The stem that stands by pressure

The pressure that comes from counting derives van 't Hoff’s law: a solution separated from pure water by a membrane that passes water but not solute draws water through it until the pressure on the solution’s side has risen by $\Pi = cRT$, a pressure set by the number of solute particles per unit volume and by nothing else about them. The same essay notes in passing that a plant cell holds a pressure of about twelve bar. The cell that bursts by geometry follows an animal cell, a red blood cell, into dilute water and finds that it has no wall to hold such a pressure; it survives only by changing shape, and when shape runs out it bursts.

A plant cell lives with the pressure. Its contents are more concentrated than the water in the soil and in the spaces between cells, so water flows into it, and it does not burst because it is enclosed in a wall of cellulose fibres that stretches a little and then holds. The pressure inside, the turgor, is the osmotic pressure difference that the wall’s tension balances. And a plant has learned to use that pressure, which would be a hazard to an animal cell, as its skeleton. A soft green plant is a structure of inflated balloons, and it stands for the same reason an inflatable boat does.

How hard a plant cell is pumped up #

The pressures involved are larger than intuition suggests, because osmotic pressures are large.

Plant cell sap typically contains three-tenths of a mole of dissolved particles per litre — sugars, potassium and chloride ions, organic acids — and against pure water that gives an osmotic pressure of three-quarters of a megapascal, seven and a half atmospheres. A cell in contact with dilute water inflates until its wall pushes back with nearly that pressure. A car tyre is held at about a quarter of a megapascal above the air. The guard cells that open and close the pores in a leaf’s surface load themselves with potassium to reach two or three megapascals. The growing tips of fungal threads, which push their way through soil and into wood, reach several. All of these are made by counting particles: no pump does the pushing, and the cell’s only work is to put the particles there and keep them in.

The pressure is not static. A cell in a leaf is in contact not with pure water but with the water in the plant’s veins, which is under tension, pulled up from the roots by evaporation at the leaf’s surface — the mechanism of the column that is pulled, not pushed. The turgor is then the osmotic pressure of the sap minus the tension of the water outside. On a hot afternoon, with the plant evaporating fast and the tension in its veins high, the turgor falls, and a plant that was crisp in the morning droops.

The tension a cell wall carries #

The wall holds the pressure by being stretched, and a thin cylinder under pressure carries a tension around its circumference that grows with its size.

For a cylindrical cell of radius $r$ and wall thickness $t$ inflated to a pressure $P$, the stress around the circumference of the wall is $Pr/t$, and along the cell’s length half that. A typical cell twenty micrometres in radius with a primary wall half a micrometre thick holds its wall at thirty megapascals. That is a substantial engineering stress, about a tenth of the strength of mild steel, carried by a wall a few hundred cellulose molecules thick. Cellulose microfibrils are among the strongest natural materials, and they are laid down in the wall in a preferred direction, around the cell’s circumference, so that the wall is strong where the stress is largest. A cell that wants to grow longer loosens the cross-links between its fibrils and lets the turgor stretch it along its length; a cell that has stopped growing stiffens its wall.

The arithmetic also explains why large plant cells are thick-walled. The stress grows in proportion to the radius at a given pressure, and a large, thin-walled cell inflated to the same turgor would exceed any wall’s strength. Turgor is roughly the same throughout a plant; the walls are what vary, matched to the cells’ sizes.

The tube that is stiff because it is full #

The pressure does more than inflate the cells. It makes them stiff, and the reason is general enough to hold for any inflated structure.

Consider a tube of membrane that has no resistance to compression at all — a soap film, a plastic bag, a thin cell wall, all of which buckle at once if pushed. Inflated to a pressure $P$, the membrane carries an axial tension of $Pr/2$ per unit length of its circumference, everywhere. Bend the tube, and the side on the outside of the bend is stretched further while the side on the inside is relieved. As long as the inside still carries some tension, the tube bends as an elastic beam. When the bending is enough to take the inside’s tension to zero, the inside goes slack and wrinkles. That happens at a bending moment of $\pi P r^3/2$; at twice that, the whole section can no longer carry the moment and the tube folds.

Both moments are proportional to the pressure and contain nothing about the membrane’s own stiffness. An uninflated tube of the same membrane carries no bending at all. This is the structural principle of inflatable boats, of air beams and inflatable antenna booms in spacecraft, of a bouncy castle, and of a soft plant. The plant’s tissue is a packed foam of inflated cells, and the same principle applies to the whole: the cells’ walls are kept in tension by their pressure, the tension lets them carry compression as a reduction of tension, and the tissue as a whole behaves as an elastic solid with a stiffness that rises roughly in proportion to the turgor. Potato tissue, measured in the 1950s, became several times stiffer as its cells were brought to full turgor.

