A fivefold axis in an ordinary crystal

A poliovirus particle has sixty identical protein subunits arranged with icosahedral symmetry: six fivefold axes, ten threefold, fifteen twofold. It crystallises readily, and the crystals are perfectly ordinary — P2₁3 in the classic case, a cubic group with threefold axes and screws and not one fivefold anything.

The two facts sit together without tension, and holding both at once is the whole subject. The restriction forbids a fivefold axis to a lattice. It says nothing whatever about what sits inside one cell. A crystal is a lattice with contents, the contents can have any symmetry they like, and the only requirement is that the lattice does not have to reproduce it.

That arrangement has a name — non-crystallographic symmetry, universally shortened to NCS — and it is not an approximation, an accident or a paradox. It is exact where it applies and absent where it does not, and both halves can be measured.

What the restriction actually forbids #

It is worth being precise, because the usual phrasing invites the wrong conclusion.

The crystallographic restriction is a statement about integer matrices: an operation that maps a lattice onto itself has an integer matrix in the lattice basis, its trace is therefore an integer, and 2cos(2π/n) is an integer only for n = 1, 2, 3, 4 and 6. Every step of that argument is about the lattice.

An operation that maps one cluster onto itself need not map the lattice onto anything. It is a symmetry of a finite set of atoms, it belongs to a finite group, and the finite groups are the cyclic and dihedral families together with the tetrahedral, octahedral and icosahedral ones, with no restriction on n at all. So a fivefold axis is available to any bounded object and unavailable to any lattice, and a crystal is a lattice of bounded objects.

The only thing the restriction demands is consistency: the space group must map the whole structure onto itself, so an operation of the space group must carry each cluster onto some cluster. It does not have to carry a cluster onto itself.

Where the copies go instead #

If a cell contains sixty copies of a protein and the space group has only twelve operations, the copies cannot all be related by the space group. So they are related by something else — and that something else is the NCS.

The bookkeeping is simple and worth stating in the site’s own vocabulary. The asymmetric unit is the part of the cell the space group repeats to make the whole; its contents are whatever they are. If the space group has order 12 and the cell holds 60 subunits, the asymmetric unit holds 5, and those five are related to one another by operations that are not in the space group. Those operations are the NCS, and they hold only inside the asymmetric unit’s neighbourhood.

That is why the phrase local symmetry is the more descriptive one. The operation is a symmetry of a region rather than of the crystal: apply it and the cluster maps onto itself, apply it far enough away and it maps atoms onto vacuum.

Proper, improper, and the pair of adjectives worth learning #

There are two kinds and the distinction matters for what can be done with them.

Proper NCS — often called closed — is when the local operations form a group: the five subunits around a fivefold axis are related by a cyclic group of order five, and applying the operation five times returns to the start. Icosahedral capsids are the extreme case, with a local group of order sixty.

Improper NCS — open — is when the operation relates two copies without closing: two molecules in the asymmetric unit related by an arbitrary rotation and translation, which composed with itself gives a third position where there is nothing. Nothing about the crystal forbids it, and it is the commoner case in ordinary protein crystallography.

The difference decides what can be assumed. A closed NCS group can be imposed exactly on a structure; an open relation can only be applied once, and applying it twice is a mistake that produces convincing density where no protein is.

The right-hand column is worth reading twice, because it is the restriction arriving from an unfamiliar direction. The usual derivation is algebraic: an integer matrix, an integer trace, five values in range. Here nothing has been said about matrices at all — points have been turned and distances measured — and the answer is the same short list. It comes out as one, two and four rather than one, two, three, four and six only because these copies sit on a square lattice, which is a choice made when the picture was drawn; a hexagonal one would return one, two, three and six. What no lattice returns is five.

Some of the local symmetry can be crystallographic, and the counting says how much #

The sharpest version of the arrangement is when part of the local group is in the space group, and it is the case virus crystallographers arrange for deliberately.

An icosahedral particle has axes of orders 5, 3 and 2. Two of those — the threefold and the twofold — are orders a lattice permits, so a particle can sit with one of its threefold axes along a crystallographic threefold. When it does, that threefold is a symmetry of the crystal as well as of the particle, the space group reproduces one third of the particle for nothing, and the asymmetric unit holds twenty subunits instead of sixty.

