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Linking hyphal growth to colony dynamics: Spatially explicit models of mycelia Graeme P. BOSWELL*, Steven HOPKINS Division of Mathematics and Statistics, Faculty of Advanced Technology, University of Glamorgan, Pontypridd CF37 1DL, United Kingdom

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abstract

Article history:

The processes responsible for the growth of fungal mycelia act over a vast range of spatial

Received 10 July 2008

scales; while nutrient uptake occurs at the molecular level, the fungal colony can expand

Revision received 3 October 2008

by the order of centimetres each day. Although experiments can provide exceptionally

Accepted 24 October 2008

detailed information on processes at specific scales, it can be difficult to relate those

Published online 10 January 2009

processes between different spatial scales. In this article a series of mathematical models

Corresponding editor:

are described that link the different spatial scales found within a mycelium. The models

Fordyce Davidson

are all closely related to each other and are applied to a range of growth environments, and the advantages and limitations of each modelling approach are discussed.

Keywords:

ª 2008 Elsevier Ltd and The British Mycological Society. All rights reserved.

Anastomosis Cellular automata Fractal dimension Hybrid model Individual-based Lattice-based Lattice-free

Introduction Fungal mycelia operate over a wide range of spatial scales, ranging from the order of microns when considering the behaviour of individual hyphae (Webster & Weber 2007) and, in extremes, up to the order of kilometres when viewing large mycelial systems on the forest floor (Smith et al. 1992). Therefore, to understand how a fungus grows and functions within its environment, it is necessary to relate to each other the many processes occurring at these different scales. This can be exceptionally difficult using experimental means alone; for example, while confocal microscopy is an exceptionally powerful tool for investigating hyphal-level behaviour (Hickey et al. 2002), it is unsuitable for analysing an entire

fungal colony. Therefore it is necessary to formulate a procedure that relates growth at the hyphal level to response and function at the colony level; mathematical modelling provides such an approach (Davidson 2007). Lord Kelvin summarised the purpose of mathematical modelling: ‘‘I am never content until I have constructed a (.) model of the subject I am studying. If I succeed in making one, I understand; otherwise I do not.’’ (Thomson 1884). The goal of mathematical modelling is not to formulate a system of equations that can account for every possible eventuality since the model would be as complicated as the biological (or physical) system from which it was derived and unusable in application. Rather, the goal is to construct a model that is as simple as possible and which captures the essential behaviour of the system. By

* Corresponding author. Tel.: þ44 1443 482180; fax: þ44 1443 482711. E-mail address: [email protected] (G.P. Boswell). 1754-5048/$ – see front matter ª 2008 Elsevier Ltd and The British Mycological Society. All rights reserved. doi:10.1016/j.funeco.2008.10.003

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focussing only on the essential components, the fundamental processes that govern the behaviour and development of a system are discovered, assisting in the understanding of the entire biological (or physical) system. Much like an experimentalist would use different tools and techniques when considering hyphal level and colony-level quantities, a mathematical modeller will also use different approaches depending on the scale. However, when formulated carefully, a common theme can underpin these approaches that enable hyphal- and colony-level dynamics to be linked. In this paper, a partial differential equation (PDE) model of fungal growth is presented. The model treats the fungus as a continuous structure comprising hyphae and hyphal tips and where a single generic substance, which is best regarded as a carbon source, is responsible for growth. The PDE model is calibrated and it is shown how behaviours at different spatial scales are related. The PDE model is then used to formulate hybrid continuum-discrete (or individual-based) models that treat the mycelium as an explicit network. These hybrid models are highly suited to capturing the intricate discrepancies that can arise when a fungus grows in nutritionally and structurally complex environments. Finally, the various benefits and limitations provided by each modelling approach are discussed.

Continuous models of fungal growth Continuous modelling approach Filamentous fungi developing in nutrient-rich conditions form highly-interconnected and dense network structures (Gow & Gadd 1995) and the resultant mycelium can be viewed as a continuous distribution of hyphae expanding upon, and interacting with, an underlying medium (e.g. Edelstein 1982; Davidson 1998; Ferret et al. 1999; Stacey et al. 2001; Boswell et al. 2003; Falconer et al. 2006). The medium must contain a suitable combination of nutrients and minerals (including carbon,

nitrogen, phosphorus, etc.), although not necessarily in the same location (Jacobs et al. 2002), since nutrients and minerals are taken up and translocated through the mycelium to enable further growth and uptake. Due to the microscopic size of the nutrients, they too are best regarded as continuous variables. Since dense mycelia can be approximated by a continuous distribution of biomass, a partial differential equation (PDE) model is appropriate to describe how the biomass is created and changed over time. Specifically, the mycelium can be regarded as comprising three component parts: (i) active hyphae, denoted by m, correspond to those hyphae involved in growth, uptake, branching and translocation, (ii) inactive hyphae, denoted by m0 , correspond to those hyphae no longer involved in growth, uptake, branching and translocation but still present in the mycelial network (e.g. those lysing), (iii) hyphal tips, denoted by p, correspond to the extending ends of active hyphae and are the primary energy sinks in the mycelium. For simplicity, a single generic growth-promoting substrate can be regarded as being responsible for fungal growth (see, for example, Davidson 1998, but see Lamour et al. 2000, for a differential equation model that accounts for multiple substrates). For calibration purposes, this substrate has often been regarded as carbon that exists in two states; internalized, denoted by si, or free in the growth medium, denoted by se. The substrate can move around (via diffusion and translocation inside the mycelium and diffusion in the growth medium), is taken up by the fungus (i.e. converted between the two states) and internalized substrate is depleted as the fungal biomass expands. The entire biomass-substrate system can therefore be modelled by the following system of five partial differential equations that encapsulate the various processes involved in mycelia growth, and a description of each variable and parameter is provided in Table 1:

