ioo6 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION II 



(a)VISKING SAUSAGE CASING 



.a. 



I5A UREA 



I5A GLUCOSE 



SUCROSE 



r-RAFFINOSE 



FILTRATION RATE Cm ^ ec x 10* 



0.2 



0.4 



0.6 



0.8 



fig. 7.4. Molecular sieving through an artificial, porous 

 membrane. The smooth curves are constructed from the theory 

 of molecular sieving, equation 7.15. The data fit pore radii in 

 the range 15-17 A. Mean pore radius for the same membrane 

 estimated from the theory of restricted diffusion was 16 A 

 and pore radius estimated from combination with Poiseuille's 

 law was 19 A. The internal consistency of these various esti- 

 mates of pore radius in artificial membranes constitutes the 

 chief evidence justifying the application of similar techniques 

 to biological membranes of comparable pore size. [Adapted 

 from Renkin (298).] 



from the size of the largest molecule which just fails 

 to pass through the membrane. A less obvious limit 

 to any assumed distribution arises from the fact that 

 filtration rate varies with the fourth power of the 

 radius so that the total fraction of large pores must 

 be limited in order to satisfy the requirements im- 

 posed by the observed filtration coefficient. If a 

 membrane contains cylindrical pores of different 

 radii then the mean equivalent pore radius, r, for 

 hydrodynamic flow is given by 



7 'Vfc+tf 



F„r; 



(7. 16) 



where F n is the fraction of total pore population 

 having a radius r n . For example, the membrane 

 illustrated in figures 7.1 and 7.4 did not allow the 

 passage of hemoglobin (a 32 A) and therefore an 

 upper limit to its pore size distribution is 32 A. How- 

 ever, less than 20 per cent of the pores could be as 

 large as 30 A otherwise 



7>yh.2x (30) 4 >20A 



which would not fit the requirement that r = 19 A 

 set by the observed filtration coefficient and diffusion 



area for water (equation 7.13). The detailed com- 

 putation of possible pore distributions which would 

 fit the data for filtration, restricted diffusion and 

 molecular sieving is possible but laborious. For the 

 membrane illustrated in figures 7.1 and 7.4 the 

 broadest Gaussian distribution of pore radii com- 

 patible with the data is defined by a mean pore radius 

 of 14 A with a standard deviation of 7 A (298). 



F. Osmotic Pressure* and Osmotic Flow Through 



Leaky Membranes; Osmotic Reflection Coefficients 



Van't Hoff's law relating osmotic pressure to 

 concentration was derived for a perfectly semi- 

 permeable membrane. Relatively little is known of 

 osmotic forces associated with diffusion and osmotic 

 flow through membranes which restrict, but do not 

 prevent entirely, the diffusion of solute molecules. 

 The quantitative significance of this problem may be 

 illustrated by a specific example. Consider a two- 

 compartment system separated by a membrane 

 containing pores of radius 30 A. Addition to one 

 compartment of an ideal solute of molecular radius 

 30 A will cause osmotic flow through the membrane 

 at a rate equal to that caused by a hydrostatic pressure 

 difference of cRT mm Hg (equation 7.12). However, 

 if the same molar concentration of a small molecule 

 such as urea (molecular radius 2.7 A) is added to one 

 compartment, it will be found that the osmotic flow- 

 is less than 5 per cent of that obtained by the hydro- 

 static equivalent (82). 



In 1 951 Staverman (349, 350) introduced the 

 expression "osmotic reflection coefficient," a, as an 

 empirical descriptive term modifying van't Hoff's 

 law for the case of leaky membranes. 



n -" CRT* (7. 17) 



The value of a ranges from unity in perfectly semi- 

 permeable membranes to less than zero when the 

 mobility of the solute exceeds that of the solvent 



(333)- 



Very small values of a have been reported for 



osmotic flow caused by small molecules diffusing 



3 "Osmotic pressure" is ordinarily defined for the case of 

 thermodynamic equilibrium across ideal semipermeable 

 membranes and the term has no equivalent meaning for the 

 irreversible process to be considered here. Possibly a different 

 term should be coined to describe the transmembrane pressures 

 arising during restricted diffusion through porous membranes. 

 "Restricted diffusion pressure" would be accurate but could 

 only be applied to the case of zero net flow of solvent through 

 the membrane. 



