The speed limit that does not care how big the molecules are

The walk that comes home was about whether a random walker returns to where it started, and it found the answer depends on the number of dimensions: in one and two it always does, in three it may never. Along the way it noticed that the same fact decides how quickly one diffusing molecule finds another. In three dimensions a sphere of radius $R$ is found by walkers diffusing with coefficient $D$ at a steady rate $4\pi DR$, the result Marian Smoluchowski derived in 1917 — a genuine rate constant, because a walker that misses is unlikely to come back, and a spreading that only runs forwards carries it steadily away. The essay pointed to this as a thread to follow and did not follow it.

This essay follows it to a conclusion that is simple enough to state in a line and surprising enough to be worth several figures: the fastest possible rate of a reaction in solution does not depend on the size of the molecules reacting. It depends on the temperature and the viscosity of the solvent. Chemists call it the diffusion limit, and a reaction that reaches it is said to be diffusion-controlled: the molecules react every time they meet, and the only thing slowing them is finding each other.

The hole a reacting sphere digs #

Put one reactive sphere into a solution of partners and let every partner that touches it disappear. At first the sphere consumes the partners that happened to be next to it, and a thin shell round it is emptied. Partners from further out diffuse into the gap, and the depleted region spreads. After a while it stops changing: partners arrive at the edge of the hole as fast as the sphere consumes them, and the concentration round the sphere settles into a profile that recovers from zero at the surface as $1 - R/r$.

That profile recovers slowly. At two radii from the centre the concentration is still only half its distant value, and at ten radii nine tenths. A reactive molecule is surrounded by a depleted halo many times its own size, which is part of why reactions between molecules that happen to be close are not independent. The profile has a familiar form. It is the electric potential round a charged sphere, and the rate at which partners flow in — the gradient at the surface times the area — is the same number as the sphere’s capacitance, with $D$ in place of the permittivity. The potential is where the wanderers stop made this correspondence exact: diffusion to an absorbing surface and electrostatics of a conductor are one problem, and how much charge a shape will hold is the same table of numbers as how fast it catches what diffuses towards it.

A rate that settles #

The steady rate is reached only after the transient, and the drawing shows how the rate approaches it: from above, as $1 + R/\sqrt{\pi Dt}$. Early on the sphere is feeding on the partners that were close at the start, which is faster than drawing them in from afar; as those are used up the rate falls towards its steady value. The approach is slow — as one over the square root of time — because diffusion’s own spreading is.

For small molecules in water the time scale is tiny. A radius of half a nanometre and a diffusion coefficient of two thousandths of a square millimetre per second give a unit of time of about a hundred and twenty-five picoseconds, and the rate is within ten per cent of steady after four nanoseconds. For any reaction a chemist could follow by mixing two solutions, the steady rate is all there is. The transient becomes visible only in experiments that start a reaction with a laser pulse and watch it with picosecond resolution — the quenching of a fluorescent molecule by a partner, for instance, where the early excess of fast quenching by nearby partners is measured and fitted with exactly this formula.

The cancellation #

When two molecules both diffuse, the rate at which they meet is Smoluchowski’s formula with the diffusion coefficients added and the radii added:

$$k = 4\pi(D_A + D_B)(R_A + R_B).$$

If the molecules were points moving at fixed speeds, bigger targets would be hit more often, as the dashed line in the drawing assumes. But they do not move at fixed speeds. How fast a sphere diffuses through a liquid is set by the liquid’s drag on it, and the jiggle that proved atoms gave the relation: $D = kT/6\pi\eta R$, the Stokes–Einstein law. A bigger sphere is dragged harder and wanders more slowly, in exact inverse proportion to its radius.

For two molecules of the same size, the formula becomes $4\pi \cdot 2D \cdot 2R$, and $DR$ is $kT/6\pi\eta$ whatever the radius. The size cancels, and the rate is

$$k = \frac{8kT}{3\eta}.$$

In water at room temperature that is $7.4\times10^9$ per molar per second. The drawing shows it as a flat line from molecules of a tenth of a nanometre to molecules of a hundred: a small ion meeting another small ion and a protein meeting another protein, if each reacted on first contact, would react at the same rate. The number contains the temperature and the viscosity of the solvent and nothing about the molecules at all. In a more viscous solvent every diffusion-limited reaction slows in proportion, which is one way chemists test whether a fast reaction is diffusion-controlled: add glycerol and see whether the rate falls with the viscosity.

