A lipid bilayer excludes almost everything a cell needs, so every useful traffic across it is a protein that had to be specified by a gene and is paid for in energy.
The previous lesson measured that exclusion: sodium crosses a bare bilayer eleven orders of magnitude more slowly than water. This lesson is about the machinery that gets around it, and about the thing the machinery produces as a by-product, which is a voltage. Nothing here assumes any biology beyond a membrane with an inside and an outside.
Three ways through, distinguished by what they cost
There are exactly three, and they are told apart by two questions: does the substance move down its own gradient or up it, and does the rate saturate.
Simple diffusion is what the previous lesson measured. Oxygen, carbon dioxide, ethanol and steroid hormones dissolve in the hydrocarbon and cross, at a rate strictly proportional to the concentration difference, with no maximum. Double the gradient and you double the flux, forever.
Facilitated diffusion goes through a protein but still only downhill. It cannot move anything against a gradient, because it supplies no energy, but it is enormously faster than the bare bilayer and it is selective and controllable. Its rate saturates, because there is a finite number of proteins and each takes a finite time per molecule.
Active transport moves a substance up its gradient and must therefore be coupled to something that releases free energy. Primary active transport spends ATP directly. Secondary active transport spends a gradient that some other pump built, which is a way of saying it spends ATP at one remove.
Saturation is the cleanest experimental signature. Plot flux against external concentration: a straight line through the origin means simple diffusion, a curve flattening to a maximum means a protein is involved and its binding sites are filling up. That single graph, taken on red cells with glucose in the 1950s, is why anyone believed in transport proteins before any of them had been purified.
Channels, and how to be fast and picky at once
A channel is a hole. When it is open, ions flow through it in single file at rates around per second, which is close to the limit set by how fast ions can arrive by diffusion. That speed makes selectivity look impossible: a hole that passes a hundred million ions a second cannot be examining each one.
Yet a potassium channel passes potassium over sodium by more than a thousand to one, and sodium is the smaller ion. A sieve cannot do that. Roderick MacKinnon's crystal structure in 1998 showed how it is done, and the mechanism is worth understanding because it is a general trick.
An ion in water is not bare. It carries a shell of water molecules whose oxygens point at it, and the energy holding that shell is what the previous lesson used to explain why ions cannot cross lipid. To enter a narrow pore an ion must shed the shell, which costs energy, and it will only do so if the pore repays it. The potassium channel's selectivity filter is lined with backbone carbonyl oxygens spaced so that they sit exactly where the water oxygens would have sat around a potassium ion. Potassium therefore exchanges one set of oxygen neighbours for an identical set and pays almost nothing. Sodium is smaller, so the same rigid cage of carbonyls cannot close around it tightly enough to compensate for the water it gave up, and a sodium ion in the filter is worse off than a sodium ion in the solution outside. Selection happens by making the right ion comfortable rather than by making the wrong ion too big to fit, and because nothing has to move or be inspected, it is compatible with the full diffusion-limited rate.
Channels are also gated. Voltage-gated channels carry charged segments that move when the membrane potential changes; ligand-gated channels open when a molecule binds; mechanically gated channels open when the membrane is stretched, which is how hearing works. A gate turns a hole into a switch, and a switch is what makes a nerve impulse possible.
Carriers, and why they are a thousand times slower
A carrier binds its cargo on one face, changes shape, and releases it on the other. GLUT1, the glucose transporter of the red blood cell, is the standard example. It never opens a continuous path from one side to the other, so it cannot leak, but it must complete a conformational cycle for every molecule and that takes about a millisecond. Its turnover is roughly a thousand molecules per second, five orders of magnitude below a channel.
The cost buys specificity of a different kind. GLUT1 has a half-saturating concentration near 1.5 mM against a blood glucose concentration around 5 mM, so it runs close to its maximum rate and is largely insensitive to blood sugar going up or down, which is what a red cell wants. The liver's GLUT2 has a half-saturating concentration near 15 mM instead, well above normal blood glucose, so its rate rises and falls in proportion to blood sugar, which is what an organ that regulates blood sugar wants. Same trick, different constant, opposite job.
