Fluids

The swelling a membrane cannot stop

Give the thing a membrane holds back an electric charge and the small ions that can cross are no longer free to distribute themselves. Two conditions — neutrality on each side, and equal chemical potential for the salt — fix where every ion goes, and they leave the charged side with more particles than the other. That excess is what holds a joint apart, and it is why a cell has to spend a third of its energy pumping.

Assumes: The pressure that comes from counting · The membrane that almost holds

The pressure that comes from counting makes osmotic pressure as simple as it can be made: a membrane holds back some particles and passes others, the ones held back have nowhere to go, and the pressure needed to stop water flowing in is the pressure those particles would exert as a gas. It depends on how many of them there are and on nothing about what they are.

Charge them and the argument acquires a second half. The particles that cannot cross now demand counterions that can, and the ones that can are no longer free to distribute themselves evenly.

The pressure a charged gel develops. The swelling pressure against the gel's fixed charge density, for several salt concentrations, on logarithmic axes. Each curve has two straight parts with different slopes, and both limits are checked against the solution rather than read off the plot. Where the fixed charge is small compared with the salt the pressure goes as the square of the charge divided by four times the salt: the bath's ions screen the fixed ones, and doubling the salt halves the pressure. Where the fixed charge dominates, every counterion it demands is an extra particle in the gel and the pressure goes as the charge itself. At 15 mM salt, a gel with 200 mM of fixed charge develops 427 kPa; At 50 mM salt, a gel with 200 mM of fixed charge develops 306 kPa; At 150 mM salt, a gel with 200 mM of fixed charge develops 150 kPa; At 500 mM salt, a gel with 200 mM of fixed charge develops 49 kPa. Cartilage carries a fixed charge of a couple of hundred millimolar from the sulphated sugars on its proteoglycans, sits in a bath of about 150 mM, and develops a swelling pressure of an atmosphere and a half — which is what holds a joint apart and carries the load across it. — Fig. 1 The swelling pressure of a charged gel against its fixed charge density, for several salt concentrations, on logarithmic axes. Each curve has two straight parts: the pressure goes as the square of the charge where the salt screens it, and as the charge itself where the charge dominates.

Two conditions, and everything follows

The whole subject is two statements about equilibrium.

Each side is electrically neutral. Inside the gel that means the counterions exceed the coions by exactly the fixed charge; outside it means the two salt ions are equal.

Each mobile species has the same chemical potential on both sides. For a salt that reduces to a single equation: the product of the two ion concentrations is the same inside as out. It is a product rather than a difference because the electrical part of the potential enters with opposite signs for the two ions and cancels when they are multiplied.

Those two are a quadratic. Solving it gives the concentration of every mobile ion on both sides, with nothing else assumed.

Where the small ions end up. The concentrations of the two mobile ions inside a gel, against how much fixed charge the gel carries, for a bath at 150 mM. Both curves are solutions of two conditions — neutrality inside, and the same product of the two ion concentrations on both sides — checked against the solution before anything is drawn. At a fixed charge of 150 mM the gel holds 243 mM of counterions and 93 mM of coions; At a fixed charge of 300 mM the gel holds 362 mM of counterions and 62 mM of coions. The counterions are drawn in and the coions are pushed out, by exactly enough to keep the product fixed. The two curves never cross and their difference is always the fixed charge, which is neutrality drawn rather than stated. What matters for everything that follows is their sum: the gel ends up holding more mobile ions in total than the bath does, and that excess is what makes water flow in. — Fig. 2 The two mobile ion concentrations inside a gel, against the fixed charge it carries, in a bath at 150 mM. The difference between the two curves is the fixed charge and their product is the bath’s — both checked against the solution before drawing. What matters is their sum, which exceeds the bath’s.

Counterions are drawn in and coions are pushed out, in the proportions that keep the product fixed. The consequence is that the charged side holds more mobile particles in total, and it is that excess — not the polymer, which contributes almost nothing on a molar basis — that pulls the water in.

The arithmetic, once

The quadratic is worth doing in full, because the answer is short and the two limits fall out of it without further work.

