Rhizosphere models

During the lifetime of a root, considerable depletion of the available mineral nutrients (MN) in the rhizosphere is to be expected. This, in turn, will affect the equilibrium between available and unavailable forms of MN. For example, dissolution of insoluble calcium or iron phosphates may occur, clay-fixed ammonium or potassium may be released, and nonlabile forms of P associated with clay and sesquioxide surfaces may enter soil solution (10). Any or all of these conversions to available forms will act to buffer the soil solution concentrations and reduce the intensity of the depletion curves around the root. However, because they occur relatively slowly (e.g., over hours, days, or weeks), they cannot be accounted for in the buffer capacity term and have to be included as separate source (dCldl) terms in Eq. (8). Such source terms are likely to be highly soil specific and difficult to measure (11). Many rhizosphere modelers have chosen to ignore them altogether, either by dealing with soils in which they are of limited importance or by growing plants for relatively short periods of time, where their contribution is small. Where such terms have been included, it is common to find first-order kinetic equations being used to describe the rate of interconversion (12). 

O. Hdjberg and J. Sprensen, Microgradients of microbial oxygen consumption in a barley rhizosphere model system, Appl. Environ. Microbiol. 59 431 (1993). 

Rhizosphere modeling remains difficult and complex, as it combines technical know-how from several fields such as plant physiology, soil physics, soil chemistry and mathematics. Mechanistic rhizosphere models do not always operate with adequate precision (Rengel, 1993 Darrah and Roose, 2001). Two main fields of application of mechanistic rhizosphere models are carbon flow in the rhizosphere and nutrient uptake by plants. While carbon flow models study the exudation of carbon compounds into the soil and its consequences on the microbial population, uptake models focus on the transport and uptake of ions by roots. In the following sections, we will concentrate on uptake models on the single root scale. 

Among the first to consider root hairs in their model were Bhat et al. (1976) and Itoh and Barber (1983), while attempting to explain experimentally obtained P uptake values exceeding those calculated by one of the previously available models. Three principal approaches to integrate root hairs into a rhizosphere model are found in the literature (1) The boundary where exudation and uptake occurs is extended by the length of the root hairs (e.g. Kirk, 1999) (2) The continuity equation for root uptake is extended with a separate sink term (e.g. Geelhoed et al., 1997) and (3) The transport equation is solved in a three-dimensional model with cylindrical coordinates (Geelhoed et al., 1997). 

The role of root exudates in the solubilization of nutrients has often been discussed (Bhat et al., 1976 Claassen, 1990). Integrating root exudation into mechanistic rhizosphere models is quite complex owing to the difficulties in quantifying the nature and role of the exudates. The effect of soluble exudates such as phytosiderophores, phosphatases, mucilage and polycarboxylic acids on the availability of P, Fe and heavy metals has been recognized (Tinker and Nye,... 

Schnepf, A., Schrefl, T., Wenzel, W.W., 2002. The suitability of pde-solvers in rhizosphere modeling, exemplified by three mechanistic rhizosphere models. J. Plant Nutr. Soil Sci. 165, 713-718. 

P. R. Darrah, Models of the rhizosphere. I. Microbial population dynamics around the root releasing. soluble and in.soluble carbon. Plant Soil 733 187 (1991). 

D. E. Crowley and D. Gries, Modeling of iron availability in plant rhizosphere. Biochemistry of Metal Micronutrients in the Rhizosphere (J. A. Manthey, D. E. Crowley, and D. G. Luster eds.), Lewis Publishers, Boca Raton, Florida, 1994, p. 199. 

Figure 3 Model of proposed interactions in the rhizosphere and in the bulk soil.

J. Swinnen, J. A. van Veen, and R. Merckx, Rhizosphere carbon fluxes in field-grown spring wheat model calculations based on C partitioning after pufse-la-belling. Soil Biol. Biochem. 26 171 (1994). 

In experiments with lowland rice Oiyzci saliva L.) it was found that roots quickly exhausted available sources of P and sub.sequently exploited the acid-soluble pool with small amounts deriving from the alkaline soluble pool (18). More recalcitrant forms of P were not utilized. The zone of net P depletion was 4-6 mm wide and showed accumulation in some P pools giving rather complex concentration profiles in the rhizosphere. Several mechanisms for P solubilization could be invoked in a conceptual model to describe this behavior. However using a mathematical model with independently measured parameters (19), it was shown that it could be accounted for solely by root-induced acidification. The acidification resulted from H" produced during the oxidation of Fe by Oi released from roots into the anaerobic rhizosphere as well as from cation/anion imbalances in ion uptake (18). Rice was shown to depend on root-induced acidification for more than 80% of its P uptake. 

Several boundary conditions have been used to prescribe the outer limit of an individual rhizosphere, (/ = / /,). For low root densities, it has been assumed that each rhizosphere extends over an infinite volume of. soil in the model //, is. set sufficiently large that the soil concentration at r, is never altered by the activity in the rhizosphere. The majority of models assume that the outer limit is approximated by a fixed value that is calculated as a function of the maximum root density found in the simulation, under the assumption that the roots are uniformly distributed in the soil volume. Each root can then extract nutrients only from this finite. soil cylinder. Hoffland (31) recognized that the outer limit would vary as more roots were formed within the simulated soil volume and periodically recalculated / /, from the current root density. This recalculation thus resulted in existing roots having a reduced //,. New roots were assumed to be formed in soil with an initial solute concentration equal to the average concentration present in the cylindrical shells stripped away from the existing roots. The effective boundary equation for all such assumptions is the same ... 

While most authors have used the finite-difference method, the finite element method has also been used—e.g., a two-dimensional finite element model incorporating shrinkable subdomains was used to de.scribe interroot competition to simulate the uptake of N from the rhizosphere (36). It included a nitrification submodel and found good agreement between ob.served and predicted uptake by onion on a range of soil types. However, while a different method of solution was used, the assumptions and the equations solved were still based on the Barber-Cushman model. 

---