Effects of Zn-complexes on zinc uptake by wheat (Triticum aestivum) roots: a comprehensive consideration of physical, chemical and biological processes on biouptake

Abstract

Commonly used equilibrium models for metal biouptake, such as the Free Ion Activity Model (FIAM) and the Biotic Ligand Model (BLM), are limited to the cases in which mass diffusive transport is not the flux-determining step. In analyses of metal biouptake from a complexing medium, all the physical (diffusion), chemical (dissociation kinetics of metal complexes), and biological (transport and internalization) processes have to be taken into account. A short-term zinc uptake by wheat (Triticum aestivum) roots from culture solutions in the absence or presence of synthetic ligands (NTA, nitrilotriacetic acid, and EDTA, ethylenediaminetetraacetate) was studied. At the same free Zn²⁺ concentration $\left( {\left\{ {{\text{Zn}}^{{\text{2 + }}} } \right\} = 1.5 \times 10^{ - 8} {\text{M}}} \right)$ , the uptake of Zn was significantly enhanced in the presence of ligands and was larger when Zn complexes have a quicker dissociation rate. The diffusional fluxes in the same culture solution were determined with the differential pulse anodic stripping voltammetry (DPASV) method, and the diffusive gels in thin film (DGT) technique. The contribution from Zn complexes to root Zn uptake was in better agreement with the degree of Zn complex labilities measured with DPASV than with DGT. The diffusion of free Zn²⁺ ion to the root surface is a rate-controlling step for Zinc biouptake when the free Zn²⁺ concentration is low. Based on the comprehensive consideration of the diffusion and dissociation processes of Zn²⁺ ion and Zn complexes and the existence of high- and low-affinity uptake systems in the root surface, a two-pathway Zn uptake model was developed to predict the resulting Zn uptake fluxes into roots in the overall range of exposure.

Appendix

Establishment of the TPUB-Model for Zn uptake

The purpose of this appendix is to derive the mathematical expression of wheat root Zn uptake model. Firstly, we shall consider the situation in which an organism is in contact with the medium, containing a bioactive free metal ion M and bioinactive metal complex ML (charges have been omitted for simplicity).

(10)

where the association and dissociation rate are k _a and k _d, respectively. Under conditions of complexation equilibrium, the stability constant, K, can be described as:

$$K = \frac{{k_a }}{{k_d }} = \frac{{c_{ML}^* }}{{c_M^* c_L^* }}$$

(11)

where ${\text{C}}_{\text{i}}^*$ denotes the concentration of species i in the bulk solution.

For a singular pathway root uptake, the steady-state biouptake flux is modeled by the Best equation, which couples interfacial Michealis–Menten (MM) kinetics with a limiting flux of mass transfer towards the biosurface (Van Leeuwen 1999). For the medium with a bioactive metal ion M, bioinactive species ML, and a planar biosurface (e.g., higher plant roots), the biouptake step can be represented by a Michealis–Menten type of flux equation:

$$J_u = J_u^* \frac{{c_M^0 }}{{K_M + c_M^0 }}$$

(12)

where J _u is the actual metal uptake flux, ${\text{J}}_{\text{u}}^{\text{*}}$ is the limiting uptake flux, ${\text{c}}_{\text{M}}^{\text{0}}$ is the concentration of the bioactive species M at the biosurface, and K _M is the characteristic bioaffinity parameters (equal to ${\text{c}}_{\text{M}}^{\text{0}}$ for J_u = 1/2J_u ^*). Mechanistically, the MM equation type steady-state flux describes a fast Langmuirian adsorption of the free metal M at the outer cell membrane followed by a first-order rate-limiting internalization step. For this uptake model, the analysis will concentrate on the ${\text{c}}_{\text{M}}^{\text{0}}$ and the supply of M from the complex medium with bulk concentrations ${\text{c}}_{\text{M}}^{\text{*}}$ and ${\text{c}}_{{\text{ML}}}^{\text{*}}$, coupled the reaction dynamics (M and ML interconversion rates governed by k _a and k _d) with diffusion process (diffusion rates governed by the diffusion coefficient D_M of the free metal M and the diffusion coefficient D_ML of the complex ML).

