Modelling the contribution of arbuscular mycorrhizal fungi to plant phosphate uptake
Department of Water, Atmosphere, and Environment, Institute of Hydraulics and Rural Water Management, BOKU – University of Natural Resources and Applied Life Sciences Vienna, Muthgasse 18, A-1190 Vienna, Austria;
Oxford Centre for Industrial and Applied Mathematics and Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK
• In this paper, we investigate the role of arbuscular mycorrhizal fungi in plant phosphorus nutrition. We develop a mathematical model which quantitatively assesses the contribution of external fungal hyphae to plant phosphate uptake.
• We derive an equation for solute uptake by a growing fungal mycelium which we couple with a model for root uptake. We analyse the model using nondimensionalization and numerical simulations.
• Simulations predict that removal of phosphate from soil is dominated by hyphal uptake as opposed to root uptake. Model analysis shows that the depletion zones around hyphae overlap within 8 h and that the transfer between fungus and root is a critical step for the behaviour of phosphorus within the mycelial phase. We also show that the volume fraction of mycelium is negligibly small in comparison to other soil phases.
• This is the first model to quantify the contribution of mycorrhizal fungi to plant phosphate uptake. A full data set for model parametrization and validation is not currently available. Therefore, more complete sets of experimental measurements are necessary to make this model more applicable.
More than 90% of all terrestrial plants form mycorrhizas, mutualistic symbiotic associations between plant roots and soil fungi (Strack et al., 2003). Mycorrhizas may offer several benefits to the host plant, including faster growth, improved nutrition, greater drought resistance, and protection from pathogens. The fungus benefits from the mycorrhizal symbiosis by receiving photosynthesis products from the plant. The two most common types of mycorrhizas are the ectomycorrhizas and the endomycorrhizas, also known as arbuscular mycorrhizas. Arbuscular mycorrhizas differ from ectomycorrhizas in that their fungal mycelium penetrates into the cortical cells of plant roots. Many experimental studies have focused on arbuscular mycorrhizal fungi and their contribution to plant phosphorus uptake (Smith et al., 2003). To understand the mechanisms that control and influence the mycorrhizal pathway of the uptake of nutrients into plants, mathematical modelling can be very useful. Many plant nutrient uptake models exist (Darrah et al., 2005; Tinker & Nye, 2000; Roose et al., 2001). It has been pointed out that it is necessary to include mycorrhizal fungi in such models of plant nutrient uptake (Tinker & Nye, 2000). It is thought that the external fungal hyphae should give mycorrhizal plants an enormous spatial advantage to access low-mobility ions such as phosphorus. However, to date no such model has been available.
In this paper, we develop a spatially explicit and dynamic model for the uptake of low-mobility nutrients, such as phosphorus, by mycorrhizal roots with a growing hyphal network (mycelium). We assume that single root models are well characterized, and aim to include in these models the uptake by fungal mycelium in the form of a sink term. For the derivation of this sink term, we consider two spatial scales: the scale of a single root surrounded by a growing mycelium and the scale of a single cylindrical hypha. We model nutrient transport towards the plant root in the soil as well as within the hyphal network. Whilst the model is applicable to general mycelial fungi and solutes, we present specific numerical simulations for phosphate uptake by the fungal species Scutellospora calospora (Nicol. & Gerd.) (Jakobsen et al., 1992).
Based on literature estimates of the necessary model parameters, we quantitatively assess the contribution of external hyphae to plant mineral nutrition and find those processes and parameter values that should be measured more accurately to improve the predictive power of this model.
For analysis and solution of mathematical models, we apply the technique of nondimensionalization (Fowler, 1997), a process of changing variables by scaling so that the new variables have no units. This procedure leads to a simpler form of equations with fewer parameters that are all dimensionless. The dimensionless parameters are useful in their own right as they describe the relative importance of different processes included in the model compared with each other. One well-known dimensionless parameter is the Péclet number, which shows the importance of convection in comparison to dispersion. In order to make the paper more accessible to a broader readership, we describe the mathematics involved in nondimensionalizing the models we derive in this paper in appendices. The resulting simplified models are presented in the main text.
