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Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References

The rhizosphere is a dynamic region where multiple interacting processes in the roots and surrounding soil take place, with dimensions set by the distance to which the zone of root influence spreads into the soil. Its complexity is such that some form of mathematical modelling is essential for understanding which of the various processes operating are important, and a minimal model of the rhizosphere must provide information on (a) the spatiotemporal concentration changes of mobile solutes in the root-influenced soil, and (b) the cumulative uptake of solutes per unit length of root over time. Both are unique for a given set of parameters and initial conditions and hence the model is fully deterministic. ‘Up-scaling’ to uptake by whole plants by integrating individual fluxes requires a measure of the growth and senescence of the root system. Root architecture models are increasingly successful in providing this. The spatio-temporal scales of the rhizosphere and roots are sufficiently different that they can be treated separately, and this greatly simplifies modelling. The minimal model has been successfully applied to the more-mobile nutrients in soil, such as nitrate or potassium, but much less successfully to less-soluble nutrients such as phosphorus, because other, undescribed processes become important. These include transfers from unavailable forms, heterogeneity of resource distribution, root competition, water redistribution and adaptive processes. Incorporating such processes into models can disrupt independent scaling. In general, scaling from the scale of the individual root to that of the whole plant does not pose insuperable problems. Paradoxically, the major challenge in introducing more complexity is that experimental corroboration of the model is required at the individual root scale.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References

The rhizosphere is the soil in the immediate vicinity of a root that is affected by root processes. It comes into being when a root tip enters a volume of soil and disappears some time after the root has died and decomposed (Figure 1). A rhizosphere evolves over time, and the conditions in a young rhizosphere surrounding a root meristem will be very different to the same soil volume in an old rhizosphere now supporting a senescing root. The length scale of the rhizosphere, i.e. the length of root generating the rhizosphere, is of the order of millimetres to centimetres. For modelling, the definition also encompasses the notion of gradients in the concentration of root-influenced solutes extending from the root surface to the ‘bulk’ soil. Gradients arise because roots abstract or deposit water and inorganic and organic solutes into the soil at their surfaces and transport processes redistribute them to form continuous gradients.

image

Figure 1. The rhizosphere is the soil surrounding a root. The intensity of carbon flow changes, as does the root hair density and mycorrhizal status, throughout the lifetime of the root. Mineral nutrient inflows tend to decrease with root age as depletion zones develop around the root. The position and sizes of the arrows indicate the direction and magnitude of the flux. Mucilage, appearing as a thin layer tightly appressed to the root, is only found in juvenile and developing rhizospheres. (From Jones et al., 2004.)

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Each solute under consideration will have a different gradient encompassing a different soil volume, and strictly the rhizosphere is the totality of all the solute gradients surrounding a root. Thus there will be gradients of NH4+, NO3, P and each of the other 13 essential mineral nutrients required by higher plants (Marschner, 1995). Most of these gradients will be depletion profiles, i.e. the solute concentrations will be smallest at the root surface. There will also be gradients of the 200 or so soluble organic solutes that roots release into the soil (Jones et al., 2004). These will typically be accumulation gradients, i.e. concentrations of each solute are greatest at the root surface. There may also be gradients in volatile compounds in the gaseous phase, e.g. O2 depletion profiles and CO2 accumulation profiles caused by root respiration, as well as gradients of compounds such as ethylene and other volatile organic compounds. Gradients in water content, microbial biomass and the abundances of individual species may also occur. We will restrict our discussion to mineral nutrients while noting that the modelling approach is applicable to other rhizosphere gradients. The radial spatial scale over which these gradients are formed varies from a few micrometres to even a few metres, but for most soluble solutes it is of the order of millimetres.

There are two main motives for modelling the rhizosphere. The first is to help understand the complex and dynamic interactions that occur in the soil adjacent to roots: such models are termed explanatory (Nye, 1992), require corroboration and are based on mechanistic equations so far as is possible (Kirk 2002). The second is to form a subprocess in some much larger model aimed at predicting larger scale responses such as crop performance or dynamics of plant communities. Such models often use a simplified approximation for the rhizosphere subprocess, and verification is at scales greater than the individual plant. They typically involve the simulation of many subprocesses, and satisfactory verification does not guarantee that the rhizosphere subprocesses have been modelled accurately. In this review we concentrate on explanatory models. Larger scale crop and water uptake models have recently been reviewed by Hopmans & Bristow (2002) and Wang & Smith (2004).

The explanatory rhizosphere model operates at the single root scale (Figure 1) and aims to simulate the dynamic behaviour of solute(s) in soil around a length of root for the lifetime of that root. The root is a dynamic component of this system as it abstracts water and mineral nutrients, detoxifies pollutants, and sustains an active rhizosphere food web via rhizodeposition. But the behaviour of the individual root also depends on events happening at the root system scale, the whole plant scale and the m2 to hectare scale as the performances of a multitude of individual rhizospheres are integrated over time into various functional responses at the coarser scale. It is therefore difficult to model the rhizosphere at the centimetric scale for any length of time without also simultaneously modelling at the scale of the root system and the whole plant as well. Many of the recent advances in rhizosphere research have come about through better understanding at the scale of the root system rather than of rhizosphere models sensu stricto. However, as the latter cannot be modelled without the former, we review both scales here.

