Lowering the hill

A catalyst changes the road, not the destination — it cannot move the equilibrium, only the time taken to reach it. What that means, why the leverage is exponential, and why enzymes are so much better at it than platinum.

1823 – 191317 min

By the end you should be able to

State what a catalyst does to the forward and reverse barriers, and why K is untouched
Use the Arrhenius exponential to turn a change in barrier height into a change in rate
Explain why the saturation of an enzyme is evidence for a discrete enzyme–substrate complex
Say what a rate enhancement of ten to the seventeen actually means for a real reaction

This subject begins with a consumer product, which is not the usual way round. In 1823, Johann Wolfgang Döbereiner, professor of chemistry at Jena, directed a jet of hydrogen at a small sponge of platinum. The platinum glowed red and lit the gas. No flint, no spark, no flame brought near it. He built it into a table lighter — Döbereiner’s Feuerzeug, a glass jar of acid, a lump of zinc to make hydrogen, and a pinch of platinum — and tens of thousands were sold across Europe. He never patented it, and he died poor. Humphry Davy had seen something similar in 1817 while working on the miners’ safety lamp: a platinum wire held in coal gas and air glows steadily without any flame at all, and goes on glowing. And afterwards the platinum is unchanged. Same mass. Same appearance. Ready to do it again tomorrow, and the day after, indefinitely. That is a scandal in the terms available in 1823. Something has made a reaction happen that was not happening, and has taken no part in it.

By the 1830s the scattered cases were piling up. Kirchhoff had turned starch into sugar with a trace of acid in 1811 — acid that was still all there at the end. Thénard had watched hydrogen peroxide decompose on contact with metals. Davy had his glowing wire. In 1835 Jöns Jacob Berzelius collected them, argued that they were one phenomenon, and gave it a name: catalysis, from the Greek for loosening. The cause, he proposed, was a catalytic force residing in certain substances. Justus von Liebig objected, and he was right to. Naming a force is not explaining anything. You can always account for an unexplained effect by positing a power that produces it, and you have added exactly nothing. It is worth being fair to both of them. Berzelius did something genuinely valuable — noticing that a lamp, a starch paste and a decomposing peroxide belong in the same box is a real discovery, and it is what made the field a field. He also did something empty: supplying a word and treating it as a cause. Both things are true, and the second took most of a century to repair.

A catalyst changes the road, not the destination

A catalyst does not push a reaction along. It offers it a different route. The uncatalysed reaction has to get over one particular pass, and the height of that pass is fixed by the chemistry of the molecules involved. A catalyst opens a second route — usually by binding a reactant, forming an intermediate, and releasing it changed — and the highest point on the new route is lower. Because the height sits inside an exponential, "lower" turns into "faster" at a ferocious rate. A reduction of a tenth of a carbon–carbon bond’s strength buys a factor of fifty. Taking off two thirds of the barrier can buy a factor of a hundred billion. But the two ends of the journey have not moved. The valley you start in is where it was; the valley you end in is where it was. The catalyst has changed the pass, and passes do not have destinations. Everything else in this lesson is a consequence of that sentence, including the thing people most often get wrong about it.

Slide the catalyst in and watch two numbers at once: the rate, which climbs by orders of magnitude, and the equilibrium constant, which does not move at all.

Loading the rate curves…

The forward barrier is measured from the reactant valley to the peak; the reverse barrier from the product valley to the same peak. Lower the peak and both come down by exactly the same amount, so their ratio — which is K — is untouched. The two dotted levels never move, whatever the catalyst does, because a catalyst has no way to change the energy of a molecule it is not part of.

The exponential is where all the leverage lives, and it was written down by Svante Arrhenius in 1889 — from data on the acid inversion of cane sugar, the same reaction Wilhelmy had used to write the first rate law thirty-nine years earlier. k = A exp(−Ea/RT) Two pieces. A counts how often the right sort of encounter happens at all, and how well aligned it is when it does. The exponential is the fraction of those encounters that arrive with enough energy to get over the pass. At 25 °C, RT is about 2.48 kJ/mol — which is a very small quantity next to any chemical barrier, and that is exactly why the exponential is so violent. Dividing a barrier of 75 kJ/mol by 2.48 gives 30, and e⁻³⁰ is about 10⁻¹³. Only one encounter in ten million million has what it takes. So the arithmetic of catalysis is simple and shocking: Every 5.71 kJ/mol you take off the barrier multiplies the rate by ten. (That is ln10 × RT at room temperature.) Take off 10 kJ/mol — less than a hydrogen bond — and the reaction runs 57 times faster. Take off 67, which is roughly what the enzyme catalase does to hydrogen peroxide, and it runs about 5 × 10¹¹ times faster. Nothing else in chemistry converts a modest change into orders of magnitude that cheaply.