Description of the model
Laboratory experiments that motivate this model
We base our model building and parametrization process on currently available experimental data. We are specifically motivated by experiments that use a two-compartment system where a root compartment is separated from a hyphal compartment by a fine mesh (Jakobsen et al., 1992; Drew et al., 2003). Jakobsen et al. (1992) measured length densities of external hyphae of three arbuscular mycorrhizal fungi at different times and distances from the membrane. The fungi considered are S. calospora, Glomus sp. and Acaulospora laevis. A. Schnepf and T. Roose (unpublished) fitted a fungal growth model to this data and derived quantitative growth parameters for the fungal species considered.
Drew et al. (2003) monitored the uptake of isotopically labelled phosphate (33P) by sub clover (Trifolium subterraneum L.) associated with the arbuscular mycorrhizal fungi Glomus intraradices and Glomus mosseae. They used a two-compartment system where the isotopically labelled phosphate was placed in the hyphal compartment. Therefore, they were able to find out what amount of phosphate was taken up by the hyphae from the hyphal compartment, translocated towards the root surface and transferred into the plant root cells. These processes determine the overall contribution of fungi to plant phosphate uptake. This experimental set-up motivates our model for nutrient uptake by a mycelial network.
Model for solute uptake by a mycorrhizal root
The model for solute uptake by a single root as described by Barber (1995), Tinker & Nye (2000) and Roose et al. (2001) includes desorption of solutes from soil solid particles, transport within the soil liquid phase towards the root surface, and uptake by the root. To model a mycorrhizal root, we include an additional pathway for the uptake of solutes into the root by developing a volumetric sink term that accounts for uptake by extraradical hyphae. Using this model, we will address two main questions. Firstly, we will assess solute removal from soil by the root and mycelium. Secondly, we will assess translocation and transfer of solutes from the mycelium into the root.
Total solute removal from soil We consider four phases in the soil: the soil liquid, solid and air phases with volume fractions θl, θs and θa, and a mycelial phase with volume fraction θm. We consider that, in addition to the solid (Cs) and liquid (Cl) phases, the solute is also present in the mycelial (Cm) phase. The volume fraction of the mycelial phase, θm, is given by:
(rm, the radius of the external hyphae; ρ, the hyphal length density.) In Jakobsen et al. (1992), the hyphal length densities were found to be in the order of 103 cm cm−3. With a hyphal radius of 5 × 10−4 cm (Ezawa et al., 2002), we find that θm is in the order of 10−4. 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). Thus, we may assume that the volume fraction of mycelium is negligibly small compared with all other volume fractions present in the soil. Hence, the overall volume conservation relation for soil phases is not significantly altered by the presence of the external hyphae, i.e.:
Therefore, we use a known single root model to describe solute movement in soil (Roose et al., 2001), and extend it with a sink term Rmyc to account for the removal of solutes from liquid phase of the soil by external fungal hyphae. Our explicit assumption is that the fungi cannot take up nutrients from direct contact with the soil particles. Instead, all the uptake is always mediated by the soil pore water. Exudation of solubilizing agents has been found to be important (Kirk, 1999). However, at this stage of modelling mycorrhizas, we concentrate on the simplest case and neglect chemical changes in the soil resulting from exudation by roots or hyphae.
The equation for Cl is given by:
[∇·, the divergence operator; ∇, the gradient operator (see list of mathematical symbols in Appendix A for more detailed explanation); Dl, the diffusion coefficient of solute in free water; Jw, the Darcy flux of water; fl, the impedance factor of solute in the liquid phase; bp, the buffer power.] The impedance factor fl can be described by the empirical relation:
To complete the model for solute transport in soil, we need to apply initial and boundary conditions. We assume that the solute concentration in soil solution, Cl, is initially constant and that the root takes up nutrients according to Michaelis–Menten kinetics. Thus, the initial and boundary conditions to our model are:
(Fmax, the maximal influx rate into the root; K, the Michaelis–Menten constant of the root; Cl,0, the initial solute concentration in the soil solution; ∂/∂n, the outward normal derivative; Jw,n, the Darcy flux of water normal to the root surface; x, the space vector; ∂Ω1, the interface between root and hyphal compartment). | x | → ∞ means that the magnitude of the vector x tends to infinity, i.e. becomes very large measured as a distance from ∂Ω1.