A minimal mechanistic model for nutrient uptake in the rhizosphere

  1. Top of page
  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References

Formulation

The minimal model requires mathematical expressions and parameters for the following.

  • 1
    The size of the potential zone of root influence and any exchanges of the nutrient into or out of it (other than root uptake).
  • 2
    The initial concentration and distribution of the nutrient in the rhizosphere.
  • 3
    The rate of uptake by a unit length of root and its regulation.
  • 4
    The rates and mechanisms of spatial redistribution of the nutrient over time.

Roose et al. (2001) re-derived equations for these based on the original models of Barber and Nye (Barber, 1995; Tinker & Nye, 2000). Two mobile forms of the nutrient are distinguished: that in the soil pore solution with concentration CL, and that associated with the solid phase with concentration CS. The only behaviours prescribed for the solutes are reaction between these phases and redistribution by diffusion and convection. This gives the general conservation equations:

  • image(1)

where θ is the volumetric water content of the soil, u is the Darcy flux of water and D is the diffusion coefficient of nutrient in the pore solution (= DLf where DL is the diffusion coefficient in water and f is the diffusion impedance factor; Tinker & Nye 2000), t is time and ds is defined below. Notice that diffusion in the solid phase is assumed to be insignificant. The terms in CS in Equation (1) can be expressed in terms of CL as follows. If the absorption and desorption of the solute on the soil solid follow first-order kinetics, which is the simplest realistic formulation, then

  • image(2)

where ka and kd are the respective first-order rate constants. If we assume that 1/kd is very small, i.e. desorption is very fast relative to the diffusional time-scale then we obtain

  • image(3)

where b is the familiar soil buffer power term. If the root can be approximated as a smooth cylinder, then Equation (1) can be written in terms of the polar radius r, and in terms of CL only by substitution of Cs from Equation (3),

  • image(4)

where V is the water flux at distance r from the root surface and a is the root radius and therefore u = −aV/r is the water flux at distance r−a from the root surface.

Most uptake models use Michaelis–Menten kinetics to describe the flux of nutrient into the root giving an inner boundary condition of

  • image(5)

where Imax andKm are the maximum rate and affinity constant for uptake, respectively. The outer boundary condition is less straightforward because the radial dimension of a rhizosphere will sometimes depend on the local root density (see later). The simplest condition is that the concentration remains constant (Dirichlet boundary) at some point, which implies that all uptake occurs from within the rhizosphere and competition between roots is not significant. When inter-root competition is significant, so that the zones of influence of neighbouring roots overlap, the zero flux condition (Neumann boundary) is often used. The initial distribution of CL in the rhizosphere is also required, and this is usually assumed to be homogeneous over the soil volume, although the soil is often divided into layers to accommodate a vertical concentration distribution (Scott et al., 1995). With a Dirichlet boundary this gives

  • image(6)

The mathematical realization of these statements and assumptions in this case is an initial-value, parabolic, second-order partial differential equation (PDE) with two boundary conditions. Note that this is not the only way to formulate the problem: a lattice Boltzmann approach has recently been developed for the simulation of convection and dispersion in porous media (Zhang et al., 2002).

Dimensionless analysis: even more minimal models

To solve the equations, changes in the variable CL must be predicted over time and space. For almost all cases of interest, one cannot do this without introducing more complexity because the solutes do more than simply move around in soil solution. Deciding which processes have to be included and which can safely be excluded from the minimal model is difficult but can be immensely aided by dimensional analysis. It is conventional in applied mathematics to non-dimensionalize any system of equations (Fowler, 1997). This simplifies the exercise of deciding which processes need to be included in the model and which can safely be excluded, which is a vital part of up-scaling. To non-dimensionlize Equations (4) to (6) we define the following dimensionless variables:

  • image

Removing the asterisks gives us

  • image(7)
  • image(8)
  • image(9)

where the parameters are the Péclet number inline image, the uptake coefficient inline image and the far-field scaled concentration inline image.

For all the essential nutrient ions, the diffusion coefficient in water, DL, is essentially the same with a value of about 10−5 cm2 s−1, and the water flux at the root surface is typically of the order 10−7cm s−1 for soils near field capacity. The impedance factor typically scales with the volumetric moisture content over a wide range of moisture content, i.e. fθ. As the soil becomes drier, the water flux will decline much faster than the impedance factor because of the typically log-linear relationship between the hydraulic conductivity and the moisture content. Therefore, the maximum value of the Péclet number will be inline image. Typical values for root radii range from 0.0005 to 0.06 cm, yielding Pe values of the order 10−7 to 10−3. Hence, Pe is always very much smaller than the dimensionless diffusion term of order 1 and it can be neglected without error in almost all cases. This in turn implies that diffusion is the overwhelmingly dominant process responsible for moving nutrient ions in the rhizosphere. This contrasts with the often cited result from Barber (1995) that mass flow contributes most to the transport of ions such as NO3, derived from model simulations. More generally, dimensionless analysis provides a robust methodology for deciding whether processes should be included on the time-scale of the simulation.