vm ¼ Dp si Vp þ vsi pVm da m; vt change in active hyphae ¼ new hyphae created by tip movement hyphal inactivation;

(1.1)

vm0 ¼ da m di m0 ; vt change in inactive hyphae ¼ hyphal inactivation hyphal collapse;

(1.2)

vp ¼ V$ Dp si Vp þ vsi pVm þ bsi m fmp; vt change in hyphal tips ¼ random and directed tip movement þ branching anastomosis;

(1.3)

vsi ¼ V$ðDi mVsi Da msi VpÞ þ c1 si se c2 Dp si Vp þ vsi pVm c4 jDa msi Vpj; vt change in internal substrate ¼ diffusion and active translocation þ uptake growth costs of hyphal extension active translocation cost; vse ¼ De V2 se c3 si se ; vt change in external substrate ¼ diffusion of external substrate uptake;

(1.4)

(1.5)

Linking hyphal growth to colony dynamics

In this model hyphal tips are assumed to move at a rate proportional to the internal substrate concentration representing the dependence of growth on the availability of nutrients. Hyphal tips primarily move in a direction that avoids nearby hyphae (Hickey et al. 2002) which is modelled using the term vsipVm. To incorporate the small and random variation in tip movement (Riquelme et al. 1998), an additional diffusive term DpsiVp is included. Thus, a single hyphal tip is assumed to move predominately in a straight line but with small variations in the growth direction. Since internal substrate is used in the extension of a hyphal tip, there is a corresponding depletion corresponding to growth costs. The model treats the formation of new hyphae as the trail left behind hyphal tips as they move and hence the amount of new hyphae created corresponds to the flux of hyphal tips. In addition to the role of internal substrate in hyphal tip growth, internalized substrate both diffuses and is translocated towards hyphal tips at a rate proportional to m, which encapsulates that the movement is through a network of that density. Since turgor pressure and the build-up of tip vesicles have been implicated in the formation of new hyphal branches (Riquelme & Bartnicki-Garcia 2004), it is assumed that active hyphae undergo branching at a rate proportional to the internal substrate concentration and is therefore modelled by the term bsim. For full details of the model, including calibrated parameter values for Rhizoctonia solani growing in a standard mineral salts medium (MSM), see Boswell et al. (2002, 2003). The modelling approach assumes that a fungus can be modelled as a continuous entity which is most applicable when the mycelium is dense, i.e. when the fungus is growing in the presence of high nutrient concentrations. This is the case considered below.

Table 1 – A description of variables and parameters used in the various models Variable/parameter m m0 p si se Di Da De Dp v b f di da c1 c2 c3 c4

Description Active biomass Inactive biomass Hyphal tip density Internal substrate concentration External substrate concentration Internal substrate diffusion coefficient Internal substrate active translocation coefficient External substrate diffusion coefficient Tip diffusion coefficient per unit substrate Velocity of hyphal tips per unit substrate Branching rate per unit substrate Anastomosis rate Rate of hyphal degradation Rate of hyphal inactivation Rate of internal substrate acquisition through uptake Cost of hyphal extension Rate of external substrate depletion through uptake Translocation costs

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Results of continuous model in uniform conditions The model equations were solved numerically with initial data representative of a plug of mycelium placed onto the centre of a Petri dish containing a uniform distribution of MSM. Substrates, biomass and hyphal tips remained strictly within the test domain and so zero flux boundary conditions were applied for all variables. The biomass expanded in a radially symmetrical manner (Fig 1A) through the extension of hyphal tips (Fig 1B) and colonised new regions from which substrate was acquired (Fig 1C, D). There was a strong qualitative and quantitative agreement between model and experimental data concerning biomass distribution and the rate of its expansion (Boswell et al. 2002). The successful simulation of biomass growth in uniform conditions supports the model structure and its calibration. For example, while hyphal tip extension is caused by the incorporation of tip vesicles released from the Spitzenko¨rper (Bartnicki-Garcı´a et al. 2000), the above approach has shown that it is not necessary to incorporate such details when attempting to model the growth of an entire colony (although hyphal growth as a consequence of tip vesicle incorporation can be modelled; Gierz & Bartnicki-Garcı´a 2001; Tindemans et al. 2006). Hence it is reasonable to view model equation (1) as comprising all the essential components required for modelling colony growth, and so the model can now be used with confidence to consider the growth and function of mycelia in more complicated environments.

