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Keywords:

  • fungal growth;
  • mathematical model;
  • pathozone profile;
  • scaling-up

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and Results
  5. Discussion
  6. Acknowledgements
  7. References
  • • 
    The transmission of many fungal soil-borne plant pathogens is mediated by the growth of the fungal colony from an infectious to a susceptible host plant.
  • • 
    Here we develop a mechanistic, spatially-explicit, mathematical model that allows us to scale from hyphal growth and branching, through colony growth to analyse and predict the transmission of infection. We derive approximate analytical solutions for colony behaviour. These are used to drive equations for the evolution of the pathozone dynamics which characterize the ability of pathogens to infect hosts from various distances in soil.
  • • 
    It is possible to scale up from hyphal behaviour to the scale of transmission of infection. We identify two key periods in pathozone dynamics: an initial period during which no transmission of infection occurs, followed by the advection of the pathozone profile away from the infectious host at an approximately constant rate.
  • • 
    The models enable the prediction of probability of transmission of infection from hyphal-scale behaviour. However, a coherent theory scaling from hyphal dynamics through colony behaviour to properties of the whole epidemic awaits further theoretical and experimental work.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and Results
  5. Discussion
  6. Acknowledgements
  7. References

Transmission of soil-borne fungal plant pathogens occurs by a range of mechanisms, including colony growth from an infected to a susceptible host plant, movement of spores in soil water, and growth and contact of root systems. In this paper we focus on the important group of soil-borne pathogens which are transmitted primarily by colony growth. For those pathogens, the rate and probability of the spread of infection depends on the distribution of hyphae within the colony. This in turn depends on microscopic parameters, for the growth, death and branching of the hyphae which constitute the colony. Environmental conditions, chemical treatments and biological control can modulate these hyphal parameters. The objective of this paper is to determine the effects of these microscopic, hyphal parameters on the behaviour of the whole colony, and hence on the transmission of infection between infected and susceptible hosts.

In order to predict the dynamics of infection, we develop a model for the growth of a fungal colony as it spreads over a soil surface from a nutrient-rich source, such as an infected host. The model is motivated for the spread of the ubiquitous, soil-borne fungus Rhizoctonia solani (Kühn) but has broad applicability to a wide range of fungi that transmit infection by colony expansion from a central site of infection. The model also applies to saprotrophic colonization involving mycelial spread between colonized and uncolonized fragments of organic matter rich in nutrients. R. solani is used here because of its characteristic growth, amenability to experimentation (Gilligan & Bailey, 1997) and economic importance in causing a variety of diseases, including damping-off, in a broad range of host species (Holliday, 1989).

The probability of transmission of infection depends on the local biomass of mycelium that reaches a susceptible host (Gilligan & Bailey, 1997). This in turn is related to the total biomass of the expanding colony as well as the mean and variance of the colony location. The model introduced here allows the derivation of colony-scale parameters that control these variables in terms of microscopic hyphal-scale parameters that control growth, death and branching of individual hyphae. There is already a considerable amount of literature on the modelling of fungal growth, ranging from the microscopic scale (Trinci & Saunders, 1977) through the hyphal scale (Yang et al., 1992) to the colony (Edelstein & Segel, 1983 and Davidson et al., 1996) and the epidemic scales (Kleczkowski et al., 1996). Although the literature provides plausible mechanisms for qualitative fungal behaviour at the various scales, such as the formation of density bands (Edelstein-Keshet & Ermentrout, 1989), the shape of the hyphal tip (Bartnicki-Garcia et al., 1989) and patterns of interaction between colonies (Davidson et al., 1996), relatively little attention has been given to determining quantitatively how hyphal parameters map onto colony-scale parameters. To address this issue, it is necessary to develop models in which the parameters concern the behaviour of hyphae, but whose predictions concern the behaviour of colonies. The models of Edelstein (Edelstein, 1982; Edelstein & Segel, 1983; Edelstein-Keshet & Ermentrout, 1989) are particularly pertinent, as the parameters relate to the growth, branching and death of hyphae and the variables (n and ρ) represent densities of hyphal apices and hyphae, respectively. These are used here as a starting point for the scaling from hyphal to colony growth.

