined where the component Rk

(∑

Rk ≤ 1) denotes the proportion of the growth domain to be removed as hexag-onal blocks of radius k, where a block of unit radius denotes a single cell. Thedistribution of the remaining habitat, which corresponds to the pore space, is thusinfluenced by R and may be connected or fragmented. Fundamental properties ofthe model pore space, such as its fractal dimension, can be determined enablingqualitative and quantitative comparisons with real soil systems.

In non-saturated soils, soluble nutrients are mainly confined to a water filmsurrounding soil particles, as described above. The effects of water surface


Fig. 7 Reactivation of model biomass. The images correspond to times (a) t = 0, (b) t = 0.25, (c)t = 0.5, (d) t = 0.75, (e) t = 1, and (f) t = 1.5 (see text for details; parameter values as in Fig. 2).


tension mean that these nutrients diffuse within the film but not across its outersurface. Such a distribution is modelled by choosing the appropriate initial dataand boundary conditions. The initial external substrate distribution is chosen suchthat se(k, 0) is zero for all cells beyond a distance ζ of a “removed” block. The non-zero initial external substrate is thus confined to a banded region (of thickness ζ )surrounding the “removed” cells, representing the water film. Zero-flux boundaryconditions are employed on the edges of this region to prevent substrate diffusingacross its surface. As before, the parameter De denotes the diffusion coefficient ofexternal substrate.

6.2. Surface tension and fungal growth

The movement of hyphal tips in the soil pore space is greatly affected by both thephysical boundaries of the soil particles themselves and also by the surface tensionof the water film surrounding these particles. Experimental investigations reportedby Schack-Kirchner et al. (2000) suggest that mycelial fungi are more commonlyfound growing on the edges of soil particles than in large air-filled pore spaces. Thisproperty is modelled by biasing the tip movement probabilities so that hyphal tipshave a tendency to stay within the water film and biased in such a way that the tipspeed is unaffected. Specifically, the tip movement probabilities are rescaled such

Fig. 8 The tip movement probabilities of Fig. 1 are rescaled to account for the effect of water ten-sion. The thick solid line denotes the location of a hypha, the thin solid lines denote the possibleextension routes of that hypha, and the dashed line denotes the edge of the water film. The prob-abilities of the five rescaled transitions during the time step �t are shown where the parameter θ

encapsulates the relative effect of the water tension (see text for details).


that the probability of exiting the water film in a time step �t is reduced and this“excess” probability is then shared between the chances of movement within thefilm and shared in such a way that accounts for the underlying biased (i.e. growthin a straight line) movement. The rescaled probabilities are then those given inFig. 8. A similar rescaling is used for branching to again account for the difficultyin exiting the water film. Notice that the rescaling does not change the total proba-bility of a tip movement after the time step �t and consequently, the rescaling doesnot alter the “speed” of hyphal tip growth. (A simple reduction of the probabilityof movement in a particular direction without changing the remaining movementprobabilities would alter the “speed” of tip extension.)

6.3. Model growth in soils

A domain exhibiting the characteristics of porous media was constructed as de-scribed above. The external substrate was confined to a region representing thewater film and the appropriate boundary conditions were applied (Fig. 9(a)). Inthe absence of any experimental data, we set θ = 10, so that it is an order of mag-nitude more difficult for a model tip to exit the water than remain within it. A star-shaped “inoculum” of biomass was placed in the centre of the growth domain andthe model system developed according to the rules specified above. Early biomassgrowth was confined to the region representing the water film and the externalsubstrate in this region was taken up and used for tip extension (Fig. 9(b)). Asmall number of tips emerged from the region representing the water film and,consistent with outgrowth experiments and simulations discussed above, extendedrapidly across the model pore space (Fig. 9(c)). By crossing the model pore space,these tips were able to locate new substrate resources which were subsequentlycolonised and exploited. The acidity produced by the biomass was modelled in asimilar manner as described above and took values between pH 3 and pH 7 (Fig. 9(d)–(f)).

The parameter corresponding to the surface tension of the water film was ob-served to play a significant role in determining biomass and acidity distribution.The reduction in the parameter θ resulted in a greater biomass distribution in themodel pore space and a faster overall biomass expansion (Fig. 9 (g)–(i)).

7. Discussion

In this work, a hybrid site-jump cellular automaton model of mycelial growth wasderived and used to investigate the development of mycelial networks in a varietyof habitat configurations including those representing homogeneous conditions,spatial and temporal nutritional heterogeneity, and soil-like structures.