How mechanistic are the models?

Whereas the minimal model is generally described as mechanistic, i.e. all processes are fully explained in terms of basic physical laws, this is probably true only at the rhizosphere scale because the processes at finer scales are not well described. Hence the theory of solute diffusion in soils does not describe the solute mobilities and concentration gradients through all the individual soil pores of different shapes and sizes, but instead uses spatial averaging to define the medium as locally homogeneous. A variety of spatially-dependent interactions between the solute in free solution and the sorbed state can be hypothesized, including transfer between inner- and outer-sphere complexes, and access to sorption sites through very narrow pores. But as long as the volumes of soil and reaction times being considered are large enough, the microscale variations can be averaged. Likewise, the Michaelis–Menten formulation has a sound mechanistic basis in terms of facilitated ion transport across membranes, where solute concentrations and concentration gradients across carriers are described with an algebraic model (Sanders et al., 1984). But, because there is a spatial component to uptake via the apoplasmic and symplasmic pathways in the root cortex (Darrah, 1993) that is not captured in the algebraic model, this should be viewed as an empirical formulation.

Parametization and solution

One solves the partial differential equations (PDEs) by integrating over time with appropriate parameter values. Methods of solution are many and various; finite-difference and finite-element methods are both widely used, and several acceptable commercial packages exist (Schnepf et al., 2002). Many approximations have also been used, generally to speed up the solution (Tinker & Nye, 2000; Darrah & Roose, 2001).

Recently, Roose et al. (2001) used the methods of asymptotics (Fowler, 1997) to derive an analytical solution to Equation (4). The method consists of deriving explicit closed formulae for different parts of the parameter space occupied by the model (e.g. λ >> 1, Cl >> 1; see section on dimensionless analysis) and then finding an expression that is valid for the whole parameter space. The final formula is fully analytical, meaning that the flux into the root at any time can be calculated directly rather than by numerical integration and so is the fastest and most accurate solution to the minimal model.

The outputs from the minimal rhizosphere model are the changes in solute concentration(s) over time and space and the influx into the root segment over time. Note, in this deterministic mechanistic model, there is one rhizosphere (r,t) solution and one influx curve.

Testing

An explanatory rhizosphere model of the type described can predict both the nutrient flux into the root and the concentration profile in the soil. To test the model these must be compared with experimental data over as wide a range of conditions as possible with all the model's input parameters obtained independently of the output. Only then can the modeller be sure that sufficient detail of the physics, chemistry and biology has been included. Nye and his coworkers (Tinker & Nye, 2000) undertook the corroboration for the minimal model of uptake for several nutrients at non-limiting concentrations. They compared experimental and predicted concentration profiles in soil using autoradiography and showed that all the relevant mechanisms had been included. However, direct experimental measurements of the concentration gradients in root rhizospheres are difficult (see later), and most models have been tested only against flux data.

At present it seems impossible to estimate the influx per root segment directly because translocation away from the segment occurs too rapidly for measurements to be made. Hence, only the cumulative flux at the whole plant scale can be measured and this requires integration of all the individual rhizosphere fluxes over time for a growing root system. Any errors in estimating root growth at any time will be compounded with errors in predicting rhizosphere influx. The errors associated with root growth will dominate because the root system typically grows quickly, even exponentially when small. Comparing observed and predicted cumulative influx is therefore rather a weak corroboration of the rhizosphere model, and new rhizosphere models must be tested at the rhizosphere scale.

Once corroboration of the underlying mechanisms for uptake in the rhizosphere has been achieved, it is then sufficient to use a weaker form of corroboration in subsequent models. Hence for growth in fertile soils, where uptake relies only on available nutrients, it is acceptable to predict the flux of nutrients into the whole plant and to compare this with observed uptake (Barber, 1995). This upscaling process is described later.

Constitutive soil and plant processes: when minimal doesn't work

The major variable in the minimal model is the amount of available nutrient in the soil, and this can vary by several orders of magnitude depending on the overall fertility and availability of mineral nutrients. Various chemical and biological processes start to become important in determining uptake at small nutrient concentrations. Some of these processes are constitutive properties of the root or soil, as in the examples below.

  • 1
    Roots show an ontogenetic change in nutrient uptake properties.
  • 2
    Roots form and lose root hairs, and epidermal and cortical cells, as part of their developmental sequence or in response to grazing damage.
  • 3
    Roots typically change the pH of the rhizosphere by imbalances in uptake of cations and anions. Soil pH is a major determinant of the availability of many soil nutrients.
  • 4
    Roots release organic acids and chelating agents that alter the availability of some nutrients.
  • 5
    Some soils fix and release nutrients to and from various well-characterized forms in response to changes in soil solution or contain precipitate forms of various nutrients that respond kinetically to soil solution changes.
  • 6
    Roots release organic compounds (0.5–5% of net fixed C) that act as substrates for microbial growth, resulting in associated changes in nutrient availabilities.

