Bioelectric Phenomena - Introduction to Biomedical Engineering - page 772

EXAMPLE PROBLEM 12.3
Consider a membrane in which there is an active K þ pump, passive channels for K þ and Cl ,

and a non-steady-state initial concentration of [ KCl ] on both sides of the membrane. Find an

K þ pump. expression for the active

Solution
From space charge neutrality,

J Cl ¼ J K ,or

m K Z K K ½ dv

dx KT

m K dK ½
dx

J K ¼ J p

q

m Cl dCl ½
dx

J Cl ¼m Cl Z Cl Cl ½ dv

dx KT

q

K þ pump. Solving for dv
dx

where J p is the flow due to the active using the J Cl equation with

Z Cl ¼ 1gives

d ½ Cl
dx

dv

dx ¼ KT

q ½ Cl

By space charge neutrality, Cl ½ ¼ K ½ , which allows rewriting the previous equation as

d ½ K þ
dx

dv

dx ¼ KT q ½ K þ

J K equation with dv
dx

At steady state, both flows are zero, and with Z K ¼ 1, the substitution is

given as

m K ½ K þ dv

dx KT

m K
q

d ½ K þ
dx

J K ¼

0 ¼ J p

d ½ K þ d ½ K þ

dx

¼ J p m K ½ K þ KT
q ½ K þ dx KT m K

q

d ½ K þ
dx

2 KT m K

q

¼ J p

Moving J p to the left side of the equation, multiplying both sides by dx , and then integrat-

ing yields

Z d Z ½ K þ o

2 KT m K

q

d ½ K þ

J p dx ¼

0 ½ K þ i

or

2 KT m K

q d ½ K þ o ½ K þ i

J p ¼

pump was present in this example, then at steady-state, the concentration on both
sides of the membrane would be the same.

Note: If no