Results of continuous model in nutritionally heterogeneous conditions Numerous experiments have considered the influence of nutritional heterogeneity on the growth of filamentous fungi (e.g. Olsson 1995; Bailey et al. 2000). Such experiments typically use tessellations of tiles or droplets of different types of media and observe how the arrangement of the tessellations affects the growth of the fungus (e.g. Ritz 1995; Jacobs et al. 2002). The model equations (1) can represent such heterogeneities by suitably defining the initial external substrate distribution. The initial distribution of external substrate was chosen to represent a square tessellation comprising nine square tiles each of side 1 cm where the central tile and the four corner tiles contained the calibrated concentrations of external substrate while the remaining four tiles were set to contain zero quantities of external substrate. Consistent with experimental protocol (Ritz 1995), the diffusion of external substrate from square tiles was not permitted. The initial distribution of biomass, hyphal tips and internal substrate was taken to represent an inoculum placed onto the centre of the central tile and the model equations were solved numerically. The biomass expanded from its initial position to cover the central tile while depleting its external substrate (Fig 2A). The biomass continued to expand with more pronounced growth on the tiles that contained external substrate but was able to expand on the zero-nutrient tiles because material acquired from the central and four corner tiles was translocated through the biomass to the zero-nutrient regions (Fig 2B, C). Since translocation of internal substrate was assumed to be directed towards hyphal tips, and the hyphal tip density

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Fig 1 – Biomass growth, simulated using the continuous model by numerically solving Eqs. (1.1)–(1.5) in a representation of an initially uniform circular environment of diameter 2 cm and cross-sections of model densities at distance x along a diameter of the domain, is shown at representations of 12 h intervals from inoculation to 2 d: (A) total hyphal density m D m0 , (B) hyphal tip density p, (C) internal substrate concentration si, and (D) external substrate concentration se. was greatest in regions corresponding to high levels of external substrate (Fig 2B, C), any movement of internal substrate to the substrate-free regions was caused solely by diffusion through the biomass network, which was reasonably modelled as a process having no metabolic cost. The simulation therefore suggests that when sufficient quantities of external substrate are available for uptake, a fungus will invest its energies in exploiting those resources through localized branching and active translocation of internalized material towards that region in preference to exploring nearby regions where limited (or indeed no) nutrients are available. The simulation has also shown that when the biomass expanded over a region devoid of substrate, the resultant density, while being in excess of what local conditions could support, was low, which corresponded to a sparse hyphal network. Thus, the treatment of the fungus as a continuous dense object, which was required in the derivation of the PDE model (1), may be no longer valid and so an alternative approach is required to consider fungal growth in low nutrient conditions.

Individual-based models of fungal growth Individual-based modelling approach When mycelia expand in low nutrient conditions, the resultant network is typically sparse and it is inappropriate to

consider densities in such circumstances. Instead the mycelium should be regarded as a discrete structure where individual hyphae and their interactions are considered giving rise to individual-based models. While the modelling philosophy of discrete and continuous systems is very different, there is a direct relationship between the two approaches that allows for consistent model construction and calibration. For example, the hyphal branching rate in the continuous model (the average number of branches per unit length of hyphae per unit substrate per unit time) can be directly converted to a probability of branching in a length of hypha that contains a specified amount of substrate over a given interval of time. Thus rates observed and calibrated at the continuum level can be mapped to probabilities at the discrete level. This approach implies that, when averaged out over a large number of simulations, mean results from discrete models will be consistent with the same properties of corresponding continuum models. However, in addition to forming explicit networks, individualbased models can also account for stochastic variations in the growth processes and so provide information beyond that captured in continuous models. Whereas the mycelium should be modelled as a discrete structure, nutrients are much smaller in scale, are more finely distributed and are hence still most suitably modelled as continuous variables, even in low nutrient conditions. Hence hybrid discrete-continuum models of fungal growth are ideal for

Linking hyphal growth to colony dynamics

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Fig 2 – Biomass growth was simulated using the continuous model by numerically solving Eqs. (1.1)–(1.5) in an initially heterogeneous environment. The total biomass density (m D m0 ), hyphal tip density (p), internal (si) and external (se) substrate concentrations are shown at times corresponding to (A) 1, (B) 3 and (C) 5 d where darker regions denote greater densities. External substrate diffusion was neglected, i.e. De [ 0 in Eq. (1.5).

modelling the growth and function of mycelia. There are essentially two related approaches to forming such hybrid models by adopting either a lattice-based or a lattice-free construction for the mycelia and the growth medium. These alternative but complementary approaches are described below.