These changes are a normal part of the development of the rhizosphere, and their incorporation causes no major modelling challenge for upscaling: although the rhizosphere model becomes more complex, involving multiple simultaneous solutions of the relevant forms of Equation (4) for the several interacting solutes (Nye, 1983), there is still only a single generic model that is relevant to all rhizospheres. Kirk & Saleque (1995) and Kirk (1999) give the theory for the simultaneous diffusion of interacting solutes in soil, and Darrah (1991) gives the theory for simultaneous microbial growth kinetics and substrate utilization in the rhizosphere. We do not consider these aspects further here.

Upscaling

  1. Top of page
  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References

Although the rhizosphere is defined with a length scale of centimetres, processes that occur at much coarser scales can affect it, for example water and solute distribution by irrigation or drainage down the profile. Also for many applications, the prediction required from the model is cumulative uptake per plant or per hectare. Hence the cumulative flux is measured on a length scale of centimetres to metres whereas the rhizosphere uses a length scale of millimetres to centimetres, and so upscaling is required. There are many ways to upscale processes, and this is reflected in the wide range of models of nutrient uptake at the plant or crop scale, recently reviewed by Tinker & Nye (2000).

Upscaling to the root system level

  1. Top of page
  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References

The minimal model

The minimal rhizosphere model outlined above predicts a single time trajectory for nutrient influx into a root segment; the precise form will vary, but the curve will tend to show a continuous decline in the rate of influx with time as the nutrient is exhausted and roots age. Conceptually, prediction of plant uptake is a two-stage process. First, the rhizosphere model is used to predict the uptake by an individual root segment over the time-course of the whole experiment. Then this curve is integrated over time with a model of root growth.

The simplest model of root growth is based on the assumption that root growth is entirely genetically encoded and that a universal law for root growth with appropriate compensations for temperature, pH, etc., might exist. The Barber–Cushman model (Barber, 1995) used this approach and imposed linear or exponential root growth kinetics. The parameters (initial root length and growth rate constant) were derived from the data used to validate the experiment, obviating the need to predict them as a function of other physical variables (but compromising the validation process). The lengths of roots falling into different age categories could then be calculated and multiplied by the cumulative uptake value for that age class to give the integrated uptake. Summation over all age classes then yielded cumulative uptake.

For relatively mobile ions such as K in fertile conditions, this approach has been very successful, with very good agreement between observed and predicted uptake in pot and field experiments (reviewed in Barber, 1995).

Lack of agreement could be due to inadequate root growth models, and clearly more realism could be built into the root growth model. The minimal model makes the following assumptions at the root system scale.

  • 1
    Roots grow at a constant rate.
  • 2
    All roots are identical.
  • 3
    Root location is unimportant.
  • 4
    Roots operate independently of one another.

Relaxing any of the first three assumptions is possible because the integration of rhizosphere fluxes simply requires that the length and age–length distribution of the root system at any time is known. However, note that the minimal model uses information from different spatial scales (rhizosphere and root system) independently, giving the spatial model useful scaling properties. Techniques do exist to deal with scaling issues (Roose & Fowler 2004a, b), but they introduce far more complexity.

Variable root growth

The minimal Barber–Cushman approach typically fails when plants are grown in nutrient-limited conditions, and models that incorporate the effect of nutrient limitation of root growth should be an improvement. For example, Darrah (1998) linked a Barber–Cushman uptake model to the Relative Addition Rate model of Ingestad & Ågren (1995), linking the exponential growth rate of the plant to the internal concentration of the growth limiting nutrient. He re-analysed a data set for uptake of P by tomato (Fontes et al., 1986) that had a 15.5% under-prediction of uptake by the classical model. Instead of the constant exponential growth rate assumed, the relative growth rates in the model were predicted to vary over a twofold range with rapid initial growth that declined markedly as depletion set in. The analysis produced better agreements with the experimental data than in the original paper.

Variable root morphology

Root systems are complex hierarchical structures that develop either by initiating zero-order roots from the seed radicle (primary roots) or from structures such as the stem or rhizome (nodal roots). In maize, for example, smaller roots, termed first-order laterals, branch off typically, at rather wide angles, and these structures iterate, e.g. second-order laterals branch from first-order ones, etc. The sensitivity analysis for the Barber–Cushman model indicated that root radius was overwhelmingly the most important parameter at the rhizosphere scale, but the classical model uses a single average radius. As described above, maize root systems are composed of three root orders with different radial dimensions: this then gives three unique rhizosphere influx curves for integration. Roose et al. (2001) incorporated zero-, first- and second-order roots into their modified Barber–Cushman model by adapting the model of Pagès et al. (1989) to generate length and age distributions for each order. They used an age-dependent population growth model to describe the birth via branching and subsequent finite elongation of roots of different orders. Essentially this allowed them to partition the root lengths by both age category and root radius when calculating the integrated fluxes. The parameters required were chosen to produce the same total root length and average radius used in the original paper. When the predicted uptake from the three-order root system was compared with the single, average root radius there was a 30% discrepancy. Root uptake is a very non-linear function of root radius, and this finding implies that the averaging procedure for estimating the average root radius may be inaccurate in the minimal model (Figure 2).

image

Figure 2. Predicted cumulative P uptake (Fc) by maize according to three models. The Barber flux curve uses the minimal model with a single average root radius and constant water content. The topmost curve introduces a root branching model into the minimal model above, which partitions the root system into three orders of root with different radii. The middle curve includes different radii and allows water content to fluctuate in the soil profile in response to rainfall and drainage. (From Roose & Fowler, 2004b.)