An important feature of the model is the simultaneous use of “cell” and “bond”structures, which allowed the explicit modelling of substrate movement throughneighbouring, but unconnected, hyphae. Clearly a similar modelling technique canbe used to simulate other branching structures where the morphology is depen-dent on the status of material inside the developing network and the explicit mod-elling of anastomoses is essential, for example, in vascular systems. Moreover, the


Fig. 9 Model mycelial growth (a)–(c) and corresponding acidification (d)–(f) in a soil-like envi-ronment at times t = 0 (a, d), t = 2 (b, e) and t = 7, (c, f). The black cells denote soil particles, thedark and light blue cells respectively denote the water film containing high and low amounts ofsubstrate, and the white regions correspond to air-filled pore spaces. The model acidity is shownusing a universal pH indicator range from pH 7 (green) to pH 3 (dark red). Parameter values arethose in Fig. 4 with θ = 10. Typical biomass growth in a similar environment but with reduced sur-face tension (θ = 1) is shown in (g)–(i) at the times corresponding to those in (a)–(c), respectively.

technique applied to model acid production may be applicable to other systems,for example, pheromone trails in insect populations (e.g. Rayner and Franks,1987).

Our model captures key experimentally observed qualitative and quantitativefeatures, such as colony radial expansion rate, biomass distribution, acidificationof the growth environment (see, for example, Sayer et al., 1995; Sayer and Gadd,1997; Gadd et al., 2001; Jacobs et al., 2002; Fomina et al., 2003) and fractal di-mension (see, for example, Ritz and Crawford, 1990). A subset of these features


has previously been modelled with some success using a continuum approach (see,for example, Edelstein, 1982; Edelstein and Segel, 1983; Davidson, 1998; Boswellet al., 2002, 2003a). However, our hybrid approach enables growth and function tobe modelled in a far more precise manner that explicitly details the developmentof the hyphal network. This detail is essential for the study of growth and functionin structured media, for example, soils.

The translocation process in the model system has a significant effect onbiomass morphology. Although this is to be expected, given the central role thattranslocation plays in tip movement, branching and uptake, it was surprising toobserve that our model allows us to conjecture that the changes to hyphal exten-sion and branching rates associated with explorative/exploitative growth, can becontrolled simply by the redistribution of internal substrate, i.e. translocation. Thesimplicity of this mechanism is attractive, as it does not require a raft of metabolicprocesses governing growth and branching to be individually controlled and sub-sequently coordinated. (We did investigate other possible control mechanisms,e.g. coordinated changes to the growth and maintenance costs in response to ex-ternal substrate levels but despite extensive investigation, we were unable to re-produce experimentally observed behaviour.) However, the identification of theproposed switch requires further investigation. In the model, this switch is inducedby external substrate levels reducing below a threshold. The existence and levelof such a threshold could be experimentally investigated by measuring the fractaldimension of colonies grown on a range of growth media. The modelling suggeststhat this threshold should be clearly marked by a significant reduction in the fractaldimension. This again requires further investigation.

In the simulations of fungal growth in porous media, the important role of sur-face tension was considered, with particular emphasis on its effect on tip move-ment. Biomass growth was concentrated in the surface film formed around soilparticles in non-saturated conditions when the relative difficulty of a hyphal tipbreaking through the film was sufficiently large. Increased surface tension result-ing in reduced mycelial growth in open pore spaces has been observed experimen-tally (Schack-Kirchner et al., 2000). This suggests surface tension has importantconsequences in controlling the growth of mycelial fungi in soils. The model pre-dicts that the potential for fungal growth in soils may be enhanced by either manip-ulating the surface tension or by simply adding further surface to the system. (Ofcourse, the latter of these would affect the distribution of oxygen within the porespace and could ultimately lead to anaerobic conditions and therefore limitationsof fungal growth.) In particular, it is predicted by the model that improved explo-ration and growth can be induced without the need to add any further nutrients,which in real systems is always associated with the negative effect of increasedcompetition from other biota. We stress, however, that the soil-water distributionis not modelled in a strictly mechanistic manner. For example, the water film maycontain flows that affect hyphal growth and, therefore, mycelial function. This pos-sibility clearly requires further investigation.

Certain fungi have significant uses as biological control agents against pestsand diseases of plants (Whipps, 2001) while others have the ability to transformtoxic metals in the context of bioremediation (see, for example, Gadd, 1999). As


nutrient recyclers, biocontrol agents and bioremediation agents, fungi are growingin environments exhibiting spatio-temporal nutritional and structural heterogene-ity. The hybrid model we have constructed is a tool by which further understand-ing, particularly at the hyphal level, of the crucial processes involved in fungalgrowth, nutrient translocation and concomitant fungal function (e.g. acidification)can be obtained. Consequently, the biotechnological applications of fungi may befurther enhanced as well as understanding of their significant roles in environmen-tal biogeochemistry.

Acknowledgments

This research was funded by the Biotechnology and Biological Sciences ResearchCouncil (BBSRC 94/MAF12243) as part of the Mathematics and Agriculture FoodSystems Initiative. SCRI receives grant-in-aid from the Scottish Executive En-vironment and Rural Affairs Department. GPB acknowledges support from theNuffield Foundation as part of the Awards to Newly Appointed Lecturers in Sci-ence, Engineering and Mathematics (NUF-NAL 04).

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