Cellular Neuroscience (207)
Ian Parker
Lecture # 3 - Membrane potential;
Nernst, Goldman equations
(How does the selective diffusion of ions across a membrane
generate a voltage)
http://parkerlab.bio.uci.edu
Diffusion
The random movements of particles (atoms, ions, molecules or larger) in solution
due to thermal energy. Each particle describes a ‘random walk’ , independent (at
least for dilute solutions) of other particles.
e.g.
Go to http://math.furman.edu~dcs/java/rw.html
for
an applet illustrating a 2-dimensional random walk
Everything is random – you can’t predict what any particular molecule will do;
BUT, on average, the mean displacement from origin increases as square root of time.
Diffusion is responsible for the net movement of a substance down its concentration
gradient. There is no ‘driving force’ on any individual molecule, but this is still a useful
concept for net movement of many molecules (increasing entropy).
Example; diffusion of a dye in water
Because diffusion varies is the square root of time, it is very fast over short distances,
but very slow over long distances. For a freely-diffusing molecule in water, mean
time t to diffuse a distance x;
x=
t=
1mm
250ms
10mm
25ms
100mm
2.5s
1mm
4min
1cm
6.5hr
If our nerves worked by diffusion, not electrically, how long would it take to wiggle your big toe?
Diffusion potentials
At the first instant solutions are added, K+ ions will move down their concentration gradient.
Cl- ions cannot move down their gradient, since the membrane is ipmermeable to ClThus, movement of K+ without corresponding movement of Cl- will set up a potential difference.
But, this potential difference will exert a force to stop net movement of further K+ ions (opposite
charges attract, like charges repel).
Equilibrium potential
K+ ions come into equilibrium when the chemical driving force (diffusion down the
concentration gradient) equals the electrical driving force.
What is the relation between concentration difference and the resulting diffusion potential?
i.e. at what voltage is equilibrium reached?
Nernst equation
E varies logarithmically with concentration ratio
E varies linearly with absolute temperature
E varies inversely with valence of ion
At room temperature and for a monovalent ion;
E = 58 mV * log10([K+]in/[K+]out)
Easiest to use the Nernst equation to figure out the absolute value of the voltage, and
then use common sense to figure out the sign (polarity)
Quiz. What are the
Nernst potentials for
each of these cases?
How many ions need to move across the membrane to
establish an equilibrium potential?
Only enough to charge the membrane capacitance
Example, consider a 1 x 1 cm square of lipid membrane, with 1 ml of 0.1.M KCl on
one side and 1 ml of 1 M KCl on the other.
E = 58 mV
C = 1 mF (remember specific capacitance of membrane ~ 1 mF/cm2)
Charge q = C*V
= 10-6 * 5.8 x 10-2
= 5.8 x 10-8 Coulombs
Faraday’s constant ~ 105 coulombs/mole
So, about 5.8 x 10-13 moles of K+ ions would need to move to establish the
equilibrium potential.
We started with 10-3 moles of K+ (I M in 1 ml)
Take-home message – The number of ions that move to establish an
equilibrium potential is vanishingly small. Concentrations of ions on either side
of the membrane remain almost constant, and the concentrations of cations
(+) and anions (-) are ALMOST equal on each given side of the membrane.
What about real cells?
For frog muscle;;
Resting potential is ~ -80 mV Thus, K+ and Cl- ions are close to equilibrium,
but Na+ and Ca2+ are far from equilibrium.
What ion(s) determine the resting potential?
Can find out by changing the extracellular concentration of different ions and measuring
effect on resting potential
Membrane potential of frog muscle behaves like a K+ electrode – Nernstian
relationship with [K+] out, except for deviation at very low [K+], owing to small
permeability to Na+.
Changing the concentration of other ions has very little effect (i.e. membrane has low
permeability at rest to Na+, Ca2+ and Cl-)
How can we deal with membranes that are permeable to more than one ion?
The Goldman-Hodgkin-Katz equation (AKA Goldman, or constant-field equation)
This is just an expanded form of the Nernst equation, with the different ions
weighted by their permeabilities.
Eg. if
PK >> PNa
PNa >> PK
PK = PNa
E ~ -101 mV (EK)
E ~ +59 mV (ENa)
E ~ -21 mV (mid-way between EK and ENa)
In practice, it is easier to measure the relative permeabilities of ions, rather than their absolute
permeabilities.
So, re-write the Goldman equation in a more useful form;
b = PNa/PK, c = PCl/PK
(Note, concentrations for Cl- are inverted, to account for difference in sign)
The permeability of nerve membrane to various ions (and hence the membrane
potential) is determined by the opening of ion channels, which may be regulated
by voltage, extracellular ligands, intracellular second messengers etc…