# Counting charges on a cell membrane

8.3.P2

 An important feature of cellular membranes is that there is an electrical potential difference across them. A typical membrane in an animal cell maintains a potential difference of 70 mV across the membrane. Many membranes are made up of pairs of lipid molecules lined up as shown in the top figure at the right, with their hydophilic ends (small spheres) on the outside and their hydrophobic ends (tails) on the inside.   Let's make a simple model of the charge distribution on the membrane that leads to the indicated potential difference. We'll approximate it as two parallel flat sheets of uniform charge with a constant field between them. The thickness of the membrane is about 5 nm.

1. Let's first ignore the fact that the space between the plates is filled with lipid molecules, and treat it as if there were a vacuum between the plates. Find the electric field between the plates and the charge density on each plate.

2. The diameter of the heads of the lipid molecules are about 0.7 nm and they are closely packed. Any extra charges on the top or the bottom of the membrane have to come in units of the charge on the electron. Estimate the fraction of lipid molecules that have an extra charge.

3. The membrane is not empty, but filled with lipids. This means the space between the plates have a dielectric constant that reduces the field produced by a given charge density. The dielectric constant of the interior of the membrane is about κ ~ 2. How does this change the result you found for part 2?

4. For a membrane with a dielectric constant, does this simple model work for what's going on in the membrane? Or do we have to consider issues like the electric polarization of the fluid in which the membrane is imbedded? For our simple "constant field" model to make sense, the separation between the charges on the membrane would have to be small compared with the thickness of the membrane. Discuss.

Joe Redish 4/19/11