We show that relatively simple assumptions about the mechanisms of hyphal growth, branching, death and infection of susceptible hosts result in two phases of the transmission of infection. In the first phase, the fungal colony increases in density (bulks up) on the infected host; in the second phase a wave of high probability of infection travels at an approximately constant rate away from the infected host. These predictions concerning the spread of infection are developed in the form of spatio-temporal pathozone profiles (Kleczkowski et al., 1996) which describe the probability that a susceptible host at a given distance from a source of inoculum is infected by a given time.

Methods and Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and Results
  5. Discussion
  6. Acknowledgements
  7. References

Development of infection model

Initially we develop a simple mechanistic model for the transmission of infection from an infected to a susceptible host. The probability of infection depends on the local biomass that contacts a susceptible host, α(x,t), and the efficiency of the mycelium in infecting that host, β(x,t). Hence for a population of hosts each located at distance x from an infected source, the probability that a host is infected in the interval δt is given by α(x,t)β(x,t)δt. If there is no decay in the susceptibility of the host, the rate at which the probability that a susceptible host has become infected changes according to the expression:

  • image(Eqn 1)

where I(x,t) is the probability that a host at location x is infected at time t. Strictly α(x,t) refers to the biomass within a small infection zone close to and in contact with the surface of the susceptible host, which is typically a root, hypocotyl or epicotyl for Rhizoctonia solani (Kühn). The infection zone is therefore a small region of the pathozone or rhizosophere surrounding the target organ. Infection is initiated by one or more hyphal apices. The efficiency variable β(x,t) for the rate of infection of a host per unit time per unit of biomass may decrease over time, due to nutrient exhaustion or to a decline in susceptibility as host tissue matures and becomes resistant to infection (Kleczkowski et al., 1996). Here for simplicity we assume that the distance between hosts is small enough that translocation of nutrients from the infected host to hyphal apices is not limiting; in addition we do not model the susceptibility of the host, and hence β(x,t) is assumed to be constant, β, for all time and space.

Given an expression for the fungal biomass, Eqn 1 can be solved numerically to predict pathozone profiles (sensuGilligan & Bailey, 1997) that give the probability of infection over time for hosts different distances apart. We now consider the evolution of that biomass, α(x,t), over time in terms of hyphal-scale parameters, in order to move quantitatively from the hyphal to the host scale. To achieve this, we develop below an analytical approximation for fungal biomass as a function of position and time in terms of hyphal growth parameters. This approximation is then linked with the infection model described above; the complete model is subsequently integrated to predict the probability that infection occurs given a dynamically changing distribution of biomass.

Development of fungal growth model

We adapt the model of Edelstein (1982) to make it possible to obtain analytical approximations for the behaviour of the colony. A comparison with the model of Edelstein (1982) and a more detailed justification of the particular form of the model is given under Fungal growth in the discussion. For convenience, the variables and parameters in the model are summarized in Table 1.

Table 1.  The variables and parameters used in the models for hyphal and colony growth and pathozone dynamics
CategoryParameter/ VariableBiological meaningDimensions
  1. For clarity and ease of reference, dimensions are scaled such that the total biomass, A, is dimensionless.

Independent variablesx,yLocation in spaceL
 tTimeT
Hyphal variablesnTip densityL−2
 ρHyphal densityL−1
Hyphal parametersrgHyphal growth rateLT−1
 rbHyphal branching rate(LT)−1
 dnDeath rate of apicesT−1
 dρDeath rate of densityT−1
Colony variablesATotal biomass 
 BMean location of colonyL−1
 CVariance of colony locationL−2
Colony parametersaRate of change of AT−1
 bRate of change of B(LT−1)
 cRate of change of CL−2T−1
Infection parametersβRate of infection per unit of biomassT−1