The arithmetic is exact and it is worth writing out. Sixty subunits per particle, divided by the order of the local group that has been made crystallographic, gives the number left for the NCS to relate. On a threefold: 60 ÷ 3 = 20 in the asymmetric unit, with a local group of order twenty relating them. On a twofold: thirty. With the particle in a general position: all sixty.

Which arrangement occurs is decided by how the particles pack, which is chemistry rather than group theory. What group theory supplies is the list of possibilities, and the list is short — only the orders a lattice admits can be shared, so a fivefold axis of the particle can never be crystallographic, whatever the packing does. The fivefold is always local, in every crystal of every icosahedral virus that has ever been solved.

The consequence for the measurement is direct. The crystallographic part shows up as systematic absences and Laue symmetry in the usual way; the local part shows up only in the self-rotation function and in the relationships between intensities that averaging exploits. Two halves of one particle’s symmetry, arriving through two different measurements, and the split between them is decided by the restriction.

Finding it, without knowing where anything is #

The measurement that reveals NCS is beautiful and it uses only intensities.

The Patterson function — the map computed from intensities with no phases at all — is the set of vectors between pairs of atoms. If a structure contains a cluster with a fivefold axis, then rotating the cluster by 72° maps it onto itself, so the set of vectors within that cluster maps onto itself too. Rotate the Patterson map by 72° and it overlaps its own copy.

So: compute the map, rotate it by every angle, measure the overlap. The angles at which the overlap peaks are the rotations relating the copies. That is the self-rotation function, and it needs no model, no phases and no idea where anything is — which is why it is one of the first things computed from a new data set.

What the doubling is, and why it matters #

The extra symmetry is worth a paragraph on its own, because it is the same fact this site meets in reciprocal space and it arrives here in real space.

A Patterson map is centrosymmetric always: the vector from atom A to atom B is present along with the vector from B to A, whatever the structure looks like. That is the real-space form of Friedel’s law, which says a diffraction pattern is centrosymmetric whether or not the crystal is. So the self-rotation function of a structure with a fivefold axis has tenfold symmetry, and a structure with no symmetry at all still gives a function symmetric under 180°.

The consequence for practice is a rule of thumb with a reason behind it: subtract the centre before reading the plot. A peak at 180° means nothing on its own; a peak at 72° means a fivefold axis; and a peak at 36° means a fivefold axis reported through a function that added a centre it always adds.

Once it is found, it is worth a great deal #

NCS is not merely a curiosity to be noted. It is the single most useful piece of information a large structure can have, and the reason is a counting argument.

A crystal with n copies of a molecule in its asymmetric unit provides n independent views of the same object, all in the same crystal, all with the same errors of measurement. Averaging them improves the map by roughly √n. More importantly, the relation between the copies is a constraint: the density around one copy must equal the density around the next after a known rotation, which is a huge number of equations relating parts of the map that phases alone would leave independent.

That constraint is strong enough to solve structures outright. The technique — NCS averaging, developed by Michael Rossmann and Gerard Bricogne in the 1970s — starts from a poor map, imposes the local symmetry, and iterates; for viruses, where n can be sixty, it routinely produces interpretable maps from phases that began as noise. The higher the local symmetry, the more the redundancy is worth, so an icosahedral virus — the object whose symmetry the restriction most emphatically forbids to a lattice — is among the easier large structures to solve.

What n copies actually buy #

The claim that n copies give n views is a good slogan and a poor accounting, and the accounting is worth doing because it says when the redundancy is large and when it is nothing at all.

A diffraction measurement gives amplitudes and not phases, so the unknowns are the phases — one per reflection. Non-crystallographic symmetry does not add measurements; it adds constraints among the unknowns. Each NCS operator says that the density at one place equals the density at another, and in reciprocal space that becomes a linear relation among the structure factors, weighted by the operator’s own transform. With n copies there are n − 1 independent relations, and the count that matters is whether those relations outnumber the phases they have to determine.

For a virus they do, overwhelmingly. Sixty copies is fifty-nine relations per reflection, and the system is so overdetermined that phases can be recovered from an almost worthless starting estimate by iterating: average the copies, back-transform, keep the observed amplitudes, average again. That is why virus structures were solved in an era when the same experiment on a protein with one copy would have given nothing.

