A map of the atoms that break the law

The map that needs no phases is the one a diffraction experiment can always compute: transform the intensities with every phase set to zero and get the set of vectors between atoms. Its difficulty is its size. A structure of n atoms has $n^2 - n$ vectors between distinct atoms, and solving from the vector set ends by naming the trick that makes the map small enough to read: feed it differences instead of intensities.

The differences in question are the ones that hide handedness when they vanish. Friedel’s law says the reflection at h and the reflection at −h have equal intensity, and it holds exactly when every scattering factor is real. An atom absorbing near its own edge scatters with a phase of its own, its factor gains an imaginary part f″, and the two halves of the pair stop being equal. Only the atoms with an imaginary part contribute to the difference, so a map built from differences shows only them.

The structure the maps are of #

The model has to be one where the difference map says something the ordinary map does not, and that is a constraint on the anomalous atoms: heavy enough to be seen in the difference, light enough not to dominate the sum.

The anomalous pair contributes 4.8 per cent of the total scattering power. That is deliberately small: with two atoms at eight times the light atoms’ weight, as a first attempt had it, the ordinary Patterson finds the same two vectors at its own strongest peaks and the difference map has nothing to add. A heavy atom is not automatically an anomalous one, and the difference map is interesting exactly when the atom it finds is not the one an ordinary map would hand over.

Every number below comes from that structure, computed exactly: structure factors summed atom by atom with complex factors, differences taken, and the map synthesised with every phase set to zero, which is what makes it a Patterson.

Why a difference keeps only the anomalous vectors #

Split the structure factor into the light part and the anomalous part, $F = F_L + F_A$, and give the anomalous part its imaginary component. The two halves of a Friedel pair then differ by

$$\Delta F = |F(h)| - |F(-h)| \approx 2\,\frac{f''}{f}\,|F_A|\,\sin(\varphi_T - \varphi_A),$$

where $\varphi_T$ and $\varphi_A$ are the phases of the whole structure and of the anomalous substructure. Squaring it gives $|F_A|^2$ — the intensity of the substructure alone, which is exactly what a Patterson of the substructure needs — multiplied by a sine squared that changes from reflection to reflection.

So the coefficients are the substructure’s intensities, scaled reflection by reflection by a factor that averages to a half. Transform them and the substructure’s vector set comes out, with the fluctuation of the sine spread over the map as background.

The squaring is not a convenience. A Patterson map is the transform of an intensity, and what makes it computable without phases is that an intensity is what a measurement gives; a map synthesised from the differences themselves would be the transform of a quantity that is sometimes negative and is not an intensity of anything, and it would not be an autocorrelation of any structure. Squaring the difference produces a quantity proportional to $|F_A|^2$, which is the intensity of a structure — the substructure — and that is the whole of why the recipe works. The price is the sine squared that comes with it, and the price is paid in the section below.

The origin peak needs a word, because it does not mean here what it means in an ordinary Patterson. There it is the sum of every atom paired with itself and weighs n, which is how a map announces how many atoms made it. In a difference map it is the sum of the squared differences over every reflection — a measure of how much anomalous signal was measured, not of how many atoms there are. It is the largest thing in the map either way, and the only use made of it here is as the scale against which the real peaks are reported.

Two atoms give two vectors, at (0.45, 0.25) and (0.55, 0.75), which are the difference of the two positions and its negative — a Patterson is centrosymmetric whatever the structure is, so the vectors come in opposite pairs. The whole structure’s map has 14,762 vectors in it. The difference map has two, and that ratio of seven thousand is the entire reason the method exists. Finding two atoms in a cell from two peaks is arithmetic; finding a hundred and twenty-two from fourteen thousand is the problem the phase problem is named after.

An average needs something to average over #

The approximation above is a statement about an average, and that is not a detail. The sine squared averages to a half over many reflections; over a few it is whatever it happens to be, and the map is then the substructure’s vector set multiplied by an unrelated pattern.