The sink term for solute uptake by the fungal mycelium Local solute uptake by the fungal mycelium depends on the hyphal surface area and influx into a single hypha. If the average radius of external hyphae is rm and the hyphal length density is ρ, then the total surface area of hyphae per unit volume of soil is A = 2πrmρ.
Based on the hyphal population growth model of Edelstein (1982), a model for hyphal length density ρ was developed (A. Schnepf and T. Roose, unpublished). The fact that arbuscular mycorrhizal fungi are obligate symbionts was considered in the model by specifying a boundary condition that represents the creation of new hyphal tips at the root surface. The simplest possible boundary condition is to assume a constant flux of tips at the root surface. In the following equation, we present the solution for one-dimensional Cartesian coordinates and constant flux of new hyphal tips at the root surface, because this is the solution that will be needed in this paper. The hyphal length density is given by:
(k, the constant tip flux at the root surface; v, the fungal elongation rate; b, the tip branching rate; d, the hyphal death rate.) From this solution we also note that the edge of the colony is located at xc = vt. A more detailed description of the fungal growth model as well as derivation of growth parameters through fitting the model to data of Jakobsen et al. (1992) is given in A. Schnepf and T. Roose (unpublished). Results are shown in Fig. 1.
To describe solute influx into a single cylindrical hypha, we assume that the soil can be regarded as homogenized medium, meaning that all soil phases are present and continuous at every point in space. For the case of phosphorus uptake from soil, the following two arguments allow us to use the well-known Michaelis–Menten equation (Eqn 9) to calculate the influx into a single hypha based on local bulk soil solution concentration. Firstly, the diffusion profile spreads over thehalf mean distance between hyphae, r1, in time Using the parameter values provided in Table 1, we find that tdiff ≈ 8 h. This means that the external hyphae start competing for nutrients after approximately 8 h. Our time-scale of interest is based on the duration of typical experiments and is in the order of weeks. Therefore, over this time-scale, we can assume that the hyphae are competing with each other. Secondly, we find that uptake of nutrients by the hyphae is much less than nutrient diffusion in soil, i.e. Fmax,m/Km << θlDlfl/rm. Therefore, the nutrient concentration gradient around each individual hypha at all times (including time < 8 h from the calculation above) is small. This implies that the solute concentration at the surface of the hyphae is approximately the same as the bulk soil concentration. Thus, the nutrient influx into a single hypha in the soil can be taken to be:
Table 1. Typical values of the parameters for the single dimensional hypha model described by Eqns 36–39
(Fmax,m, the maximal influx rate into mycorrhizal hyphae; Km, the Michaelis–Menten constant of mycorrhizal hyphae; Cl, the bulk concentration of solutes in the soil.) A mathematically detailed development of this expression is shown in Appendix B. Using the parameter values provided in Table 1 and assuming a low soil solution concentration of Cl = 10−4 µmol cm−3, the value of the flux is Fmyc = 5.53 × 10−8µmol cm−2 s−1. If these assumptions do not hold for a different fungus or a different solute, then we have to use more complex expressions such as the time-dependent approximate analytical solution for solute uptake by a single root derived by Roose et al. (2001).
