Permeability and reflection coefficient for a hard spherical solute, radius rs,through a cylindrical pore, radius rpore. Hydrodynamic calculation from Bassingthwaighte 2006, corrected 2012

Description

OBJECTIVE: The equations for transport of hydrophilic solutes through aqueous pores
 provide a fundamental basis for examining capillary-tissue exchange and water and
 solute flux through transmembrane channels, but the theory remains incomplete for
 ratios, alp, of sphere diameters to pore diameters greater than 0.4. Values for
 permeabilities, P, and reflection coefficients, sigma, from Lewellen [18], working
 with Lightfoot et al. [19], at alp = 0.5 and 0.95, were combined with earlier values
 for alp < 0.4, and the physically required alp < 1, to provide accurate expressions
 over the whole range of 0 < alp < 1. (alp =rs/rpore & stands for the greek alpha in the text.)

 METHODS: The "data" were the long-accepted theory for alp < 0.2 and the computational
 results from Lewellen and Lightfoot et al. on hard spheres (of 5 different alp's)
 moving by convection and diffusion through a tight cylindrical pore, accounting
 for molecular exclusion, viscous forces, pressure drop, torque and rotation of
 spheres off the center line (averaging across all accessible radial positions),
 and the asymptotic values at alp = 1.0. Coefficients for frictional hindrance to
 diffusion, F(alp), and drag, G(alp), and functions for sigma(alp) and P(alp), were
 represented by power law functions and the parameters optimized to give best
 fits to the combined "data".

 RESULTS: The reflection coefficient sigma = {1 - (1 - (1-PHI)^2)*G'(alp)} + 2*alp^2*PHI*F'(alp),
 and the reloative permeabiltiy P/Omax = PHI*F'(alp)*(1 + 9* alp^5.5 * (1 -alp^5)^(0.02)],
 where PHI is the partition coefficient or volume fraction of the pore available to solute.
 The new expression for the diffusive hindrance is
 F'(alp) = (1-alp^2)^(3/2)*PHI / (1 + 0.2 alp^2 *(1 - alp^2)^16), and for the drag factor is
 G'(alp) = (1-(2/3)*alp^2 - 0.20217*alp^5) / (1 - 0.75851*alp^5) - 0.0431*[1 - (1 - alp^10)].
 All of these converge monotonically to the correct limits at alp = 1.

 CONCLUSION: These are the first expressions providing hydrodynamically based estimates
 of SIGMA(alp) and P(alp) over 0 < alp < 1. They should be accurate to within 1 t0 2%.

Figure: Permeabilities and reflection coefficients as a function of alp, the ratio of solute radius to pore radius. phi is the solute partition coeffecient, Fjbb is the new fractional hindrance factor, Prel is Relative pore permeability due to exclusion, Prelcorr is the ratio of pore permeability to maximum pore permeability, sigjbb is the new reflection coefficient and sigBean is Bean's calculation of the reflection coefficient. Parameter set 'Par1' of JSim model PoreTransport.proj was used to obtain the above figure.

Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.