Most reactions are slower, and the limit still governs them #

Few reactions run at the diffusion limit, because most need more than a meeting. Two molecules that touch must also have enough energy between them to cross a barrier, and the fraction of encounters that do is the Boltzmann factor for the barrier’s height — tiny for most reactions at room temperature. The observed rate is then the encounter rate times the chance an encounter reacts, and it is limited by chemistry rather than by transport. Frank Collins and George Kimball wrote the combination in 1949: the rate is the product of the two steps divided by their sum, the same way two resistances add in series. Whichever step is slower dominates, and the diffusion limit is the ceiling that the combination approaches as the chemistry becomes easy.

The ceiling matters even for reactions far below it, through the other direction. A reaction and its reverse are tied by the balance at equilibrium: the ratio of the forward and backward rates is fixed by the free-energy difference. For two molecules binding, the forward rate can be no faster than the diffusion limit, so a binding that is very strong — whose equilibrium lies far towards the bound pair — must get its strength from a slow reverse rate, a complex that lasts a long time once formed.

Why tight binding means long binding #

That consequence has practical weight in pharmacology. A drug that binds its target protein with a dissociation constant of one nanomolar — the concentration at which the target is half occupied, a site filling like an energy level as the chemical potential rises — has an on-rate that cannot exceed about $10^9$ per molar per second. Its off-rate is the on-rate times the dissociation constant, so at most about one per second: once bound, the drug stays for a second or more. A picomolar binder must stay for tens of minutes; a femtomolar one, for days. The strongest non-covalent binding in biology, between the protein streptavidin and the small molecule biotin, has a complex that lasts for days, and its on-rate is not unusually fast. The strength is almost entirely in the staying.

Drug designers have come to treat residence time as a property worth optimising in its own right, because a drug that stays bound after its concentration in the blood has fallen keeps working. The arithmetic behind that is two lines long: the on-rate has a ceiling set by the viscosity of water, so the only knob left for binding strength is how long the partners stay together.

Unequal partners meet faster #

The cancellation is exact only for equal partners. For unequal ones the product $(D_A + D_B)(R_A + R_B)$ becomes proportional to $2 + \rho + 1/\rho$, where ρ is the ratio of the radii, and it is smallest when ρ is one. A small molecule finding a large one does better than either pair of equals: the large molecule contributes a big target, the small one contributes speed, and the product of the sum of their speeds and the sum of their sizes gains from both. A partner a hundred times smaller than its target meets it twenty-five times as fast as two equal molecules would.

This is the regime of a small ligand finding a protein, or a protein finding a much larger complex, and it is why the fastest biological binding reactions, between small molecules and large enzymes, can exceed the equal-partner limit even before any other effect is counted.

Why enzymes rarely reach it, and some beat it #

An enzyme that turned every encounter with its substrate into product would run at the diffusion limit. The best enzymes come within a factor of ten of it — the ratio of their turnover to their binding constant, the measure of their efficiency, reaches $10^8$ to $10^9$ per molar per second — and they are described as catalytically perfect: faster catalysis would not help, because the substrate cannot arrive any faster. But most fall short, and the first reason is geometry. An enzyme’s active site is a small patch of its surface, and a substrate that arrives anywhere else does not react. If a patch covering one per cent of a sphere reacted only with the one per cent of encounters that land on it, reactions would be a hundred times slower than the limit.

They are not, and the reason is the same property of three-dimensional diffusion that makes the steady rate exist: a molecule that touches the surface in the wrong place does not immediately wander off. It is close, and it makes many further brief contacts nearby before it escapes, so it has several chances to find the patch. Both partners are also rotating as they diffuse, turning different parts of their surfaces towards each other. The combined effect is that a reactive patch catches a far larger share of encounters than its share of the surface.

Some enzymes beat the limit outright, and they do it with charge. Superoxide dismutase and acetylcholinesterase have fields round their active sites that steer oppositely charged substrates towards them, making the effective target larger than the molecule. The diffusion limit assumes the partners feel nothing until they touch; a long-range attraction changes the capture rate the way a charged sphere’s field changes the orbits of the charges near it, and these enzymes run at rates above $10^9$.