Write the fixed charge inside as $Z$ , the bath as $c_0$ , and the two mobile ion concentrations inside as $c_+$ and $c_-$ . Neutrality gives $c_+ = c_- + Z$ . Equal chemical potential gives $c_+c_- = c_0^2$ . Substituting the first into the second gives $c_-^2 + Zc_- - c_0^2 = 0$ , whose positive root is

c_- = \frac{-Z + \sqrt{Z^2 + 4c_0^2}}{2}

and $c_+$ follows. The total inside is $c_+ + c_- = \sqrt{Z^2 + 4c_0^2}$ , and the excess over the bath’s $2c_0$ is the difference of those two.

Every number in this essay is that expression evaluated somewhere. The pressure is the excess times $RT$ ; the potential is the log of the counterion ratio times $RT/F$ ; the cell’s instability is the observation that the excess is positive for every positive $Z$ . One quadratic, three consequences — which is the reason the subject has a name and a single equation rather than a collection of results.

The two regimes

The excess is $\sqrt{Z^2 + 4c_0^2} - 2c_0$ , with $Z$ the fixed charge and $c_0$ the bath, and it has two limits that behave completely differently.

At high salt it approaches $Z^2/4c_0$ . The bath’s ions screen the fixed ones, the partition is nearly even, and doubling the salt halves the pressure. This is the same screening that the long-range force that does not reach is about, arriving as a thermodynamic quantity rather than as a length.

At low salt it approaches $Z$ itself. Every fixed charge demands a counterion, that counterion is an extra particle inside, and the pressure is simply the fixed charge counted as a gas. A polyelectrolyte in pure water is an extremely powerful osmotic agent for that reason: its own charge multiplies its effective particle count by the number of charges per chain.

The crossover between the two is at $Z \approx 2c_0$ , and it is where most things of interest sit. Cartilage at 200 mM of fixed charge in a 150 mM bath is close to it; a hydrogel in distilled water is well into the low-salt limit; a protein in blood is well into the screened one, which is why the colloid osmotic pressure of plasma — about 3.5 kPa, and the quantity that keeps fluid inside a capillary — is far smaller than the protein’s charge alone would suggest, and is calculable only with the screening included.

The polymer’s own concentration never appears. A gel of one long chain and a gel of a thousand short ones with the same total charge give the same pressure, which is a good check that the effect is about the counterions.

The tissue this holds up

What a salt bath does to a charged tissue. The swelling pressure of a gel carrying 200 mM of fixed charge, against the salt concentration of the bath it sits in. At 50 mM the pressure is 306 kPa; At 150 mM the pressure is 150 kPa; At 300 mM the pressure is 80 kPa; At 600 mM the pressure is 41 kPa; At 1000 mM the pressure is 25 kPa, a fall of a factor of 12.4 across the range. This is a laboratory experiment rather than a calculation: soak a piece of cartilage in concentrated salt and it shrinks and softens, put it back in a physiological bath and it recovers. Nothing has been done to the collagen or to the sugars; the bath's ions have screened the fixed charges and taken away the osmotic excess that was holding the tissue apart. The same measurement is how the fixed charge density of a tissue is determined, and how the loss of it in osteoarthritis is followed — a cartilage that has lost its proteoglycans carries less charge, swells less, and bears load through its collagen instead, which it is not built to do. — Fig. 3 The swelling pressure of a gel carrying 200 mM of fixed charge, against the salt in its bath. Soaking a piece of cartilage in strong salt makes it shrink and soften; returning it to a physiological bath makes it recover. Nothing has happened to the tissue.

Cartilage is a mesh of collagen filled with proteoglycans — huge molecules carrying sulphate and carboxyl groups, giving a fixed charge density of a couple of hundred millimolar. In a physiological bath that produces a swelling pressure of about an atmosphere and a half, and the collagen mesh is in tension holding it in.

That is how a joint carries load. The pressure is not generated when the joint is loaded; it is there all the time, and loading it squeezes water out until the fixed charge is concentrated enough for the pressure to match the load. The tissue is a hydrostatic structure whose pressure adjusts itself, and the adjustment is slow — which is why cartilage creeps under a sustained load and recovers overnight.

The salt experiment is the measurement. Soaking cartilage in strong saline and watching it shrink is how the fixed charge density is determined, and the loss of that charge is the earliest measurable change in osteoarthritis: a cartilage with fewer proteoglycans swells less, carries less of its load hydrostatically, and passes the difference to a collagen mesh not built for it.

What the pressure does mechanically

A swelling pressure is only useful if something is holding it in, and the arrangement is the same one a pressurised vessel uses.

