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solute (B) across the interfaces of a lipid membrane. The solid lines illustrate concentration profiles in the two solutions and within the lipid membrane.

aqueous layer and are unaffected by the presence of the barrier; in other words, they are not significantly influenced by either the openings or the walls of the pores through which they pass. The diffusion coefficient of a solute within the disk is the same as its diffusion coefficient in a continuous aqueous system (free solution), and the concentration of the solute just within the disk at each interface is the same as its concentration in the immediately adjacent aqueous solution. Thus, the sintered glass disk is a nonselective barrier because the properties of the solute within the disk are the same as those in the surrounding solutions.

If, as illustrated in Fig. 4, the sintered glass disk is replaced by a lipid membrane, the description of diffusion becomes slightly more complicated. First the concentration of the solute just within the membrane at each interface, in general, will not be equal to the concentration in the immediately adjacent aqueous solutions. If the solute is hydrophilic (or lipophobic; e.g., an ion), it will prefer the aqueous phase to the lipid phase, and its concentration just within the membrane will be less than that in the adjacent aqueous solutions. On the other hand, if the solute is lipophilic (e.g., fats, sterols, phospholipids), it will distribute itself so that the concentration at the interfaces just within the lipid barrier will exceed that in the adjacent aqueous solutions. The distribution of a given solute between the aqueous phase and the adjacent oil or lipid phase is described, quantitatively, by a unitless number termed a partition coefficient (also, a distribution coefficient). The partition coefficient of a given solute i between two solvent phases is determined in the following manner: The two immiscible solvents, for example, olive oil and water, together with an arbitrary amount of solute are placed in a separately funnel. The funnel is then stoppered and shaken until an equilibrium distribution of solute i in the two phases is achieved. The olive oil-water partition coefficient is defined as:

In this simple way, partition coefficients between lipid and aqueous phases have been determined for a large variety of solutes.

Returning to the example illustrated in Fig. 4A, it follows that if the solute in the two compartments is hydrophobic or lipophilic (Ki > 1), the concentration just within the membrane at the interface with compartment o will be KfiO, and the concentration just within the membrane at the interface with compartment i will be KCi. Thus, the concentration difference within the membrane is K, AQ, so that the partition coefficient has essentially amplified the driving force for the diffusion of the solute across the membrane. The flux of solute through the lipid membrane due to diffusion is now given by:

where the diffusion coefficient Di now reflects the ease with which the solute can move through the lipid and will differ from the value of Di in an aqueous solution. (We are assuming that diffusion through the membrane is slow, or rate-limiting, compared to partitioning into and out of the membrane at the two interfaces.) Conversely, if the solute is highly water soluble (i.e., hydrophilic or lipophobic), then K < 1 and its concentration difference within the membrane will be attenuated.

Now, in general, K, Di, and Ax cannot be readily determined. For this reason, these unknowns have been lumped together to give a new term, the permeability coefficient, Pt, which is defined as:

Because K is unitless, when Dt is given in cm2/hr and Ax in cm, then Pi has units of cm/hr. The permeability coefficient of a given membrane for a given solute is simply:

and can be readily determined experimentally. Clearly, the permeability coefficient of a membrane for an uncharged solute is the flow of solute (in moles per hour) that would take place across 1 cm2 of membrane

Equilibrium concentration of i in olive oil Equilibrium concentration of i in water when the concentration difference across the membrane is 1 M.

We now recapitulate some of the laws that apply to diffusional movements across artificial as well as biologic membranes. First and foremost, the driving force or sine qua non for the net diffusion of an uncharged solute is a concentration difference. In the absence of a concentration difference, a net flux due to diffusion is impossible. In the presence of a concentration difference, diffusion will take place spontaneously and the direction of the net flux is such as to abolish the concentration difference. In other words, diffusion only brings about the transfer of net uncharged solutes from a region of higher concentration to a region of lower concentration; the reverse direction is thermodynamically impossible! For this reason, transport due to diffusion is often referred to as downhill (because the flow is from a region of higher concentration to one of lower concentration) or passive transport (because no additional energy need be supplied to a system to enable these flows to take place; the inherent thermal energy responsible for random molecular motion is sufficient). As we shall see, there are numerous biologic transport processes that bring about the flow of uncharged solutes from a region of lower concentration to one of higher concentration. These uphill, or active, transport processes cannot be due to diffusion and are dependent on an energy supply in addition to simple thermal energy.

Within the membrane, the link between the driving force, Ac,, and the flow, Ji, is the diffusion coefficient, Di. This is the factor that determines the flow per unit driving force, and it is determined by the properties of the diffusing solute and those of the membrane through which diffusion takes place. At the molecular level, the diffusion coefficient is a measure of the resistance offered by the membrane to the movement of the solute molecule.

At the turn of the century, Einstein (1905) proposed that the resistance experienced by a diffusing particle results from the frictional interaction between the surface of the particle and the surrounding medium because the two, in essence, are moving relative to each other. Drawing on Stokes' law, which describes the friction experienced by a sphere falling through a medium having a given viscosity, Einstein demonstrated that the diffusion coefficient of a spherical molecule should be inversely proportional to both the radius of the molecule and the viscosity of the surrounding medium. Thus, when one is concerned with the diffusion of a variety of solutes through a single medium (fixed viscosity), the relative diffusion coefficients should be inversely proportional to their molecular radii. Einstein's contributions to our understanding of diffusion and the nature of the diffusion coefficient have been repeatedly confirmed, and the Stokes-Einstein equation has proved to be a valuable approach to the calculation of molecular dimensions from measurements of diffusion coefficients.

We can now make several educated guesses about the permeability of biologic membranes to uncharged solutes. First, because biologic membranes are primarily composed of lipids, Eq. 5 suggests that the permeability coefficients of solutes having approximately the same molecular dimensions (same diffusion coefficients) should vary directly with their partition coefficients (Ks); that is, the more lipid soluble the molecule, the greater its permeability coefficient. This deduction has been verified repeatedly and is a statement of Overton's law. In fact, as already noted, Overton's observation that lipid-soluble molecules penetrate biologic membranes more readily than water-soluble molecules of the same size predated the chemical analyses of membranes and was the first clue that biologic membranes are made up of lipids.

Second, Eq. 5 also predicts that the permeability coefficients of solutes having the same lipid solubilities (same Ks) should vary inversely with their molecular sizes; that is, the larger the molecule, the lower its permeability coefficient. It should be noted that Ks can vary over many orders of magnitude, whereas Ps generally do not differ by more than a factor of five. Thus, the permeability coefficients for simple diffusion of uncharged solutes across biologic membranes are more strongly influenced by differences in lipid solubility than molecular size.

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