Pumps, and the bill they run up
Jens Christian Skou, in Aarhus in 1957, was studying nerve membranes and found an enzyme that hydrolysed ATP only when both sodium and potassium were present. It is the sodium potassium ATPase, and it is present in every animal cell.
Per molecule of ATP it moves three sodium ions out and two potassium ions in. It does this by phosphorylating itself: ATP transfers its terminal phosphate onto an aspartate residue of the pump, the pump changes shape and releases sodium outside, then potassium binding triggers dephosphorylation and the pump returns. The cycle is slow by channel standards, around a hundred ions per second, and there is a great deal of it.
The stoichiometry is not balanced. Three positive charges leave and two enter, so every cycle moves one net positive charge out of the cell and the pump is electrogenic: it contributes directly to the membrane voltage as well as to the concentration gradients.
Example. Human cells hold sodium at about 12 mM inside against 145 mM outside, potassium at 140 mM inside against 5 mM outside, and sit at about 70 mV negative inside. Work out whether one ATP is enough to run one cycle of the pump at 37 degrees Celsius.
Moving one mole of sodium out costs concentration work kJ, plus electrical work kJ against the inside-negative potential, giving 13.2 kJ per mole of sodium and 39.5 kJ for three. Potassium is cheaper: the concentration work is kJ against it, but the electrical term now helps, at kJ, so the net is 1.83 kJ per mole and 3.7 kJ for two. The total is kJ per mole of ATP hydrolysed. The next lesson shows that ATP hydrolysis inside a cell releases about 50 kJ/mol, so one ATP is enough, with roughly 7 kJ to spare. That margin is the interesting part. It is small, about fifteen per cent, which means the pump runs near thermodynamic reversibility, and it predicts that if the ATP supply falls or the gradients steepen the pump should stall or even run backwards and synthesise ATP. It does exactly that in the laboratory, which is a much stronger confirmation of the mechanism than the forward reaction is.
Now you. Ouabain, a plant glycoside, blocks the sodium potassium ATPase. Digoxin, a close relative from foxglove, has been used to treat heart failure since William Withering described it in 1785, and it makes the heart beat more strongly. Given that cardiac muscle also has a transporter that swaps three sodium ions in for one calcium ion out, and that contraction strength depends on internal calcium, explain the drug.
Answer
Blocking the pump partially lets internal sodium rise. The sodium calcium exchanger is powered by the sodium gradient, not by ATP, so a shallower sodium gradient means less free energy available to push calcium out, and internal calcium rises too. More calcium in the cell means more available to the contractile machinery, so each beat is stronger. The chain is worth noticing because nothing in it is a direct effect on contraction: the drug hits a pump that has nothing to do with calcium, and the effect arrives through a shared gradient two steps later. That is what a cell's economy looks like once one currency powers several machines. It also explains why digoxin is dangerous. The therapeutic and the toxic doses are close, because pushing the same lever further raises calcium into the range that causes arrhythmia, and this is one of the few drugs still in use whose blood level is routinely measured.
The pump is expensive. In a resting cell it typically consumes twenty to thirty per cent of all ATP produced, and in neurons, which are constantly discharging and refilling their gradients, estimates run to half or more of the brain's entire energy budget. A large fraction of what you eat is spent keeping sodium out of your cells.
Water, and why cells burst
Water crosses membranes freely, so a cell cannot control its water content directly. It can only control its solutes, and water then follows.
The osmotic pressure of a solution of total solute concentration is . Cytoplasm and blood plasma both run at about 300 milliosmoles per litre, which is Pa, or 7.6 atmospheres. Put a cell in pure water and that is the pressure difference driving water in. A red cell swells, becomes a sphere, and lyses at about 1.4 times its normal volume, which is why intravenous fluid is 0.9 per cent sodium chloride: 9 g/L divided by 58.44 g/mol is 154 mM, and sodium chloride dissociates into two particles, so mOsm, matching plasma.
