Membrane Biophysics

A barrier that good is only useful if it can be opened selectively, and the hardest thing a channel does is discriminate. Potassium channels pass K⁺ at near the diffusion limit and exclude Na⁺ by a factor of about ten thousand, and sodium is the smaller ion, so a simple size filter would do the opposite.

Roderick MacKinnon's 1998 structure shows the trick. An ion in water is surrounded by a shell of water molecules whose oxygens coordinate it, and pulling it out of that shell costs energy — more for sodium, because it is smaller and holds its water more tightly. The channel's selectivity filter is lined with backbone carbonyl oxygens held rigidly at spacings that reproduce almost exactly the coordination geometry that water provided for potassium. A potassium ion therefore pays nothing on balance to enter. A sodium ion, being 0.3 Å smaller in radius, cannot reach those oxygens at the right distances; the filter cannot repay its larger dehydration cost, and it stays out. Selectivity comes from a structure rigid enough to be wrong for the smaller ion.

Peter Agre's aquaporin solves a related problem in the opposite direction: pass water at enormous rates while blocking protons, which travel through hydrogen-bonded water chains faster than any ion diffuses. The pore's answer is a positively charged arginine and a geometry that forces one water molecule in the middle to break the chain by reorienting — so there is no continuous hydrogen-bonded path for a proton to hop along.

By counting molecules. A human red blood cell has a surface area of about 140 µm². A phospholipid headgroup occupies about 0.65 nm² in a packed bilayer, so one leaflet needs

$$\frac{140 \times 10^{-12}\ \mathrm{m^{2}}}{0.65 \times 10^{-18}\ \mathrm{m^{2}}} = 2.2 \times 10^{8}\ \text{molecules},$$

and both leaflets about $4\times10^{8}$ lipid molecules per cell. That figure is what Gorter and Grendel's monolayer area amounts to, and it is the sort of number that makes the bilayer concrete: four hundred million molecules, held in place by nothing but their dislike of water.

By electricity. A bilayer separating two salt solutions is a capacitor: conductor, insulator, conductor. Its specific capacitance is

$$\frac{C}{A} = \frac{\varepsilon_0 \varepsilon_r}{d}.$$

Hydrocarbon has a relative permittivity of about 2.2, and the measured capacitance of every biological membrane and every artificial black film is close to 1 µF/cm², or $10^{-2}$ F/m². Solving for the thickness of the insulating layer:

$$d = \frac{(8.854\times10^{-12})(2.2)}{10^{-2}} = 1.9 \times 10^{-9}\ \mathrm{m} \approx 2\ \mathrm{nm}.$$

Two nanometres of hydrocarbon — the thickness of two lipid tails meeting tail-to-tail, with the headgroups and their water outside the dielectric. An electrical measurement, with no chemistry in it, reproduces the structural answer. This is why the capacitance figure appears in every textbook of membrane physiology: it is a thickness measurement disguised as a circuit parameter.

By the field it sustains. A resting cell holds about 70 mV across a membrane some 4 nm thick overall, so the electric field inside it is

$$E = \frac{0.07\ \mathrm{V}}{4\times10^{-9}\ \mathrm{m}} = 1.8\times10^{7}\ \mathrm{V/m},$$

or 18 megavolts per metre. Dry air breaks down at about 3 MV/m, so the membrane holds six times the field that would arc across a spark gap, and does so continuously, for the life of the cell, in a film that is held together by no covalent bonds at all. It is among the best insulators per unit thickness in nature, and the entire electrical activity described under electrophysiology consists of proteins briefly and selectively spoiling that insulation.

The same numbers set the scale of what a channel must do. A potassium channel passing $10^{7}$ ions per second carries a current of

$$(10^{7})(1.6\times10^{-19}) = 1.6\ \mathrm{pA},$$

which is a picoampere — measurable, one molecule at a time, which is what the patch clamp does.

A bilayer is fluid in its own plane — lipids diffuse through it like molecules in a two-dimensional liquid — and it has no shear rigidity at all. What it does resist is bending, and Wolfgang Helfrich wrote down the energy for that in 1973 by analogy with liquid crystals: an energy per unit area quadratic in the curvature, with a bending modulus of about 20 $k_BT$.

This converts the shape of a cell into a variational problem. Minimise bending energy over all closed surfaces with a given area and a given enclosed volume, and the solution for a red blood cell's ratio of the two is the biconcave disc it actually has. Changing the volume moves the solution through a family of shapes that matches those observed in vesicles under osmotic stress. The equations that result are fourth-order and belong to the differential geometry of surfaces, close relatives of the Willmore functional — one of the clearest cases in biology where a shape is explained by a minimisation rather than by a mechanism.

The mechanics is not only descriptive. Ardem Patapoutian's Piezo channels, found in 2010, are gated by tension in the surrounding bilayer: each is a three-bladed propeller that dimples the membrane locally, and pulling the membrane taut flattens the blades and opens the pore. Touch, hearing and the sensing of blood pressure all run through a protein that reads the mechanical state of a sheet two molecules thick.

What remains unsettled is how the hundreds of lipid species in a real membrane are arranged, and whether the ordered domains seen readily in model bilayers exist in living cells as anything more than fleeting assemblies at the edge of detectability. That argument has run since 1997 and is the open problem above.