A Continuum Model for Volume and Solute Transport in a Pore

As Gene Surdutovich and I prepare the 6th edition of Intermediate Physics for Medicine and Biology, we have to make many difficult decisions. We want to streamline the book, making it shorter and more focused on key concepts, with fewer digressions. Yet, what one instructor may view as “fat” another may consider part of the “meat.” One of these tough choices involves Section 5.9 (A Continuum Model for Volume and Solute Transport in a Pore).

Neither Gene nor I cover the rather long Sec. 5.9 when we teach our Biological Physics class; there just isn’t enough time. So, at the moment this section has been axed from the 6th edition. It now lies abandoned on the cutting room floor. (But, using LaTex’s “comment” feature we could reinstate it in a moment; there’s always hope.) Russ Hobbie would probably object, because I know he was fond of that material. Today, I want to revisit that section once more, for old times sake.

The section develops a model of solute flow through a pores in a membrane. One key parameter it derives is the “reflection coefficient,” σ, which accounts for the size of the solute particle. If the solute radius, a, is small compared to the pore radius, R_p, then solute can easily pass through and almost none is “reflected” or excluded from passing through the pore. In that case, the reflection coefficient goes to zero. If the solute radius is larger than the pore radius, the solute can’t pass through (it’s too big!); it’s completely blocked and the reflection coefficient is one. The transition from σ = 0 to σ = 1 for medium-sized solute particles depends on the pore model.

The fifth edition of IPMB presents two models to calculate how the reflection coefficient varies with solute radius. The figure below summarizes them. It is similar to Fig. 5.15 in IPMB, but is drawn with Mathematica as many of the figures in the 6th edition will be. 

The blue curve shows σ as a function of ξ = a/R_p, and represents the “steric factor” 2ξ − ξ². It arises from a model that assumes there is plug flow of solvent (usually water) through the pore; the flow velocity does not depend on position. The maize curve shows a more complex model that accounts for Poiseuille flow in the pore (no flow at the pore edge and a parabolic flow distribution that peaks in the pore center), and gives the reflection coefficient as 4ξ² – 4ξ³ + ξ⁴. (Is it a coincidence that I use the University of Michigan’s school colors, blue and maize, for the two curves? Actually, it is.) Both vary between zero and one.

You can consult the textbook for the mathematical derivations of these functions. Today, I want to see if we can understand them qualitatively. For plug flow, reflection occurs if the solute is within one particle radius of the pore edge. In that case, the number of particles that reflect grows linearly with particle radius. The steric factor 2ξ − ξ² has this behavior. For Poiseuille flow, the size of the particle relative to the pore radius similarly plays a role. However, the flow is zero near the pore wall. Therefore, tiny particles adjacent to the edge did not contribute much to the flow anyway, so making them slightly larger does not make much difference. The reflection coefficient grows quadratically near ξ = 0, because as the particle radius increases you have more particles that would be blocked by the pore edge, and because the larger size of the particle means that it experiences a greater flow of solvent as you move radially in from the pore edge. So, the relative behavior of the two curves for small radius makes sense. In fact, for small values of ξ the two functions are quite different. At ξ = 0.1, the blue curve is over five times larger than the maize curve.

I find explaining what is happening for ξ approximately equal to one is more difficult. For plug flow, when the solute particle is just slightly smaller than the pore radius, it barely fits. But for Poiseuille flow, the particle not only barely fits, but it blocks all the fast flow near the pore center and you only get a contribution from the slow flow near the edge. This causes the maize curve to be more sensitive to what is happening near ξ = 1 than is the blue curve. I don’t find this explanation as intuitively obvious as the one in the previous paragraph, but it highlights an approximation that becomes important near ξ = 1. The model does not account for adjustment of the flow of solvent when the solute particle is relatively large are disrupts the flow. This can’t really be true. If a particle almost plugged a pore, it must affect the flow distribution. I suspect that the Poiseuille model is most useful for small values of ξ, but the behavior at large ξ (near one) should be taken with a grain of salt.

If find that it’s useful to force yourself (or your student) to provide physical interpretations of mathematical expressions, even when they’re not so obvious. Remember, the goal of doing these analytical toy models is to gain insight.

For those of you who might be disappointed to see Section 5.9 go, my advice is don’t toss out your 5^th edition when you buy the 6^th (and I’m assuming all of my dear readers will indeed buy the 6^th edition). Stash the 5^th edition away in your auxiliary bookshelf (or donate it to your school library), and pull it out if you really want a good continuum model for volume and solute transport in a pore.