Every cell has to solve this, and there are three solutions. Animal cells pump: the sodium potassium ATPase is also an osmotic device, because it keeps sodium out and sodium therefore behaves as an impermeant external solute, balancing the impermeant proteins and nucleic acids trapped inside. Stop the pump and an animal cell swells and dies, which is one of the fastest consequences of losing ATP. Plants, fungi and most bacteria build a rigid wall instead and let the pressure rise: a plant cell sits at 5 to 20 atmospheres of turgor pressure pushing outward against its wall, and wilting is that pressure falling. Freshwater protists such as Paramecium use a contractile vacuole, collecting water and expelling it several times a minute, a bailing pump running continuously against an ocean.
Example. A patient is given an intravenous drip of pure 5 per cent glucose rather than saline. Glucose is 180 g/mol, so this is 278 mM, close to plasma osmolarity. Why does this behave completely differently from saline once it is in the body?
Because osmotic effect depends on the solute staying where it was put. Sodium is held outside cells by the pump, so saline stays in the extracellular space and expands it. Glucose is taken up by cells through GLUT transporters and metabolised, so within a short time the glucose is gone and what was infused is effectively pure water, which distributes through the whole body and enters cells. A 5 per cent glucose drip is a way of giving water without giving salt, and giving too much of it drops plasma sodium and swells cells, including brain cells inside a rigid skull, which is a real clinical hazard rather than a theoretical one. The general point is that "isotonic" is not a property of a solution alone. It is a property of a solution and a membrane together, and it lasts only as long as the solute cannot cross or be consumed.
Now you. Wilting is loss of turgor pressure. Explain why a plant with a strong cell wall wilts at all, when an animal cell with no wall keeps its shape perfectly well.
Answer
The two get their mechanical stiffness from opposite sources. An animal cell is held in shape by an internal protein scaffold, the cytoskeleton, which works whether or not the cell is full of water, and by neighbouring cells and extracellular matrix. A plant cell wall is strong in tension but is a thin shell, so on its own it is floppy in the way an empty tyre is floppy; its rigidity comes from being inflated. Turgor is the inflation, and it exists only while the vacuole is drawing water in osmotically, which requires water to be available outside. Lose water faster than the roots supply it and the pressure falls, the wall goes slack, and the tissue droops even though not one wall has broken. It is worth noticing that this makes a plant's mechanical state a direct readout of its water balance, which is why wilting is such a fast and reversible signal, and why a cut flower recovers within minutes of being put in water.
The voltage that arrives for free
Now the result that everything later depends on. Separate charge across an insulator and you have a capacitor, and a membrane is an excellent insulator 5 nm thick. So the moment ions are unevenly distributed and can move at all, there is a voltage.
Consider a membrane permeable to potassium alone, with 140 mM inside and 5 mM outside. Potassium leaves down its concentration gradient, carrying positive charge with it, so the inside becomes negative, and that negative charge pulls potassium back. Equilibrium is where the two exactly balance. Writing the electrochemical potential of an ion as and setting it equal on both sides gives
which is the Nernst equation. At 310 K, mV, so mV. Do the same for sodium, at 145 mM outside and 12 mM inside, and mV.
A real cell sits between those, at about mV in a neuron and mV in skeletal muscle, and the reason is that a resting membrane is not permeable to potassium alone. It is perhaps twenty to a hundred times more permeable to potassium than to sodium, so the resting potential lies close to but is pulled a little towards by the sodium leak, and the pump has to keep bailing that leak out. Changing the permeability ratio moves the voltage between the two Nernst values, and doing that abruptly by opening sodium channels is a nerve impulse.
How much charge is involved is the surprise. A spherical cell of radius 5 µm has cm² of membrane, and biological membranes have a capacitance near 1 µF/cm², so F. To reach 70 mV requires C, which at C per ion is ions. The same cell contains M L potassium ions. The fraction that has to move is , three ions in every hundred thousand.
That reconciles two things that look contradictory. The voltage is set by the concentration ratio through the Nernst equation, yet establishing it changes the concentrations by a third of a thousandth of a per cent. Electrical and chemical bookkeeping are almost completely decoupled, which is why a neuron can fire hundreds of times before its gradients measurably run down.