Lattice-based models A lattice-based model is essentially a cellular automaton. Hyphae are confined to the edges of a regular lattice that is typically either square (Ermentrout & Edelstein-Keshet 1993) or triangular (Boswell et al. 2007), but other lattices are possible (Boswell 2008). The regular arrangement of the network on these lattices allows the explicit modelling of branching and anastomosis since the modelled hyphae can only intersect at specific locations (the vertices of the lattice). Moreover, in the cases of Boswell et al. (2007) and Boswell (2008), other processes crucial to the mycelial growth habit, such as nutrient uptake and translocation, can be incorporated. The entire system develops by updating its state at regular and discrete time intervals according to a combination of deterministic and/or stochastic rules. In Boswell et al. (2007) the growth domain comprised an array of hexagonal cells containing external substrate that diffused between neighbouring cells at a rate proportional to the gradient of the respective substrate concentrations. Thus the change in the external substrate concentration in a hexagonal cell k due to diffusion over time Dt was Ddiff se ðkÞ ¼ De

X Dt se ðjÞ se ðkÞ ; Dx connected cells j Dx

(2)

where Dx was the distance between the centres of adjacent hexagonal cells and De the diffusion coefficient. This form of

substrate diffusion is essentially the finite-difference discretization of the corresponding term in Eq. (1.5). A triangular lattice was embedded on top of the hexagonal lattice (where the vertices of the triangular lattice were located at the centres of the hexagonal cells) and was used to model the fungal network. The edges of this triangular lattice could either be occupied by active hyphae, inactive hyphae or be available for hyphal growth. External substrate was internalized at a rate equivalent to that of the PDE model (1) and then transported through the active hyphal network using a combination of diffusion and metabolically-driven translocation. The diffusive component of internal substrate translocation was modelled using a similar form to Eq. (2), where adjacent cells were connected only if there were active hyphae directly connecting them, and the active translocation component was modelled by ( UðkÞUðjÞ X Dt si ðjÞ; if UðkÞ>UðjÞ; Dx Dactive si ðkÞ¼Da Dx cells j joinedto k byactivehyphae UðkÞUðjÞ si ðkÞ; otherwise; Dx (3) where U(k), defined as a negative linear function of the distance from cell k to the nearest hyphal tip, denoted the demand for internal substrate. Consistent with the PDE model, the definition of U corresponded to the hyphal tips being the greatest energy sinks and therefore had the greatest demand for internal substrate in the system. The cost of active translocation was accounted for by a local subtraction of a multiple of the amount of substrate moved from that cell by that means. The movement of hyphal tips was modelled by a biased random walk on the vertices of the triangular lattice where the bias incorporated the directed motion and the randomness related to the diffusive aspect of tip movement. The

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Fig 3 – Lattice-based and lattice-free mycelia networks obtained using a hybrid approach. (A) Lattice-based simulation of planar growth in an initially uniform environment. The model pH of the growth medium ranged between pH 4 (dark red) to pH 7 (green) and the model biomass and acidity are shown at times representing (i) t [ 7, (ii) t [ 14 and (iii) t [ 21 h. (B) Lattice-based simulation of planar growth across a tessellated domain at times representing (i) t [ 1 d, (ii) t [ 3 d and (iii) t [ 5 d. White regions denote areas with no external substrate; light blue denotes areas having less than 30 % of the calibrated external substrate level; and dark blue denotes regions containing in excess of 30 % of the original external substrate level. (C) Lattice-based simulation of growth in a soil-like structure formed by randomly removing hexagonal blocks from the growth domain surrounded by (i) a narrow (width 2 cells) and (ii) a broad (width 5 cells) water film with

Linking hyphal growth to colony dynamics

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probability of directed and diffusive growth at the end of a time step Dt was, respectively, Dt P ðdirectedÞ ¼ vsi Dx ; Dt P ðdiffusiveÞ ¼ Dp si Dx 2;

it no longer played any role in substrate uptake and translocation. Inactive hyphae degraded with probability diDt during the time interval Dt.

(4)

where the directed component acted only in the existing growth direction and the diffusive component acted in each of the three directions at acute angles to the existing growth axis. (Since hyphae do not exhibit any sharp bending when growing in unconstrained domains, the probability of such sharp bending was set to be zero.) The probability that a hyphal tip did not move during the time interval was simply one minus the probability that it did move. (In the simulations, the time step Dt was sufficiently small so that the sum of the movement probabilities was less than unity.) If a hyphal tip moved during the time step then a hyphal segment was created that connected the current and previous position of the hyphal tip and so generated a new connection of length Dx in the biomass network. To represent the energy costs of forming a length of hypha, the internal substrate concentration in the originating cell was depleted by a constant amount. If the hyphal tip moved into a hexagonal cell that already contained other hyphal segments, the tip was assumed to anastomose with those segments and a further connection in the network was then established and the hyphal tip was removed from the model. Active hyphae were assumed to branch at a rate dependent on the internal substrate concentration and hence the probability that a hyphal segment underwent branching during the time interval Dt was bsiDt. If the hyphal segment branched, the direction of the resultant new hypha was chosen to be in either of the two directions at an acute angle to the existing growth direction. The underlying triangular lattice thus imposed a constant branching angle of 60 . While this angle is different to that often measured for R. solani (typically approximately 90 , e.g. Stacey et al. 2001), it is representative of many other filamentous fungi (e.g. Hutchinson et al. 1980 quantified branching angles as normally distributed about 56 in Mucor hiemalis). However, since certain processes act independently of the explicit network structure (e.g. the model construction assumed that gradients of internal substrate between connected hyphal segments were independent of their alignment), it was reasonably assumed that the impact on the results of the model if a branching angle of 60 was imposed would not be significant. (The alternative approach of simulating the mycelium on a square lattice would result in unrealistic orthogonal growth; the use of the triangular lattice is therefore viewed as a suitable compromise.) A hyphal segment became inactive (e.g. lysis) if its internal substrate concentration was below a critical concentration and, while it continued to form part of the network structure,