The diffusional transport of metal in the medium can simply be written as the flux J_diff that only related to M:

$$J_{diff} = \frac{{D_M }}{{\delta _M }}\left( {c_M^* - c_M^0 } \right) = J_{diff}^* \left( {1 - \frac{{c_M^0 }}{{c_M^* }}} \right)$$

(13)

where ${\text{J}}_{{\text{diff}}}^{\text{*}}$ is the limiting diffusion flux for the case of ${\text{c}}_{\text{M}}^{\text{0}} = 0\,\left( {{\text{i}}{\text{.e}}{\text{.,}}\,{{{\text{D}}_{\text{M}} {\text{c}}_{_{\text{M}} }^* } \mathord{\left/ {\vphantom {{{\text{D}}_{\text{M}} {\text{c}}_{_{\text{M}} }^* } {\delta _{\text{M}} }}} \right. \kern-\nulldelimiterspace} {\delta _{\text{M}} }}} \right)$, δ_M is the steady-state diffusion layer thickness.

For wheat roots, there is a cylindrical diffusion formula (Crank 1964) for calculating the diffusional flux of the free ion M towards the roots.

$$J_{diff} = 2\pi D_M \left( {c_M^* - c_M^0 } \right) \cdot \log \left( {\frac{{r_{root} }}{{r_{root} + \Delta }}} \right) = J_{diff}^* \left( {1 - \frac{{c_M^0 }}{{c_M^* }}} \right)$$

(14)

where r_root is the radius of root and Δ is the width of the boundary layer at the root surface. Here,

$$J_{diff}^* = 2\pi D_M \log \left( {\frac{{r_{root} }}{{r_{root} + \Delta }}} \right)$$

(15)

Inclusion of Eq. 13 or 14 into Eq. 12 to eliminate ${\text{c}}_{\text{M}}^{\text{0}}$ yields a resulting biouptake flux expression model in terms of two most elementary parameters.

$$J = J_u^* \cdot Q = J_u^* \frac{{\left( {1 + a + b} \right)}}{{2b}}\left\{ {1 - \left[ {1 - \frac{{4b}}{{\left( {1 + a + b} \right)^2 }}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\}$$

(16)

where Q is known as the Best equation (Bosma et al. 1997), “a” is the relative bioaffinity parameter, and “b” is the limiting flux ration. Low “a” values (a≪1) correspond to a high affinity of the organism for uptake, and large “b” values correspond to limiting diffusional transport conditions, in which the metal diffusion flux towards the root is lower than the biouptake flux. The ratio between “a” and “b” is the famous bioavailability number B_n (Bosma et al. 1997).

$$a = \frac{{K_M }}{{c_M^* }}$$

(17)

$$b = \frac{{J_u^* }}{{J_{diff}^* }}$$

(18)

When the characteristic lifetimes of M and ML in reaction (Eigen 1965) are much shorter than the effective time scale of the experimental conditions (dynamic condition: k _ac_L ^*t and k _dt ≫1), and the kinetic flux of M resulting from net dissociation of ML (J_kin) is much greater than the purely diffusional flux of ML (J_diff) (Van Leeuwen 1999), the complex ML is then said to be labile. Thus, the resulting diffusion flux of the free metal M can be written as (Degryse et al. 2006b):

$$J_{diff} = \frac{{D_M }}{\mu }\left( {c_M^* - c_M^0 } \right) = J_{diff}^* \left( {1 - \frac{{c_M^0 }}{{c_M^* }}} \right)$$

(19)

where μ is the reaction layer thickness, a well-known concept in electrochemistry, which is a thin layer adjacent to the electrode surface, or in this case the root surface, in which the rate of dissociation of the metal complex is not fast enough to follow the depletion of the free metal ion (Van Leeuwen et al. 2005). The thickness of this reaction layer can be seen from Eq. 20

$$\mu = \sqrt {\frac{{D_M }}{{k_a \cdot c_L^* }}} = \sqrt {\frac{{D_M }}{{k_d }} \cdot \frac{{c_M^* }}{{c_{ML}^* }}} $$

(20)

