Where symmetry stacks the vectors

A Patterson map of a structure with a hundred atoms has nearly ten thousand peaks in a cell that holds a hundred. Read as a map it is a hill with texture on it, and no individual vector can be picked out.

Symmetry changes that, and it does so in a way that is exactly computable. If a crystal has a two-fold screw axis, then every atom has an image under it, and the vector between an atom and its own image is not free: it is confined to a plane. All of the thousands of such vectors — one for every atom in the structure — land on that one plane, and searching a plane is a completely different task from searching a volume.

Those planes are Harker sections, they were pointed out by David Harker in 1936, and they are the reason the heavy-atom method worked at all in an era of hand computation.

Why an operation confines its vectors #

The derivation is three lines and it is entirely mechanical.

An operation of a space group sends a point x to Mx + t. The vector between the atom and its image is

$$\mathbf{u} = \mathbf{x} - (M\mathbf{x} + \mathbf{t}) = (I - M)\mathbf{x} - \mathbf{t}$$

As x ranges over the cell, u ranges over the image of the matrix I − M, shifted by −t. So the question is whether I − M is invertible.

If it is, the image is everything, the vectors fill the map, and the operation confines nothing. If it is singular, the image is a plane or a line, and every vector produced by that operation lands on it, whatever the structure is. That is the whole of it: a Harker section is the image of I − M, and its position is set by the translation part.

The site’s habit is to compute rather than to inspect, so the section is found as the left null space of I − M in integers — the row vectors f with f(I − M) = 0. Each such row gives one linear condition f·u = −f·t, and the conditions cut out the section exactly. No geometry, no cases, no table of space groups.

Which operations do it, and which do not #

The test sorts a space group’s operations into two kinds, and the sorting has a satisfying reading.

Rotations, screws, mirrors and glides confine. A rotation about an axis leaves that axis fixed, so I − M kills the axis direction and the vectors lie in the plane perpendicular to it. A mirror leaves its own plane fixed, so I − M kills two directions and the vectors lie along a line.

Inversions and general rotations do not. For an inversion, M = −I, so I − M = 2I is invertible: the vector from x to its image is 2x − t, which can be anywhere at all. That is not a defect — it is why an inversion centre gives no Harker section and why a centrosymmetric structure gets no help from its centre in the Patterson map, having to make do with its screws and glides.

Two coordinates for free #

The reason a section is worth so much is that reading a peak off it gives coordinates, not merely a distance.

Take P2₁ again. An atom at (x, y, z) has an image at (−x, y + ½, −z), so the vector between them is (2x, ½, 2z). A peak on the section v = ½ at position (U, W) therefore says x = U/2 and z = W/2 immediately. Two of the three coordinates, from one peak, with no model and no phases.

The third coordinate is not determined, and cannot be: the group has no operation that fixes y, so shifting the whole structure along b changes nothing observable. That is an origin choice rather than a piece of missing information, and the convention is to set y = 0 for the heavy atom and let everything else be measured against it.

In P2₁2₁2₁ the three sections give the three coordinates twice over, which is a genuine consistency check: the value of x read off one section must agree with the value read off another, and a disagreement means the peak was not what it was taken for.

Sections of higher order, and why a fourfold gives three #

A screw of order n has n − 1 non-identity powers, each of which is its own operation with its own translation — so it produces n − 1 sections rather than one, and the pattern of where they sit is the pattern of the screw’s pitch.

P4₁ is the clean example. Its quarter-turn advances by a quarter of a cell, so the vectors it relates lie on w = ¼. Its square is a half-turn advancing by a half, so those vectors lie on w = ½. Its cube advances by three quarters. Three sections at three heights, from one axis, and each carries a different subset of the vectors.

That has a practical consequence worth naming: the sections of a high-order screw are closer together than a low-order one’s, so their contents overlap more in a map of finite resolution. A fourfold screw gives more information in principle and can be harder to read in practice, which is a trade this subject makes repeatedly and rarely mentions.