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Root turnover

Roots have a finite lifetime, although the factors affecting root death and turnover are complex. Incorporating root turnover into the minimal model destroys the assumption of initially isotropic distribution of nutrient in the soil because roots growing in layers where other roots have already grown and died encounter different initial nutrient concentrations. This can give rise to an infinite range of starting conditions, an infinite range of rhizosphere uptake curves and a major scaling problem. Darrah & Staunton (2000) solved this problem by incorporating root turnover via polynomial interpolation from a small subset of different initial-value solutions.

Root architecture models

Increases in computing power have allowed the simulation and visualization of plant architecture at high spatial resolution (Godin, 2000). Several root architecture models have been developed over the last decade or two and use various developmental rules to generate two- or three-dimensional representations of root systems. Depending on the underlying genetic architecture, the response to the gravitational vector field and the degree of root plasticity, many architectures can be constructed, ranging from tap-rooted to fibrous. RootMap (Diggle, 1988) used order-specific root elongation rate, branching density and time between branchings to build root systems with stochastic, gravitropic and deflection rules determining how roots explored the soil. Such models contain many parameters, e.g. a similar model to RootMap developed by Pagès et al. (1989) required the specification of growth rates, length of non-branching apical and basal zones, interbranch distances, number of xylem poles, insertion angles and gravitrophism indices for each of three orders of maize root; this meant the provision of more than 30 parameter values for the default case, with the possible provision of more than 170 values if there are substantial differences between different internodes. This French group has recently further developed their modelling approach with RootTyp, which can simulate many different root forms (Figure 3), but at the expense of more root types and more growth and directionality rules (Pagès et al., 2004). However, the group have for the first time introduced root death allowing the long-term simulation of architecture.

image

Figure 3. Observed and simulated root systems of (a) Arabidopsis thaliana, (b) Lolium multiflorum and (c) Achillea millefolium. Simulations (on the right-hand side) were generated by RootTyp. (From Pagès et al. 2004.)

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Models built using similar concepts but differing slightly in their implementation are available for crops, e.g. wheat (Diggle, 1988), maize (Pagès et al., 1989), pea (Tsegaye et al., 1995) and bean (Lynch et al., 1997), and tree species, e.g. palm (Jourdan & Rey, 1997) and plum (Vercambre et al., 2003). It is not always clear how such models are tested and ‘eyeballing’ often seems the only criterion (Pagès et al., 2004). Walk et al. (2004) have used fractal analysis to characterize roots generated by SimRoot.

The linking of growth rules to the underlying biology is not yet strongly developed, and much more mechanistic development is necessary. However, Bidel et al. (2000) have linked root growth to the transport and partitioning of carbon from a source of assimilate. Segments with individual sink and transport properties represent root axes, each terminated by a meristem. The availability of photosynthate to each meristem determines the growth of that axis. Genetic information is now being incorporated; for example a model of how three key regulatory genes (ClAVATA1, ClAVATA3 and WUSCHEL) are maintained in a three-dimensional model of the shoot apex of Arabidopsis (Prusinkiewicz, 2004).

The initial motivation in constructing such architectural models was to understand the structure and development of root systems and the early models did not deal very much with nutrient uptake. A partial exception was SimRoot (Lynch et al., 1997), which estimated the extent of rhizosphere depletion from the soil diffusion properties. SimRoot held the age description of each node in its Extensible Tree data structure, allowing the depletion zone of each root segment to be approximated from r + 2(Det)½; the depletion zones were then painted on to the root visualization to show where nutrients had been extracted. However, as noted by Tinker & Nye (2000), this gives only an approximate indication of the zone of influence of a root and is too simplistic as an accurate measure of uptake.

Root architecture models and competition

The minimal model assumes that the outer boundary of the rhizosphere is of infinite extent or imposes a zero-flux boundary at some radial position calculated as a function of root density at harvest. The latter condition implies that roots are competing with each other for nutrients, and for a growing, spatially structured root system this competition causes considerable complications because the radius of each rhizosphere becomes a variable that depends on the spacing and growth of other roots.

