Liquid Membrane Transport Phenomenon

Mechanisms of Transport

You have already seen, perhaps, that there are several ways in which we can set up a liquid membrane system that will accomplish what we need it to do. The next question is, why do those methods work?

If you actually followed the stages of the cation transport, you notice that there are two stages involving diffusion. That's probably the best place to start because there are in fact two major categories of transport, active and passive transport, but in understanding those, you should also understand the rules governing diffusion.

I know you've gotten used to my writing style by now, so you know what my next sentence is going to say. And the next sentence is ... "therefore, I'm going to cover the following."

Rules Governing Diffusion

Simple Diffusion

The flux of a gas through a membrane under a concentration gradient along the x axis is dictated by Fick's First Law,

If one happens to apply a potential across a system, things become a little shaky if you continue using Fick's Law. Therefore, we resort to the Nerst-Planck equation,

If there is no electrochemical potential, we can use the Einstein equation to predict the diffusivity.

Constant Concentration Gradient

If you assume there is a constant concentration gradient, dC/dx over the entire diffusing region, you can integrate the dC/dx term as a function of the distance, x. In doing so, you realize that at steady state, the flux is constant over the diffusing distance, independant of x. The Nerst-Planck Equation, as shown before, is suddenly rewritten as

And as the electrochemical potential goes to zero, Fick's First Law becomes

Active Transport

When talking of active transport, you are most often referring to carrier kinetics and chemistry which allow for transport against a concentration gradient, or, uphill transport. To do so requires a slight shift in thinking. If you rewrite Fick's First law and begin to define the flux in terms of (a) the conjugate forces, (b) the coupled flux, and (c) the coupled chemical reactions, then you end up with a function looking somewhat like this

In work done by Kedem and Caplan, it was shown that if the sum of Ji*Xi + Jr*Xr is positive, the flux may be negative and uphill transport is possible.

For more information, it will be necessary to look through a book on membrane transport.

Antiport Mechanics

In the antiport typical model, the dissociation constants for the complexes are defined by the equation

where i and j are components in the phases, and X refers to the carrier. If there is no j in phase I, then the equations for flux of species i from phase I to II and from II to I, as well as the overall flux are

The above equations are not really helpful for most people, so I'll explain a little. The really interesting thing is the point at which the overall flux becomes zero, which is when

Thus, you now have a range over which the uphill transport and downhill transport is possible. If the left hand side of the above equation is greater than the right hand side, then you have uphill transport. If the left hand side is less than the right, you have downhill transport, so long as the left is above zero. If you really look at the equation, though, you'll note one more thing. Since there is no j in phase I, the flux of i from Phase I to II is uninhibited by and competition with j, and is therefore independant of j. However, the reverse flux, Ji from II to I, requires teh competition of i with j for the complexation with the carrier. In this case, the energy required for the uphill transport is supplied by the downhill transport of j.

Symport

I am not going into the equations as I did with the above setup; if you really want to know, look into one of the books on membrane transport. What you really need to know is this; in the antiport setup, having a low concentration of a species in one phase caused for the uphill transport of the other species; in this configuration, it is the exact opposite. If species j has a low concentration in phase I, then the transport of i is done according to the concentration gradient, or, spoken differently, it occurs downhill. If phase I has a high concentration of j, the uphill transport of i occurs from Phase I to II. The reason for this is because in antiport, the competition for carriers was the cause of downhill transport. In symport, the species cooperate, and in fact it is needed in order for the coupling to occur.

Passive Transport

Passive transport is basically transport with the concentration gradient, driven by a difference in chemical potential. The flux for a species i is therefore following Fick's Law, and can be expressed in terms of the diffusion coefficient, the concentrations, and the thickness of the membrane.

There is also about fourteen different models for relating the flux of a metal ion through a membrane, but there is no way I will cover these; the math involved would take up too much room on my account, so I'll break down a few into a sentence or so. If you couldn't tell by that wonderful lead-in, these are really general break-downs. One of these is the case where M + L --> ML, which is diffused, and the same equilibrium constant is applied to both interfaces. Essentially what you get is that the transport ceases when the concentration in both phases becomes the same, and, more importantly, flux is directly proportional to the total carrier concentration in the membrane. Okay, so I lied. I'm only breaking down that one, since everything else requires a more extensive preface.

A Few Words on Fick's Law

Essentially, Fick's Law breaks down like this.

This law work for systems with neutral species and any system with charged species but no applied potential.

A Few Words on the Nerst - Planck Equation

The Nerst-Planck equation refers to several new variables.