The hyphal mycelium comprises a large number of interconnected hyphae. The hyphae extend at their apices, and form new apices by subapical branching. Here we consider the growth of hyphae in two dimensions, with Cartesian co-ordinates x and y, in contrast to the model of Edelstein (1982) in which hyphal density is expressed as a function only of distance from the centre of the colony. We define the variables n(x,y;t) and ρ(x,y;t) such that n(x,y;t)∂x∂y is the number of hyphal tips in the rectangular region between (x,y) and (x + x, y + y) at time t, and ρ(x,y;t)∂xy is the total length of hypha in that region (∂x and ∂y are small). For convenience, the variables n and ρ are referred to below as ‘hyphal apex density’ and ‘hyphal density’, respectively. We distinguish four types of hyphal tip, growing in the positive x, negative x, positive y and negative y directions, denoted as nx+, nx−, ny+ and ny, respectively (so that total tip density n = nx+ + nx−+ ny+ + ny−). Similarly we distinguish two types of hypha, parallel to the x and y axes, denoted by ρx and ρy, respectively (hence total hyphal density ρ = ρx + ρy).

Hyphal growth causes the movement of the hyphal apices, n and the increase of hyphal density, ρ. As described in Edelstein (1982), the movement of tips can be represented mathematically as a flux (or advective) term. Here, the direction of the flux depends on the types of apex involved; hence, considering growth only,

  • image
  • image
  • image
  • image

where rg is the hyphal growth rate. The generation of hyphal density as a result of growth is given by:

  • image
  • image

that is, apices growing in the positive or negative x direction cause the production of hyphal density aligned with that axis, and similarly y-aligned apices produce hyphal density parallel to the y-axis. Hyphal branching results in the formation of new hyphal apices, at a constant rate per length of hypha. We assume here that hyphal branches are formed perpendicular to the direction of growth of the parent hyphae; this occurs in R. solani and also certain other fungi such as Coriolus versicolor (Aylmore & Todd, 1984). This assumption can be relaxed, however, whilst still retaining analytical tractability for certain other branching angles (for example, 60 degree branching). Branching is assumed to occur equally in either of the two possible directions; hence, considering branching only,

  • image
  • image

where rb is the rate of branching per unit length of hypha. Finally, we assume that the rates of decay (either through quiescence or death) of the hyphal tips and hyphal density are the constants dn and dρ, respectively.

The full model can therefore be stated in matrix form as:

  • image(Eqn 2)

where the vector w is (nx+, nx−, ny+, ny−,ρx,ρy)T, T denotes the transpose of the vector; the matrix M is given by:

  • image(Eqn 3)

Model analysis

In this section we derive colony-scale parameters from hyphal-scale parameters for the rate of hyphal growth (rg), branching (rb), apex death (dn) and hyphal death (dρ). We show that the shape of the colony tends towards a Gaussian distribution; its evolution can be completely described by three parameters, the rate at which the biomass increases, the rate at which the mean location of the biomass moves in space, and the rate at which the variance of the biomass location increases. Initially we consider the evolution of the total biomass of the colony by considering a non-spatially dependent form of the model; we then use Fourier analysis to obtain estimates for the position and spread of the colony.

Numerical solution of fungal growth model

To test the accuracy of the analytical approximations presented below the model was solved numerically, using the method of characteristics (Zwillinger, 1992) in which the complex spatial dependence of the variables in the model is removed by tracing their evolution in a moving frame of reference. The characteristics of Equation 2 travel at a constant rate in the positive and negative x and y directions. Numerical solution of the model can therefore be facilitated by tracing the development of the system along these characteristics. Typical output from numerical solution of the model is given in Fig. 1 showing the development of a Gaussian shape, and the amplification and spreading of the colony. A two dimensional Gaussian curve was fitted by the method of least-squares to the hyphal density predicted from the numerical simulations at two successive times, using the Nelder-Mead simplex algorithm (Press et al., 1989) to optimize the fit. The first of these times was selected such that transient behaviour of the system due to initial conditions had decayed sufficiently for the relative error of the fit at this time to be < 1%. A number of independent fits were performed ending with different times to verify that the estimated rates of change of the properties of the Gaussian curve (total area under the curve, mean and standard deviation) were independent of the interval between the two successive times.

image

Figure 1. Typical output from numerical solution of the model for hyphal growth and colony expansion, with initial conditions corresponding to a small amount of mycelium at x  =  y  = 50 for rg  = 0.3, rb  = 0.02, dn = dρ =  0.05 for three times (A: t  = 200, B: t  = 400, C: t  = 600 units).