Using Eqns 8 and 9, the total uptake of solutes by fungi at any given position and time in one-dimensional Cartesian coordinates is given by:
Solute transport within the mycelial phase The fate of solutes within the mycelial phase varies greatly for different solutes, as it is dependent on its biological function and biochemical form (Beever & Burns, 1980). Phosphate is thought to be taken up in excess of fungal need and transported towards the root surface where it is transferred into the root cells (Smith et al., 2003). To model the transport of phosphorus along the hyphae, we use the information that phosphate is thought to be transported through cytoplasmic streaming (Bago et al., 2002). Cytoplasmic streaming is a bulk flow of the cytoplasm, the semifluid matter contained within the plasma membrane of the cell, which is caused by cytoskeletal forces. Hence, we assume in our model that solute transport within the mycelial phase in addition to passive diffusion can also be by convection, so that the equation for the mycelial phase is:
(Rmyc, the source term to account for solute uptake by the mycelium from the soil solution given by Eqn 10; vm, the velocity of the cytoplasm.) Because hyphae grow in soil pores, the tortuosity of the mycelium has to be higher than the tortuosity of the soil pore space. This implies that the impedance factor for diffusion within the mycelial phase should be less than that for the soil liquid phase, i.e. fm ≤ fl. Therefore, the soil impedance factor in this system is an upper bound for the impedance factor in the mycorrhizal phase. A general relationship between the impedance factor in porous media and the volume fraction has been found empirically in the form of a power law, also called Archie's law (Kume-Kick et al., 2002):
When there are no external hyphae present initially, we have ρ = 0 at t = 0 and therefore:
The mechanisms of solute transfer across the fungus–root interface are not very well understood. It is thought to be the limiting step for phosphate transfer into plant root cells, rather than the translocation within the mycelium. Ezawa et al. (2002) found that efflux of phosphate from the hyphae may be limited by the activity of the phosphate transporters or channels. A time lag has been observed between uptake of 32P by hyphae and its arrival in plant shoots, and the delay was found to occur at the root–fungus interface (Peter Schweiger, pers. comm.). As a first attempt, we assume that the flux at the fungus–root interface is linearly dependent on the concentration in the mycelial phase at the interface, i.e. we take:
(kh, the mycelium root transfer rate.) The second boundary condition has to be applied at the edge of the colony, which consists of tips only. This boundary, ∂Ωm, is moving forward into the soil with time, and therefore we have to consider it as a moving boundary. We assume that hyphae take up solutes everywhere except exactly at the tips, so that we have at the edge of the colony:
Model summary for one-dimensional Cartesian coordinates
In summary, we developed a model for the fate of solutes in the soil as well as in the mycelial phase. It considers solute transport in the soil towards the root surface and uptake by the root as well as by a growing fungal mycelium. In particular, external hyphae and the root are competing with each other for phosphate in the soil solution. Total solute removal from the soil is therefore the sum of solute uptake by the root and by the external fungal hyphae. Within the mycelial phase, we assume that solute transport towards the interface between mycelium and root is by diffusion and convection. Whilst we calculate solute movement inside the mycelium, the internal solute concentration and amount transferred to the plant root do not influence the solute uptake by the mycelium. We could couple the model for hyphal uptake of solutes from the soil solution with the internal solute status inside the fungus. At present, we are unable to find suitable experimental data for parametrization of the way in which mycelial uptake depends on the solute concentration inside the mycelium. However, it is clear that the mycelial uptake might depend on the solute uptake inside the mycelium, and therefore more experimental investigations in this area are needed.
In order to apply this model to an experiment such as that by Drew et al. (2003), we need to consider the experimental set-up of the two-compartment systems. If the roots in the root compartment are tightly packed, we assume that the dynamics within this compartment does not matter. In this case, we can model the membrane separating the root and hyphal compartments as a boundary condition with same parameters as the root surface. Furthermore, the results are only dependent on the horizontal distance from this membrane, so that one-dimensional Cartesian coordinates may be used. Assuming that the interface between root and hyphal compartments is located at x = 0 and the edge of the colony is located at xc as described in Eqn 8, and taking the parameters Dl, Dm, Jw, θl, bp and fl to be constant, the model equations become:
Equations for the solutes in soil:
Equations for the solutes in the mycelial phase:
(Jw, the water flux towards the root; vm, the velocity of the cytoplasm towards the root; xc = vt, the position of the front of the fungal colony; xmax, the length of the hyphal compartment; Rmyc(x, t; Cl) is given by Eqn 10.)
