EFFECTS ON PERMEABILITY AND ACTIVE TRANSPORT 173 



At equilibrium the steady state may be represented by: 



v„^ = v,^ = PAC, (1-1) 



where v^ is the equilibrium active transport rate, v^^ is the equilibrium 

 outward diffusion rate, P is a permeability constant, and AC^ is the con- 

 centration difference between the inside and the outside of the cell. During 

 active uptake of the substance the rate of accumulation {v^.) is given by: 



V, = Va- va = KfiJC) - PAC (1-2) 



where v^^ and v^ are the active inward and passive outward rates, respec- 

 tively, Z is a constant characterizing the active transport system, AC is the 

 concentration gradient, and f{AC) represents some function of the concen- 

 tration gradient. As accumulation proceeds, v^ decreases and eventually 

 comes to zero at equilibrium; this results certainly from an increase in v^i 

 and probably also from a decrease in v^. An inhibitor may reduce accumula- 

 tion by either decreasing. iiC or increasing P, i.e., slowing active transport 

 or increasing the permeability. 



In order to determine the behavior of such a system we must assume 

 some relationship between active transport and AC. It is likely that the 

 rate of transport declines as the accumulation proceeds and is minimal at 

 equilibrium, inasmuch as oxidative metabolism in a tissue such as nerve 

 is usually low in a resting state but increases if a series of impulses alters 

 the gradients of Na+ and K+. The simplest and most logical function would 

 be v„ = K{A — AC), the rate of transport slowing as AC increases and 

 stopping when it reaches some value designated as ^, so we may now write: 



"^^^'^ K{A - AC) - PAC (1-3) 



dt 

 which may be integrated to: 



(C, 



[^p+Hh'^'^*"'l "■'' 



where {C ,) and {C,,) are the concentrations inside the cell and in the me- 

 dium, respectively. As accumulation proceeds and t becomes larger, the 

 equilibrium state is eventually reached, at which time the concentration 

 gradient is 



The rate of accumulation and the equilibrium gradient thus depend on the 

 relative values of K and P, and hence the effect of an inhibitor, acting on 

 either active transport or the permeability, will be determined by both. 