One can gauge the importance of competition by comparing the length scale of root separation with the diffusive length scale of different nutrients. Roose et al. (2001) estimated from literature sources that the radii of maize roots were in the range 0.001–0.05 cm, with inter-root separations of the order of 1 cm. Time-scales of interest will vary, but typical pot experiments last about 1 month, and arable and other annual plants about 4 months, whereas for perennial plants, root longevities vary from a few weeks to many years (Eissenstat et al., 2000). The uptake length scale depends on diffusion, diffusive length scales of 2(Det)½, and diffusion coefficients range from 10−6 to 10−8 cm2 s−1. Hence, ions with small mobilities, such as P, always have a diffusive length scale shorter than the inter-root distance, and their roots will not compete significantly. Ions with moderate mobilities, such as K+ and NH4+, will not compete in short-term experiments, whereas maize roots will always compete for the highly mobile NO3. However, Roose et al. (2001) noted a simplification in the case of NO3 and SO42–: because both diffuse rapidly and are taken up relatively slowly, the nutrient profiles in the rhizosphere are fairly flat, and uptake into the root can be accurately approximated by an analytical equation.

RootMap was recently extended to include root uptake (Dunbabin et al., 2002). The spatial information for each node of the root architecture model was used to define the average root density in rectangular volumes of soil arranged in a three-dimensional grid pattern. Each volume corresponded to a ‘scoreboard’ holding data about the volume. The scoreboards are initialized with initial solute concentrations allowing for depth-dependent or three-dimensional spatially heterogeneous solute distributions. In some ways the scoreboards correspond to the elements or nodes in finite-element or finite-differencing methods. The model itself consists of a computational ‘engine’ linking the scoreboard array to different process-based modules. The engine controls an asynchronous clock allowing modules to operate in different time frames or in response to specific stimuli. Within each rectangular volume, a solute module uses an analytical steady-state upscaling approximation to predict nutrient uptake.

The major assumption in their upscaling approximation is that the concentration profile in the rhizosphere develops in a step-wise fashion such that at each time-step it remains at steady state. This is equivalent to saying that the quantity entering the root is the same as that crossing any radial boundary within the rhizosphere. The Michaelis–Menten relation for root uptake, Equation (5), is also simplified to Imax/KmCL, implying that Km >> CL at the root surface. The simplification allowed the generation of a first-order time-differential equation as an approximate solution to uptake in the volume. The root system in the rectangular volume is assumed to be distributed randomly and new roots entering the zone also enter randomly.

Recently, asymptotic approximation methods have been used to generate analytical solutions to the minimal model where the concentration term forming the Michaelis–Menten boundary condition is cast in the form of the far-field concentration rather than the concentration at the root interface (Roose & Fowler, 2004b). This advance essentially eliminates the spatial dependency of root distribution and allows root competition to operate by depleting a common far-field pool of nutrients and thereby preserving the excellent upscaling properties of the minimal model.

Upscaling to the soil profile

  1. Top of page
  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References

A major disadvantage of the minimal model is that it cannot deal with water movement at the profile scale, which affects both the diffusion properties of solutes and also the concentration profiles in the rhizosphere. Several models deal with water and solute movement at the profile scale with simultaneous treatment of rhizosphere uptake. Somma et al. (1998) simulated water and nutrient flow on a profile basis using finite-element methods to solve the three-dimensional equations. They also included water and nutrient uptake by roots, root and shoot growth as influenced by water and nutrient uptake, and allocation of resources between root and shoot. The upscaled nutrient uptake model considered roots to be a locally distributed sink with no consideration of rhizosphere transport limitations. RootMap (previous section) includes water flow and nutrient leaching in its modular formulation as well as transport-limited rhizosphere uptake. Roose & Fowler (2004b) combined a water movement and uptake model with an asymptotic approach to preserve the scaling properties of the minimal model (Figure 2).

Dealing with heterogeneity

  1. Top of page
  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References

Models often assume homogeneous distribution of nutrients, whereas soils are often biologically, chemically and physically heterogeneous over a range of length scales. Jackson & Caldwell (1993) found that all the variation in nutrient availability in a 120-m2 plot occurred within the rooting zone of a single plant, implying a spatial autocorrelation of the order of 1 m. This heterogeneity is often fractal over several length scales, implying that absolute heterogeneity decreases as the intervening distance decreases.

Heterogeneity at the root system scale

In general, nutrient concentrations vary markedly with depth. At the rhizosphere level, this means that roots experience different initial concentrations depending on the depth at which they form. This is easily dealt with by the look-up table approach of Darrah & Staunton (2000).

However, plants show strong plastic responses to exogenous conditions (Hodge 2004), and the relation between root density and depth is likely to be effected by the nutrient gradient down the profile (Alpert & Simms, 2002). To a large extent, this response to vertical heterogeneity is already included in existing models of root architecture that derive their rules by simulating the appearance of root systems that have presumably developed in soils with vertical gradients (Pagès et al., 2004). But there is a danger of circularity if models that have been constructed to mimic the appearance of a root system that is already modified by an external, unknown gradient, are then used to predict the response to nutrient gradients.

The roots of agricultural crops can also respond to local nutrient patches (Drew, 1975), whereas wild plants show a range of responses that are partially correlated with their intrinsic growth rate (Robinson & van Vuurren, 1998). These local responses are not generally built into the design of architectural models. The morphological responses of roots to nutrient gradients and patches may actually be maladaptive in agricultural monoculture (Maina et al., 2002), and competition between species might be required to achieve an adaptive advantage of plasticity (Robinson et al., 1999). We know that different species show very different morphological and physiological responses to nutrient heterogeneity (Hodge 2004), although the significance for nutrient capture is often uncertain.

