One experiment gives the cosine, the other gives the sine

The phase problem states the loss: a detector records |F| and the phase of F is gone, and the phase is where the structure is. The law that hides handedness states the exact fact that makes half the recovery possible — with real scattering factors I(h) = I(−h) exactly, and with an anomalous scatterer that equality fails by a measurable amount.

Putting the two together recovers the phase itself. That step is what turned crystallography from a method for salt into a method for haemoglobin, and the arithmetic in it is two lines.

Two lines, and each of them is one experiment #

Write the structure as a light part F_A, of unknown phase φ, plus a heavy atom F_H whose position — and therefore whose phase θ — is known, because a Patterson map of the differences finds it.

The first line is the law of cosines applied to a triangle of complex numbers:

$$|F_A + F_H|^2 \;=\; |F_A|^2 + |F_H|^2 + 2|F_A||F_H|\cos(\varphi - \theta).$$

Measuring the intensity with the heavy atom present and again without it gives cos(φ − θ). That is the isomorphous replacement experiment, and it leaves exactly two candidates, φ = θ ± Δ, because a cosine does not know a sign.

The second line is what happens when the heavy atom scatters anomalously. The Friedel pair splits, and

$$I(h) - I(-h) \;=\; 4\,|F_A|\,f''\sin(\varphi - \theta),$$

which gives sin(φ − θ) — the very quantity the first experiment could not supply. Two candidates from a cosine, one sign from a sine, one phase.

The pleasing part is the reversal. Friedel’s law holding exactly is what makes the phase unreachable from one measurement; its breaking is what hands the missing half over.

What is checked, and how far the trip goes #

A small structure is built. Its intensities are computed. The phases are then recovered from those intensities without the structure, by the two lines above, and compared with the truth.

The cosine leaves the truth among its candidates, always. Over every reflection in a window the nearer of the two candidates is the true phase to within 6 × 10⁻¹⁵ — which is the floating-point floor rather than a residual.

The sine picks the right one, always. The chosen candidate agrees with the truth to the same precision, on every reflection. Neither number is fitted; both are computed from intensities the structure produced and from the heavy atom’s position.

Then the recovered phases are put back into a Fourier synthesis, and the peaks of the resulting map are required to sit on the atoms. All four do, each within a grid spacing.

The refusals, which are the point of the figure #

The three panels above are the phase problem stated as a picture, and the two failures matter more than the success.

The wrong branch gives a structure, and it is the wrong one. Taking the other candidate everywhere is not noise: it is a systematic transformation, and it produces the structure inverted through the heavy atom. The map has peaks, they are as sharp as the right map’s, and none of them is where an atom is. A method that produced this and had no way to tell would be worse than useless.

Random phases give a map with nothing in it. Same amplitudes, no structure — which is the standard demonstration that the phases carry the information, made here on the same data as the other two panels rather than on a separate example.

Both refusals had to be sharpened once. The first version picked peaks by taking the four hundred highest cells of the map, and at that generosity every panel contained a cell near every atom, so all three “found” the structure. A peak is a local maximum with its neighbours suppressed, and once that was fixed the recovered map found four atoms of four and the other two found none. A check whose threshold is generous enough tests the threshold rather than the claim.

How small the signal is #

Only the sign of the anomalous difference is used, so with exact data the method works for an arbitrarily small f″ — which is precisely the claim an experiment refuses.

So the differences are measured as a fraction of the mean intensity, and noise of a stated size is added before the sign is read. At f″ = 0.4 on a structure of light atoms the mean Bijvoet difference is 1.3%, and the phases come out right for 98% of reflections at 0.2% noise, 84% at 1%, and 58% at 5% — barely better than a coin. At a larger f″ the differences are 11% and 5% noise still leaves 90% correct.

That is the honest shape of the method. The information is there at any f″; whether it survives the measurement is a different question, and it is the question that kept anomalous phasing from being routine for thirty years after it was understood. What changed was not the arithmetic but synchrotron sources and area detectors — the ability to measure a one per cent difference reliably.

The plot is worth a sentence about what it is not. The recovery obtains its sine by dividing the anomalous difference by a constant, so plotting one against the other would be plotting a number against itself; the horizontal coordinate here is computed from the true phases instead, which the recovery never sees, and the agreement is therefore a comparison of two independent quantities rather than an identity redrawn. The magnitudes span a factor of a hundred and sixteen, and the smallest of them is the one an experiment has to measure reliably before any of this is available at all.

Where the heavy atom’s position comes from #

The whole construction assumes θ is known, which means the heavy atom’s position is known, and that step is the map that needs no phases.

The Patterson function of the differences between the derivative and the native data has peaks at the vectors between heavy atoms and almost nowhere else, because the light structure cancels. With one heavy atom per cell there is one vector — the one from the atom to its own symmetry-related copies — and reading its position off the Harker section is a hand calculation.