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Evolution of colony biomass

The spatially independent form of the model with equal hyphal growth in all directions can be written as:

  • image(Eqn 4)

where w = (n*,ρ*), in which n* = nx+ = nx = ny+ = ny and ρ* = ρx = ρy. The solution to Equation 4 is:

  • image(Eqn 5)

where w1 and w2 depend on the initial conditions. λ± are given by:

  • image(Eqn 6)

As time increases, the term in λ+ comes to dominate in Equation 5; therefore, for sufficiently large time, n and ρ both increase exponentially at rate λ+. Thus the rate of increase of the fungal biomass has been expressed in terms of the hyphal parameters rg, rb, dn and dρ (Equation 6).

Evolution of colony distribution in space

We now extend the analysis to the spatially dependent form of the model. The mycelium is assumed to be initially confined to a small region of space, corresponding to an infected host. It is convenient to express properties of the whole colony (for example, how spread out the colony is in space) in terms of the Fourier transform of the variables. Hence, to analyse the spatial form of the model, we first take the Fourier transform of Equation 2 with respect to the x and y co-ordinates successively; this reduces Equation 2 to a linear system of ordinary differential equations:

  • image(Eqn 7)

where ŵ denotes the vector of double-Fourier-transformed elements of w and the matrix M^ is given by:

  • image(Eqn 8)

where j and k are wavenumbers in the x and y directions, respectively;inline image. The terms in the matrix in Equation 8 correspond with those in the expression for w in Equation 3, but with the spatial dependence in the diagonal terms in x and y represented by the wavenumbers j and k, respectively. The solution to Equation 7 can be expressed in the form

  • image(Eqn 9)

where λs(j,k) are the eigenvalues of M^ and the functions ŵs depend on the initial conditions. For large time, the value of ŵ becomes increasingly dominated by the λ with the largest real part, λmax; hence, Equation 9 reduces to:

  • image(Eqn 10)

where ŵmax is the eigenvector associated with λmax. Given that λmax is an eigenvalue of &#x004d;̂,

  • image(Eqn 11)

where I is the identity matrix. Hence, from the definition of &#x004d;̂ (see Equation 8) the function λmax is of order k2 (or j2) for large λ. We therefore approximate λmax by the first five terms in its Taylor series in j and k; hence

  • image(Eqn 12)

where the factors i and −1/2 are introduced into bij and c(j2 + k2)/2, respectively, for convenience below. The coefficients a, b, c and d can be obtained from the expression for λ (Equation 11); the same coefficient, c, is used for j2 and k2 due to the symmetry between the x and y directions implicit in the model. Substituting the expression for λmax (Equation 12) into Equation 11 and discarding terms of higher than order j2 or k2, we obtain:

  • image(Eqn 13)
  • image(Eqn 14)
  • image(Eqn 15)
  • image

whereinline image. Substituting the expression for λmax into Equation 10, noting in particular that b = d = 0 (d = 0, corresponds to a lack of interactions between the x and y axes, and b = 0 shows that the whole colony does not move in an advective way) we obtain an approximation for the evolution of the Fourier-transformed system,

  • image(Eqn 16)

The mappings from the hyphal- to colony-scale parameters are summarized in Table 2.

Table 2.  The relationship between the hyphal and the colony-scale parameters for the model of colony growth
ParameterBiological interpretationValue
  1. Here,inline image.

aRate of change of biomassinline image
bRate of change of colony location0
cRate of change of colony spreadinline image

Performing the back-transform of Equation 16 to Cartesian co-ordinates (see Honerkamp, 1994) gives an expression for the long-time behaviour of the original system (Equation 2):

  • image(Eqn 17)

The function N(ν; µ,σ2) here denotes the value of the Gaussian probability density function with mean µ and standard deviation σ evaluated at ν. a0, b0x, b0y, c0x and c0y are constants which depend on the initial conditions; in Equation 16 these are contained in ŵ. If initially there is a small amount of biomass approximately normally distributed close to the origin, a0, c0x and c0y are small, and b0x and b0y are zero, giving:

  • image(Eqn 18)

Hence, from Equation 17, exp(at) can be interpreted as the total biomass of the colony, and ct as the variance of the colony location in the x and y directions, at time t.