The combination of Eqs. 12 and 19 results in a resulting biouptake flux expression, in the form identical to Eq. 16

$$J = J_u^* \cdot Q = J_u^* \frac{{\left( {1 + a + b} \right)}}{{2b}}\left\{ {1 - \left[ {1 - \frac{{4b}}{{\left( {1 + a + b} \right)^2 }}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\}$$

(16)

where

$$a = \frac{{K_M }}{{c_M^* }}$$

(17)

$$b = \frac{{J_u^* }}{{J_{diff}^* }} = \frac{{J_u^* }}{{\frac{{D_M }}{\mu }c_M^* }}$$

(21)

Based on the principle above and considering wheat roots with two different affinity uptake systems in the complex medium, we put forward a two-pathway uptake Best equation model (TPUB-Model). Here, the two pathways uptake MM equation can be written as:

$$J_u = J_{u,1}^* \frac{{c_M^0 }}{{K_{M,1} + c_M^0 }} + J_{u,2}^* \frac{{c_M^0 }}{{K_{M,2} + c_M^0 }}$$

(1)

where K _M,1 and K _M,2 denote the characteristic high- and low-affinity parameters, respectively; ${\text{J}}_{{\text{u,1}}}^{\text{*}}$ and ${\text{J}}_{{\text{u,2}}}^{\text{*}}$ denote, respectively, the limiting uptake fluxes of high- and low-affinity uptake systems. The diffusional flux expression of the free metal ion is the same as the above diffusional fluxes. The combination of the Eq. 1 with the diffusional flux expression yields a two-pathway uptake resulting flux expression: (we divided ${\text{c}}_{\text{M}}^{\text{*}}$ as ${\text{c}}_{\text{M}}^{\text{*}} \ll K_{{\text{M,2}}}$ and ${\text{c}}_{\text{M}}^{\text{*}} \gg 2K_{{\text{M,1}}}$ in order to easily resolve the equation by simplifying Eq. 1)

• (1) When ${\text{c}}_{\text{M}}^{\text{*}}$ is much smaller than K _M,2 (i.e., ${\text{c}}_{\text{M}}^{\text{*}} > K_{{\text{M,2}}}$), Eq. 1 can be simplified as:

  $$J_u = J_{u\,,1}^* \frac{{c_M^0 }}{{K_{M\;,1} + c_M^0 }} + J_{u\,,2}^* \frac{{c_M^0 }}{{K_{M\,,2} }}$$

  (22)

  The two pathways uptake resulting flux expression can be seen as:

  $$J_u = J_{u\,,2}^* \frac{{\left( {a_2 b_1 + a_1 b_2 + 2b_2 - a_1 a_2 - a_2 } \right)}}{{2_{b2} \left( {a_2 + b_2 } \right)}}\left\{ {1 - \left[ {1 - \frac{{4\left( {a_2 b_1 + a_1 b_2 + b_2 } \right)\left( {a_2 + b_2 } \right)}}{{\left( {a_2 b_1 + a_1 b_2 + 2b_2 - a_1 a_2 - a_2 } \right)^2 }}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\}$$

  (2)

• (2) When ${\text{c}}_{\text{M}}^{\text{*}}$ is larger than 2K _M,1 (i.e., ${\text{c}}_{\text{M}}^{\text{*}} \ll 2K_{{\text{M,1}}}$), Eq. 1 can be simplified as

  $$J_u = J_{u\,,1}^* + J_{u\,,2}^* \frac{{c_M^0 }}{{K_{M\;,2} + c_M^0 }}$$

  (23)

  Therefore, the two-pathway uptake resulting flux expression can be written as:

  $$J_u = J_{u\,,2}^* \frac{{\left( {b_1 + a_2 + b_2 + 1} \right)}}{{2b_2 }}\left\{ {1 - \left[ {1 - \frac{{4\left( {b_1 + b_2 + a_2 b_1 } \right)}}{{\left( {b_1 + a_2 + b_2 + 1} \right)^2 }}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\}$$

  (3)

  where a₁, b₁ and a₂, b₂ denote the a and b values of the high- and low-affinity uptake pathways, respectively, and their meaning is the same as defined above.