Zhang & Forde (1998) identified components of the underlying sensory mechanism. The signal transduction pathway remains uncertain, however, with some evidence for the involvement of auxin in the responses to both N and P (Forde, 2002; Al-Ghazi et al., 2003), whereas Linkohr et al. (2002) found that auxin-deficient Arabidopsis mutants behave like wild-type plants. Encouragingly, Arabidopsis does show a proliferation response (Casimiro et al., 2003), and the sequencing of the genomes of rice and maize should allow rapid progress on this topic (Hochholdinger et al., 2004). The signalling pathways controlling root development in Arabidopsis are complex (Casson & Lindsey, 2003). However, both short-range (e.g. local NO3 availability) and long-range (e.g. N status of the plant) stimuli are important in modifying root architecture (Forde, 2002; Lopez-Bucio et al., 2003). The development and regulation of root system architecture is further complicated by the involvement of microorganisms (Persello-Cartieaux et al., 2003) and soil animals (Bonkowski, 2004), which can markedly change root morphology indirectly or directly via phytohormones.

Models of root architecture have not yet led to a full mechanistic understanding of how heterogeneity in nutrient distribution translates into heterogeneity of root distribution. However, they can help to quantify the theoretical costs and benefits of different degrees of plasticity (DeWitt et al., 1998). It is encouraging in this regard that predictions from some theoretical models have subsequently been borne out by experimental evidence. For example, Baldwin et al. (1975) predicted that lateral root development would be important for uptake of P but not of NO3, and Fitter et al. (2002) used the axr4 mutant of Arabidopsis thaliana, which causes reduced number of lateral roots, to confirm this. Similarly, Gahoonia & Nielsen (2003) isolated a barley mutant that lacked root hairs and found that the wild type acquired twice as much P in root mat experiments, confirming the theoretical predictions about the importance of root hairs.

Jackson & Caldwell (1996) used the minimal model to investigate the importance of heterogeneous distribution at several spatial scales. At the plant scale, three- and fivefold increases in the initial nutrient concentration increased predicted uptake by 3.2–4.3-fold and 4.9–6.6-fold, respectively, for P. The range resulted from the simulation time used (2 or 10 days) and whether morphological and physiological plasticity was allowed for. The response was to be expected given the known sensitivity of the minimal model to the initial nutrient concentration (Barber, 1995). More revealing was their model of the response to spatial heterogeneity, in which they simulated areal heterogeneity by dividing the soil into 25 volumes and running independent minimal models in each. They modelled plasticity by changing the Michaelis–Menten parameters (defining the physiological response) and the root growth rate (defining the morphological response) in two steps associated with medium and high concentrations of nutrient relative to the control. For P, allowing plasticity increased the uptake by 28% in 10 days, with approximately equal contributions from morphological and physiological plasticity. Note that plasticity in this case imposed an additional cost in new roots (and presumably increased numbers of transporters) without any compensatory decrease elsewhere in the system. Plastic responses to NO3 were larger but were almost entirely due to physiological up-regulation. Finally, Jackson and Caldwell investigated an extreme case of heterogeneity by imposing a hypothetical patch structure on the 25-cell array: the same amount of nutrient was either distributed uniformly or heterogeneously. Plasticity was advantageous in these circumstances, with less P being extracted in all heterogeneous simulations compared with homologous conditions without plasticity. With plasticity included, results favoured growth in homogeneous or heterogeneous conditions depending on the nutrient.

More recently, the effect of root architecture on nutrient acquisition has been investigated with architectural models. Ge et al. (2000) used SimRoot to investigate the effect of competition by altering the branching angle of basal roots (roots emitted from the stem), which was equivalent to changing the strength of response of the roots to the gravitational field. The effect was to generate root systems with different depth-density distributions. Examination of the figures reveals that competition was significant only for simulated root systems with very strong gravitational responses so that basal roots were growing parallel to one another in a very narrow cylinder or for solutes that diffused rapidly. In a second study, Ge et al. modified SimRoot to allow multiple root systems sharing the same soil domain to be simulated (Rubio et al., 2001) and concluded that competition between roots from different plants was always more intense than competition between roots of the same plant with the same architecture and that shallow and deep rooting architectures in neighbouring plants led to better resource partitioning. However, the limit to the accuracy of the estimation of inter-root competition has to be borne in mind.

Although more complicated, RootMap is conceptually reminiscent of the multiple-minimal model approach of Jackson & Caldwell (1996). Dunbabin et al. (2003) used RootMap to simulate different architectures by changing the structural laws underlying the construction of the root system. Each simulation was assigned the same amount of carbon so that the volume of each root system was comparable at each time, allowing the efficiency of nutrient capture to be assessed in terms of construction costs. The simulations were for NO3 uptake on sandy soils with the potential for large leaching losses if roots failed to intercept the NO3. The coarse herringbone form that had been considered optimal for uptake of NO3 as a result of its long diffusional distance (Robinson et al., 1999) wasn't in this case, and an intermediate, more dichotomous form was favoured as this allowed the roots to intercept NO3 more effectively. The model included plasticity by linking the signal representing plant demand to a signal representing the capacity of a root segment to supply NO3 via a positive feedback: thus when demand for NO3 was large, more of the root growth was allocated to segments in local patches of NO3 (Dunbabin et al., 2004).

