242 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



Nernst equation will give the transmembrane poten- 

 tial at which the distribution will be at equilibrium. 

 Physically, 8g is the potential which would be gener- 

 ated across a membrane permeable only to S. This 

 follows because the movement of the permeable ions 

 through the membrane down their concentration 

 gradient constitutes the separation of charge across 

 the membrane. In turn, this potential gradient op- 

 poses further ionic penetration. This process continues 

 until there is no net flux of the ion, i.e., equilibrium 

 [see (135) for a more detailed treatment]. 



The Nernst equation is an expression of the condi- 

 tion for electrochemical equilibrium and so is immedi- 

 ately derivable from energetic considerations. At 

 equilibrium for S, the sum of the concentration and 

 electrical potential (free) energy differences per mole 

 must be zero; the concentration energy is RT In 

 [S]i/[S]o and the electrical is FZggg. 



The summation of concentration and electric 

 potential energy difference for any [S] and £ is called 

 the electrochemical potential difference (Ams) between 

 the two solutions: 



Ams = FZse -f- RTln [S]i/[S]o 



(4a) 



From Nernst's equation (3a) it can be seen that 

 RT In [S]i/[S]o = — FZsSg. Thus the electrochemical 

 potential difference for S can also be written in a 

 more meaningful and convenient form as 



Ams = FZs(S - Ss) 



(4b) 



The variation of electrochemical potential with dis- 

 tance, the electrochemical gradient, is a measure of the 

 total electrical and concentration forces acting on an 

 ionic species. Across the membrane, the electrochemi- 

 cal gradient is approximated by Ams/5. Therefore, 

 to the approximation used in defining perme- 

 ability (equation i), transmembrane flux is propor- 

 tional to Ams- If Ms is expressed in the form of current 

 density (Ig), the flux equation becomes Ohm's law 

 for a single ion : 



Is = gs(£ - Es) 



(4c) 



The proportionality constant (gg) has the dimensions 

 of conductance per unit area (mho/cm-) and is 

 called the specific conductance of S. In general, gg 

 varies with the voltage across the membrane and with 

 the concentrations of ions in the vicinity of the mem- 

 brane. Nevertheless, this equation is a useful repre- 

 sentation of the fluxes through a membrane and, in 

 particular, shows the way in which the concentrations 

 of S on either side of the membrane alter the effective 



transmembrane voltage, i.e., the driving force is 

 £ — Sg, not £. 



Permeability and conductance are closely related 

 but not identical quantities. Both are measures of the 

 ease with which an ion can penetrate the membrane 

 but Pg is proportional to Dg and gg to mg (mobility). 

 The variation of Pg with fi in a membrane with fixed 

 properties can be determined by integrating the one- 

 dimensional form of equation 2 b. The solution be- 

 comes more symmetrical if the potential is taken as 

 — 82 outside and £,2 inside the membrane: 



Mg = Ps{[S]ie^^S8/2RT _ |-sj_^e-^2s£/2RTj 



(2C) 



where 



^■^=°^{£, 



gFZgS/RT d^ 



Mg can be factored out of the definite integral because 

 flux in the steady state does not vary with distance 

 through the membrane. The permeability is deter- 

 mined by the value of the integral and hence depends 

 on the variation of £ with x in the membrane. 



The foregoing discussion has been based upon the 

 assumption that each particle's movement through the 

 membrane is uninfluenced by the movement of anv 

 other particle. Although this assumption appears to 

 be valid for transmembrane ionic movements during 

 electrical activity (33, 58), there is experimental evi- 

 dence that at least three K+ interact as they move 

 through the membrane (65) and other evidence (85) 

 that the efflux of Na+ is influenced by [Na+]o (see 

 below). 



The Nernst equation can be used to determine 

 whether an ion is distributed at equilibrium across the 

 cardiac cell membrane. To the approximation that 

 gg is independent of £, an ion is equilibrated if £g is 

 equal to the average transmembrane potential (S). 

 Values of the equilibrium potentials for Xa+, K"*", and 

 Cl^ ions are given in tlie right-hand column of table i . 

 At first glance it appears that none of these ions is 

 equilibrated. However, the value for internal chloride 

 concentration ([Cl^]i) given in the table was calcu- 

 lated from the Nernst equation with the assumption 

 that the effective average membrane potential for Cl~ 

 of a rhythmically beating heart is about —50 mv as 

 compared with the diastolic or "resting" potential 

 £r = —80 mv. There is considerable evidence that 

 Cl^ distributes according to the membrane potential 

 in both skeletal (55, 74) and cardiac muscle (73). 

 Thus, [Cl~]i is lower in a quiescent than in an active 

 heart, and some of the changes in the electrical and 