We have therefore shown: after initial transients have decayed, the fungal colony tends to a two dimensional Gaussian distribution in space, as suggested by the numerical solution of the model; the mean location of the colony remains constant; the variance of the colony increases at a constant rate, given by c in Equation 15; and the total amount of fungal material increases exponentially with a rate given by a in Equation 13; the Fourier analysis gives an identical result for the increase in biomass as the nonspatial analysis above (see Equation 6). We have therefore moved from the hyphal scale, in which the behaviour of the mycelium is expressed in terms of the growth (rg), death (dn and dρ) and branching (rb) of hyphae to a description of fungal growth at the colony scale which is characterized by colony-scale parameters for the rate of biomass increase, a, the rate of colony movement, b, and of colony spread, c; see Table 2.

Hyphal growth

Moving from hyphal to colony scale

We have shown that colony growth can be characterized by the total biomass, the mean location, and the spread of the colony. The total biomass, A, increases exponentially, according to the expression A = A0 exp(at) where A0 depends on the initial conditions and the rate a can be expressed in terms of the growth, branching and death of the hyphae. The mean location of the colony remains constant (that is, b = 0 in Equation 14). The variance of the location of the biomass, C, increases linearly; that is C = c0 + ct (from Equation 17) where c can be expressed analytically in terms of the hyphal parameters (see Table 2).

Accuracy of analytical solution of the model

The model was solved numerically for an extensive region of parameter space (rg = 1, 100rb > rg > rb for 0 < dn < rg and 0 < dρ < rg but the results can clearly be generalized to any other value of rg) over a period of 35 time units (sufficient for the decay of transients resulting from the initial conditions), and the rate of biomass increase, a, and colony expansion, c were estimated over the period from 25 to 35 time units. For dn = dρ = 0, the estimated rates corresponded to within 10% for 0.5rb < rg < 50rb (that is, over a range of two orders of magnitude of rb). In all these cases, the correlation between the analytical approximation and numerical simulation was consistently better than r > 0.95. Similarly, when dn = dρ = 0.2 the estimated values of a and c are accurate to within 10% (for the same time period). For dn = 0.1, but dρ = 0 the correlation between the estimated and actual shape of the colony is less good, and in particular the estimated value for the colony variance is accurate only for rb > 0.2rg.

Fig. 2 illustrates the typical evolution of the relative error between the analytical approximation for biomass distribution and the numerical solution of the model. Initially, the error is relatively large, and increases due to the choice of initial conditions, but as the transients decay, the relative error also decays towards zero, and is negligible by t  = 35.

image

Figure 2. The evolution of relative error of the analytical approximation for biomass distribution for rg  = 1, rb  = 0.1, dn  =  dρ  = 0, showing the decay of transients which are due to the initial conditions (Gaussian distributions of n and ρ centred at the origin with standard deviations equal to 0.01 spatial units, and with an integral under the Gaussian curves of 0.1).

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Disease transmission

Equation 1 was solved numerically under the assumption that there is a single infectious host at the origin (I(x,0) = 0 for all x) to provide predictions for the evolution of the pathozone profile over time; a typical solution is illustrated in Fig. 3. The predicted profile has two key qualitative properties: first, there is very little change in the pathozone profile initially, as the mycelium bulks up; and second, the probability of infection depends on time and distance of the susceptible host from the infectious host. In particular, after an initial lag, a wave of high probability of infection moves out at an approximately constant rate from the infectious host.

image

Figure 3. Typical output of the model for the evolution of the pathozone profile ( rg  = 1, rb  = 0.015, dn = dρ  = 0, β  = 1). The intensity of the grey-scale corresponds to the probability of infection. Note in particular there is little activity before time t  = 15 , after which a wave of infection spreads from the infected host, at position x  = 0.

