A Mathematical Description of Pollutant Uptake in Plants by Single Cylindrical Root

The classical model of plant root nutrient uptake given by Nye, Tinker and Barber is modified and extended for pollutant uptake in plants. An explicit closed mathematical description is given for the uptake, by a single cylindrical root for all cases of practical interest, by solving the absorption-diffusion equation for the soil pollutant concentration asymptotically in the limit of large time. This single root model can be used as a building block to construct a model for multiple root branching structure in a more realistic plant root system. The theoretical results derived analytically are compared with numerical results and experimental studies.


INTRODUCTION
The high concentration of pollutants in soils is reflected by higher concentrations of pollutants in plants, and consequently in animal and human bodies. The ability of some plants to absorb and accumulate xenobiotics makes them useful as indicators of environmental pollution. The study of excessive concentrations of pollutants in biological matrices has been reported in numerous publications (Jensen et al., 1992;Lynch et al., 1997;Marschner, 1995). Pine and spruce needles, mosses and grasses are widely used as a specific indicator in the study of a geographical and temporal pattern of pollutants. Streit and Strum et al. (1993) classified the exchange of chemicals between soil and plants and

Research Article
divided the most common methods of assessing toxicity to plants from soil into three categories in the conditions closed system: (i) monitoring of the presence or absence of specific plant ecotypes and/or plant species (indicator plant), (ii) measurements of pollutant concentration in tissues of selected species (accumulative bio-indicators), (iii) recording of physiological and biochemical responses (biomarkers) in sensitive bioindicators.
The aim of the paper is to investigate relations between pollutant gradient in soil and its accumulation in plants through cylindrical roots.
The classical model for uptake developed by Nye and Tinker (1997) and Barber (1984) supposes a single cylindrical root to be surrounded by an infinite extent of soil, with prescribed far field soil water concentration. The pollutant diffuses through the soil water (via the pore water), and its uptake at the root is given by a Michaelis-Menten dependence on concentration (Jungk et al., 1997;Jensen, 1992). This absorption-diffusion model thus consists of a linear diffusion equation with the nonlinear root surface absorption condition. In this paper we deal with this problem by providing a fully explicit approximation to the basic Nye-Tinker-Barber model.

Model Formulation
We assume that the soil consists of a solid phase, a liquid phase and a gas phase, and that the volume fraction of each phase stays constant, so, 1 s l α θ θ θ + + = (2.1) where θ s is the volume fraction of the soil solid phase, θ l is the volumetric water content and θ a is the volume fraction of the air phase. The sum of θ l and θ a is the soil porosity Φ. Typical values of Φ are 0.3-0.6 and typical values for θ l in soils are 0. 15 -0.4 (Richardson, 1995). We assume that pollutant is present in the solid (C s ) and liquid (C l ) phase and the total concentration of pollutant in soil (C T;soil ) is 2) Assuming equilibrium sorption according to a linear Freundlich isotherm and neglecting intraparticle diffusion, the rate of change of C s with respect to t is (Tinker, 1975, Morton et al. 1994) where b p is the buffer power of the soil. For a volume of soil V without internal sources or sinks, the conservation law gives where the flux q is derived by assuming diffusive and convective transport and given by Fick's law that describes the movement of solutes in the direction of decreasing concentration gradient: where D l is the diffusion coefficient of the solute in free water, f l (θ l ) is the impedance factor of solute in the liquid phase and v is the Darcy flux. Hence The rate of change of C T; soil with time is given by: Taking initial concentration C l,0 (µmol cm -3 ) in soil solution as constant and assuming the uptake of pollutant at the root surface follows Michaelis-Menten kinetics , the conditions at the root surface will be is the operator for the outward normal derivative, v n is the Darcy flux of water normal to the root surface, F m (µmol cm -2 s -1 ) is the maximal root uptake rate and K m (µmol cm -3 ) is the Michaelis Menten constant.
If there is no competition between roots then concentration far away from the root surface stays constant. So, the boundary condition far away from the root surface will be ,0 l l C C at x = → ∞ For simulating radial flow into a cylindrical root, the model is where r 0 is the radius of the root, r 1 is the half distance between two roots (r 1 =1/√πρ, where ρ is root length density) and v 0 is the Darcy flux of water into the root surface (Fowler, 1997;Varney et al. 1993).

Let
Then the non-dimensional model is (dropping asterisks) Hence dimensionless parameters are Peclet number,

APPROXIMATE SOLUTIONS
At P e << 1, neglecting convective transport, the model becomes

Numerical Solution
For the numerical solution of Equations (3.1)-(3.3), we use a finite difference scheme with a centered discretisation in space and the θ-method in time (Ezawa et al., 2009). Let h be the step size of a uniform mesh in space, ∆t be the time step and J +1 be the number of grid points. If n j n C (jh, n t) U and C (n t) V ∞ ∆ ≈ ∆ ≈ . Then the numerical scheme is given by The scheme can be rewritten as We solve the tri-diagonal system of equations T and D is the differential operator, a tridiagonal (J + 1) × (J + 1) matrix whose entries are zero except for The first and last rows of matrices ( ) are modified according to the boundary conditions. The numerical scheme for the initial condition and for flux equation (3.2) will be and at every time step, we approximate C ∞ in equation (3.3) with an explicit Euler Scheme.