In cartilage the proteoglycans supply the pressure and a network of collagen fibrils supplies the tension. The fibrils are stiff in tension and useless in compression, exactly like the fabric of a balloon, so the tissue’s compressive stiffness comes entirely from the fluid pressure — and its stiffness in tension comes entirely from the fibrils. A material stiff in two different ways for two different reasons is a composite, and this one is a composite whose pressurised phase is made by chemistry rather than by a pump.

The same design appears in a plant. A non-woody plant stands up because each of its cells is inflated against a cellulose wall, and wilting is the loss of that pressure rather than any change in the wall. How high water will climb and the column that is pulled, not pushed are about getting the water there; this is about what it does when it arrives.

Two numbers describe such a structure: the pressure inside and the stiffness of the envelope. Where the envelope is fabric, one of them is doing all the work in compression and the other all of it in tension, and neither substitutes for the other.

The voltage that comes free

The voltage across a membrane that nothing is pumping. The electrical potential of the gel relative to its bath, against the fixed charge it carries, at 150 mM salt. It is negative because the gel's own charges are, and it is what the ions arrange themselves against: the counterions are held in by it and the coions kept out. At a fixed charge of 150 mM the potential is -12.4 mV; at four times that it is -37.1 mV. These are tens of millivolts, which is the scale of every biological membrane potential, and they arise here with nothing pumping anything: the potential is a consequence of equilibrium rather than of work being done. That is worth separating from the resting potential of a nerve cell, which is not an equilibrium at all — it is maintained by pumps against a continual leak, and its size happens to be similar for a different reason. — Fig. 4 The potential of the gel relative to its bath, against the fixed charge. It is tens of millivolts, negative because the fixed charges are, and it arises at equilibrium — no current flows and no work is being done to maintain it.

The partition of the ions and the potential are the same fact stated twice: the ions are distributed as they are because of the potential, and the potential is what it is because of the distribution.

It is worth separating this from the resting potential of a nerve or muscle cell, which is a similar number arrived at differently. That one is not an equilibrium: sodium leaks in continuously and is pumped out continuously, and the potential is the steady state of a driven system. Switch off the pump and it decays over minutes. A Donnan potential has nothing to switch off.

Why the potential and the pressure are not independent

There is a temptation to treat the potential as an electrical effect and the pressure as an osmotic one, and they are the same effect seen from two directions.

Consider what would happen if the potential were somehow removed while the ion concentrations were left alone. The counterions inside would be at a much higher concentration than outside with nothing holding them, so they would leave — and as they left they would carry charge, which would rebuild the potential. The equilibrium is the state in which those two tendencies balance, and both the partition and the potential are that balance described in different units.

The pressure is a third description of the same thing. It is the mechanical work per unit volume needed to stop water following the ions, and it is fixed by the partition, which is fixed by the potential.

So there is one degree of freedom here and three names for it, and the reason to keep all three is that each is what a different instrument measures: a pressure transducer reads the third, an electrode reads the second, and a chemical assay reads the first. A measurement that gets two of them and finds them inconsistent has found something — usually that the membrane is passing something it was not supposed to.

Why a cell has to pump

Why a cell cannot simply sit there. The osmotic pressure a cell would develop from its own impermeant contents alone, against how much fixed charge those contents carry, in a bath of 150 mM. 50 mM of fixed charge gives 10 kPa; 100 mM of fixed charge gives 40 kPa; 150 mM of fixed charge gives 88 kPa; 250 mM of fixed charge gives 224 kPa. Every value is positive, and that is the problem: a cell containing charged proteins and nucleic acids that cannot leave, in a bath of salt that can, is never in osmotic balance. Water enters, the cell swells, the fixed charge is diluted, the excess falls — and it does not reach zero, because a smaller fixed charge still gives a positive excess. The cell swells until it bursts. What saves it is that one of the mobile ions is made effectively impermeant. A pump that throws sodium out as fast as it leaks in makes sodium behave, from the osmotic point of view, like something the membrane does not pass — and a second impermeant species on the outside is exactly what is needed to balance the first. A third of a resting cell's energy goes on that pump, and this is what it is buying. — Fig. 5 The osmotic pressure a cell’s own impermeant contents would develop, against how much fixed charge they carry. Every value is positive, and diluting the charge by swelling reduces it without reaching zero.