Eight answers to one peak, and why none of them is wrong #

The arithmetic that gives two coordinates for free gives them with a doubling in it, and the doubling is not free.

A peak on P2₁’s section sits at (U, W) and says 2x = U and 2z = W — as fractional coordinates, which is to say modulo one. So x is U/2 or U/2 + ½, and likewise z. One peak, four positions. In P2₁2₁2₁ the same doubling runs in all three directions and the single heavy atom has eight candidate positions.

The reason none of them is wrong is that the eight are related by the shifts of a half along each axis, and those are exactly the alternative origins the group permits: P2₁2₁2₁’s symmetry elements are unchanged by moving the origin half a cell along any axis, so the eight solutions are one structure described eight ways. Choosing between them is choosing an origin and nothing more, and the convention is simply to take the first.

The ambiguity becomes real the moment there is a second site. Two heavy atoms are only useful if they are referred to the same origin, and the section that finds the second one offers its own eight positions with no indication of which of them belongs with the first. The vector between the two sites settles it — it appears in the map as an ordinary non-Harker peak, and only one of the eight pairings puts a peak where the map has one. So the second site costs a cross-check the first did not.

And the whole set has a mirror image the map cannot see. A Patterson is centrosymmetric whatever the structure is, so a constellation of heavy atoms and its inversion through the origin produce identical maps. With one site that is another origin choice; with two or more it is a genuine ambiguity between a structure and its other hand, carried forward into two mirror-image phase sets and two mirror-image electron-density maps. Nothing in the intensities decides it, because Friedel’s law hides exactly that. What decides it is anomalous scattering, which breaks the law, or the crystallographer noticing that one of the two maps has left-handed α-helices in it.

Sharpening, and the peak that is always in the way #

A section is a plane through a map, and how readable it is depends on how sharp the map’s peaks are — which is a property of the coefficients rather than of the symmetry.

A Patterson peak is the convolution of two atoms’ electron densities, so it is broader than either atom, and at the resolution a protein crystal gives, neighbouring peaks on a section merge into a ridge. The standard repair is to sharpen: divide each intensity by the mean intensity at its own scattering angle, which replaces the real atoms by point atoms and narrows every peak. The coefficients are then the normalised |E|² rather than |F|², and the same substitution is what makes a Patterson section of a difference map legible at all.

Sharpening is paid for twice. It multiplies up the weak high-angle terms, which are the ones measured worst, so noise rises along with resolution. And a sharpened series truncated at finite resolution rings: every peak acquires a set of ripples around it, and a ripple crest sitting on a Harker section is indistinguishable, to a peak search, from a small real peak.

The origin peak makes both worse. Every atom contributes a vector to itself, so the map’s tallest feature by a wide margin sits at u = 0, and its ripples reach across the whole cell. Subtracting the mean of |E|² from the coefficients removes it — the map is then computed on |E|² − 1 — and that single subtraction does more for the readability of a Harker section than any amount of care with the peak search afterwards.

None of this is symmetry, and that is why it sits here rather than in the derivation. Which plane the vectors lie on is decided by an integer matrix and is exact. How tall a peak on that plane has to be before it is worth believing is decided by the coefficients, the resolution and the errors of measurement, and every one of those is a property of the experiment.

What a section costs as well as gives #

Two hazards, and both were well known by 1950.

Every vector on a section is a candidate and most are accidents. A section is a plane cut through a map that is dense everywhere, so it carries ordinary peaks — vectors between unrelated atoms that happen to lie in that plane — along with the ones the symmetry put there. A tall peak on a Harker section is evidence of a heavy atom; it is not proof, and the standard discipline is to check that the position it implies is consistent with the other sections and with the non-Harker peaks the atom would also produce.

And a special position produces a peak that means nothing. An atom sitting on a symmetry element is its own image, so its vector to that image is zero and contributes to the origin peak rather than to the section. Worse, an atom on a special position has fewer images than a general one, so the expected peaks are missing — and a search that assumes a general position looks for something that is not there.