Approximate values for the parameters in the model are given in Table 1. The values of the soil parameters represent the case of phosphate in the soil. The fungal growth parameters are derived by A. Schnepf and T. Roose (unpublished). They show that the solution to the linear model (Eqn 8) can be fitted to the fungal species S. calospora, and hence the parameter values for this species will be used for all subsequent calculations. The parameters of most uncertainty are those describing solute transport in the hyphae, Dm and vm, and transfer between the mycelium and the root, kh. We will now discuss how we can estimate these parameters. Given that the fungal cell mainly consists of water, we assume that the diffusion coefficient in the mycelial phase is at most equal to the diffusion coefficient in water, i.e. Dm = Dl = 10−5 cm s−1. To calculate the impedance factor in the mycelial phase, fm, we use Eqn 12 and take the parameter P to be p = 1 (Tinker & Nye, 2000; Roose et al., 2001). We estimate the parameter vm from the velocity of cytoplasmic streaming. Data for the velocity of cytoplasmic streaming are scarce. Bago et al. (2002) gave estimates for cytoplasmic streaming of 4.0 × 10−4 to 1.1 × 10−3 cm s−1. Nielsen et al. (2002) estimated the net phosphate flux within arbuscular mycorrhizal hyphae towards the root surface to be Fp = 1.5 × 10−1 µmol cm−2 s−1. Assuming that the concentration of P in the hyphae is Cm = 10.0 µmol cm−3 (Ezawa et al., 2002), and that the translocation results from convection only, the estimated velocity is vm = Fp/Cm = 1.5 × 10−2 cm s−1. This value is an upper bound for the range given by Bago et al. (2002), because it neglects diffusion completely. Another similar method to estimate vm is as follows. Assuming that the solute is transported away through the cross-sectional area of the hypha at the same rate at which it is taken up per unit length of hypha, we have vm = (2/rm)[(Fmax,mCl)/(Km + Cl)](1/Cm). Assuming that the soil solution concentration is 10−4µmol cm−3, and that the uptake properties of the hyphae are the same as for the root, the velocity is vm = 2.21 × 10−5µmol cm−2 s−1. This is lower than the range given by Bago et al. (2002), possibly because the experiments by Bago et al. (2002) and Nielson et al. (2002) were conducted under nonlimiting phosphate concentrations.
The transfer at the root–fungus interface may be a limiting step in the overall contribution of hyphae to root uptake (Ezawa et al., 2002). However, no experimental data to estimate the parameter kh were available. In our model, the parameter kh accounts for the transfer of solutes from the mycelial phase into the root. Because the hyphae enter the root cells at so-called ‘entry points’, the value of kh is likely to be dependent on the number of entry points per unit root surface area, i.e. is the total number of entry points at the root surface and n is the number of entry points at the surface at any given time. It could be that we simplify the transfer of solutes between root and fungus too much by specifying a simple boundary interface. A more detailed morphological model of the interface between root and fungus would need to include arbuscules, fungal structures inside the root cells where the transfer of certain solutes such as phosphorus from fungus to root is thought to occur.
The model described by Eqns 16–23 is nondimensionalized and simplified by neglecting small terms in Appendix C. We show that, for the parameters given in Table 1, diffusion in soil dominates convection. Furthermore, it was found that equilibrium inside the mycelium is achieved rapidly in comparison to replenishment from the soil. These two facts lead to the simplified model described by Eqns 24–30, where we neglect solute convection in the soil as well as the dynamics of the solute inside the mycelium.
Simplified equations for solutes in the soil:
Simplified equations for solutes in the mycelial phase:
This is the simplest model for the most general fungal and root uptake parameter values. Further simplification is possible when one assumes large hyphal uptake parameters. In this case, we show in Appendix C that the mycelium quickly depletes the soil so that only the region near the front of the fungal colony takes up solutes as it propagates away from the root surface (results are shown in Fig. 2b).
In the next section, we present numerical simulations to quantitatively assess the potential contribution of external hyphae to root solute uptake. All calculations were performed in nondimensional form and the results were back-transformed to their dimensional form.
