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.
Evolution of colony biomass
The spatially independent form of the model with equal hyphal growth in all directions can be written as:
- (Eqn 4)
where w = (n*,ρ*), in which n* = nx+ = nx− = ny+ = ny− and ρ* = ρx = ρy. The solution to Equation 4 is:
- (Eqn 5)
where w1 and w2 depend on the initial conditions. λ± are given by:
- (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:
- (Eqn 7)
where ŵ denotes the vector of double-Fourier-transformed elements of w and the matrix M^ is given by:
- (Eqn 8)
where j and k are wavenumbers in the x and y directions, respectively;. 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
- (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:
- (Eqn 10)
where ŵmax is the eigenvector associated with λmax. Given that λmax is an eigenvalue of M̂,
- (Eqn 11)
where I is the identity matrix. Hence, from the definition of M̂ (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
- (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:
- (Eqn 13)
- (Eqn 14)
- (Eqn 15)
where. 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,
- (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
|a||Rate of change of biomass|
|b||Rate of change of colony location||0|
|c||Rate of change of colony spread|
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):
- (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:
- (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.