At 81, 121 and 169 reflections the two vectors are nowhere near the top of the map — at 121 they rank sixth and fifth, which for a map with two atoms in it is a failure rather than a near miss. At 289 reflections they are first and second after the origin, and from there on more data changes the height by a few per cent and nothing else.

The threshold is not a constant of nature; it depends on how well the light structure randomises $\varphi_T$. The same computation with ten, twenty or forty light atoms instead of a hundred and twenty needs 841 reflections rather than 289 to put the vectors on top. Fewer atoms make a less random set of phases, the sine squared correlates with the substructure’s own intensities, and the average takes longer to arrive. A difference map is a statistical instrument, and the statistic it relies on is the randomness of the rest of the structure.

The size of the signal is not what matters #

The obvious expectation is that a stronger anomalous signal gives a better map. It does not — at least, not in the absence of measurement error.

Computed at three strengths of f″, giving mean Bijvoet differences of 16.4, 4.25 and 1.42 per cent of $|F|$, the difference map’s peaks come out at exactly the same height relative to its origin: 0.47 in all three cases. Every coefficient is multiplied by the same factor, and a Patterson of coefficients scaled by a constant is that Patterson scaled by a constant. The map has no idea how strong the signal was.

Error is where the strength comes back. A relative error of one per cent on each measured magnitude is nothing against differences of 16 per cent and fatal against differences of 1.4 per cent, and the curves show it: plotted against the ratio of error to difference, the three collapse onto one line. The peak is at about half its clean height when the error equals the difference, and the vectors drop out of the map entirely between two and three and a half times it.

That ratio is the number an experiment is designed around. It is why anomalous work waited for synchrotrons and good detectors rather than for a better theory, the same reason the recovery of a phase from an intensity waited: the arithmetic was known and the measurement was not good enough.

Three atoms, six vectors, and where it stops #

Two anomalous atoms is the easy case, and the interesting question is how far the map carries.

Three atoms give six vectors, and the map shows all six within the first six peaks after the origin. Their height has fallen from 0.47 to 0.29. Four atoms give twelve vectors, the height falls again to 0.23, and only seven of the twelve are in the map at all: five have sunk into the background that the fluctuating sine leaves behind.

The arithmetic behind the fall is simple enough to state. The origin peak collects the whole of the anomalous signal; each of the n(n − 1) off-origin vectors collects the part belonging to one ordered pair, and with n atoms of equal weight that is about one part in n of the total. A half, a third, a quarter — which is what the three bars measure. The background does not fall with it, because it comes from the sine’s fluctuation rather than from the substructure, so the peaks and the noise converge from opposite directions and meet at a few atoms.

That is why a real substructure of ten or forty sites is not solved by looking at a difference map. The map is computed, and then handed to the same direct methods that would be hopeless on the whole structure but are comfortable on a few dozen atoms — the difference map’s job being to reduce a structure of thousands of atoms to a problem of tens. In a space group rather than in p1 there is one more piece of help: the vectors between symmetry-related copies fall on the sections symmetry stacks them on, so a search reads a line or a plane of the map rather than the whole of it.

Two maps that are exactly zero #

Between the statistics sit two statements that are exact, and both are checked here at machine precision rather than argued.

With every f″ set to zero the difference map is identically zero — not small, zero, at every point of the map, because Friedel’s law holds term by term in the structure-factor sum. That is the sense in which the map contains only anomalous information: remove the anomalous scattering and there is no map at all, whatever the atoms are doing.

A centrosymmetric structure gives a zero map too, however strong its f″. The largest value anywhere in it is 10⁻²⁸, which is arithmetic noise. With a centre of symmetry every structure factor is real apart from one overall phase, the two halves of a pair have equal magnitude again, and every difference vanishes. So the method needs a structure without a centre — and since whether there is a centre is a statistic rather than something read off a picture, that condition is worth checking before the experiment rather than after. Proteins, being built of one-handed amino acids, cannot crystallise in a centrosymmetric group at all, which is why this method belongs to them.