The volume at which a leaf goes limp #

How much water a cell can lose before its pressure is gone depends on how stiff its wall is.

The diagram, drawn first by Karl Höfler in 1920, has three curves. The osmotic pressure of the sap rises as the cell loses water and its solutes are concentrated. The turgor exerted by the wall falls as the cell shrinks, roughly in proportion to the shrinkage, and reaches zero when the wall is just slack. Their difference, turgor minus osmotic pressure, is the water potential, which decides which way water moves: a cell with a water potential below its surroundings’ takes water in. In pure water the cell fills until the water potential is zero, which for a wall of volumetric stiffness ten megapascals is at a volume seven per cent above the slack volume, with three-quarters of a megapascal of turgor.

The stiffness of the wall means that a small loss of water causes a large loss of pressure. Losing less than seven per cent of its water takes this cell from full turgor to none, and below that point its tissue is limp. A plant therefore wilts when its cells have lost only a few per cent of their water, long before they are dry. Below the turgor-loss point the cell can lose more water only by concentrating its sap, and many plants survive drought by doing that deliberately — accumulating sugars and ions to raise their osmotic pressure so that they can keep drawing water from drying soil and keep some turgor. The turgor-loss point read off a Höfler diagram is now one of the standard measures of a plant’s drought tolerance, and across species it correlates with the dryness of the places they live.

The height a soft stem loses as it wilts #

A stem standing upright is a column under its own weight, and a column that is too tall for its stiffness buckles sideways.

The tallest a uniform column can stand under its own weight before buckling was found by Alfred Greenhill in 1881: it is $(7.84\,EI/wg)^{1/3}$, where $EI$ is the column’s bending stiffness and $w$ its mass per unit length. Greenhill’s motivating question was how tall a tree could grow, and for a stem of wood the answer is tens of metres. For a stem of soft tissue whose stiffness is proportional to its turgor, the buckling height goes as the cube root of the turgor. In the model drawn — a solid stem of tissue six times as stiff as its turgor, which is the order measured in soft plant tissue — a stem two and a half millimetres in radius could stand eighteen centimetres at full turgor and fourteen at half.

The cube root makes the dependence gentle, and that is the danger. A stem growing to the height its full turgor allows has no margin, and when its turgor falls on a hot afternoon its buckling height drops below its actual height and it bends over. Real herbaceous stems are much more sophisticated columns than the model: hollow, ribbed, with strands of stiffer tissue near the surface, where they do the most good in bending, and with the softer turgid tissue inside pushing those strands outward and putting them in tension, so that the whole stem is prestressed. The combination stands far higher than the model’s eighteen centimetres. But the drooping of a thirsty seedling, and its recovery within minutes of watering, is the model’s prediction made visible.

Motion made of pressure #

A pressure that a cell can change is a pressure that can move something. The leaf pores called stomata are each bounded by two guard cells whose walls are reinforced by cellulose fibres running around them like the hoops of a barrel. When the guard cells pump potassium in, their osmotic pressure rises, water follows, the turgor rises to several megapascals, and the hooped cells, unable to grow fatter, grow longer and bow apart, opening the pore. Pumping the potassium out closes it. The plant regulates its gas exchange and its water loss by adjusting the turgor of a few thousand cells per square millimetre of leaf.

The sensitive plant Mimosa pudica folds its leaves within a second of being touched, by a sudden loss of turgor in cells at the base of each leaflet when an electrical signal opens channels in their membranes and lets ions and water out. The Venus flytrap closes faster still, by snapping a curved leaf through an elastic instability that a change of turgor triggers. In each case the motor is the same: osmosis supplies the pressure, the wall’s geometry turns the pressure into a shape, and a change of solute changes the shape. The swelling a membrane cannot stop follows the same physics into gels, which swell and shrink by tens of times as their internal solute changes; plants use it with walls that stop the swelling at the shape they want.

The fungus that punches through a leaf #

The largest turgor pressures in biology are not in plants but in fungi, and they are used as a weapon. The fungus that causes rice blast, one of the most destructive diseases of the world’s staple crop, lands on a leaf as a spore, grows a short thread and then swells its tip into a dome-shaped cell called an appressorium, pressed against the leaf’s waxy surface. The appressorium lays down a layer of melanin in its wall, which makes the wall nearly impermeable to everything but water, and then fills itself with glycerol, to a concentration of more than three moles per litre. Water floods in. The turgor rises to about eight megapascals, eighty atmospheres, ten times a plant cell’s and forty times a car tyre’s.