A cell contains proteins and nucleic acids that cannot cross its membrane and carry a large net negative charge. Around it is a salt solution whose ions cross freely. That is exactly the arrangement above, and it has no stable state.

Water enters, the cell swells, the fixed charge is diluted — and the excess falls but stays positive, because a smaller fixed charge still gives a positive excess. There is no volume at which the cell is in balance. It swells until the membrane fails.

The way out is to make one of the mobile ions behave as though it cannot cross. A pump that ejects sodium as fast as it leaks in produces, from the osmotic point of view, an impermeant species outside — and one impermeant species on each side is exactly what is needed for a balance to exist. Stop the pump and a real cell swells and lyses, on a timescale of tens of minutes, which is the experiment.

There is a second reading of the same fact that is worth having. What the pump manufactures is not a concentration difference but an impermeant species, and impermeance is a property of a rate rather than of a barrier: sodium crosses the membrane freely and is removed as fast as it arrives, so on any timescale longer than the pump’s it behaves as though it did not cross at all. The same trick appears wherever a steady removal makes something look absent — a reaction product swept away, a heat load carried off by a coolant, or the entropy a living thing exports to stay ordered, which is what a system actually minimises read at a boundary.

A third of a resting cell’s energy budget goes on that pump, and this is what it buys: not a concentration gradient for its own sake, but the ability to have a volume at all.

The same two conditions, elsewhere

The arrangement is not special to biology, and three other places it appears are worth knowing because they make the generality obvious.

An ion-exchange resin is a gel with fixed charges by design, and everything above describes its behaviour: it excludes coions, which is why it can be used to strip a salt; its capacity is set by the fixed charge density; and it is regenerated by soaking in strong brine, which screens the charges and releases what was bound. A water softener runs this cycle a few times a week.

A clay soil carries a fixed negative charge on its particle surfaces, holds cations against being washed out, and swells when rained on — because rain is low in salt and the low-salt limit is where the pressure is largest. Sodium-rich clays swell so much that they close their own pores and stop draining, which is a serious problem in irrigated agriculture and is fixed by adding calcium, whose double charge partitions differently.

A polyelectrolyte hydrogel is the working part of a disposable nappy, and it absorbs several hundred times its own weight of water for exactly this reason. Its capacity in urine is a fraction of its capacity in distilled water, and the ratio is the screening in the second figure.

In each case the useful quantity is the same one, and the design lever is either the fixed charge or the salt in the bath. There is nothing else in the expression to change.

Where the model stops

The van 't Hoff expression is used for the pressure, which is the ideal-gas form and is accurate only at low concentration. At the hundreds of millimolar reached inside a charged gel there are real corrections — from ion size, from ion pairing, from the polymer’s own excluded volume — and a careful treatment of cartilage carries all of them.

The fixed charges are treated as smeared out. They are on chains, at specific places, and near a chain the local ion concentration is far higher than the average. That matters for the potential, and for anything that binds to the chain, more than it does for the total pressure.

Only one salt, with singly charged ions. Calcium, which is doubly charged, partitions much more strongly, and its presence changes the pressure out of proportion to its concentration. Real tissue sits in a mixture.

And the membrane is assumed to reflect the polymer perfectly and the salt not at all. The membrane that almost holds is where the intermediate case is worked out, and a real cell membrane is intermediate for a good many species.

The number that makes it urgent

The instability is not slow. It is worth putting a time on it, because “the cell would eventually swell” and “the cell swells in twenty minutes” are different statements.

A red cell’s membrane passes water fast enough that a cell dropped into distilled water swells and bursts in seconds, which is a demonstration anybody with a microscope has seen. The osmotic excess a cell would develop from its own contents unopposed is a fair fraction of that driving force, so the volume changes on a timescale of minutes rather than hours.

That is why the pump cannot be intermittent and why its failure is quick. Cooling a cell to stop the pump, or poisoning it with ouabain, produces visible swelling within tens of minutes — a standard demonstration, and one whose timescale is set by the water permeability rather than by anything about the pump.

A steady state maintained against a fast process needs continuous work, and the size of the work is set by how fast the process is rather than by how far from equilibrium the state is. That is the general shape of every homeostatic mechanism.