You might think

A catalyst pushes the reaction forward, so it increases the yield and shifts the equilibrium in the direction you want.

Actually

It cannot, and the reason is not a technicality — it is a thermodynamic prohibition with teeth. A catalyst works by lowering the transition state, and the transition state sits between the reactants and the products, so lowering it lowers the forward barrier and the reverse barrier by exactly the same amount. The equilibrium constant is the ratio of the two rate constants, and in that ratio the change cancels identically. Now suppose it did not. Suppose some substance could shift an equilibrium and come out unconsumed. Add it, let the mixture settle at the new position, remove it, let the mixture settle back at the old one, and repeat — each cycle harvesting the difference, from a substance you never use up. That is a perpetual motion machine of the second kind, built out of a catalyst. Wilhelm Ostwald made exactly this argument in the 1890s and wrote the restriction into the definition. What a catalyst changes is the time: a reaction with a perfectly favourable equilibrium constant that would take three hundred years now takes an afternoon. Every industrial process that looks like a counterexample has had its equilibrium moved by something else — pressure, temperature, or continuously removing the product — with the catalyst supplying the speed and nothing else.

The experiment

Hydrogenation over finely divided nickel

Paul Sabatier and Jean-Baptiste Senderens · 1897 · Université de Toulouse

The question: Nickel carbonyl had just been discovered — a metal binding a small molecule reversibly. Would nickel do the same with ethylene, giving an isolable nickel–ethylene compound?
The apparatus: Nickel oxide reduced in a stream of hydrogen to give a black, finely divided metal, then ethylene passed over it at a couple of hundred degrees. The intent was preparative: make a compound, isolate it, weigh it.

Theory predicted

A nickel–ethylene complex, analogous to nickel carbonyl, that could be collected and characterised.

They measured

No complex. Ethane came out of the far end of the tube, and went on coming out for as long as ethylene and hydrogen were fed in. The nickel was not consumed, and would hydrogenate almost any carbon–carbon double bond put over it: alkenes, aromatics, oils, and eventually most of the organic chemist’s catalogue.

Why it mattered

Catalysis stopped being a curiosity of platinum sponges and became a general method. It is why margarine exists, and why a great deal of modern synthesis is possible at all. Sabatier shared the 1912 Nobel Prize with Grignard, and left behind the principle that still guides the design of catalysts: the surface must bind the intermediate neither too weakly nor too strongly. Too weak and nothing sticks long enough to react; too strong and the product never leaves and the surface poisons itself. The best catalyst for a reaction sits at the top of that compromise, and plotting activity against binding strength gives a curve shaped like a volcano.

A metal surface catalyses by adsorption. A molecule landing on the metal shares electrons with it; the bonds inside the molecule are weakened in the process, sometimes to the point of breaking altogether. Two fragments held on adjacent sites are then held still, close together, and in the right orientation — which is a large part of what the catalyst is for. In free solution they would have to find each other, and arrive facing the right way, at the same instant. For a hundred years nobody knew which atoms on the surface did the work, and it was tempting to assume all of them did. They do not. In 1925 Hugh Stott Taylor argued that catalysis happens at a small minority of positions — active centres — where the geometry is unusual: a step in the crystal face, a kink in the step, a corner, a defect. That was a hypothesis from indirect evidence, and it took another fifty years and a great deal of vacuum equipment to confirm it directly. Gerhard Ertl’s Nobel Prize in 2007 was for the surface-science methods that finally made individual steps on a single crystal face watchable. The practical consequence is uncomfortable: a catalyst’s performance depends on features that occupy a tiny fraction of its area, which is why catalyst preparation was, and to a real extent still is, a craft. Two batches of the same metal, prepared slightly differently, are not the same catalyst.

Problem

What sixty-seven kilojoules buys

Hydrogen peroxide decomposes on its own with an activation energy of about 75 kJ/mol. In the presence of catalase — the enzyme in your blood, in yeast, and in every aerobic cell — the barrier is about 8 kJ/mol. Assuming the pre-exponential factor is unchanged, by what factor does the rate constant increase at 25 °C?

