Lecture 3: Biophysical basis of excitability

1. THE NERST EQUATION

The membrane: contains channels which can open and close

In the sea -> salty water -> high concentrations of Na+

Nerst potential: 2 opposite forces:

  • diffusion
  • electrical force

You can find the equilibrium which gives you the Nerst potential

 

 

 

 

 

 

 

 

 

 

Diffusive flux: (Fick's first law)

D= diffusion coefficient

[D] = m^2/s

C = concentration

[C] = mol/m^3

J = diffusive flux

[J] = mol/(s m^2)

Take a uniform electric field:

concentration

[C] = mol/m^3

Electric field

[E] = V/m

mobility

[μ] = m^2/(Vs)

J = flux due to electric field

[J] = mol/(s m^2)

Diffusive flux: (Fick's first law)

D= diffusion coefficient

[D] = m^2/s

C = concentration

[C] = mol/m^3

J = diffusive flux

[J] = mol/(s m^2)

Take a uniform electric field:

concentration

[C] = mol/m^3

Potential

[ϕ] = V

mobility

[μ] = m^2/(Vs)

J = flux due to electric field

[J] = mol/(s m^2)

Diffusive flux: (Fick's first law)

Take a uniform electric field:

Einstein relation

In equilibrium, J = 0

i: inside the cell

o: outside the cell

Potassium K+:

  • room temp: RT/F = 25 mV
  • [K]i = 140 mM
  • [K]o = 2.5 mM

 

                                =>    V = -101 mV

Nerst potential: 2 opposite forces:

  • diffusion
  • electrical force

You can find the equilibrium which gives you the Nerst potential

 

 

 

 

 

 

 

 

 

 

Nerst potential: 2 opposite forces:

  • diffusion
  • electrical force

You can find the equilibrium which gives you the Nerst potential

 

 

 

 

 

 

 

 

 

 

2. GOLDMAN-HODGKIN-KATZ

This time we assume that the flux is not zero, we are in a non-stationairy situation

We assume that    is a linear function of x

X-axis: goes from inside to outside

inside

outside

The homogeneous equation:

A particular solution:

inside

outside

inside

outside

Denote P = D/L as the permeability of the membrane to the substance.

[J] = mol / s m^2

To find the electric current, we multiply by the amount of charge conveyed by a mole (zF): [zF] = C/mol

This is the famous GHK equation, which gives the relation between the value of the current and the concentrations.

For z < 0

Thus for z = +1

Thus for z = -1

Just change c0 and ci

3. GOLDMAN-HODGKIN-KATZ IN A MIXTURE OF IONS

  • Now we can account for the situation where we have a mixture of several ions.
  • In that case the total current through the membrane must be zero, so the potential does not change.
  • Assume ions with charge +1 (Na+, K+) and -1 (Cl-)

Thus for z = +1

Thus for z = -1

Consider K+ and Na+:

V = -88.53 mV

V = -88.53 mV

This is already a big difference with previous value of only considering K+ (-101 mV)

Real value V = -90 mV -> so this is a good estimate!

4. Electrodynamics of a cell

Text

1. GHK expression

2. Linear approximation

= 0