So the sequence is: differences → Patterson → heavy atom position → θ → cosine and sine → φ → map. Every step is computed from measured amplitudes, and the only thing put in by hand is which reflections belong to which data set.

What the ambiguity looks like reflection by reflection #

The geometry of the two candidates is worth following, because it says which reflections are phased well and which are not.

The candidates sit at θ ± Δ, where Δ = arccos of the isomorphous term. When the cosine is near ±1 the two candidates nearly coincide and the phase is almost determined by the first experiment alone — but those are the reflections where the derivative and the native differ most, which is where non-isomorphism does most damage. When the cosine is near zero the candidates are ninety degrees either side of θ and the sine is doing all of the work.

So the two experiments are strong in complementary places, which is why the combination is so much better than either. A reflection badly determined by the cosine is well determined by the sine, since sin and cos of the same angle cannot both be small.

That complementarity is the practical argument for MAD over MIR, and it is a fact about a right-angled triangle rather than about chemistry.

Two ambiguities, and they sit at right angles #

The essay above resolves one ambiguity with the other, and it is worth drawing the two separately, because they are not the same shape and the difference decides what a single experiment can do.

The isomorphous ambiguity is a reflection about θ. The cosine fixes |φ − θ| and leaves the sign, so the two candidates are θ + Δ and θ − Δ: mirror images in the heavy atom’s own phase.

The anomalous ambiguity is a reflection about θ + 90°. The Bijvoet difference fixes sin(φ − θ), and a sine is unchanged by replacing its argument with π minus it — so the two candidates are θ + Δ′ and θ + 180° − Δ′, mirror images in the direction perpendicular to the heavy atom’s phase.

Two mirrors at right angles, and each experiment’s pair is the other’s blind spot. That is the geometric content of the complementarity the essay states in terms of strength, and it says something sharper than “they help each other”: the two ambiguities can never coincide, so the intersection of one pair with the other is a single phase for every reflection except where a candidate lands exactly on a mirror.

It also explains what a single anomalous experiment leaves. SAD gives a map and its mirror image about the heavy-atom direction — which, taken over all reflections at once, is a structure and something that is not quite its inversion, and the two are separated in practice by which one has flat solvent regions. That is a criterion about the content of the map rather than about the measurement, which is why it works for a protein and not for a small molecule.

The reflections that have no ambiguity at all #

There is a set of reflections where both mirrors are the same mirror and the phase is not ambiguous but determined, and they carry a load out of proportion to their number.

If a projection of the structure down some axis is centrosymmetric — which happens in every group with a two-fold axis, a mirror or a glide, for the zone perpendicular to it — then the structure factors of that zone are real, and their phases are 0 or π. The heavy atom’s phase θ is 0 or π too, so Δ is 0 or π and both candidates coincide.

The restriction is a fact about the group rather than about the crystal, which is why it is available before anything is known about the structure. An operation taking h to −h makes the structure factor of a reflection its own conjugate up to a fixed phase factor, and 2φ is then pinned modulo a turn — leaving two values half a turn apart, both read off the operation’s translation part.

Those centric reflections are what the heavy-atom parameters are refined against, because for them the isomorphous difference is exact rather than a projection: |F_PH| − |F_P| is ±|F_H| with no cosine in the way. A heavy-atom position that does not predict the centric differences correctly is wrong, and it will still fit the acentric ones tolerably — which is the standard reason a derivative that looked usable turns out not to be.

They also fix the hand. The heavy-atom constellation and its mirror image give identical isomorphous differences, as the Patterson always does, so the choice between them has to be made somewhere else — and the anomalous differences of the acentric reflections are where, since Friedel’s law breaks only for those.

Where the exactness stops #

The arithmetic is exact and the experiment is not. Everything here is computed on a structure with no noise, no absorption, no scaling error and no non-isomorphism, and every one of those is a reason real phasing is hard. What is demonstrated is that the information is present and where it sits; what is not demonstrated is that it survives an experiment.

Non-isomorphism is the omitted difficulty and it is the biggest one. The two lines assume the derivative is the native structure plus a heavy atom and nothing else. A real heavy-atom soak moves the surrounding atoms, and the difference then contains a contribution from the movement as well as from the heavy atom — which is indistinguishable from the signal being measured. Every practical treatment weights the phases by an estimate of how badly this holds.

The scale factor is assumed. Both intensities have to be on the same scale before their difference means anything, and putting them there is itself a fit. The sign of an anomalous difference is robust to a scale error and the magnitude of an isomorphous difference is not, which is one reason the anomalous half of the method has aged better.

And the ambiguity is genuinely two-fold, not approximately so. The two candidates are exact reflections of each other about the heavy atom’s phase. Nothing about the amplitudes distinguishes them, and nothing ever will — the second measurement is not a refinement, it is the missing information.

The same arithmetic, under four names #

The construction appears in the literature under four names that describe four experiments and one calculation.