Results and Discussion
Using the fungal growth parameter values for the fungal species S. calospora derived by A. Schnepf and T. Roose (unpublished) and assuming that the soil is initially free of external hyphae, the model result for the hyphal length density at three different times is presented in Fig. 1. The final time for all subsequent calculations is taken to be 14 d. The figure shows that within that time the mycelium has grown more than 3 cm away from the root surface, and we can expect that it will contribute considerably to the removal of solutes from the soil.
We shall consider the model results for the case where the hyphal uptake parameters per unit surface area are equal to those of the root, and where they are larger than the root values (Schweiger & Jakobsen, 1999). Figure 2a and b show the solute concentration in solution, Cl. We can see by comparing Figs 1 and 2a,b that the depletion zone reaches almost as far as the mycelium grows. For both uptake parameter regimes, the mycelium has depleted the soil around it almost completely. In the time-scale of interest, the width of the depletion zone without mycorrhizas is at most 2 mm. This value is typical for solutes of low mobility such as phosphorus. This corresponds to a relatively small soil volume from which the plant can access this nutrient. Inclusion of mycorrhizas results in a depletion zone that is determined by the size of the mycelium. Thus, the volume from which the plant can potentially access the solute increases enormously by at least 10-fold. This finding is supported by data from Li et al. (1991). They grew white clover (Trifolium repens L.) infected with arbuscular mycorrhizal fungi in a two-compartment system and found that the Olsen-P (potentially plant-available soil phosphorus) was depleted up to the end of the hyphal compartment. Direct sampling of soil solution at different distances from the root surface would be possible with the microsuction-cup design of Puschenreiter et al. (2005). Our current simulations may overestimate the degree of depletion, because we assume that the whole mycelium has the same capacity to take up nutrients everywhere. Practical constraints to this may be the ability to maintain continuity of mycelium from tips to root over long distances or the fact that only parts of the mycelium are actively involved in the uptake process.
Figure 3a and b show the sink term Rmyc for the two parameter regimes considered. In both cases, we see that the mycelium depletes the soil completely and hyphal uptake occurs mostly near the front of the colony where there is still undepleted soil. The larger the uptake paramaters of the mycorrhizal hyphae, the smaller is the width of the region behind the front of the colony where uptake occurs.
The difference between the two parameter regimes can be seen when comparing the influxes of solutes into the root (Figs 4, 5, 6b). This will be explained in detail in the following paragraphs.
We wish to assess the contribution of external fungal hyphae to the total influx of solutes into the root. In Appendix C, we show that the total removal of solutes from the soil is given by:
In this model for total removal of solutes from the soil, the hyphae are included in the form of a volumetric sink term. A physical interpretation of this is that the hyphae are treated as long and thin extensions of the root, similarly to the way in which root hairs would be treated.
Note that, in this model, the value of kh does not influence the influx into the root from the mycelial phase. This means that, at any given time, our model and its parametrization imply that the 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 much faster in comparison to uptake. If we wish to assess whether the transfer of solutes at the root–fungal interface is a limiting step in the contribution of fungi to root solute uptake, either a more sophisticated model is needed, more accurate measurements of movement within a single hypha are required, or both.
Figure 4 compares the influxes of solute into the root that are attributable to the external hyphae for the two uptake parameter regimes considered. In both cases, an initially steep increase of influx levels off when the mycelium has depleted all the available solutes and can access new solutes only as it grows further. In the case of the larger uptake parameters after Schweiger & Jakobsen (1999), this situation is reached more quickly.