Lattice-based models in unstructured environments Since the lattice-based model described above was derived from a previously calibrated continuous model, the parameter values were carefully mapped across and the model was simulated in what represented an initially uniform growth medium. The biomass expanded in an approximately radially symmetric fashion and generated a highly branched and interconnected network (Fig 3A). Key properties of the simulated biomass, such as its growth rate and distribution, clearly agree with the results of the continuum model (cf. Fig 1) but the discrete modelling approach allowed for stochastic variations in the biomass density, an aspect that cannot be obtained from a purely continuous approach. In addition to modelling the growth of a mycelium, Boswell et al. (2007) considered the functional consequences of such growth, focussing on the acidification of the growth environment. Various enzymes are secreted as part of the process of nutrient uptake that allow fungi to solubilize otherwise insoluble compounds and Jacobs et al. (2002) showed that R. solani produced acidity only in the presence of a utilizable carbon source. Hence, the resultant acidification of the growth environment was modelled by assuming acid production was proportional to the internal substrate concentration and the pH of the growth medium obtained through the well-known logarithmic relationship. A new variable was introduced to model the acidity and was assumed to diffuse through the growth medium with the same diffusion coefficient as the external substrate and modelled using a similar form to Eq. (2). On running the simulation it was seen that associated with the biomass expansion was an increase in the model acidity (Fig 3A). Since the acidity was allowed to diffuse, there was a small reduction in the pH of the growth environment that preceded the leading biomass edge (shown in Fig 3A by a colour change surrounding the biomass), but the bulk of the acidification occurred in the regions occupied by established biomass. The well-known hyphal growth unit (HGU), i.e. the length of hyphae associated with each hyphal tip when viewed over the lifetime of the mycelium, of the model biomass can be calculated throughout its expansion by dividing the biomass length (i.e. the product of number of line segments and their length) by the number of hyphal tips (Fig 4). There is a transient phase in which the model HGU initially increases before declining towards a limiting value, which in the calibrated model was between 130 and 140 mm. This qualitative and quantitative change in the HGU has been obtained experimentally in

external substrate concentration as shown. (D) Lattice-based simulation of mycelial growth in a three-dimensional soil-like environment where 16 % of the environment was removed as ‘‘soil particles’’ and are coloured according to their z-coordinate. The simulated biomass is shown at times representing (i) 12, (ii) 18 and (iii) 24 h. (E) Lattice-free simulation of outgrowth from the origin shown at regular time intervals. (F) Lattice-free simulation of growth in an initially uniform distribution of external substrate. The green region denotes the calibrated value of the external substrate while dark blue denotes those areas where the external substrate concentration was depleted to zero at times representing (i) 8, (ii) 16, and (iii) 24 h after initiation.

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Time (days) Fig 4 – The model hyphal growth unit (HGU) for the latticebased model simulated in initially uniform conditions. Solid line denotes HGU for growth in the calibrated conditions while the dashed line denotes HGU when external substrate was reduced to one tenth of the calibrated value. a number of classic studies (e.g. Trinci 1974) for a selection of fungal species. There was no concept of an HGU incorporated in the model and instead it arose as a consequence of local branching and anastomosis rules. The modelling thus suggests that while the HGU may be a useful statistic from which to quantify and describe the response of a mycelium growing in a particular environment, it is not a sufficient statistic to describe the dynamics of a fungus as a whole. Furthermore, because the growth medium impacts on branching rules, the model HGU can change considerably between different environments. For example, a 10-fold decrease in the calibrated external substrate concentration reduced the branching frequency and so increased the model HGU (Fig 4). The lattice-based model was also used to simulate growth over the square tile tessellation described above. The central tile was ‘‘inoculated’’ with a star-shaped biomass that quickly expanded and took up much of the substrate (Fig 3B). Further expansion occurred with the biomass density greatest in those regions that corresponded to tiles with nutrients. Biomass growth occurred in the nutrient-free regions as a result of substrate translocation through the biomass network from the substrate-rich tiles. Unlike the continuous model, there were small variations in the time at which the four corner tiles were ‘‘colonised’’ by the biomass. These small variations are consistent with experimental observations on such systems (Ritz 1995) and cannot be captured in a continuum (i.e. PDE) model because stochastic effects are neglected.