Heterogeneity at the rhizosphere scale: resolution and downscaling issues

The responses of some plants to nutrient heterogeneity, toxicity or deficiency are inducible, involving, for example, organic acid production (Jones et al., 2003), exudate production (Dakora & Phillips, 2002) or pH change (Hinsinger et al., 2003). This means that many different rhizospheres could develop, depending on the local context, with root responses perhaps ranging from minimal behaviour in nutrient-rich patches through enhanced exudation at intermediate patches and major pH modifications in poor patches. Mycorrhizal status is also correlated with nutrient deficiency. Other responses might give different root behaviour over time; for example, nutrient deficiency experienced as the plant matures might translate into extra exudation flow into the rhizospheres of young tips.

An extreme response to this downscaling feedback would be to model the rhizosphere at the root scale and to simulate the soil around all segments of root simultaneously. Conceptually, this would be straightforward: sophisticated finite-element packages already exist that could simulate the simultaneous rhizospheres around the static root system of a maize plant. However, it is probably not feasible computationally. A more promising solution would be to define a finite range of behaviours and use both spatial and temporal averaging to allow for simultaneous up- and downscaling.

Measurement and corroboration

  1. Top of page
  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References

The rhizosphere is difficult to experiment on: it is cylindrical, small, difficult to localise and composed of opaque, abrasive and porous material. The networks of pores in which biological and chemical activity occurs and through which solutes and water move are complex three-dimensional structures (Young et al., 2001; Young & Crawford, 2004). Modelling is generally limited by the difficulty of conducting experiments to generate and corroborate mechanisms rather than theoretical constraints. Most corroborative experiments rely on the growth of root mats on soil columns, where the roots are separated from the soil by a mesh screen. Paradoxically, this approach, which spatially averages the response from all types and ages of root, is not appropriate where the roots of different types or ages behave differently, and experimental downscaling to the scale of the individual root is required. Imaging various reporter constructs (Leveau & Lindow, 2002; Li et al., 2004), e.g. arabinose detection in barley rhizosphere (Casavant et al., 2002) or NO3 reductase activity in sludge samples (Bothe et al., 2000), can be used to follow the fate of individual solutes or functions around single roots. In Vivo Expression Technology (IVET) can be used to determine individual genes that are induced only in particular rhizosphere contexts (Rainey, 1999). Soil solution concentrations around the root can be monitored by microsampling of rhizosphere solutions (Tomos & Sharrock, 2001). Most of these techniques rely on the growth of plants in laboratory microcosms, which makes the study of all but small plants (e.g. Arabidopsis) or seedlings of crop plants difficult. Consequently, our fundamental knowledge of many aspects of mature plants (e.g. nutrient uptake kinetics, rates of root death) is very small. On a plant scale, X-ray computer tomography has the potential to trace changes in root architecture over time non-invasively (Gregory et al., 2003).

Conclusions

  1. Top of page
  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References

Models of the rhizosphere are usually constructed to simulate plant uptake of nutrients by well-fertilized arable crops, and generally it is found that the majority of uptake can be accounted for by the movement of nutrients to exponentially increasing root surfaces by physical transport processes. This minimal model has ideal scaling properties because uptake at the rhizosphere scale operates independently of growth at the root scale: a unique curve for nutrient uptake is combined with a unique growth curve to predict the integrated uptake by different-aged surfaces over time.

The minimal model tends to fail when nutrient concentration levels are so small as to limit plant growth, and plants deploy adaptations to increase rhizosphere uptake or to increase root surface area. Many adaptations at the rhizosphere scale can increase uptake. But provided these are deployed deterministically and predictably, they will increase the complexity of the model but without changing scaling properties. Similarly, foraging responses to nutrient heterogeneity at the root scale might lead to a greater uptake surface in particular layers of soil, but we can deal with the resulting competition between roots either by increasing the number of rhizospheres considered, or by reference to a common far-field concentration, or by local spatial averaging of root uptake. Most of the adaptations of plants to poor nutrient availability can be incorporated within the general framework of the minimal model.

One exception to this might be the microbiology of the rhizosphere, where estimates of taxonomic diversity have suggested a tonne of soil could contain several million different bacterial taxa. The apex of the root is largely sterile, suggesting that each rhizosphere is colonized from the surrounding soil. Thus each new section of root would represent an independent, random sampling of the bacteria present, and each microbial rhizosphere could be unique.

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  2. Summary
  3. Introduction
  4. A minimal mechanistic model for nutrient uptake in the rhizosphere
  5. Upscaling
  6. Upscaling to the root system level
  7. Upscaling to the soil profile
  8. Dealing with heterogeneity
  9. Measurement and corroboration
  10. Conclusions
  11. References
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