The appressorium then grows a narrow peg from its base, and the whole of that pressure is focused onto the peg’s tip, which punches straight through the leaf’s outer wall. In the laboratory such appressoria punch through sheets of plastic. The force is osmotic from start to finish: the fungus does no mechanical work beyond synthesising glycerol and keeping it in, and the melanin layer is essential, because without it glycerol leaks out and the pressure never builds. A mutant fungus that cannot make melanin cannot infect rice. The leak that the membrane that almost holds measures with a reflection coefficient — the fraction of a solute’s osmotic pressure a real membrane manages to develop — is here the difference between a disease and a harmless spore.

Measuring a pressure inside one cell #

Turgor in a single cell can be measured directly. The pressure probe, developed in the 1970s, is a glass capillary drawn to a tip a micrometre across, filled with oil and connected to a pressure sensor. Pushed into a cell, its tip lets cell sap into the capillary, forming a meniscus with the oil; a plunger then pushes the meniscus back to the tip, and the pressure needed to hold it there is the cell’s turgor. The same probe, pushing a small volume in or out and recording the pressure change, measures the wall’s stiffness — the slope of the turgor curve in the Höfler diagram — and how fast the pressure relaxes measures how readily water crosses the membrane.

For a whole leaf or shoot the instrument is the pressure chamber, introduced by Per Scholander in 1965: the cut shoot is sealed into a steel chamber with its cut end outside, and the gas pressure in the chamber is raised until sap just appears at the cut. That pressure balances the tension the water was under in the intact plant — the tension at the heart of the column that is pulled, not pushed — and so measures the plant’s water potential. Taking a series of readings as the shoot dries gives the plant’s whole Höfler curve, and with it the turgor-loss point at which the shoot will wilt.

Why animals do not do this #

An animal cell has no wall and cannot hold a pressure, so animals keep the solutes inside and outside their cells balanced and their cells at almost no turgor; their skeletons are bone, cartilage or chitin. A plant’s choice is the other one: a wall strong enough to hold the pressure, and a cytoplasm deliberately more concentrated than its surroundings, so that the pressure is always positive. The cost is that a plant must keep water flowing to its cells, and the turgor it stands by is held only as long as the water lasts. The water a plant loses through its stomata to keep its leaves supplied with carbon dioxide is the water its structure depends on, and the closing of stomata in drought is a trade between growth and standing up.

The same counting rule that what it costs to take the salt out turns into the minimum energy of desalination is, in a plant, the source of its rigidity. The plant pays for it by spending energy to accumulate solutes against their gradients, just as a desalination plant pays to remove them; the osmotic pressure is the price per unit volume either way.

A wall treated as a membrane #

Turgor is computed from van 't Hoff’s law, which holds for dilute ideal solutions and overestimates slightly at the concentrations of real sap, where the pressure stops counting molecules exactly. The cell is treated as a thin-walled cylinder and its wall as a membrane in tension; real walls are layered composites of oriented fibres with stiffness that depends on direction. The Höfler diagram assumes a linearly elastic wall and a fixed amount of solute; real cells have walls that stiffen as they are stretched and adjust their solute over hours. The membrane-tube moments are exact for an ideal membrane with no compressive stiffness. The stem model is a solid uniform column with a tissue stiffness taken as six times the turgor, a stated figure of the right order for soft tissue rather than a property of any plant, and it ignores the stem’s taper, its stiffer outer strands and its prestress.

Still open: how a growing cell controls its wall #

Turgor provides the force that expands every growing plant cell, but the wall decides where and how fast the expansion happens, by loosening its cross-links in some places and not others. How the cell controls that loosening — through enzymes, through acidification of the wall, through the orientation of new cellulose laid down by enzyme complexes that move along the cell’s internal skeleton — is understood in outline and not in detail, and it is the central question of how plant shapes are made. The physics is simple: a uniform pressure, a non-uniform wall, and growth wherever the wall yields. The biology that decides where the wall yields is not.

The habit worth carrying away is to ask what is holding a soft structure up. A plant cell with 0.3 osmoles of solute per litre fills until its wall pushes back with 0.74 MPa, three times a car tyre, and a tube of membrane under pressure resists bending in exact proportion to that pressure whatever the membrane is made of — so a soft stem’s stiffness is its turgor, and a few per cent of lost water, which takes the turgor to zero, is enough to bring it down. A lettuce is an inflatable, and wilting is a puncture made of evaporation.