The plane case, where a section is a line #

The same argument runs in two dimensions with one dimension less to spare, and it is worth drawing because the whole calculation fits in a picture.

The plane case also makes a point about what the condition is. The line is not where the glide is; it is where the vectors between glide-related atoms are. Those are different objects in different spaces, and confusing a Patterson map with a picture of the structure is the commonest mistake made with one — the map is a map of vectors, and a peak in it is a separation rather than a position.

The difference map, which is where this is actually used #

The map a protein crystallographer reads is not the Patterson of the structure but the Patterson of a difference, and the distinction is what makes the method work on something with ten thousand atoms.

Prepare two crystals: the native protein, and the same protein with a heavy atom bound at one site. Measure both. Subtract the intensities. The difference is dominated by the heavy atoms — everything else is common to the two and cancels to first order — so the Patterson map of the difference is, approximately, the Patterson map of a structure containing only the heavy atoms.

That structure has two or three atoms rather than ten thousand, so its map has a handful of peaks rather than a hundred million, and its Harker sections can be read directly. The whole apparatus above is applied not to the protein but to the two or three atoms bolted onto it, and their positions are what phase everything else.

The approximation in “cancels to first order” is where the difficulty lives: the two crystals must be genuinely isomorphous, the binding must be at one site rather than smeared over several, and the intensity differences are small compared with the errors of measurement. Every one of those is a practical problem rather than a symmetry one, and none of them is visible in the arithmetic on this page.

Harker, and the era this belongs to #

David Harker published the sections in 1936, two years after Patterson’s function appeared, and the timing is not accidental: the function was new, everybody could see it was the only map a measurement gave, and the problem of the moment was how to extract anything from it.

The context is worth remembering, because it explains why the idea mattered so much. A Fourier synthesis in the 1930s was done by hand, with Beevers–Lipson strips — cards printed with the values of cosine terms, added up in rows. A three-dimensional map was out of the question for most laboratories, and a section was affordable: a two-dimensional array of numbers on a chosen plane. So Harker sections were not merely a convenience for interpretation, they were the only part of a three-dimensional map many people could compute at all.

The technique’s second life came with proteins. Isomorphous replacement needs the positions of the heavy atoms before it can phase anything, and those positions come from a difference Patterson map read on its Harker sections. That is how myoglobin and haemoglobin were solved in the 1950s, and it is still how heavy-atom sites are found when a structure resists everything easier.

Where the exactness stops #

Three limits.

The sections are exact and their contents are not. Which plane the vectors lie on is integer arithmetic with no tolerance in it. Whether a given peak on that plane is a heavy-atom vector is a judgement about intensities, made against a background of accidental peaks and truncation ripple, and it is the part of the method that requires experience rather than a computation.

A section is only useful if the group has one. P1 has none. P1̅ has none. A structure in either gets no help of this kind, and its Patterson map has to be interpreted whole — which is one of several reasons triclinic structures were disproportionately hard before direct methods.

A section is a plane of the Patterson cell, not of the crystal. Nothing sits on it; no atom has coordinates on it; it is a locus in vector space. A reader who takes the section u = ½ for a plane through the structure at u = ½ will look for atoms there and find the question meaningless, and the confusion is common enough that the Tables are careful to use different letters — u, v, w for the Patterson and x, y, z for the structure.

And the whole apparatus assumes the space group is known. The sections are computed from the operations, so a mis-assigned group puts them in the wrong places, and a search on a wrong section finds nothing and says nothing about why. Assigning the group first, from the absences and the Laue class, is not optional.

Where the ladder goes next #

The Patterson map’s other use is comparative rather than interpretive: rotate it, lay it on itself, and read off the rotations that map the vector set onto itself. That finds the symmetry a crystal has locally — the axes relating copies of a molecule within one asymmetric unit — and this site has already used it once, to find a fivefold axis in a crystal that cannot have one.