What the map is for #

A difference map with two peaks in it is not a structure; it is a substructure, and the substructure is a starting point. Its positions give the phases of the anomalous atoms, those phases give an approximation to the phases of everything else, and the map computed with those phases shows enough of the structure to improve them. That is the shape of every route out of the phase problem: find a few atoms exactly, and let them phase the rest.

What makes the substructure worth having is that it phases everything else. The anomalous atoms’ positions give their own structure factors exactly, in magnitude and phase; the measured magnitude of the whole reflection then restricts the total to a circle about that known contribution, and the Bijvoet difference restricts it again. Two restrictions leave a small ambiguity rather than a free choice, and a map computed from the survivors shows enough of the structure to improve them. Every modern phasing pipeline is that loop, and its first step is the map above.

The alternative route, isomorphous replacement, computes the same kind of difference between two crystals rather than between two halves of one pair, and it has the same shape and a harder experiment — two crystals that differ in one atom and in nothing else. Two structures, one Patterson is the reminder that neither route is an inversion: the vector set does not determine the structure, and what these methods do is reduce the number of candidates to something a chemist can choose between.

There is also a limit worth naming. The difference map holds n(n − 1) peaks for n anomalous atoms — two for two, six for three, ninety for ten — so the method scales the way the ordinary Patterson does, just from a much smaller starting number. A protein with forty selenium sites gives 1,560 vectors, which is again a map nobody reads by hand, and the substructure is then solved by the same direct methods that would not work on the whole structure.

What the map cannot settle #

A Patterson map is centrosymmetric, and this one is no exception: its two peaks are a vector and its negative, and they would be exactly the same two peaks for the substructure reflected through the origin. So the map determines the anomalous substructure only up to inversion, and the two choices are not equivalent: one of them is the structure and the other is its mirror image.

That is the same ambiguity two structures, one Patterson is about, arriving at the one step of the method where it does the most damage. Choosing the wrong hand and phasing with it gives a map of the mirror image of the true structure — for a protein, a map full of left-handed helices, which is how the mistake is usually caught. The principled way to decide is to phase both and keep the choice that gives the more interpretable map; the arithmetic will not decide it, because the data the map was built from is invariant under the choice.

It is worth being precise about which invariance this is. Inverting the whole structure changes the sign of every Bijvoet difference, which is the measurement that fixes an absolute configuration; inverting only the substructure, with the light atoms left alone, is not a symmetry of anything and is not what the map is blind to. What the map is blind to is the sign of the vector set it was computed from, which is a property of Patterson maps and not of anomalous scattering.

The checks, and what they refuse #

The refusal that keeps the rest honest is the last. The ordinary Patterson of this structure does not show the anomalous vectors among its strongest peaks — checked, not assumed — so the difference map is doing work that the intensities alone do not do. On a structure with two very heavy atoms that test fails, and the failure is the reason the model here is built the way it is.

Who worked it out #

Johannes Bijvoet used the inequality of the pair to settle an absolute configuration in 1951, which is the measurement the differences were invented for. Using them as Patterson coefficients came shortly after, with Yoshiharu Okaya, Yoshio Saito and Ray Pepinsky, whose $P_s$ function of 1955 is this map; Michael Rossmann’s difference Pattersons of the early 1960s put the same idea to work on the heavy atoms of protein derivatives, and the substructure solution at the start of every modern phasing pipeline is its descendant. The arithmetic above is theirs; what is added here is that the statistical step is measured rather than taken on trust, and that the two exact zeros are checked as zeros.

Where this goes: more atoms, and two wavelengths #

In three dimensions the map is a volume and the peaks are vectors in it, but nothing about the argument changes. What does change the argument is measuring at more than one wavelength: the real part f′ varies near an edge as well, so two wavelengths give a dispersive difference between datasets alongside the Bijvoet difference within each, and the two together separate the substructure’s contribution more sharply than either alone. Whether the ratio found here — error against difference, with the threshold between two and three and a half — is the same for dispersive differences, where the two measurements come from different datasets and their errors are not independent, is the question the next step of this argument would have to compute.