But the relations are only independent if the copies are in different orientations. An NCS operator whose rotation part is a crystallographic operation of the space group adds no constraint at all, because that relation is already imposed. And an NCS operator that is a pure translation — copies in identical orientation, shifted — adds almost nothing either: its relation multiplies each structure factor by a phase and leaves the amplitudes alone, so what it produces in the data is a systematic pattern of weak and strong reflections rather than a constraint on any individual phase.

So the useful measure is not how many copies there are but how many general rotations relate them. Two copies related by a general rotation are worth more than four related by a crystallographic axis, and a crystallographer choosing between two crystal forms of the same protein is often choosing exactly that.

The averaging is a filter, and it can be run backwards #

There is a hazard in the method that follows from the same arithmetic, and it is the reason NCS averaging is reported with a figure of merit rather than as a step that was performed.

The averaging operation replaces the density in a region by the mean of its n images. Applied to a map, that suppresses whatever differs between the copies and keeps whatever they share — which is the intention when what differs is noise. It is not the intention when what differs is the structure. Copies of a molecule in one asymmetric unit are related by an approximate symmetry, and the places they genuinely differ are often the interesting ones: a bound ligand in one copy and not another, a loop ordered in one environment and disordered in the next. Averaging removes exactly that.

The second hazard is subtler and is a property of any iteration that imposes a constraint the data does not support. If the NCS operators are slightly wrong, averaging with them produces a map that agrees with them better than the crystal does, and the next cycle refines the operators against that map. The procedure converges — to the symmetry it was told to look for. A five-fold axis put in by hand at a position two degrees away from the true one can be made to look convincing, and the check that catches it is the ordinary one: hold back a fraction of the reflections, and ask whether the averaged map predicts amplitudes it never saw.

The mask is the third. Averaging needs a region to average over — the envelope of one copy — and that envelope has to be determined from the map it is meant to improve. A mask drawn too large averages parts of neighbouring molecules together; drawn too small, it leaves the surface unimproved. Neither error announces itself, and both look like a slightly worse map rather than a wrong one.

Where the exactness stops, and it stops sooner than for the rest of this site #

Three limits, and the first is the most important on the page.

Real NCS is approximate. The figures above have an exact local axis because they were built with one. In a real virus the sixty subunits are chemically identical and physically not: each sits in a slightly different environment, the packing differs from one to the next, and the “exact” fivefold axis is exact to a fraction of an ångström rather than to the arithmetic’s precision. So the local operation is a measured symmetry, with a residual and a tolerance — the same situation near-symmetry describes, and unlike almost every other claim on this site.

The NCS operators have to be found before they can be used, and the finding is a fit. The self-rotation function gives angles; the translations come from other means; and the resulting operators are refined against the data. Nothing here is decided by integers, and there is no round trip available.

And a local axis is not a crystallographic one however good it is. However exact the fivefold relation, it does not enter the space group, it produces no systematic absences, and it constrains nothing about the lattice. A structure report gives the space group and the NCS separately for that reason, and running the two together is a category error.

The other resolution, and why it took a decade #

There is a second way for a fivefold axis to appear in something that diffracts sharply, and this site has written about it at length: the object can fail to be periodic.

What Shechtman measured was tenfold diffraction from a solid, and the initial explanations were all attempts to keep the lattice — multiple twinning of ordinary crystals in five orientations was the leading one, and it is exactly the NCS idea pushed as far as it will go. It failed on the evidence, and the correct resolution was that the material had no lattice at all.

The two resolutions are worth holding side by side. In a virus crystal the lattice is real, the fivefold axis is local, and the restriction is untouched. In a quasicrystal the fivefold axis is global, the lattice does not exist, and the restriction never applied. Both are consistent, both were proposed in 1984 for the same observation, and telling them apart took the sharpness of the peaks.

Where the ladder goes next #

The instrument this essay leans on is the self-rotation function, and its parent is the Patterson map — the map computable from intensities alone, whose peaks are interatomic vectors rather than atoms. That map has a great deal more to say than the rotations of a cluster, and it is the next thing worth building.