Appendix A: List of Mathematical Symbols
∇· divergence operator. In Cartesian coordinates, the divergence of a vector field F = (Fx, Fy, Fz) is given by ∇· F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z. In cylindrical polar coordinates, it is given by ∇· F = (1/r) (∂rFr/∂r) for F = (Fr, 0, 0).
∇ gradient operator. The gradient of a real valued function f(x, y, z) in Cartesian coordinates is given by ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z). In cylindrical polar coordinates, it is given by ∇f = ∂f/∂r for f = f(r).
∈ ‘element of’ symbol used to characterize those elements that are contained in a set (e.g. a ∈ A).
|| magnitude operator. The magnitude of a real number is its absolute value; the magnitude of a vector of real numbers in Euclidian three-dimensional space is most often the Euclidian norm given by
Appendix B: Solute Uptake on the Single-Hypha Scale
In this appendix, we derive Eqn 9 for hyphal solute influx and show under which conditions it is applicable. On the scale of a single hypha, we model solute transport in the soil and uptake at the hyphal surface. We assume that the soil consists of solid, liquid and air phases and that the volume fraction of each phase stays constant, i.e.:
(θs, the volume fraction of the soil solid phase; θl, the volumetric water content; θa, the volume fraction of the air phase.) The sum of θl and θa is the soil porosity Φ. Typical values of Φ are 0.3–0.6 and typical values of θl in soils at field capacity are 0.15–0.4 (Scheffer et al., 2002). We assume that solute is present in the solid (Cs) and liquid (Cl) phases. As is conventional, Cs is expressed in moles per unit total soil volume, whereas Cl is expressed in moles per unit volume of liquid phase. Like Darrah & Staunton (2000), we consider the sum of the concentrations in solution and adsorbed onto the solid phase as the diffusible concentration of solute present in soil (Cd):
In the following paragraphs, we develop equations for the rate of change of concentration in each soil phase. We assume that the soil air phase is not taking part in any transport or reaction mechanisms. Assuming equilibrium sorption according to a linear isotherm, and neglecting intraparticle diffusion, the rate of change of Cs with respect to t is given by (Atkins, 1998; Barber, 1995; Darrah & Staunton, 2000):
(bp, the buffer power of the soil.) Considering an arbitrary volume of soil V with concentration θlCl, the rate of change of θlCl in V must be equal to the flux of solutes, q, across its boundary ∂V plus the contribution from any internal sources, sinks or both. From now on, we assume that there are no internal sources or sinks. The conservation of mass leads to the following equation:
We assume that solutes can move by diffusion and convection. Fick's law describes diffusion as the movement of solutes in the direction of decreasing concentration gradient, i.e.:
qdiff = −Dlθlfl∇Cl,
(Dl, the diffusion coefficient of the solute in free water; fl, the liquid phase impedance factor for solute diffusion.) The impedance factor fl can be described by the empirical relation fl(θl) = 1.6θl − 0.172 for most agricultural soils (Barber, 1995). The convective movement of solutes is given by:
qcon = JwCl,
(Jw, the Darcy flux of water.) With q = qdiff + qcon, the equation for movement of solutes towards the hypha is:
The rate of change of the diffusible solute concentration Cd with time is given by the sum of Eqns 34 and 35, i.e. we have:
We assume that the concentration in soil solution is initially constant, so that the initial condition is given by:
(Cl,0, the initial concentration of solute in soil solution.) We also assume that the uptake of solute at the hyphal surface follows Michaelis–Menten kinetics (Barber, 1995), so that the boundary condition at the hyphal surface is given by:
(∂/∂n, the operator for the outward normal derivative; Jw,n, the Darcy flux of water normal to the hyphal surface; Fmax,m and Km, the maximal uptake rate and Michaelis–Menten constant of mycorrhizal fungi, respectively.
Assuming that the hyphae are parallel and evenly distributed in the soil, we calculate the average half distance between two hyphae as (Reginato et al., 2000):
(ρ, the hyphal length density.) According to Jakobsen et al. (1992) we can assume that the hyphal length density is at most 3000 cm cm−3. Hence r1 is in the order of 0.01 cm or more. Other published values of hyphal length densities are 16–60 m g−1 (McGonigle & Miller, 1999) and 2–45 m g−1 (Smith et al., 2004). [Assuming a soil bulk density of 1.02 g cm−3 (Table 1), these values convert to 1632–6120 and 204–4590 cm cm−3, respectively.] At this density, hyphae are likely to compete with each other. Therefore, to complete the model, we apply a zero-flux boundary condition at the half distance between hyphae r1.