Lattice-based models in structured environments The above model considered only nutritional variation in the growth environment and not structural variation, such as that arising in soil systems, which form the natural growth environment for many fungal species (Webster & Weber 2007). In soils the fungus is restricted by soil particles to grow in the

soil-pore space (although over time fungi can decompose organic matter, affecting soil aggregation, Ritz & Young 2004). The discrete nature of the lattice-based model means that it is straightforward to incorporate such structural heterogeneities by appropriately formatting cells in the hexagonal array corresponding to the growth domain. Specifically, ‘‘soil’’ slices (which have been the subject of numerous experimental investigations, e.g. Otten et al. 2001) can be represented by removing collections of cells from the hexagonal array and restricting biomass expansion to the remaining elements. Boswell et al. (2007) used such an approach to model growth in non-saturated soils where a film of water containing external substrate surrounded each ‘‘soil particle’’. Substrate (and acid) diffusion was confined to the water film and to encapsulate the effects of surface water tension, the movement probabilities of hyphal tips were biased so as to reduce the likelihood of hyphal tips crossing the water film (see Boswell et al. 2007 for details). While this approach did not model soils in a strictly mechanistic sense, it did allow the study of fungal growth and function in qualitatively similar conditions. Simulations performed in soil structures show that the nature of biomass expansion is highly dependent on the water film, i.e. the soil saturation. A soil-like structure with a narrow water film has external substrate distributed along a tortuous path; the biomass expanded along this path because of the inclusion of surface water tension effects (Fig 3C). The same soil-type structure, but with a broader water film and having the same total amount of external substrate, had the resource being distributed more uniformly since the ‘‘air pockets’’ were smaller and less frequent and so the biomass expanded along a less tortuous route. There was therefore a complex relationship between biomass expansion in soil-type systems and the thickness of the water film. While the effects of surface water tension in narrow films channel biomass growth resulting in a faster tip expansion, broader water films lead to a less fragmented resource distribution (Fig 3C) and provide the biomass a more direct route to reach distant resources. Furthermore, for a given amount of external substrate, the amount of biomass was greater in broader water films (Fig 3C, ii). The model so far described can only consider the growth of mycelia in planar environments, and does not best represent the growth of fungi in soils which are structurally complex three-dimensional systems. However, the lattice-based model can be extended to a fully three-dimensional model by carefully choosing the lattice on which the biomass expands. Boswell (2008) applied the model described above to a facecentred cubic (FCC) lattice, which can be thought of as a volume densely packed by balls of equal radii (Conway & Sloane 1999). The FCC lattice comprises horizontal and various diagonal layers that can be regarded as equivalent to the hexagonal lattice previously considered. A domain exhibiting a structure similar to soil systems can therefore be formed by randomly removing collections of cells from the FCC lattice representing soil particles, the remaining cells representing the soil-pore space and contain external substrate. While the basic rules that govern the expansion and development of the model biomass can be carried across from the planar to the three-dimensional system without any significant changes, their implementation is far more complex. For example, the movement probabilities of hyphal

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Fig 5 – The lattice-based model was simulated in three-dimensional ‘‘soil-like’’ volumes representing 1 cm3 and approximations of the fractal dimension of the biomass structures were calculated over the duration of the simulations. Thick solid line represents expansion in a volume comprising no soil particles and where the fractal dimension of the soilpore space was Dpore [ 3; dot-dashed line denotes soil-particle density of 16 % and Dpore [ 2.92; dotted line denotes soilparticle density of 32 % and Dpore [ 2.85; dashed line denotes soil-particle density of 64 % and Dpore [ 2.55; thin solid line denotes soil-particle density of 80 % and Dpore [ 2.19.

tips continue to comprise a directed and a diffusive component as in Eq. (4), and while the directed component acts only in the existing growth direction, the diffusive component now acts in all of the six directions that are at acute angles to the existing growth axis. The subsequent model was simulated in what represented different soil structures by removing a range of collections of cells from the FCC lattice. The remainder of the domain initially contained a uniform distribution of external substrate and represented the soilpore space that was ‘‘inoculated’’ at its centre with biomass. The simulation ran using the previously described rules and the biomass expanded through the soil-pore space and around the ‘‘soil particles’’ (Fig 3D). The impact of the soilstructure on the resultant model biomass was quantified by calculating its approximate box-counting fractal dimension (Fig 5). The box-counting dimension of the model biomass increased to a limiting value (cf. Ritz & Crawford 1990; Boddy et al. 1999) that was determined by the box-counting dimension of the soil-pore space. Hence, the modelling predicted that there is a strong and quantifiable relationship between the structure of a soil system and the fungi that develop within it (Ritz & Young 2004).

Lattice-free models in planar environments Lattice-based mycelia growth models, such as those described above, allow the explicit simulation of branching, anastomosis and translocation, and can include their dependence on locally changing substrate concentrations because of the regular grids on which those processes occur. However, when growth is confined to a lattice, a regular geometry is imposed on the simulated biomass network. This regular structure offers computational benefits, particularly for translocation and anastomosis. However, the imposed geometry compromises the form of the modelled mycelium which, in turn,

could impact on its functionality (e.g. acidification of growth environment). Thus, it is desirable to relax or, better still, completely remove the regularity of the simulated network. An alternative to lattice-based models are lattice-free models that treat the mycelial network as a collection of line segments that are free to take any orientation on a planar surface. Such models generate significantly more realistic networks than lattice-based approaches, since there are no geometrical constraints to the location of hyphae. While this approach has been adopted previously (e.g. Cohen 1967; Hutchinson et al. 1980; Bell 1986; Mesˇkauskas et al. 2004a, b), due to extensive computational difficulties, these models have hitherto all ignored anastomosis and translocation (but see Yang et al. 1992). Since those two processes are central to mycelial growth, many previous lattice-free models were therefore incapable of forming quantitative predictions for fungal growth and function in heterogeneous environments.

