CELLULAR ELECTROPHYSIOLOGY OF THE HEART 



field. The form of this term follows from Ohm's law 

 written in the vector form: 1 = —c grad S; where I is 

 the current density (amp/cm-), a is the conductivity 

 of the medium (mhos/cm) and S is the electric poten- 

 tial.^ Current density can be converted to ionic flux by 

 dividing I by the valence (Zg) of the ion carrying the 

 current and by the Faraday (F), which converts 

 coulombs to moles: Mg = I/FZs- The conductivity 

 of an ion species in solution depends directly on the 

 concentration of the ion, on its mobility (m^) and on 

 the square of the valence: as = Zs[S]F-mg. Mobility 

 is directly proportional to the diffusion constant, since 

 it is a measure of the same property of the ion. The 

 diffusion constant is a measure of the ease with which 

 an ionized substance can move through the solution 

 when it is driven by a concentration gradient. Mobil- 

 ity is the measure of ease of movement when the ion 

 is driven by an electrical gradient; hence, from 

 energetic considerations, rUg = Dg/RT, where R is 

 the universal gas constant and T the absolute tempera- 

 ture. Ohm's law can now be written as Mg = 

 — (DgFZg/RT)[S] grad S, the flux of an ion due to a 

 potential gradient. With the addition of the two fluxes, 

 equation i becomes 



-Ms = Ds grad [S] + (DsFZs/RT)[S] grad S (2a) 



Since [S] and S are independent variables, this 

 equation cannot be integrated to give Mg directly in 

 terms of S and [S], which are experimentally measur- 

 able, without another relationship between tliem. 

 This other relationship is Poisson's law: the divergence 

 of —grad 8 is proportional to the charge density, the 

 difference between the cation and anion concentra- 

 tions in any small volume. The resulting set of differ- 

 ential equations is so difficult to solve for any real 

 system that some assumption is usually made concern- 

 ing the variation of voltage or concentration with 

 distance. Substitution of the assumed relationship in 

 equation 2a permits its direct integration. Goldman's 

 assumption {49) that the electric field is constant 

 throughout the membrane is perhaps the most useful 

 for calculating the fluxes through relatively imperme- 

 able membranes. This assumption is equivalent to 

 assuming that the charge density within the mem- 



^ The use of the symbol £ for the electric potential or po- 

 tentizil difference is not common, the symbols V and E being 

 most frequently used. All three are listed as symbols for po- 

 tential or emf in the Handbook of Chemistry and Physics (67). 

 S is used here because it seldom denotes other quantities, 

 whereas V frequently symbolizes volume or velocity, and E 

 nearly always stands for the magnitude of the electric field 

 intensity | — grad sl. 



brane is negligible. The resulting flux and voltage 

 equations describe quite accurately a number of ex- 

 perimental phenomena (33, 47, 55, 63, 74). 



There are two convenient ways of approaching the 

 contributions of voltage and concentration differ- 

 ences across the membrane to the transmembrane 

 flux. The voltage can be considered to modify the 

 concentrations of the ion or, conversely, the concen- 

 trations to modify the voltage difference. These 

 approaches lead to the idea of eflective concentrations 

 or effective voltage differences. The manner in which 

 a transmembrane voltage contributes to the effective 

 concentration can be determined by multiplying 

 both sides of equation 2a by the integrating factor 

 exp(FZg6/RT). The equation becomes 



- M seFZsS'RT = Ds grad [S]ef Zs^/RT 



(2b) 



This is similar to equation i if [S] exp (FZgS/RT) is 

 replaced by [S]. Thus, voltage affects concentrations 

 exponentially insofar as transmembrane fluxes are 

 concerned. 



In the converse situation, the contributions of the 

 inside and outside concentrations of an ion to the 

 eff"ective transmembrane voltage follow from the 

 condition for ionic equilibrium across the membrane, 

 i.e., the relationship between the concentrations and 

 the voltage at which the net flux of the ion is zero. 

 Any membrane, no matter how convoluted, is so thin 

 that it can be considered a plane; hence, grad [S] and 

 grad £ are equal to their respective derivatives with 

 respect to x, the distance through the membrane 

 measured from inside to outside. Setting Mg = o, 

 taking £ = o and [S] = [S]o outside the cell and 

 taking [S] = [S]; and 8 = £g inside the cell, integra- 

 tion of equation 2 b gives, when solved for £g. 



£s ■■ 



RT [S]„ 



(3a) 



This is Nernst's equation. With logarithms to the 

 base 10, a temperature of 20°C (293° absolute), 

 values of R and F in appropriate units and £g ex- 

 pressed in millivolts, equation 3a becomes 



r 58, [S]„, . 



Es = — log — - (mv) 

 ^s \i>\\ 



(3b) 



fig is called the equilibrium potential of the ion, S. 



The condition for an uncharged substance to be 

 equilibrated across the membrane is that the concen- 

 trations on either .side be equal. However, for an 

 ionized substance, the ion concentrations on the two 

 sides can be arbitrarily chosen and solution of the 