The fastest reaction in water #

The fastest bimolecular reaction known in water is the one that makes it: a hydrogen ion meeting a hydroxide ion. Manfred Eigen measured its rate in the 1950s, by disturbing water’s equilibrium with a sudden jump in an electric field or temperature and timing the return, and found about $1.4\times10^{11}$ per molar per second — twenty times the equal-sphere limit. Both of the exceptions just described contribute. The two ions carry opposite charges, which attract them from well beyond contact and enlarge the effective target. And a hydrogen ion in water does not diffuse as a single particle: it hands its extra proton along chains of hydrogen-bonded water molecules, each handover moving the charge by a molecule’s width without any molecule moving far. The ion effectively travels faster than anything its size could diffuse. The measurement, and the relaxation method it was made with, earned Eigen a share of the 1967 Nobel Prize in chemistry, and the reaction remains the reference against which “diffusion-controlled” is judged.

A few thousand traps on a cell #

Howard Berg and Edward Purcell made the patch argument quantitative in 1977, for a cell sensing molecules in its surroundings through receptors scattered over its surface. With $N$ absorbing patches of radius $s$ on a sphere of radius $R$, the cell catches a share $Ns/(Ns + \pi R)$ of what it would catch if its whole surface absorbed. The share is a half when $N = \pi R/s$: for a bacterium-sized cell five micrometres across and receptors a nanometre in size, about sixteen thousand receptors, covering less than a fiftieth of one per cent of the surface. Ten times as many, still covering under a fifth of a per cent, catch ninety-one per cent.

The result goes as the number of receptors times their size, not their area, because what matters is how far a molecule near the surface has to wander to reach one, and that is set by the spacing between receptors compared with the cell’s size. It has a striking consequence for cells: a cell can devote a tiny fraction of its surface to receptors for any one molecule and sense that molecule nearly as well as if it were covered in them, so it can carry receptors for dozens of different signals at once without any of them suffering. Berg and Purcell drew from the same arithmetic the limits on how precisely a bacterium can measure a concentration by counting arrivals, which is the physics under a bacterium’s ability to swim up a gradient of food.

Where the Smoluchowski picture stops #

Continuum diffusion. Everything here treats the partners as diffusing smoothly right up to contact. On the scale of a solvent molecule the motion is jerky, the solvent forms shells round each reactant, and two reactants that meet are held together for a few collisions in a solvent cage before they separate. The cage is why a pair that meets gets several chances to react before diffusing apart, and it makes the diffusion-limited rate an upper bound that real reactions approach from below.

Stokes–Einstein. The cancellation rests on $D \propto 1/R$, which holds well for molecules much larger than the solvent’s molecules and increasingly poorly for molecules comparable to them. Small ions diffuse somewhat faster than Stokes–Einstein predicts, and the universal number is really a band a factor of a few wide.

Dilute, unhindered solution. Inside a cell the medium is crowded with proteins occupying a fifth to a third of the volume, diffusion is slower and not always simply diffusive, and the effective limit is lower. In two dimensions — on a membrane — there is no steady rate at all, as the walk that comes home found, and reaction rates depend logarithmically on the system’s size.

What the profiles do not show #

The profiles are averages over many encounters. A single encounter is a random walk that touches, separates, touches again, and either reacts or wanders off, and the distribution of times between encounters is broad. For reactions involving very few molecules — a single gene switched by a single regulator in a single cell — it is the distribution, not the average rate, that matters, and the time a regulator takes to find its target on a strand of DNA is shortened by a combination of three-dimensional diffusion and one-dimensional sliding along the strand, a mechanism the simple sphere model does not contain.

Still open: how fast is diffusion inside a cell #

The Smoluchowski limit is well tested in simple solvents. Inside living cells it is not simple to apply, because the cytoplasm is crowded, structured and in parts almost gel-like, and measured diffusion coefficients of proteins there are several times lower than in water and depend on the protein’s size in ways Stokes–Einstein does not predict. Whether the cell’s interior behaves as a viscous liquid, a porous solid, or something whose apparent viscosity depends on the size of the thing moving through it, and what that does to the rates of the reactions that run the cell, is measured with single-molecule tracking and argued over case by case.

The habit worth carrying away is to look for a cancellation when a quantity seems as if it ought to depend on size. A diffusion-limited rate is a target size times a speed, and in a viscous liquid speed falls in exact proportion as size grows — so the fastest any reaction in water can run is set by the water, and a cell’s receptors can cover almost none of its surface and still catch almost everything that arrives.