The movement of solutes in plants can take place via the bulk flow of solutions, as in the xylem and phloem, or via specific transporters, where a protein is involved in the flux of a specific substance or group of substances. For neutral solutes the same driving forces occur as for water: the solute moves down gradients of its free energy (chemical potential) determined largely by gradients of activity (concentration). The effects of pressure (Nobel, 2005) and gravity are trivial in the context of cellular solute movements. For ions, however, there is an additional force that plays a very important role in their net movement - electric charge. As ions are charged particles, their movement is influenced by the presence of electric fields.
3.6.1 Chemical, electrical and electrochemical potentials and gradients
Unlike the case with water, differences in electrical potential (E) have a major influence on the movement of ions as ions carry a net charge (z, positive for cations and negative for anions; the charge carried by 1 mole of protons is 9.65 x 104 C or Faraday's constant). The chemical potential of an ion j is given by:
where z is the valency, F is the Faraday and E is the electrical potential (cf. Eq. 3.2 for water).
3.6.2 Diffusion - Fick's first law
Although in the phloem and the xylem solutes move in a mass flow of solution, in many situations the movement of solutes depends upon diffusion. Diffusion results from random movements of solute molecules. Where there are differences in concentration between two sites, there is a greater statistical probability of movement from a region of high concentration to a region of low concentration as there are more molecules in the high concentration region than in the region of low concentration. The flux (J) of a solute j (viz. jj, the quantity of j crossing a unit area per unit time e.g. mol m-2 s-1) is directly proportional to the concentration gradient of j, viz.:
where Scj/Sx is the concentration gradient of j along the distance axis x and Dj is the diffusion coefficient of j (a coefficient rather than a constant as the value varies with temperature and concentration of j). This relationship is commonly known as Fick's first law of diffusion, after its discoverer. Diffusion coefficients of common solutes in water at 25°C have different values, examples of which are 0.52 x 10-9 m2 s-1 for sucrose and 1.9 x 10-9 m2 s-1 for K+ (with Cl-). It is informative to note that the time taken for a little over a third (36.8%, 1/e) of a population of K+ ions to diffuse across a cell (50 ^m) is 0.6 s, while the time taken for the same proportion of these K+ molecules to diffuse over the distance of 1 m would be about 8 years (Nobel, 2005). Diffusion is not a process suited to long-distance transport in biological systems. In cells, there is a bulk movement of the cytoplasm, known as cytoplasmic streaming, which results in mass movement of solution and so reduces the time taken for solutes to move between parts of a cell.
Where diffusion takes place across a barrier, such as a membrane or cell wall, the concentration gradient -Scj/Sx can be represented by the difference in average concentration across the barrier divided by its effective width, i.e. the difference of concentration between the outside (o) and the inside (i) across the distance x, viz. (co - cj )/Ax. The distance over which solutes diffuse is the width of the barrier, plus any unstirred layers on either side of that barrier (layers where, because of friction between the barrier and the bulk solution, the bulk flow of solution is reduced to zero). Unstirred layer can be greater in width than the thickness of the barrier itself. Because the barrier, the membrane or cell wall, is not of the same chemical
composition as the bulk solution, the concentration of solute in the barrier depends on the partition between the two phases. This means the effective concentration difference across the barrier is Kj(co - cj), where Kj is a partition coefficient, a dimensionless ratio of the concentration of the solute in the barrier and in an aqueous solution.
The flux across the membrane is given by:
where the permeability coefficient Pj is:
The permeability coefficient for K+ in a cell membrane is about 10-9 ms-1, typical of small charged ions.
In cell walls, ions will diffuse through aqueous channels, but these are relatively small in proportion to the unit area over which diffusion is occurring. For a cell wall whose thickness is 1 ¡m, the permeability coefficient would be 1 x 10-3 m s-1, considerably greater than that of a cell membrane. However, the permeability coefficient of the same K+ ion in an unstirred layer of 30 ¡xm would be 3 x 10-5 m s-1, lower than that for the cell wall per se. For larger solutes, whose molecular dimensions are similar to the pore size in the walls, the walls can act as a 'membrane' with a reflection coefficient (see Section 3.2.2) less than 1 and osmotic water withdrawal can occur across the walls, as has been demonstrated for tissues where the cells have been disrupted by freezing and thawing (Flowers and Dessimoni Pinto, 1970). The limiting size of molecules that cross cell walls is 3.5-5.2 nm according to species (Carpita et al., 1979).
Although the values of D, Kj and Ax may be uncertain, Pj is a readily measurable quantity. Provided the flux of a substance can be measured and the internal and external concentrations are known, Pj can be calculated. The chief problem in estimating permeability coefficients is in obtaining the values of c° and cj at the membrane surface, since it is only possible to measure these in the solution on either side of the membrane. The boundary layer, in which the concentration varies with distance from the membrane, confounds estimation of the actual concentrations at the membrane surfaces themselves. The boundary layer cannot be entirely removed by rapid stirring of the solution, especially if intact plant cells are used, as this layer will be located within the cell wall where stirring is not possible. Values of Pj for small molecules such as glucose, glycerol and urea lie in the range from 0.01 to 3 x 10-9 m s-1. It is difficult to estimate absolute permeabilities for ions, but PNa/PK is about 0.2 and Pci/Pk, 0.003 (see Nobel, 2005).
If a salt, such as potassium chloride, is added as a solid to a beaker of water, the ions dissolve in the water and a concentration gradient is established within the beaker, which leads to the diffusion of potassium and chloride ions from high to low concentration. If the ions are of different sizes, they will have different mobilities in the solution and diffuse at a slightly different speeds so that a difference in charge develops, which is known as a diffusion potential. In a solution, this occurs over microscopic distances. Such diffusion potentials arise where microcapillaries filled with concentrated electrolyte as a conductor are inserted into cells to determine the potential across a membrane (cf. Section 2.6.3). For an electrode filled with 3 M KCl and inserted into a cell, the diffusion potential is about -2 mV.
If a cell is placed in a dilute solution of a salt and allowed to equilibrate such that the diffusion of ions into the cell balances the diffusion out of the cell, there is no net movement of ions and the electrochemical potential of, say, K+ inside is equal to that outside the cell. In general terms, at equilibrium, xo = P-), whence, expanding using Eq. 3.16:
H*j + RT ln a° + ZjFEo = n * + RT ln aj + ZjFEi (3.20)
The potential at equilibrium ENj (named after Nernst, who first derived this relationship) is given by ENj = Ei - Eo. Rearranging Eq. 3.20,
ZjF ( E1 - Eo) = RT( In a° - In a)) = RT In -j whence aj
ENj = E1 - Eo = — ln -j = —2.303 log -j j ZjF aj ZjF aj
At 25°C for a monovalent cation, this reduces to:
Equation 3.21 demonstrates the poise between electrical potential and chemical concentration in systems at equilibrium: a difference in concentration of a monovalent cation of tenfold across a membrane is balanced by a difference in potential of 59 mV when the temperature is 25°C (at 20°C, the balancing potential is about 58 mV).
Cell walls offer yet another level of complexity to the diffusion of ions in that walls carry a net negative charge. In such a system, at equilibrium, the Nernst equation can be applied, but here the concentration of anions and cations is different. For example, with K+ at a concentration (and activity) of 1 mM and Ca2+ at 0.5 mM outside a cell and with a cell wall with a concentration of fixed anions equivalent to 100 mM, it is possible to calculate the concentration of potassium and calcium ions in the cell wall and the difference in potential between the outside solution and the cell wall (Briggs et al., 1961):
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