Uncatalysed barrier: Ea = 75 kJ/mol
With catalase: Ea = 8 kJ/mol
Temperature: T = 298 K
Gas constant: R = 8.314 J mol⁻¹ K⁻¹

The experiment

The kinetics of invertase

Leonor Michaelis and Maud Leonora Menten · 1913 · The Städtisches Krankenhaus am Urban, Berlin

The question: An enzyme speeds up a reaction enormously. Is it a surface that makes collisions more effective, or does it take hold of the substrate as a definite compound?
The apparatus: Invertase splitting cane sugar into glucose and fructose, followed with a polarimeter — Wilhelmy’s instrument, on Wilhelmy’s reaction, sixty-three years later. Two controls did the real work: the acidity was held fixed with a buffer, which had only just become possible after Sørensen introduced the pH scale in 1909, and only the initial rate was used, before enough product had accumulated to interfere.

Theory predicted

If the enzyme merely helps molecules collide, the rate should stay proportional to the sugar concentration however much sugar is present. There is no reason for it to stop rising.

They measured

It stops rising. At low sugar the rate is proportional to concentration; the curve then bends over and flattens onto a horizontal ceiling, beyond which adding sugar changes nothing at all. The ceiling is proportional to how much enzyme is present. The concentration at which the rate reaches half the ceiling is not.

How sure could they be? The pH control is what makes the data usable, and it is the reason this paper is remembered rather than Henri’s. Invertase activity varies sharply with acidity, and the reaction itself changes the pH as it proceeds, so unbuffered runs drift in a way that looks like ordinary scatter and is not. Holding one variable still is often worth more than measuring another one better.

Why it mattered

Saturation is not a detail of the curve — it is the result. A fixed ceiling means a fixed number of places for the substrate to go, each of which must be occupied, used and vacated in turn. That is an argument for a discrete enzyme–substrate complex, made from a polarimeter reading, forty years before anyone could see a protein.

Push the substrate up and watch the curve leave the straight line. Then halve the enzyme and see which of the two numbers moves.

Loading the rate curves…

The red dashed line is what a catalyst with nowhere to bind would do: rate proportional to substrate, for ever. The real curve leaves it and flattens. Halving the enzyme halves the ceiling — the ceiling counts sites — while K_M, the concentration at which the rate is half its maximum, does not move at all, because it is a property of one enzyme molecule rather than of how many there are.

The shape of the curve is the evidence

It is easy to read Michaelis–Menten as a formula to be fitted. It is better read as an argument, and the argument is entirely in the shape. A catalyst that only made collisions more effective could not saturate. More substrate would mean more collisions would mean more product, without limit. The rate would climb along a straight line as far as solubility allowed. A catalyst that has to bind must saturate. There are only so many enzyme molecules. Each takes hold of one substrate, does the chemistry, and releases the product before it can take another. Once nearly every one of them is occupied at any given instant, adding substrate simply lengthens the queue. So the ceiling is a count of sites, and the bend is where the queue begins. And K_M — the concentration at which the rate is half of V_max — is read straight off the graph without fitting anything, which is precisely why it became the number everyone quotes. It is roughly the concentration at which half the enzyme is busy.

Two things the story rounds off

The equation was not theirs. Victor Henri derived it in 1903, ten years earlier, in his doctoral thesis in Paris, and Michaelis and Menten said so plainly in their paper. What they added was experimental control — the buffer, and the use of initial rates — which turned an equation that fitted nobody’s data cleanly into one that fitted theirs beautifully. That is the difference between an equation and a measurement, and it is the reason the names stuck. It is also worth saying that Henri’s priority is not in dispute among people who have read the paper; it is only rounded off in the retelling. Worth knowing about Menten herself: she was Canadian, one of the first Canadian women to earn a medical doctorate, took a year in Berlin to do this work because Canadian institutions would not let a woman do research, and spent most of her career as a pathologist in Pittsburgh, where she co-invented the azo-dye coupling method still used to stain enzymes in tissue. And the Michaelis constant is not a binding constant. It is usually taught as one, and under one special assumption it is. In general it is not. Briggs and Haldane showed in 1925 that if you drop the assumption that the enzyme–substrate complex is in equilibrium with free enzyme, and instead just require its concentration to be steady, then K_M = (k₋₁ + k_cat) / k₁ — which contains the rate at which the complex turns over, not only the rate at which it falls apart. An enzyme that binds its substrate weakly but processes it very fast can have a small K_M. Reading K_M as affinity is safe only when the chemistry is slow compared with the unbinding, and that is an assumption to be checked rather than assumed.