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Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and Results
  5. Discussion
  6. Acknowledgements
  7. References

The models introduced here have shown that it is possible to scale up analytically from the hyphal to the colony behaviour. The rates of colony expansion, movement and of increase in colony biomass can all be expressed in terms of the growth, death and branching of the individual hyphae (Table 2). This enables the colony scale to be understood quantitatively in terms of the properties of its constituent hyphae. The model can be used to interpret pathozone behaviour in relation to hyphal dynamics. Kleczkowski et al. (1996) showed experimentally that the development of the pathozone profile occurs in three stages; initially, there is a lag, which is followed by a constant wave of infection moving out from the infectious host; finally, the infection ceases to spread before all hosts have been infected. Here we have illustrated that the simple mechanistic model replicates the first two of these stages. The third component could easily be incorporated by allowing the transmission parameter β to decline with time, but neither nutrient limitation nor a change in host susceptibility are incorporated into the current model.

Fungal growth

The model for colony growth is an extension of that of Edelstein (1982). Common to both models are the advection of hyphal apices, the deposition of hyphal density, hyphal branching and death of hyphae. Two key differences between the models concern the inclusion of anastomosis in the model of Edelstein (1982) and the assumption that hyphae only grow away from the centre of the colony, implicit in Edelstein (1982) but relaxed in the current paper.

Inclusion of anastomosis has the advantage of limiting colony growth (Edelstein, 1982) by providing nonlinear, density-dependent removal of hyphal apices. It is more likely, however, that nutrient availability is the key factor that restricts colony expansion, especially for many soil-borne fungal plant pathogens, where the mycelium derives nutrients from an infected host at the centre of the expanding colony rather than from the medium over which it is growing. Here we have assumed that anastomosis has a negligible effect on colony growth in relation to the spread of infection. Anastomosis requires interaction between hyphal apices and hyphal density, and therefore is likely to be of importance only when the densities of both hyphal apices (n) and the hyphae (ρ) are large. This occurs only in the centre of the colony, away from the focus in this paper on the transmission of infection, which occurs predominantly towards the colony margin where density is sparse. The introduction of an anastomosis term here would have introduced convolutions into the Fourier-transformed form of the model, thus preventing analytical solution. Hence omission of anastomosis had the considerable advantage of enabling an explicit representation of colony parameters in terms of hyphal parameters, while not compromising the representation of hyphal dynamics within the zone of principal interest towards the edge of the colony.

Edelstein (1982 ) used a polar co-ordinate system to facilitate the mathematical description of hyphae advecting away from the centre of the colony. It is possible that for growth of mycelium on an agar plate, hyphal growth towards the colony centre will occur only slowly due to the depletion of nutrients in that region. This pattern of nutrient depletion does not necessarily occur when mycelium grows out from an infected host. The model of Edelstein-Keshet & Ermentrout (1989 ) relaxes the assumption of centrifugal growth. They succeed in showing how symmetry-breaking can occur in fungal colonies. The focus here, however, is on the quantitative mapping from the microscopic to the macroscopic scale; it has been convenient to use the Cartesian co-ordinate system as no assumptions are made concerning the direction of hyphal growth relative to the centre of the colony. Analytical approximation imposed a cost in restricting hyphal growth to orthogonal branching along the x and y axes, as is characteristic of R. solani . The model can easily be extended to incorporate non-orthogonal branching, typical of many other fungi.

Discrepancies between numerical solution of the fungal growth model and the analytical approximations are attributable to three sources of error: the analytical approximations require that initial transient behaviour has become negligible; the numerical solution of the model adds an error which depends on the time-step used in the solution and the precision of the calculations; and the optimization of the fit of the model may introduce some error. If transient behaviour has not become negligible, the colony shape will still be largely influenced by the initial conditions. Thus during early colony growth the patterns of biomass distribution fail to conform to a Gaussian shape, and the analytical approximation fails to predict the colony-scale parameters. A stochastic model, such as that developed by Yang et al. (1992), may be more appropriate for investigating and predicting the properties of the colony under these circumstances. In this paper, we focus primarily on the later growth of the colony and the consequences of that growth for transmission of disease; we therefore do not consider the early growth in depth here.

When d≠ dρ the time taken for the transients to decay is large (for initial conditions corresponding to a small amount of biomass in a small region of space). Thus the analytical approximation may be less useful for these regions of parameter space. However, if the same mechanisms are responsible for the decay of apices and density (for example, programmed cell death, or damage by environmental factors) it is reasonable to postulate that d≈ dρ. In a rapidly expanding colony with an abundant nutrient supply, the rates of death may, in any case, be negligible compared with the rates of growth and branching.