Example. A neuron's resting potential is close to at mV. Predict what happens to the voltage if the extracellular potassium concentration is raised from 5 mM to 10 mM, and say why this is a clinical emergency.
Doubling the outside concentration halves the ratio in the Nernst equation, so mV instead of mV, a shift of 19 mV towards zero. Since the resting potential tracks closely, the cell depolarises by something of that order. That sounds modest and it is not, because voltage-gated sodium channels open in a narrow range and then inactivate: a partial depolarisation first makes cells hyperexcitable and then, as the sodium channels inactivate at the new resting voltage, makes them unexcitable. In cardiac muscle the sequence is arrhythmia followed by arrest, which is why a plasma potassium of 7 mM is treated as an emergency and why potassium is the agent used in judicial execution. The general point is that the Nernst equation is logarithmic, so a doubling of a small extracellular concentration moves the voltage as much as a very large change in the internal concentration would, and the extracellular pool is the small one. A body regulates plasma potassium to within about half a millimole for exactly this reason.
Now you. Given the same equation, explain why changing extracellular sodium from 145 mM to 135 mM has far less immediate electrical effect than changing potassium from 5 mM to 10 mM.
Answer
Two reasons, and both are visible in the equation. The first is proportional size: sodium has moved by seven per cent while potassium has moved by a hundred per cent, and the Nernst potential depends on the ratio, so shifts by mV while shifts by 19 mV. Small absolute changes in a large concentration are small ratios. The second is weighting: a resting membrane is far more permeable to potassium than to sodium, so the resting voltage sits close to and is pulled only slightly towards , which means a change in influences the resting potential far less than an equal change in . This is why hyponatraemia, low plasma sodium, presents as a problem of water and cell swelling rather than as an electrical problem, while hyperkalaemia presents as an electrical problem almost immediately. Two ions, the same equation, and completely different clinical pictures, decided by which one the resting membrane can pass.
One honest caveat about that capacitance. Treating the membrane as a parallel plate capacitor with relative permittivity 2 and thickness 5 nm gives µF/cm², three times smaller than the measured value. The discrepancy is real and the resolution is that only the hydrocarbon core is a low-permittivity insulator, while the head group region is hydrated and behaves electrically much more like water, so the effective insulating thickness is closer to 2 nm than 5. It is a good example of a number that is worth reproducing from first principles precisely because it does not quite work.
Spending a gradient
A gradient built with ATP is a store of free energy, and a cell can spend it on things other than voltage.
The sodium glucose cotransporter SGLT1, in the lining of the small intestine, binds two sodium ions and one glucose molecule and lets them cross together. The sodium runs downhill, at 13.2 kJ per mole as computed above, so two of them deliver 26.4 kJ. Concentrating glucose fourfold, from 5 mM in the gut lumen to 20 mM in the cell, costs kJ. There is energy to spare by a factor of seven, which is why the intestine can absorb glucose down to almost the last molecule rather than stopping when the concentrations equalise.
This is not a laboratory curiosity. Oral rehydration therapy, which treats cholera and other severe diarrhoeal disease with a drink of salt and sugar in water, works because SGLT1 keeps running when much else has failed, and water follows the absorbed sodium and glucose osmotically. It was developed in the 1960s and deployed at scale during the 1971 refugee crisis in Bengal, where it cut cholera mortality in the camps from around thirty per cent to a few per cent. Either half alone does far less: salt without sugar, or sugar without salt, does not drive the cotransporter.
The same principle appears everywhere. Nerve terminals reload neurotransmitter using a proton gradient across the vesicle membrane. Bacteria run flagellar motors on protons rather than on ATP. And mitochondria, as a later lesson shows, turn the entire logic around: instead of spending ATP to build a gradient, they spend a gradient to build ATP.
Every mechanism in this lesson ended at the same place. Channels are free, but the gradients they discharge were not; carriers are free, but they only equalise; pumps cost ATP outright; and cotransporters spend gradients that a pump paid for. The whole traffic across a cell membrane is financed by one molecule, and the next lesson asks what that molecule actually is, why its hydrolysis is worth 50 kJ/mol rather than the 30.5 kJ/mol in the tables, and how a cell keeps the account from ever going empty.