SIR — single isomorphous replacement — is the cosine alone, with the two-fold ambiguity left standing and resolved by density modification or by luck.

MIR — multiple isomorphous replacement — is the cosine twice with two different heavy atoms, whose candidate pairs intersect in one phase. This is what Perutz used on haemoglobin from 1953, and it is the method that made protein crystallography possible.

SAD — single-wavelength anomalous diffraction — is the sine alone, which leaves a different two-fold ambiguity, resolved in modern practice by the fact that one of the two maps has flatter solvent than the other.

MAD — multiple-wavelength anomalous diffraction — measures at two or three wavelengths near an absorption edge, where f′ and f″ change sharply, so the same atoms provide both the isomorphous and the anomalous differences. That is this essay’s calculation with the two experiments done on one crystal, and it is why a synchrotron with a tunable beamline changed the field.

All four are one calculation because all four supply the same two numbers. SIR and MIR supply cosines; SAD and MAD supply a sine and, at more than one wavelength, a cosine as well. What distinguishes them is not the arithmetic but where the second constraint comes from — a second crystal, or a second wavelength on the same one — and the whole practical history of the field is the story of that difference. A second crystal has to be isomorphous with the first and no crystal quite is; a second wavelength is the same crystal at a different colour, and it is isomorphous with itself.

Who found it, and when #

Bijvoet did the decisive experiment in 1951, using the anomalous scattering of zirconium radiation by rubidium in sodium rubidium tartrate to determine an absolute configuration — settling a convention that had been arbitrary since van 't Hoff, and doing it with the sine above.

Perutz solved the isomorphous half in 1953, after fifteen years, by getting mercury into haemoglobin without destroying the crystal — and the arithmetic he then used is the first line of this essay.

Harker gave the geometrical construction of intersecting circles in 1956, which is what the first figure draws, and Blow and Crick in 1959 gave the probabilistic treatment that turns two intersecting circles with errors into a phase with a figure of merit. Every modern phasing program is a descendant of that paper rather than of the exact construction here.

The exact construction is still the right thing to look at first, because it says what is being estimated. A probability distribution over phases is only interesting once it is clear that the phase is there, in the amplitudes, waiting to be extracted by two measurements that supply a cosine and a sine.

What a map made from recovered phases is, and is not #

The recovered map deserves one paragraph of caution, because it is a better object than the phases that made it and a worse one than a structure.

It is better than its phases because a Fourier synthesis is a sum: an error in one phase spreads over the whole map and is diluted, so a map from phases that are individually mediocre can still show the structure. That is why figures of merit are reported per reflection and maps are judged as a whole.

It is worse than a structure because the peaks are not atoms. They are maxima of a truncated Fourier series, displaced by series-termination ripples and by every unphased reflection, and turning them into coordinates is a fit rather than a reading. In this essay’s map the four peaks sit within a grid spacing of the four atoms, which is as close as a 48 × 48 grid can report; on a real map the step from peaks to coordinates is refinement, and refinement is where the remaining error goes.

Why the heavy atom has to be heavy #

One assumption deserves a sentence of its own, because it is where the method’s practical limits come from.

The isomorphous difference is |F_PH|² − |F_P|², and the term carrying the phase information is 2|F_A||F_H|cos(φ − θ). That term is linear in the heavy atom’s scattering while the intensities themselves go as the square of the whole structure — so as the light structure gets larger the signal falls relative to the measurement it sits in.

That is why proteins need mercury and small molecules do not need anything. A structure of thirty light atoms has a mercury signal of tens of per cent; a structure of three thousand has one of a few per cent, and one of thirty thousand has one that a single crystal cannot supply reliably. The number of heavy atoms has to grow with the structure, which is why large-complex crystallography uses clusters of heavy atoms rather than single ones.

The anomalous route scales the same way and has one advantage: f″ is a property of an atom rather than of a difference between crystals, so it does not suffer non-isomorphism at all. That is the reason MAD displaced MIR once tunable sources existed, and it is a fact about experiments rather than about the arithmetic, which is identical in both.

All of it rests on a measurement of intensities, and it is worth remembering that the intensities carry a symmetry the crystal need not have — which is why the anomalous half of this construction is the only part that reaches information Friedel’s law would otherwise delete.

Where the ladder goes next #

This rung recovers a phase from amplitudes. Two rungs sit above it.

Absolute configuration, which is the same anomalous difference read for a different purpose: not to phase a structure but to decide which of two enantiomers is on the diffractometer. The law that hides handedness sets that up, and the number crystallography quotes for it — the Flack parameter — is a refinement against the Bijvoet differences directly.

Direct methods, which need no heavy atom at all. The phases of a structure of atoms are not independent: triple products of normalised structure factors have a distribution concentrated near zero phase, and for small structures that is enough to bootstrap the whole set. It is the statistical counterpart of this essay’s exact construction, and it works for the same reason the intensity statistics work — the amplitudes carry more than one number at a time.