The relative contributions of solute uptake by root and fungi are shown in Fig 5a and b for the case in which the hyphal uptake parameters are the same as those for the root. In Fig 6a and b we show the results for when the hyphal uptake parameters are larger, as estimated by Schweiger & Jakobsen (1999). For both uptake parameter regimes, cumulative influx is dominated by the mycelium after an initial period in which the mycelium is still too small to exert its influence. The time at which cumulative influx attributable to external hyphae becomes greater than influx attributable to the root is approximately 2 or 0.5 d, depending on the fungal uptake parameters used (Figs 5b, 6b). After 14 d, cumulative influx is almost 1.5 times larger when the larger uptake parameters (Schweiger & Jakobsen, 1999) are used. These results are true for a colonized root. However, the root system as a whole is only partly colonized. Thus, depending on the degree of colonization, the contribution of root uptake will be more significant on a whole-plant basis. Published values of root phosphate inflow on a whole-plant basis range from 1 to 37 fmol cm−1 s−1 for nonmycorrhizal plants and from 3 to 120 fmol cm−1 s−1 for mycorrhizal plants (Tinker & Nye, 2000). McGonigle & Fitter (1988) presented a range for mycorrhizal plants of 35–110 fmol cm−1 s−1. Assuming a root diameter of 450 µm, the influx shown in Fig. 4 gives a root inflow of approximately 10 fmol cm−1 s−1. Hence, it is at the lower end of the range of values presented by Tinker & Nye (2000) and approximately 3-fold smaller than the lowest value presented by McGonigle & Fitter (1988). In a growing root system, roots continuously grow into new, as yet unexploited soil. Therefore, on a whole-plant scale, our simulation results could be quite different from results on a single-root scale. In further work, we will upscale our single-root model to a whole-plant scale to quantitatively address this question.
In summary, we have developed a model for solute uptake by a mycorrhizal root. Our main goal was to estimate the contribution of external fungal hyphae to total uptake by a single root. Based on the parameter estimation that transport within the mycelial phase is dominated by convection and is very fast, we arrived at a model in which the solutes that have been taken up by the hyphae are almost instantaneously translocated to the root surface and transferred into the root. The fast translocation is in accordance with experimental evidence. However, the transfer from fungus to root requires a more detailed model. This question will be addressed by us in future publications.
For both hyphal uptake parameter regimes considered, we found that, after an initial period of 0.5–2 d, uptake was completely dominated by the fungi. This supports a recent article by Smith et al. (2003), who claim that, under certain conditions, plants may depend completely on the mycorrhizal pathway for their phosphorus nutrition. One way to improve this model would be to replace the boundary condition for hyphal tips (now assumed to be constant) at the root surface with a model for root infection. Use of the logistic equation (McGonigle, 2001) or Lotka–Volterra type models (Neuhauser & Fargione, 2004) has been suggested. Such models would reflect temporal patterns in root colonization and modify the contribution of uptake via the mycelial pathway.
In conclusion, this is the first model to quantify the contribution of mycorrhizal fungi to plant solute uptake in a mathematically detailed manner. It predicts that the uptake of low-mobility solutes from the soil will be dominated by the fungal mycelium as opposed to the root. This model could be fitted to data on solute uptake as well as solute concentration in the soil and thus help to find the important processes in specific situations. However, a full data set for model parametrization and validation is not currently available. We will pursue this question in a close collaboration between modellers and experimentalists to make this model even more informative.
We acknowledge the FWF Austrian Science Fund (P15749) and the British Council (ARC June 2004) for financial support. We further thank Dr Peter Schweiger for constructive advice about mycorrhizas. AS thanks the Oxford Centre for Industrial and Applied Mathematics and the Centre for Mathematical Biology for hosting her stay at the Mathematical Institute, University of Oxford. TR is a Royal Society University Research Fellow.
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:
(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):
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:
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):
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:
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*
Uptake parameters same as for root vm = 1.50 × 10−3 (vm = 2.21 × 10−5)
Values in brackets refer to a different value of vm. Changing vm only affects the dimensionless parameter γ.
We do not estimate λ2, because the value of kh is unknown. However, the value of kh does not contribute to the solution for the influx of solutes into the root from the mycelial phase.
5.23 × 10−3
5.23 × 10−3
3.36 × 10−6
3.36 × 10−6
7.00 × 10−6
7.00 × 10−6
6.31 × 10−2 (9.30 × 10−4)
6.31 × 10−2 (9.30 × 10−4)
2.93 × 10−2
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:
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:
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:
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.