Approximate Analytic Solutions
3.2.1 Analytic solutions for the limit c ∞ < 1

Approximations for λ >> 1
(1) When λ >> 1 i.e., pollutant uptake is high, we expect the gradient of pollutant concentration near the root surface to be big i.e.
Thus, there is a boundary layer near r = 1 and by rescaling the independent variables r and t to stretched variables R and T, i.e., r = 1 + R/λ and t = T /λ 2 , the problem reduces to and the initial condition is C = 1 at T = 0 for 0 < R < ∞. The solution to this leading order problem (3.10)-(3.11) is given by (2) For (3.16) and the flux at the root surface r = 1 is According to (Richardson, 1993) α is some positive number. Though it is not unique, but for the purposes of current research possibly quite satisfactory.
is Euler's constant. Hence at the leading order without singularity at γ ) the approximation can be written as and the all time solution can be found by taking the uniform interpolation in the form is some positive number. Though it is not unique, but for the purposes of current research and the all time solution can be found by taking the uniform interpolation in the form (3.21) is some positive number. Though it is not unique, but for the purposes of current research

Approximations for λ <<1
For λ<<1, we expect the region (where C is less than order 1) to be larger. The asymptotic behaviour of this system turns out to be mathematically similar to that of Oseen type of problems of flow past cylinder (Greenwood et al. 1997, Richardson 1995 and is derived in terms of two length scales r ≈1 and r >>1. In the first case i.e. the inner (near root surface) expansion has ln r terms which make it impossible to satisfy the boundary condition. This is overcome by considering the variables of second case (Outer expansion) such that to the first approximation the radius of the root is negligible. Hence the first term in the outer expansion is the solution in the absence of the disturbance from the root. The second term in the outer expansion represents the disturbance to the far-field due to a thin root. By matching the inner and outer solutions, a uniformly valid approximation can be derived.
At the root surface r = 1 gives R = σ << 1, we find that far away from the root, the root acts as an infinitely thin line sink at the origin. This suggests that we should look for a similarity type solution with similarity variable η = R 2 / (4τ). The solution satisfying the far-field boundary condition C → 1 as η → ∞ is the outer expansion of the problem and given by: 2. For inner expansion we again rescale the space variable near root surface region, i.e., r = Rσ, and keeping long time-scale variable τ, we have The leading O(σ 0 ) solution satisfying the root surface boundary condition at r = 1, is given by In order to satisfy the boundary condition at the root surface we choose F (τ ) = λC(1, τ ). C(1, τ ) and F (τ) can be determined via matching with the outer solution.
To match the above solutions in the overlap region, we expand the outer solution at the small, near root surface, limit and the expansion at small argument η limit will be Thus, the outer solution at the inner limit will be 1 2 2 which is equal to the inner solution. So by, matching the O(ln R) terms in the expansions we find that and matching the remaining leading order terms we get The flux F in terms of root surface pollutant concentration can be estimated as F(τ) = λC(1, τ), so the root surface Pollutant concentration is given by To obtain the approximation of the flux we use this approximation to the root surface concentration and substitute it into the full non-dimensional flux condition F= λC 1 /(1 + C 1 ) where C 1 = C ∞ C(1, t). Hence we obtain 2 ( ) 1 (4 1) The results of numerical experiments in comparison to the approximations derived, are shown in Fig. 3. The solution agrees reasonably well with the numerical experiments conducted with the full non-linear model. The slight error at larger times is due to the leading order approximation of the non-linear boundary condition.

C ∞ ≥
We will now investigate the case when the Pollutant far-field concentration is large, i.e., and thus C 0 >> K m . we rescale C =C ∞ C, so the equation and boundary conditions become with small time approximation given by and Large time approximation is where γ ≈ 0.57722.. is Euler's constant.  (3.40) and the initial condition C = 1 at T= 0 for 0 < R < ∞.

Approximate solution for λ <<1
For the case λ<< 1we have ∂C/∂r| r=1 ≈ λ<< 1, we expect the region, where C is less than order 1, to be larger. Similar to Section 3.1.2 we use the Oseen type expansion to find the approximate solution.
Thus, after rescaling to large time and space variables τ and R, i.e., t = τ /σ 2 and r = R/σ with σ << λ << 1 and σ = e −1/λ , we find that the outer similarity solution satisfying boundary  The approximation to the flux, which is given in this approximation by F(t) = λC ∞ , is even worse than the approximation to the root surface concentration (Fig. 5). This is due to the leading order approximation of the nonlinear boundary condition.

CONCLUSION
In this paper the derivation of the model for pollutant uptake, initially developed by Nye, Tinker and Barber is presented together with its non-dimensionalisation and mathematical analysis at different dimensionless parameter limits. We estimated the critical time t c after which the pollutant flux into the root starts decreasing due to the deficiency of pollutants at the root surface. This critical time is relatively large for the case when the far-field concentration c 0 is much larger than the Michaelis-Menten uptake coefficient K m , and also when the dimensionless pollutant uptake coefficient 1 m m F a DK λ φ = is very small compared to the far-field concentration. When the pollutant uptake coefficient λ is large compared to the far-field concentration C ∞ , the flux starts decaying very rapidly after a short transition time, which is approximately of order 1/λ 2 , and in the case of the small uptake coefficient λ, the decrease in root surface Pollutant concentration takes place over the exponentially large time-scale, more specifically of order e 2/λ corresponding to times larger than the average vegetative period of agriculturally important plants.