We are interested in simulating radial flow into a cylindrical hypha. In radial polar coordinates, the model is:
(Eqn 40) (Eqn 41) (Eqn 42) (Eqn 43)
(rm, the radius of the hypha; Jw,0, the Darcy flux of water into the hyphal surface.) The convection term is due to conservation of water. We need 2rπJw(r) constant, and hence 2rπJw(r) = 2rmπJw,0. Note that the change of sign is a result of the fact that Jw,0 represents the flux of water into the hypha. For nondimensionalization, let and . Then the nondimensional model is (dropping asterisks):
(Eqn 44) (Eqn 45) (Eqn 46) (Eqn 47)
The dimensionless parameters are , , and where Pe is the Péclet number, λ is the dimensionless uptake parameter, is the dimensionless half distance between hyphae and c∞ is the dimensionless initial concentration of solutes in soil solution.
Typical values for the dimensional parameters are given in Table 1. Based on these values, the dimensionless parameters are Pe = 5.41 × 10−5, λ = 0.30 and c∞ = 1.72 × 10−2. Because Pe << 1, we can, at the leading order, neglect convective transport and the model becomes:
(Eqn 48) (Eqn 49) (Eqn 50) (Eqn 51)
The behaviour of this model can be described using the following two arguments. Firstly, solutes spread over a distance in time . Relevant experiments have durations in the order of weeks, which corresponds to a dimensionless time in the order of 1000. Therefore, on the time-scale of such experiments, competition between individual hyphae becomes important. In dimensional form, the diffusion time for the distance r1 is ≈ 8 h. If the competition between hyphae is strong, the solute concentration profile between hyphae is flat. This implies that at times larger than 8 h we are able to neglect, at the leading order, concentration gradients between individual hyphae. In this case, the influx into hyphae is determined only by the local bulk soil solution concentration and is simply given by:
Secondly, when λ << 1 and c∞ = O(1) or less, the concentration gradient near hyphae is small even for times less than 8 h. Therefore, Eqn 52 holds at all times. In order to obtain the dimensional influx, we multiply Eqn 52 by the scale for the flux, , and obtain:
If the approximations of the diffusion time-scale and uptake properties do not hold for a different nutrient or fungus, then we can use the time-dependent approximate analytical solution for solute uptake by a single root derived by Roose et al. (2001).
Appendix C: Analysis and Simplification of the Full Model on the Single-Root Scale
In this appendix, we nondimensionalize the model given by Eqns 16–23. For nondimensionalization, we express ρ(x, t), given by Eqn 8, as the product of a constant and a function with values between 0 and 1. Taking:
we can write ρ(x, t) as:
(0 ≤ g(x, t) ≤ 1; xmax, the maximal distance of fungal growth in the experiment.)
To nondimensionalize our model, we introduce the scaled variables x = [x]x*, t = [t]t*, and . The planar geometry lacks a characteristic length scale. Therefore we can choose it from the model equations. We are interested in solute diffusion in the soil. Because the duration of relevant experiments is in the order of weeks, we choose the time-scale [t] = 1 wk (= 604 800 s) and the length scale such that the coefficient in front of the diffusion term becomes 1, i.e. the length scale is
the diffusion length scale, . The
nondimensional model equations are (dropping asterisks):
(Eqn 56) (Eqn 57)
where the dimensionless parameters are Pe = Jw[x]/(θlDlfl), , , , , κ = Km/K. The Péclet number Pe is a measure of the relative importance of advection in comparison to diffusion. The parameter S compares fungal solute uptake time to diffusion time in soil solution. The parameters ɛ and γ are measures for diffusion and convection, respectively, in the mycelial phase in comparison to solute diffusion in soil solution. The parameter δ in front of the time derivative measures how fast equilibrium in mycelial transport is obtained in comparison to solute replenishment from the soil. The parameter κ compares the Michaelis–Menten constants of root and hypha. If we assume that the uptake properties of the two are the same, then we have κ = 1. The function g(x, t) describes the growth of the mycelium and has values between 0 and 1. It is given by:
where x and t are now dimensionless. The dimensionless boundary conditions are:
(Eqn 59) (Eqn 60) (Eqn 61) (Eqn 62)
and the dimensionless initial conditions are
(; ; and .) compares the size of the fungal mycelium to the length scale of diffusion [x], and λ1 and λ2 are measures for the relative importance of diffusion in soil in comparison to root uptake from the soil and mycelium, respectively. The parameter c∞ shows how the initial concentration of solutes in soil solution compares to the Michaelis–Menten constant. The coefficients of the dimensionless model, computed based on the parameter values listed in Table 1, are given in Table 2.
