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14.1 Permeation without channels generate, create none none with none projectscode 39 printing To gain an app none none reciation of how essential channels are to ion permeation, we first examine permeation (not necessarily of ions) through a membrane when there are no channels. The earliest insight into membrane permeation dates back to the nineteenth century. The theory of Overton relates the permeability of a substance to its tendency to partition between water and hydrophobic solvents.

Since the interior of a membrane is hydrophobic,. .NET Framework 3.5 ION PERMEATION AND CHANNEL STRUCTURE hydrophobic su none for none bstances partition into the oily interior of a membrane and cross more easily. To develop this idea we make use of the partition coefficient, , which is the ratio of concentrations of a dissolved substance in water and a hydrophobic solvent, when the two solvents are in contact and at equilibrium. Taking these concentrations as cw and ch, respectively, we have.

ch cw (14:1). Partition coef none for none ficients such as these allow one to estimate the free energy of transfer of a substance between the two environments. This was used to quantitate the hydrophobic effect (Section 2.8).

If a solute dissolves in water and then equilibrates at each face of a membrane, then a concentration gradient between the two aqueous solutions will produce a proportional gradient inside the membrane between its two surfaces. Equation (14.1) thus implies that a concentration difference within the membrane is proportional to the difference in the two aqueous concentrations; cm cw.

If the flux through the membrane is proportional to the internal driving force, cm, then the flux will be proportional to cw. Thus, a membrane s permeability to a substance is directly related to the substance s partition coefficient. Experiments have confirmed this relation for organic molecules.

To take into account a molecule s mobility, a diffusion coefficient, D, can be factored in. For 16 substances a log log plot of D versus permeability is linear over a six-order of magnitude range, and the slope is close to one (Finkelstein, 1987). Inorganic ions generally have extremely low values of ; they are essentially insoluble in hydrophobic solvents.

Thus, we can account for the low permeability of inorganic ions through lipid bilayer membranes within the framework of Overton s simple theory. The free energy difference of an ion in water versus the membrane interior is accounted for through the self energy and the difference in dielectric constant (Section 2.2).

A calculation of the image force gives the following result for the work necessary to move a charge from water, with its high dielectric constant of "w $ 80 to the middle of a membrane, where the dielectric constant is much lower, "h $ 2 (Eq. (2.6); see Fig.

2.3a). G     q 2 1 1 q2 2"w ln 2a "h "w "h l "w "h (14:2). where l is the none for none thickness of the membrane, a is the radius of the ion, and q is the charge. The variable G in Eq. (14.

2) is the height of an energy barrier seen by an ion as it crosses the membrane. If we envision the flux of the ion through a membrane as a barrier crossing process, then the rate will be proportional to an exponential factor ( 7). J / e G=KT (14:3). 14.1 P E R M E A T I O N W I T H O U T C H A N N E L S Anything that reduces G will increase the flux across the membrane. This theory makes two clear predictions: (1) for ions of the same charge, the larger ones cross the membrane more rapidly (by reducing the first term of Eq. (14.

2)); and (2) thinner membranes are easier to cross than thicker membranes (by making the second term more negative). Both of these predictions are testable. The first prediction was confirmed by the demonstration that large organic ions such as tetraphenylphosphonium, tetraphenylboron, and dipicrylamine cross much more rapidly than small inorganic ions.

These are all monovalent ions with effective radii several times larger than those of monovalent inorganic ions such as Na and Cl . The second prediction of how flux changes with membrane thickness has also been tested. The thickness of an artificial bilayer membrane can be varied by forming membranes from lipids with different chain lengths.

Furthermore, for reasons that are not clear, when the lipids are dissolved in alkanes with longer chains, the bilayers formed from these solutions are thicker. The thickness was determined by measuring the capacitance, which is inversely proportional to thickness. In studies of the large organic cation dipicrylamine, the barrier-crossing rate was measured from current relaxation experiments.

A plot of rate versus thickness agreed well with Eqs. (14.2) and (14.

3) (Fig. 14.1).

These experiments illustrate the electrostatic nature of the barrier to ion flux across the hydrophobic core of a membrane. As the radius of the ion gets smaller, the first term of Eq. (14.

2) becomes very large; so the major biological ions such as Na , K , Cl , and Ca2 see insurmountable energy barriers. One essential function of ion channels is to reduce this energy..

20 18 16 14 12 none none J / J0 10 8 6 4 2 0 2.5 3.0 3.

5 l(nm) 4.0 4.5 5.

0. Fig: 14:1: The rate of dipicrylamine movement across a bilayer as a function of its thickness. The flux, J, was normalized to J0, the value at l 5. The solid curve is e 3.

69 17.8/l, based on Eqs. (14.

2) and (14.3) (data from Benz and Lauger, 1977). .

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