Hi Gihan, and anyone else who finds this interesting.  Sorry for the
late response:

> how should one determine a point group and the space group of an
> unknown crystal?
> 
> I have a protein crystal with know unit-cell parameters. (these are
> XFEL data so indexing wouldn't give the point and space groups). I
> checked the PDB, but no luck the PDB structures have the different
> space group assigned, no definitive answer hopefully, somebody can
> point me in the right direction 

Space group determination using serial crystallography data is
different to rotation crystallography because you have to start from
the highest possible symmetry and work downwards by finding and
resolving ambiguities, instead of merging in the lowest possible
symmetry and working upwards by looking for possible symmetries.

Just as with any data, the golden rule is that the space group is
only a hypothesis until the structure is solved (and even then...).

Here's a very brief step by step guide.  Start by determining the cell
parameters, which you've done already.

Say the cell parameters look like a hexagonal P lattice.  Proceed for a
while on the assumption that it really is hexagonal P, but keep the
golden rule in mind.  In this case it might be, amongst others,
monoclinic with two axes similar in length and an angle close to 120
degrees.  Use your crystallographic knowledge to spot centering
possibilities, for example a cubic F lattice might look rhombohedral
with angles of 60 degrees (however, the indexing program should spot
these for you).

Merge the snapshots according to the highest point symmetry permissible
by the lattice.  You can look this up in many places including the
symmetry chart distributed with CrystFEL:
https://www.desy.de/~twhite/crystfel/twin-calculator.pdf
It's always the point group with a grey background in the bottom left
corner of the individual table for the lattice type.  For the example
hexagonal P lattice, it's point group 622.

Do the standard tests on the data, particularly twinning tests
including an L-test.  Whenever you see apparently twinned data with
serial crystallography, either the crystals are physically twinned or
the true symmetry is lower and you need to resolve an indexing
ambiguity and merge again.  You will need to try rounds of ambiguity
resolution until the tests are clean and the structure can be solved.

The difficult part is finding your way through the maze of possible
symmetries.  You can attempt a resolution into any subgroup of the
current symmetry, provided that the "ambiguity operator" is just a
rotation (no reflections/inversions).  The CrystFEL table shows the most
obvious subgroups (the ones with the same lattice type).  The
comprehensive map of possibilities can be found in International Tables
A, Fig 10.1.3.2 (in 5th edition).  There are many special cases, and the
find_ambi tool from the CrystFEL extra programs repository can help you
find cases of "accidental" ambiguities due to the particular values of
the lattice parameters:
https://www.desy.de/~twhite/crystfel/programs.html

If there are signs of twinning, try resolving the ambiguity into each
of the subgroups.  If the ambiguity resolves nicely (one way to tell is
by the correlation coefficient graph, which should separate nicely:
http://journals.iucr.org/j/issues/2016/02/00/zd5001/zd5001fig3.html),
try merging the reindexed patterns in the lower symmetry point group,
and check the twinning tests again.

Once you have a merged set of reflections with no apparent twinning,
things are the same as rotation crystallography.  Examine the systematic
absences to see if they suggest any screw axes.  Try molecular
replacement in all the possible space groups.  Consider revisiting the
earlier steps if there are problems.

It's not easy, but it's also not that difficult, just different to
usual.  I would almost go as far as saying it's fun, like cracking a
secret code.  This kind of situation is my personal favourite part of
crystallography (!)

This recent paper embodies a similar workflow in a nice algorithm:
https://journals.iucr.org/d/issues/2018/05/00/rr5155/

Hope that helps,

Tom

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Thomas White <[log in to unmask]> <[log in to unmask]>
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