Outgrowth from an isolated nutrient supply using a lattice-free model Recently, Carver & Boswell (unpublished) used a lattice-free approach to model a mycelium by representing hyphae as a series of connected line segments nominally of length Dx on a planar surface and whose orientation was unrestricted (i.e. the line segments were not confined to a regular grid). Central to this approach was the development of a powerful and convenient method that stored the position and properties of all the line segments throughout the mycelial network. In particular, by systematically storing the line segments in an array, the connections between different line segments were easily accessible allowing for efficient modelling of translocation. Each line segment contained a quantity of internal substrate and the hypha extended through the formation of

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a new line segment at its growing end. The probability of hyphal extension over a time interval of Dt was taken to be identical to the probability of tip movement in the calibrated lattice-based model described above. To encapsulate the straight line growth habit of fungal hyphae, if a new line segment was created at the end of a time step, its direction of growth was selected from a normal distribution with mean being the same direction as the previous direction of growth and whose variance corresponded to the diffusive component of tip movement in the lattice-based model. If a new line segment intersected an existing line segment then the new segment was assumed to fuse at the point of intersection and a new network connection was established (and the length of the line segment was reduced accordingly). For simplicity and consistent with a variety of fungi including Allomyces macrogynus and Galactomyces geotrichum (Webster & Weber 2007), Aspergillus niger (Reynaga-Pen˜a et al. 1997), Scutellospora calospora (Schnepf & Roose 2006) and Neurospora crassa (Riquelme & Bartnicki-Garcı´a 2004), only apical branching was considered and the probability of a tip branching at the end of a time interval Dt was modelled as

The most important factor determining the structure of the biomass networks formed using this approach was the rate at which substrate acquired at the origin was translocated to the biomass periphery. Low translocation rates gave rise to sparse networks with few branches and interconnections, since the amount of substrate transported to the hyphal tips was small, resulting in minimal branching. However, networks created with larger translocation rates had more branches, more interconnections and a greater capacity for internal substrate and were therefore denser. The relationship between the modelled biomass and its translocation rate was quantified by calculating the box-counting fractal dimension of biomass, and demonstrated a direct relationship between the translocation rate and the fractal dimension of the resultant biomass (Fig 6). Similar experimental data may assist in the quantification of nutrient translocation processes in fungal mycelia.

(5)

The simulated outgrowth experiment described above assumed that the external environment was constant and there was limited interaction between the fungus and its environment. We have extended the same modelling philosophy to simulate the growth of mycelia that do interact with their environment and which changes as a consequence. Effectively this treatment combines the lattice-free approach with features of the continuum model. The growth environment was modelled as comprising a continuous distribution of external substrate that was subject to standard diffusion. While the external substrate was regarded as a continuous distribution, the mycelium was modelled as a collection of discrete line segments which were

where the constant b was (potentially) different from the calibrated values used previously, since an alternative branching physiology was assumed. If a line segment underwent branching, then the direction of the two new branches was selected from a normal distribution whose mean was the current growth direction and variance consistent with that of hyphal tip movement used above. The rules that governed the translocation of internal substrate between connected line segments were identical to those specified above for the lattice-based model but, for simplicity, the active translocation component was neglected. As a first step in the simulation of a lattice-free model of mycelia growth in heterogeneous environments and representing typical experimental protocol (e.g. Persson et al. 2000; Davidson & Olsson 2000), Carver & Boswell (unpublished) considered the outgrowth of a mycelium from a nutrient source into a nutrient-free domain. Thus, substrate uptake was only possible at a point source and was chosen to represent a continually replenished resource of external substrate. The modelled mycelium could, therefore, only expand because material acquired at the substrate source was transported through the biomass network to the extending hyphal tips. Since fungi acquire resources in excess of local needs and that the substrate source was continually replenished, fixed boundary conditions were used to represent the internal substrate concentration at the nutrient source. The model was simulated with initial data representing an inoculum positioned directly over the nutrient source and the network developed according to the rules specified above (Fig 3E). Substrate acquired at the colony centre diffused through the modelled network towards newly-formed hyphal segments at the colony periphery. These segments underwent branching and extension, further propagating colony growth. Since the modelled network was not confined to a lattice, it developed using its own geometry and hence more closely resembled real fungal mycelia than either of the lattice-based or continuum models.

