Understanding the Nernst equation

Understanding the Nernst equation:

The Nernst equation can seem arbitrary and complicated however conceptually it is quite simple. Learning where the equation comes from can be greatly helpful in understanding and remembering it.

Membrane potential:

Nerve cells have a potential difference between the voltage inside the cell, and the voltage outside the cell.

As the barrier allowing this difference is the cell membrane, it is called the membrane potential.

 It is given the symbol:  V_m

 It is calculated by simply working out the difference between the voltage inside and the voltage outside.

o   Vm = membrane potential

o   Vin = voltage inside the cell

o   Vout = voltage outside the cell

Maintaining a membrane potential:

The membrane potential is maintained by 3 principle ions;

•  Potassium (K+)
•  Chlorine (Cl-)
• Sodium (Na+) 

At the resting potential:

• Higher concentration f K+ inside cell
• Higher concentration of Na+ and Cl- outside cell

This is largely produced by the sodium potassium pump (Na+/K+ ATPase),

Which pumps two 3 sodium ions out for every 2 potassium ions pumped in

However the membrane also contains sodium and potassium leakage channels which allow ions to move freely across the membrane.

Two opposing forces:

This means there are two forces affecting the movement of ions across the membrane:

• The difference in concentration (diffusion gradient)
• The difference in potential (electrostatic gradient)

Ions want to move down their concentration gradient, away from areas where they are in high concentration.

And

Ions want to move down their electrochemical gradient, away from areas high in the same charge.

For example:

• Potassium is found in greater concentration in the cell, therefore it wants to move down its concentration gradient across the membrane and outside the cell. 

• However as it is a positive ion it also wants to move away from areas of high positive charge, and so wants to move inside of the cell where it is more negative.

 

The membrane potential at which electrostatic forces equal the action of diffusion for a particular ion is known as the:

 Nernst equilibrium

Deriving the equation:

As we have seen the Nernst equation needs to model the opposing actions of the concentrating gradient, and the electrochemical gradient.

• [C] (x)is the concentration of an ion, at position (x) along the membrane
• [V](x) is the potential at some point along the membrane, (x)

Flicks law of diffusion:

Flicks law of diffusion allows us to model the diffusive flux.

“flux” means the number of molecules flowing through a certain area in a certain time.

o   C = concentration difference

o   X = distance to diffuse

o   D = diffusion constant

For example, Calculate the flux of oxygen across a membrane segment with area 2x10 ^-6m²,  if the concentration on the right hand side of the membrane is 4mL/L and on the left side is 2mL/L. molecular diffusion constant for oxygen = 3x10^-5

 

• The difference in concentration = 2ml/l (4 – 2)
• Molecular diffusion constant for oxygen = 3x10^-5
• Length of membrane = 2x10 ^-6m

So:                  

= 30 molecules/sec  

Ohms law:

We can model the electrostatic flux with a similar equation: using a version of ohms law.

μ = motility  

z = valence of ion, eg +1, +2

[C] = concentration

[v] = potential difference

X = distance, length of membrane.

Total flux:

Therefore the full movement of ions can be models by the sum of the equations:

The diffusion constant in flicks equation can be model more accurately and related to the motility through Einstein’s relation;

K  = Boltzmann constant

t = temperature

q = charge

u = motility

Thus we can replace D and change the equation to:

From molecules to moles:

As we have seen this equation will give the diffusive flux in molecules.

However often in science it is more useful to use moles, and we change the equation to reflect this.

We can do this simply by taking some of our constants, which are in terms of molecules and multiplying them by Avogadro’s number (6.022 x 10²³ the number of molecules in a mol)

·         The Boltzmann constant (K) is related to the energy of a particular particle, we can simply times it by Avogadro’s number to work in term of moles.

                K x Avogadro’s constant = R

               This number is called the gas constant, and given the symbol R.

        So;  to work in terms of moles instead of particles, we simply need to replace the Boltzmann         constant with the gas constant.

          

·         We must also do a similar thing to the electrostatic aspect of the equation.

Currently we are working with the charge of an individual electron.

Again we can times this by Avogadro’s number to work in terms of moles

               q x Avogadro’s constant = F

                F is the charge of a mole of electrons, and is known the faraday constrant.

        So; again to work in terms of moles instead of particles, we simply need to replace the individual charge with Faradays constant.

   

So we can ultimately write the molar form of the equation:

From Flux to charge:

Flux is the flow of molecules, and current is the flow of charge.

So by simply multiplying the flux, by the total charge of all the ions we can work out the current going through the membrane.

The total charge per mol of electrons will simply by faradays constant (F) multiplied by the valence of the ions (z)

So if we multiply the whole equation by Fz, we can see the current through the membrane is:

This simplifies to the famous Nernst equation:

Although we have missed out a lot of the maths here, we can see that the Nernst equation is equal to the equilibrium potential.

·         This potential is a balance between the diffusion flux and the electrostatic flux

·         Both are dependent upon the concentration of ions inside and outside the cell,

·         Both can accurately be model using molar constants related to

o   The energy each particle has to diffuse (RT)

o  The charge acting on each particle (zF)