What the pictures cannot show

Every figure here is an equilibrium, and the interesting behaviour of a charged tissue is its approach to one. Cartilage loaded suddenly does not reach its new water content for hours, because the water has to be squeezed out through a mesh with a very low permeability, and the time constant depends on the square of the thickness. The pressure the figures draw is where it ends up, not what it does.

Nor do the figures show what the gel is made of. The whole argument depends on the fixed charge and not at all on what carries it, which is the result and is also a warning: two tissues with the same charge density behave identically here and quite differently in every mechanical respect, because the network holding the pressure in is doing the other half of the job.

Where the ladder goes next

The osmosis ladder began with the pressure that comes from counting, where the pressure depends on the number of particles and nothing else, went on to what it costs to reverse — the thermodynamic minimum for taking salt out of seawater — and then to the membrane that almost holds, where the membrane’s selectivity is a number rather than a promise. This rung adds charge to the thing being held back and finds that the counting still works, applied to particles nobody was counting.

The rung after it is the case where the fixed charges are close enough together for the counterions to be genuinely condensed onto the chain rather than distributed through the gel, which changes what “fixed charge” means and is the point at which the smeared-out picture stops. The habit worth carrying is the one this rung is built on: when an argument counts particles, check what the particles it holds back have brought with them.

Part 4 of 6

This essay is one argument about Osmosis. The others:

$The fraction of the pressure a membrane can hold. The osmotic pressure actually developed across a membrane against a 300 mol/m³ solution at 298 K, as the reflection coefficient runs from a membrane the solute passes freely to one it cannot pass at all. The line is straight with the van 't Hoff pressure 743.69 kPa as its slope — checked against the drawn line — because the coefficient enters as a simple factor. At σ = 1 the pressure is 743.69 kPa; At σ = 0.6 the pressure is 446.21 kPa; At σ = 0.2 the pressure is 148.74 kPa. Van 't Hoff's law is the ceiling rather than the answer, and a membrane's coefficient against a given solute is as much a property of the pair as the concentration is of the solution.$ The membrane that almost holds Part 3 — Van 't Hoff's law gives the osmotic pressure a perfectly selective membrane would develop, and no membrane is. What a real one develops is a fraction of it — a number between zero and one that belongs to the membrane and the solute together, and that decides whether a solution is isotonic in effect or only on paper. $Most of a chain's counterions never leave it. The fraction of a charged rod's counterions lying within a distance r of it, against r in rod radii on a logarithmic axis, for a charge parameter ξ = 4.2 — the Bjerrum length of water, 0.7135 nm, over a charge spacing of 0.17 nm, which is DNA's. Each curve is the Poisson–Boltzmann solution for the rod at the centre of a cell of radius 10², 10⁴, 10⁶ rod radii, with its counterions checked to neutralise it exactly. Diluting the solution widens the cell by four decades at a time, and a counterion free to go anywhere in the cell ought to spread with it; instead each curve keeps a plateau near the rod whose height does not change. At the inflection of every curve the enclosed fraction is 0.762, which is Manning's 1 − 1/ξ, and the plateau sits there: 76 per cent of the counterions stay bound to the chain however dilute the solution, and only 24 per cent spread through it.$ The counterions that never leave the chain Part 5 — Dilute a solution of DNA a million times and its counterions ought to scatter through the whole volume. Three quarters of them do not. A line of charges closer together than the Bjerrum length — 0.71 nm in water — holds on to its counterions however much room they are given, until the chain's charge is cut back to one per Bjerrum length, and every osmotic pressure, swelling gel and packed virus built from such chains is set by that length rather than by the chemistry.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The like charges that pull together

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Biological physicsChemical potentialElectroneutralityElectrostatics EquilibriumGelIonMembrane Osmosis Osmotic pressure ScreeningSwelling

The film that goes black before it bursts equilibrium, screening
The gas that nobody counted chemical potential, equilibrium
The part of the wrap that is actually gripping equilibrium, screening
Why the triple point is a point chemical potential, equilibrium

Two conditions, and everything follows #

The arithmetic, once #

The two regimes #

The tissue this holds up #

What the pressure does mechanically #

The voltage that comes free #

Why the potential and the pressure are not independent #

Why a cell has to pump #

The same two conditions, elsewhere #

Where the model stops #

The number that makes it urgent #

What the pictures cannot show #

Where the ladder goes next #