Everyone knows that there are 230 space groups, and these belong to one
of 32 point groups, which, in turn, belong to one of the 14 Bravais
lattices, and 7 crystal "systems": triclinic, monoclinic, orthorhombic,
tetragonal, hexagonal, rhombohedral and cubic.
Or are there? If you look in ${CLIBD}/symop.lib of your nearest CCP4
Suite install, you will find not 230 but 266 entries for "space groups",
and 43 different kinds of point groups. And those so-called
"rhombohedral" systems can apparently be represented as hexagonal, so
maybe there are only six crystal systems?
Blasphemy! (I can almost hear the purists now) But, the point I am
trying to make here is that there is a disconnect between the
traditional way that crystallography is taught (aka Chapter 1: crystal
symmetry) and the pragmatic practice of crystallography (aka "what
MOSFLM is doing"). It is ironic really that the first thing you must
decide for a new crystal is its "space group" when in reality it is the
last thing you will ever be certain about it. Probably one of the most
common examples of this is the P2221 and P21212 space groups.
Technically, P2122, P2212, etc are NOT space groups! However, given
that orthorhombic unit cells are traditionally sorted a<b<c, simply
giving such a unit cell with the space group P2221 is not enough
information to be sure which axis is the "screwy" one. Also, I'm sure
many of you have noticed that for any trigonal/hexagonal crystal there
is always a C222 cell that comes up in autoindexing? This is because
you can always index a trigonal lattice along a "diagonal" and that
makes it look like centered orthorhombic. But, if you try going with
that C222 choice you find that it doesn't merge ... most of the time.
The fact of the matter is that all autoindexing algorithms give you is a
unit cell, and that is just six numbers. The cell dimensions generally
allow you to EXCLUDE a great many symmetry operations, but they can
never really INFER symmetry. Except, of course, in the special case
where all three angles of the reduced cell are not 90 (or 60) degrees,
then the only possible space group is P1. On the other hand, it is
perfectly possible to have P1 symmetry with all three cell edges the
same length and all angles 90 degrees. It just isn't very "likely" (in
the "posterior probability" sense). This is why MOSFLM and other
autoindexing programs pick the highest-symmetry lattice and give you a
"space group" consistent with that lattice, even though there are plenty
of other possibilities. This is why you should always take the "space
group" that comes out of autoindexing with a grain of salt. Do NOT make
the mistake of classifying your crystals by the result of autoindexing
alone!
Something similar is true for point groups. A high Rsym for a given
symmetry operator (like you will see in the output of "pointless") means
that there is NO WAY that the given symmetry operation is part of the
space group. A low Rsym, however, does not mean that you have a given
symmetry. Could always be some kind of "twinning" or
nearly-crystallographic NCS (NCNCS?). Twinning is relatively rare, and
gets increasingly rare as you get into the non-merohedral stuff, but it
is always a possibility. Yes, intensity statistics can tell you
something is twinned, but if you have just the right mixture of twinning
and pseudotranslation, then the twinning can go undetected. So, in
general, you can always have _less_ symmetry than you think, but proving
the existence of a symmetry operation is hard.
Space groups, or narrowing down the screw vs rotation nature of various
axes generally requires phasing and looking at a map. The one with
right-handed alpha helices is the correct one. Yes, there are plenty of
"tricks" like systematic absences, native Pattersons and the like but
there are a lot of false positives and false negatives possible with
each of these. In fact, you tend to throw out more "rejects" in scaling
than you ever have observations of systematic absences, so why trust
those "absent" spots so much? In fact, sometimes you need to even go
all the way to the end of refinement to settle the space group. It is
possible to get "stuck" with R/Rfree too high because the crystal very
slightly violates the symmetry you think it has. ("NCNCS" again)
Whatever you do, don't forget to try all the possible P2122-like space
groups if you are searching for heavy atoms or doing MR with a primitive
orthorhombic crystal. Far too many people have missed solving their
structure because they didn't know to do this! Fortunately, modern
computers tend to have 8 or so CPUs in them, and there are never more
than 8 space groups possible on any given point group. So, you might as
well launch 8 parallel MR or heavy-atom site-finding jobs in different
space groups, since it will take just as long to run 8 jobs as it will
take to do only one. Well, okay, some of the non-protein ones have more
than 8 choices for a given point group, but I don't generally care about
those.
One important pitfall to NOT get stuck in is dropping a real symmetry
operator and replacing it with a twin operator. This will always lower
R/Rfree because you compress the dynamic range of your data. Best
"control" I can think of is to take an operator you don't think is
twinning and "twinify" that one to see what kind of R drop you expect
from doing the "wrong thing". If you get the same drop for all symmetry
operators (and if the L-test does not say "twinning") then you will have
a hard time defending your hypothesis that the crystal was twinned.
Another unfortunate pitfall is exchanging an actual crystallographic
operator for "NCS". This sounds innocuous at first, because there
really isn't a difference between these two ways to represent the same
symmetry operation. In fact, many frustrated crystallographers at one
time or another have asked the obvious question: why can't we just do
everything in P1? The problem arises when you start thinking about
choosing your "random" Rfree set. For example, you could take a crystal
that actually belongs to P212121 and "drop the symmetry" to P21 (with a
90-degree beta angle) and then apply an "NCS" operator between the
monomers in your new "asymmetric unit". Problem is, if you just pick a
"random" Rfree set in this P21 cell, then pretty much every hkl in your
"free" set will have an "NCS symmetry mate" in the working set. All you
have to do is turn up the x-ray weight and voila! You have a reasonably
low Rfree and R-Rfree gap for what could be a completely wrong structure.
Other space groups have similar traps, and there is no good tabulation
of all the ways a "random" Rfree set can go wrong if you are applying
NCS or de-twinning operators. This is because there are actually a
large number of ways that a given "free" reflection can be "correlated"
to one in the working set. Thin shells don't always do the trick. In
fact, the mere fact that protein crystals are ~50% solvent actually
creates "local" correlations in reciprocal space. This is why solvent
flattening works, but it also produces some "bias" in the Rfree. How
much bias? It's actually hard to say. I know of a few people who have
written programs for picking free-R flags in an "unbiased" way, but as
far as I can tell nobody is distributing such programs as "software",
perhaps due to fear that some clever idiot will find a way to abuse it.
Personally, I try to stay away from NCS, unless I'm really sure that
there is nothing "funny" going on in the crystal symmetry.
-James Holton
MAD Scientist
On 11/13/2012 1:55 AM, vincent Chaptal wrote:
Dear all,
I am not sure I understand point groups and relations between groups
and subgroups anymore, and would appreciate some guidance.
I was under the impression that all point groups were related to an
original P1 cell, and that by applying specific lattice symmetries,
one could "get" higher point groups. Thus, if one knows the symmetry
operators, one could jump from one point group to another. Inspection
of the reflections can then determine the "real" point group and space
group.
At least that's what I thought Mosflm was doing? Am I correct?
P1 +(symm-opp)>C2 + (symm-opp2)>P3
same P1 +(symm-opp3)> P2 + (symm-opp4)>P222 ....
If that's the case, could someone point to me where to find these
symmetry opperators (International tables?), because it's not obvious
to me.
Or are these relations between groups and subgroups only true for
certain crystals where the cell parameters are specific, and allows a
symmetry operator to generate a higher symmetry point group?
Thank you for your help.
vincent
