The cell that bursts by geometry

The pressure that comes from counting found that a membrane permeable to water and not to solute develops a pressure proportional to the number of dissolved particles, seven and a half atmospheres for blood plasma, and noted in passing that a cell dropped into pure water bursts. The membrane that almost holds and the swelling a membrane cannot stop followed what real membranes and charged contents do to that pressure. None of them asked how the bursting happens.

The usual picture is of a balloon: water flows in, the cell inflates, its skin stretches thinner and thinner, and at some pressure it gives way. For a red blood cell that picture is wrong in every particular but the last. The cell does not inflate and its skin does not thin. It swells by changing shape, its membrane slack the whole time, until it reaches the one shape that holds the most volume for the membrane it has — a sphere — and then it bursts at once, at a pressure a hundred times smaller than the osmotic pressure doing the pushing. The concentration at which this happens is a fact about geometry, and medicine measures it.

A bag with too much skin #

A human red cell is a biconcave disc about eight micrometres across, thicker at the rim than at the centre, with a volume of about ninety femtolitres and a membrane area of about 135 square micrometres. A sphere of the same area would hold about 147 femtolitres. The cell is, in that sense, a bag with far too much skin for its contents, and that is what lets it do its job. To reach every tissue a red cell must squeeze through capillaries narrower than itself and through slits in the spleen two or three micrometres wide, and only a bag with spare membrane can fold into a sausage or a cup without stretching. A sphere of the same volume could not change shape at all without stretching its surface.

The membrane’s mechanical properties are lopsided to a degree that makes the geometry decisive. It is a lipid bilayer, held from beneath by a network of spectrin filaments. Shearing it — changing its shape at constant area — costs almost nothing: its shear modulus is a few micronewtons per metre, and a cell can be pulled into a long thin shape by the forces of a flow. Changing its area is quite different. A lipid bilayer has a fixed area per molecule, and stretching it means pulling molecules apart against their attraction, so its area-expansion modulus is about 450 millinewtons per metre, a hundred thousand times larger. And it cannot stretch far: at a tension of about ten millinewtons per metre, a stretch of about two per cent, the bilayer opens and the cell’s contents escape.

So to a very good approximation a red cell’s membrane has a fixed area and no preferred shape. That is the opposite of a balloon, whose rubber has a natural size and stretches easily away from it. It is also different from the surface of a drop, which the skin that is not a skin found to have a constant tension and to shrink its area whenever it can. A drop wants a sphere because a sphere has the least area for its volume; a red cell becomes a sphere only because a sphere has the most volume for its area.

Water in, shape out #

The water that flows into a cell follows the counting rule. Water crosses the membrane quickly through channels, and the cell’s interior comes to the same concentration of dissolved particles as the solution outside. If the outside is diluted by a factor of two, the inside must be too, which means the water inside must double. Not all of the cell’s volume is water free to change, though: about two-fifths of it is haemoglobin and other solids that take up room without being diluted. The cell’s volume therefore follows the Boyle–van 't Hoff law,

$$V = V_b + (V_0 - V_b)\,\frac{c_0}{c},$$

where $V_b$ is the inactive part and $c_0$ the normal concentration, with 0.9 per cent salt by weight being isotonic for human blood. In a stronger solution the cell shrinks and puckers into a spiky shape with membrane to spare; in a weaker one it swells.

As it swells, it changes shape: the dimples fill, the disc becomes a cup, the cup a sphere, all at constant area and at almost no cost in tension. The pressure inside stays nearly equal to the pressure outside, because a slack membrane cannot hold a difference of pressure, and the osmotic balance is struck by water and not by force. Then the cell is a sphere. Its volume cannot grow without its area growing, and its area can grow by only two per cent. For a 90 femtolitre cell with 135 square micrometres of membrane, that point arrives at 0.44 per cent salt. Below it, the cell bursts.

The answer contains nothing about the membrane’s strength. It contains the cell’s volume, its area, its inactive fraction and the counting law. The same cell with a membrane twice as strong would burst at almost exactly the same concentration, because the two per cent of stretch it buys is worth very little volume.

The same arithmetic sets a rule every hospital follows. Fluids run into a vein are made isotonic with blood, normal saline at 0.9 per cent or a glucose solution of equal strength, and pure water is never infused, because the blood near the needle would be diluted past the bursting point of its cells before it mixed. Half-strength saline, 0.45 per cent, sits almost exactly at the onset of haemolysis in a test tube and is nonetheless given intravenously, slowly: the stream entering a large vein is diluted by the blood flowing past it many times over within a few centimetres, and the cells it meets never see a concentration far below their own. What the clinician calls tonicity and what the physicist calls osmotic pressure are the same counted particles; the bursting point is where the counting meets the geometry.