So why is an enzyme so much better than a platinum sponge? Not because it lowers the barrier by some ordinary amount very well. Because of what it binds. Linus Pauling put it in a sentence in 1948. An enzyme is not shaped to fit its substrate. It is shaped to fit the transition state — the strained, half-broken, fleeting arrangement at the top of the pass. A pocket that grips the top of the hill more tightly than it grips the bottom is, by that fact alone, pulling the top of the hill down. The tighter the preference, the lower the pass. That is why the numbers are so hard to take seriously: Catalase processes about 4 × 10⁷ molecules of hydrogen peroxide per enzyme molecule per second. Forty million. It is close to the limit set by how fast substrate can diffuse to it, which means the chemistry is no longer the slow part — finding the enzyme is. Orotidine 5′-monophosphate decarboxylase, an ordinary enzyme in the pathway that makes your pyrimidines, accelerates a reaction whose uncatalysed half-life Radzicka and Wolfenden measured, in 1995, at 78 million years. With the enzyme it takes about 18 milliseconds. That is a rate enhancement of about 10¹⁷, achieved by a protein with no metal, no cofactor, and nothing in it but the ordinary amino acids. A platinum surface offers a site. An enzyme offers a site that was built, by selection, to be complementary to a shape that exists for a hundred femtoseconds.

I think that enzymes are molecules that are complementary in structure to the activated complexes of the reactions that they catalyze, that is, to the molecular configuration that is intermediate between the reacting substances and the products of reaction for these catalyzed processes.
Linus Pauling"Chemical Achievement and Hope for the Future", American Scientist, 1948. It was a conjecture at the time — no enzyme structure had been solved, and would not be for another eleven years — and it has held up better than almost anything else said about enzymes in that decade.

1811Kirchhoff turns starch into sugar with a trace of acid, and gets the acid back.
1817Davy finds a platinum wire glowing indefinitely in coal gas and air, with no flame.
1823Döbereiner sells a lighter that works on a sponge of platinum. Tens of thousands are made; he patents nothing.
1835Berzelius names the phenomenon catalysis, and attributes it to a catalytic force. Liebig objects that this explains nothing.
1889Arrhenius writes k = A exp(−Ea/RT), from data on the acid inversion of cane sugar.
1894Ostwald defines a catalyst by what it does to rate — and argues that it cannot alter the equilibrium, on pain of perpetual motion.
1897Sabatier and Senderens fail to make a nickel–ethylene compound and discover catalytic hydrogenation instead.
1903Victor Henri derives the saturation equation in his Paris thesis.
1913Michaelis and Menten measure it properly, with a buffer and initial rates, on invertase.
1925Briggs and Haldane replace the equilibrium assumption with a steady state. Taylor proposes that catalysis happens at a few active centres, not across the whole surface.
1948Pauling: an enzyme is complementary to the transition state, not to the substrate.
1995Radzicka and Wolfenden measure an uncatalysed half-life of 78 million years, and an enzyme that beats it by 10¹⁷.

Where this leaves us

A catalyst offers a different road. It lowers the pass, in both directions equally, and because the pass sits inside an exponential, a modest reduction buys an absurd increase in speed. It cannot move the destination, and anything that appears to has had its equilibrium moved by something else. Enzymes do it best because they are built to bind the top of the hill rather than the bottom — a pocket complementary to an arrangement of atoms that exists for a hundred femtoseconds, produced by selection, made of nothing but amino acids, and good for a factor of 10¹⁷. And now notice what every argument in this act has had in common. Heats measured in a calorimeter. Positions of equilibria measured by analysis. Orders measured with a clock. A ratio of two, measured with a Geiger counter and a polarimeter. A curve that bends, measured with a polarimeter again. Every one of them is an inference about the arrangement of atoms in space, made from a measurement on bulk matter. Not one of them looked at a molecule. Pauling could say that an enzyme is complementary to a transition state. He could not show you the pocket. Nobody could show you a pocket, or a bond length, or which of two proposed structures was the real one, except by argument from what a flask did. That is about to end, and it ends the way most things on this site end: with a photographic plate. Shine something with a wavelength the size of an atomic spacing at a crystal, and the spots that come out are not a picture of the crystal. They are something better — a set of numbers from which the coordinates can be calculated.