Table 2. Values of the dimensionless parameters for the mycorrhizal root model described by Eqns 56–63 for different hyphal uptake parameter regimes*
|Pe||5.23 × 10−3||5.23 × 10−3|
|δ||3.36 × 10−6||3.36 × 10−6|
|ɛ||7.00 × 10−6||7.00 × 10−6|
|γ||6.31 × 10−2 (9.30 × 10−4)||6.31 × 10−2 (9.30 × 10−4)|
|κ||1.00||2.93 × 10−2|
|c∞||1.72 × 10−2||1.72 × 10−2|
We can simplify the model equations if some of the dimensionless parameters turn out to be sufficiently small. We can see in Table 2 that the Péclet number Pe is small, implying that diffusion is the dominant mechanism for solute movement in the soil. Hence we may neglect the convection term in Eqn 56 describing solute movement in the soil. At the leading order, it can be simplified to give:
(Eqn 64) (Eqn 65) (Eqn 66) (Eqn 67)
We solve Eqns 64–67 numerically, using a finite difference scheme with a centred discretization in space and the θ-method in time (Morton & Mayers, 1994). The model is nonlinear due to the sink term and the boundary condition Eqn 65. It is solved implicitly using fixed-point iteration (Suli & Mayers, 2003). The overall removal of solutes from the soil by the root and by the external hyphae is calculated taking into account the solute uptake by the root at the root surface plus the solute uptake by the hyphae. Hence the removal of solutes from the soil Ftot(t) is:
(Cl, the local bulk concentration of solutes; Froot, given by Eqn 65.)
Equation 64 can be further simplified if the fungal uptake parameter S is large, i.e. S >> 1. Dividing Eqn 64 by S, we obtain:
Neglecting O(1/S) terms, we have at the leading order that:
(, the dimensionless speed of the fungal front propagation away from the root surface.) Rescaling time so that and neglecting the O(1/S) diffusion term, we obtain that:
and so the solute concentration in soil solution becomes:
The time-scale over which the uptake stops is given by t 1/S << 1 and uptake occurs in a small region behind the front of the fungal colony of width << 1. This result holds only for the case where the fungal uptake parameters are large. Using the values according to Schweiger & Jakobsen (1999), the dimensionless width of this interval is 1.01 and the time when uptake stops is 0.3. In dimensional terms, this corresponds to a region of 0.5 mm behind the front of the fungal colony where uptake occurs, and a time of ≈ 5 h over which uptake occurs at a given position. The depletion zone propagates away from the root surface at the speed of growth of the front of the hyphal colony.
Solute movement inside the mycelial phase is described by Eqn 57. On the time-scale considered here ([t] = 1 wk), the smallest parameter in Eqn 57 is δ (Table 2). The parameter δ shows how fast an equilibrium value inside the mycelium is obtained in comparison to replenishment from soil. Note that, on smaller time-scales, the value of δ would be larger, for example on the time-scale of convection in the mycelial phase, which is [x]/vm ≈ 32 s, we would have ɛ << δ. However, 32 s is clearly too short a time to be distinguishable in experiments. Neglecting the time derivative in the equation for Cm, the model for the concentration in the mycelial phase becomes at the leading order:
(Eqn 73) (Eqn 74) (Eqn 75)
The total influx of solutes into the root from the soil and the mycelial phase is calculated taking into account the solute uptake by the root from the soil at the root surface plus the solute uptake by the root from the mycelial phase at the root surface. Integrating the right hand side of Eqn 73 from 0 to the dimensionless position of the moving boundary condition, , we obtain an expression for the influx of solute into the root from the mycelial phase. Hence the total influx of solutes that is effectively transferred into the root, Feff(t), is:
which we can compute from the numerical solution of Eqns 64–67, for example using the composite trapezium rule (Suli & Mayers, 2003).
Solutes may accumulate inside the mycelial phase, for example because they are used for fungal metabolism or because solutes are transferred into the root according to plant demand. Therefore, we expect that the influx of solutes into the root from the soil and the mycelial phase is smaller than or equal to the total removal of solutes from the soil, i.e. Ftot ≥ Feff. Note, however, that Eqn 76 is equal to Eqn 68. This means that, at any given time, this model and available parameter values imply that solutes that are taken up by the external hyphae are almost instantaneously transferred into the root. They do not accumulate in the mycelium, because cytoplasmic streaming is fast in comparison to uptake from the soil. The cytoplasmic streaming has to be of the order of 10−8 cm s−1 or lower for the solutes to start accumulating. This velocity is clearly not within the range of values estimated from experimental data. Under current assumptions, the transfer of solutes is not rate limiting. If we modelled the transfer at the interface between root and fungus in greater detail, it might become a rate-limiting step. What we calculate with this model is the potential maximal uptake, i.e. the total removal of solutes from the soil.