Six kilopascals against seven hundred #

Once spherical, the cell can resist a little more inflow by stretching, and how much pressure that buys follows from Laplace’s law for a sphere, the relation the small bubble blows up the big one used: a tension τ in the surface holds a pressure difference $2\tau/R$. At the tearing tension of ten millinewtons per metre and a radius of 3.3 micrometres, that is six kilopascals.

The osmotic pressure of the cell’s contents is about 720 kilopascals, seven atmospheres. The membrane can hold back a difference of under one per cent of it. In terms of dilution, once the cell is a sphere, diluting the outside by a further one per cent bursts it. That is why the onset of haemolysis is sharp for each cell: there is a long range of concentrations over which the cell changes shape at no cost, and then an interval of about one per cent in which it goes from slack to torn.

This contrast — a pressure of seven atmospheres held off by geometry, not by strength — is the reason every animal cell regulates its volume actively, pumping ions out to keep its interior from drawing in water. A plant cell survives dilute water because it has a rigid cellulose wall that can take the full osmotic pressure. An animal cell has no wall, and a membrane that can hold a hundredth of the pressure; it survives by never letting the pressure develop.

Walls, and an antibiotic that removes one #

Bacteria live with much the same problem the other way round. A bacterium’s cytoplasm is concentrated, and in ordinary pond water or in a body’s tissues the osmotic pressure difference across its membrane is several atmospheres. Its membrane, like a red cell’s, could hold a hundredth of that. What holds it is the cell wall outside the membrane, a mesh of sugar chains cross-linked by short peptides into a single molecule the shape of the cell, which takes the whole of the turgor pressure as a tank takes the pressure of its gas.

Penicillin and the antibiotics related to it kill bacteria by exactly the mechanism of this essay. They block the enzymes that make the cross-links, and a bacterium that goes on growing while it cannot complete its wall develops weak points in it. The membrane bulges through a weak point and, holding no pressure of its own, bursts. A growing bacterium treated with penicillin in a solution made as concentrated as its own contents — sucrose is the usual choice — does not burst; it loses its wall and rounds up into a sphere bounded by membrane alone, and survives as long as the solution keeps the osmotic difference at zero. The drug does not attack the membrane at all. It removes the one structure that let the membrane avoid carrying a pressure.

The failure of the membrane itself, when it comes, has more in common with the film that goes black before it bursts than with a balloon popping. A bilayer under tension opens by a hole that appears through the fluctuations of its molecules and grows if it is larger than a critical size set by the tension and the energy of the hole’s edge, as a soap film ruptures from a nucleated hole. Below the critical tension holes close as fast as they open; above it the first one large enough runs away.

Fragility that is a ratio #

Real red cells are not identical. Their volumes vary by ten per cent or so about the mean and their areas by a few per cent, and each cell reaches its own sphere at its own concentration. A sample of blood dropped into a series of salt solutions therefore haemolyses over a range: the first cells burst near half a per cent, half have gone by 0.42 per cent in the model, and nearly all by a third of a per cent. Measured by the haemoglobin released into the solution, that sigmoid curve is the osmotic fragility test, a standard investigation since the 1930s.

What the test reveals is visible in the other two curves. In hereditary spherocytosis, a group of inherited defects in the proteins that tie the bilayer to its spectrin skeleton, the cell sheds small vesicles of membrane over its life and ends up with less area for the same volume. With an eighth less membrane the modelled cells burst at 0.54 per cent, well into the range a normal sample survives. Cells that are small and thin, with less haemoglobin and the same membrane, as in iron deficiency or thalassaemia, have area to spare and resist down to a third of a per cent. Clinicians speak of increased and decreased fragility, but in both cases the membrane is about as strong as a normal one. What has changed is the ratio of membrane to contents.

How much membrane a cell can afford to lose #

The curves steepen sharply as the area falls towards the smallest sphere that holds the cell’s contents. A 90 femtolitre cell that loses a tenth of its membrane moves its bursting point from 0.44 to 0.55 per cent; one that loses a quarter is close to isotonic, which is the same as saying it would barely survive in its own plasma. The steepness is the reason a modest membrane defect has large consequences, and it is also how spherocytosis does its damage in the body, where nobody dilutes the blood.