1.3

1.25

Fractal dimension

P ðbranchingÞ ¼ bsi Dt;

Simulating growth in changing environments using a lattice-free model

1.2

1.15

1.1

1.05

0

5

10

15

20

25

Iterations Fig 6 – The lattice-free model was simulated to represent outgrowth from a continuously replenished substrate source and the mean (and standard deviation) of the fractal dimension from 20 realisations are shown for different translocation parameter values B, D [ 101; D, D [ 103; ,, D [ 105. There were significant differences in the fractal dimension ( p < 0.01) between the translocation parameters at the 20th iteration.

Linking hyphal growth to colony dynamics

each capable of substrate uptake. The uptake process was modelled using a similar functional form as in both the latticebased model and the continuum model and hence the change in the external substrate distribution was modelled by Eq. (1.5). Internal substrate was translocated through the network using the same rules as above where, for simplicity and following experimental data of fungal growth in uniform conditions (Olsson & Jennings 1991), only diffusive translocation was considered. Consistent with both the continuum and the lattice-based model, it was assumed that line segments branched from existing hyphal segments at a normally-distributed angle of mean 60 (randomly chosen to be on either side of the existing line segment) and with a probability proportional to localized internal substrate concentration. Thus lateral branching was modelled and generated new line segments that represented new hyphae. Hyphal tip movement and any subsequent anastomosis were modelled using the rules described above. The model was simulated on what represented an initially uniform external substrate distribution. An ‘‘inoculum’’ of biomass, comprising a collection of connected line segments containing equal amounts of internal substrate, was positioned at the centre of the domain and the system developed according to the rules specified above. The biomass expanded in an approximately circular manner (Fig 3F) with biomass density greatest at the colony centre, and generated structures that were significantly more realistic representations of fungal mycelia than those obtained through the lattice-based approach (e.g. Fig 3A). Indeed, the biomass density profile closely matched that of the continuum model (and corresponding experimental data, e.g. Boswell et al. 2002), which was expected given the derivation of the lattice-free model from the continuum model. However, unlike either the continuum model or the lattice-based model, the fungal network was represented in its most appropriate form, i.e. as an unconstrained structure. Furthermore, as the biomass expanded, it depleted the external substrate also in an approximately circular manner (cf. depletion of external substrate in the continuous model, Fig 1D), but with variations that arose because of the stochasticity in the construction of the biomass network. This external substrate distribution is again consistent with the continuum model (cf. Fig 1D) but crucially the lattice-free modelling approach has demonstrated the existence of small pockets of external substrate that are not encountered or exploited by the expanding biomass network.

Discussion Using a common approach, three complementary mathematical models of the growth and function of filamentous fungi (continuum, lattice-based and lattice-free) have been derived. Each of these models can be immediately compared because they share common parameters enabling their consistent calibration. The partial differential equation (PDE) model treated the mycelium as a continuous structure and proved to be an invaluable starting point for the modelling of filamentous fungi at the mycelium scale. Indeed, the approach only

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failed when growth was considered in complicated environments (e.g. structurally heterogeneous conditions, such as soils) or when finer details were to be examined (e.g. microscopic variations in substrate distributions). Crucially, by forming and calibrating a PDE model, related hybrid continuum–discrete models were derived that considered those instances where a continuum approach was not desirable. The lattice-free hybrid model generated structures that were far more reminiscent of fungal mycelia than those obtained using a lattice-based model. However, the simulation of network growth in lattice-free models is computationally more expensive than in lattice-based models, particularly for large fungal colonies. This expense arises because at each step of the simulation all new hyphae must be compared against existing hyphae to locate any intersections that result in anastomosis. In terms of computational effort, the remaining processes, including hyphal branching, translocation and nutrient uptake, were similar between latticebased and lattice-free models. Thus, while lattice-free models are ideal for modelling the growth and function of smaller fungal colonies, a lattice-based approach is preferable for modelling larger colonies. However, a more efficient algorithm, or possibly even parallel implementation, might offer a compromise. Effectively a large colony can be divided into smaller ‘‘zones’’. Since distant hyphae on opposite sides of a large colony would not encounter one another, all anastomoses would be within a zone or involve those hyphae crossing into neighbouring zones. Such an implementation would significantly reduce unnecessary computation time in assessing properties of new line segments and so allow faster simulation of large fungal colonies. Structural heterogeneities, e.g. soils, were only considered in the lattice-based model. In that model, the regular geometry of the network was closely allied to that of the growth environment resulting in a computationally efficient simulation. Indeed, the incorporation of structural heterogeneities in the lattice-free model would significantly increase the computation time and potentially render the model unusable in application. Thus, the most suitable modelling approach clearly depends on the application; a lattice-free approach is desirable at very fine scales in unconstrained environments, a lattice-based model is essential when considering fungal growth and function in structurally complex environments, and a PDE model is most suited to large scale problems. The mathematical models of the growth and function of fungal colonies presented above relate behaviour at the individual hyphal level to responses at the colony level. The models all strongly agree with available experimental data and so suggest the core processes that underpin mycelia growth and function. By focussing attention on each of these processes, the models can link the different components together enabling a more complete understanding of how fungi grow and interact with their environment.

Acknowledgement SH would like to thank the University of Glamorgan for a research studentship.

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