The model developed here is restricted to two-dimensional growth of the colony. Certain soil-borne plant pathogens, such as R. solani, grow predominantly along the surface of the soil; hence it is natural to formulate the model for fungal growth in two dimensions. The model can be easily extended to a three-dimensional equivalent for systems in which the third dimension (in this case, depth in the soil) has a significant impact on the probability of transmission; this will be discussed in more detail in a subsequent paper.

Infection

Kleczkowski et al. (1996 ) used a partially mechanistic model to describe the evolution of the pathozone for the pathogen R. solani growing out from primary particulate and secondary inoculum. They postulated that the rate of formation of new infections can be described by three factors, a spatial decay due to the distance of the inoculum from the host, a temporal decay due to changes in host susceptibility and inoculum infectivity over time and a delay in the onset of infection. These three factors were sufficient to obtain a close fit with experimental data for the evolution of the pathozone surrounding primary and secondary inoculum of R. solani using radish as a host plant. Gilligan & Bailey (1997 ) developed a stochastic model for conditional probabilities of germination of inoculum, growth to a host and subsequent infection. The density functions were based on empirical quantification of fungal colonies. The probability of contact was estimated in two ways; firstly, as the probability that a threshold distance is exceeded using the distribution of furthest extent of hyphal growth and secondly as the probability of contact from the radial distribution of hyphal density, characterized by a negative binomial distribution for n , with parameters changing empirically with distance. The second form of this model was derived for sparse, radially asymmetrical colony expansion while the first form assumed radial symmetry and a comparatively rich source of nutrient. The models in Kleczkowski et al. (1996 ) and Gilligan & Bailey (1997 ) capture certain mechanistic elements of the two-dimensional growth of R. solani in relation to pathozone behaviour and infection, but each involves a largely arbitrary choice of functions for the temporal and spatial dependence on infection rate and neither effectively scales from hyphal to colony behaviour. The model introduced in this paper presents a mechanistic justification for the spatial part of the Kleczkowski et al. (1996 ) model for secondary infection: we have shown that the colony can be approximated by a Gaussian distribution, which corresponds to the spatial dependence included in Kleczkowski et al. (1996 ). In addition, we have identified and quantified a possible cause for the ‘delay in the onset of infection’ as the time required for the fungal mycelium to bulk up prior to growth out from the colony centre.

Several simplifying assumptions have been used in deriving models for hyphal and colony growth. More work needs to be done in devising a coherent theory that scales from hyphal dynamics through colony behaviour to epidemic behaviour. We have not considered the limitation of growth due to the exhaustion of the nutrient supply from the infected host. Competition for nutrients by opportunistic organisms may also limit growth. Similarly, the dynamics of the susceptible host which can lead to both decreased susceptibility (due to the development of resistance) or increased susceptibility (as the amount of infectible tissue increases) has not been incorporated into the current model. We have assumed throughout that fungal density is sufficiently large for the hyphae to be treated using a continuum approximation, whereas in reality stochastic fluctuations, particularly at the margin of the colony, modulate the probability of infection as described in Gilligan & Bailey (1997). These differences may be amplified by spatial heterogeneity in environmental conditions and together with the dynamics of nutrient exploitation require further theoretical and experimental work.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and Results
  5. Discussion
  6. Acknowledgements
  7. References

We gratefully acknowledge funding from the Wellcome Trust (AJS), the Ministry of Agriculture, Fisheries and Food (JET), and the Royal Society and Leverhulme Trust (CAG). In addition we would like to thank the two referees for a number of helpful comments on the original manuscript.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods and Results
  5. Discussion
  6. Acknowledgements
  7. References
  • Aylmore RC, Todd NK. 1984. Hyphal fusion in Coriolus versicolor. In: JenningsDH, RaynerADM, eds. The ecology and physiology of the fungal mycelium. Cambridge, UK: Cambridge University Press.
  • Bartnicki-Garcia S, Hergert F, Gierz G. 1989. Computer simulation of fungal morphogenesis and the mathematical basis for hyphal (tip) growth. Protoplasma 153: 4657.
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