A spherical cell has no spare area, so it cannot fold to pass through a narrow slit. The spleen filters the blood through slits between the cells lining its sinuses, about two micrometres wide, and a normal red cell squeezes through them several times a day for four months. A spherocyte cannot, and is held back and destroyed. The anaemia of hereditary spherocytosis is caused by the spleen, not by bursting, and removing the spleen cures most of it while leaving the cells exactly as spherical as before. The same ratio that the fragility test reads in a row of test tubes decides, in the body, whether a cell can still pass through a gap narrower than itself, as it must to flow through capillaries and give blood the viscosity that belongs to the tube.

The counting law at its edge #

The Boyle–van 't Hoff law in the figures is the counting rule of the pressure that comes from counting applied to a container that can change its volume freely. It treats the inside as an ideal dilute solution, with all its osmotically active particles counted as independent and the solids as a fixed excluded volume. A red cell is far from dilute: it is about a third haemoglobin by weight, packed so closely that the molecules interact, and its osmotic coefficient is not one. Measured red cells do follow a straight line against the reciprocal of the outside concentration over a wide range, but with a slope that corresponds to an apparent inactive fraction, about 0.4, that includes those non-ideal effects rather than only the volume of the protein itself.

Nor are cells perfectly semi-permeable to the salt in the solution. Over the minutes of a fragility test sodium and potassium leak slowly, which is why the test is read after a fixed time, and why an incubated version of it, after a day at body temperature during which the cells’ pumps run down, shows the difference between normal cells and spherocytes more strongly. The model draws only the geometric part of the answer, with the cells’ leaks and pumps switched off.

A geometry the cell keeps #

The deeper question is why a red cell has so much spare membrane in the first place, and why its shape is a biconcave disc rather than any other shape of the same area and volume. The answer that has emerged since the 1970s is that the shape is the one that minimises the membrane’s bending energy for the given area and volume, with a small contribution from the spectrin skeleton’s elasticity and from the difference in area between the bilayer’s two leaflets. Models built on that principle reproduce not only the disc but the series of shapes cells take as their volume, salt or membrane composition change: the cup, the spiky echinocyte of a cell in strong salt, the sphere.

What the shape buys is the reserve the swelling figure shows: a cell at rest uses about sixty-four per cent of the volume its membrane could enclose. That reserve is the room for deformation in a capillary and the margin against a fall in the tonicity of the plasma, and it is spent quickly by any loss of membrane. A red cell does not make new membrane — it has no nucleus and nothing to make it with — and as it ages it loses a little area to vesicles, its ratio of area to volume falls, and old cells are cleared by the same splenic filter that clears spherocytes.

Where the picture is honest #

The cell drawn here is the Evans–Fung average shape, the population model scatters volume and area independently and normally, and the bursting rule adds a fixed two per cent of stretch to the sphere. Real cells’ areas and volumes are correlated, and the distribution of each has tails that the normal distribution misses, so the modelled fragility curves are the right shape and in the right place but not a fit to any patient’s. The pressure figure uses a single area modulus and a single tearing tension, measured on cells held in micropipettes, where both vary from cell to cell and with how fast the stretch is applied. And the membrane’s two per cent of stretch is not a property of the bilayer alone: lysis begins with transient pores whose opening depends on time and temperature.

The domain of the argument is any cell whose membrane resists stretch far more than it resists bending and shear, that is close to osmotic equilibrium with its surroundings, and whose wall cannot carry the osmotic pressure. That includes most animal cells, and many have a smaller reserve of area than red cells, which is one reason they regulate their volume so actively.

Still open: how a membrane decides its shape and keeps its area #

The equilibrium shapes are well described by bending-energy models, but how the cell maintains its area-to-volume ratio over its 120-day life, how quickly it loses membrane and why, and how the shedding of vesicles is controlled, are not fully understood. Nor is the process of lysis itself: whether the membrane fails by a single pore that grows, by many small ones, or by the skeleton letting go of the bilayer in one place, and how the answer depends on the rate of swelling, are studied with fast imaging and simulation of single cells. Measuring area and volume cell by cell, rather than inferring them from a population’s fragility, is becoming routine with microfluidic devices, and it may turn the test into a direct measurement of the ratio it always read.

The habit worth carrying away is to ask which way a soft object is soft. A red cell’s membrane changes shape for a few micronewtons per metre and changes area by two per cent before it tears, so it swells by giving up shape, from a 94 fL disc to the 146 fL sphere its membrane can enclose, and bursts at about 0.44 per cent salt whatever its strength. A test named for fragility reads a ratio of area to volume, and the spleen reads the same ratio in a slit two micrometres wide.