The more flexible a molecule, the more difficult it is to get a crystallization that actually tells you something about the molecule's free-phase arrangement/behavior. Basically (the best) crystallography requires you get a molecule to grow (crystals) in the same conformation.
The hinge in IgG1 for instance appears to be able to rotate from 0 to 180 degrees (although range may be more limited in actual molecules, depending on the "bulk" of what's attached to the hinge). If you somehow manage to grow a crystal of a full IgG you'll just get a snapshot of that at one angle. (That 2011 paper says there was still a single such image for a full-length [human] IgG1 for instance--the one binding HIV (IgG1 b12) found in your 1HZH file.)
According to a 2018 review only 4 full structures of IgG antibodies had been determined; by PDB entry with year of paper publication:
- 1IGT  -- "a mouse IgG2 with 3 hinge disulfide (SS) bonds, while human IgG2 has 4 SS bonds"
- 1IGY  -- a mouse IgG1 with 2 SS bonds
- 1HZH  -- a human IgG1 with 2 SS bonds
- 5DK3  -- a human IgG4; "the hinge SS was more stabilized due to the conformational alteration of S228P mutation"
Of these, only the last one, pembrolizumab, is used therapeutically, as far as I know, but the first one, was (interstingly) a [proposed] veterinary product at one point (Mab231--against canine lymphoma); the USDA suspended its license after it failed in larger trials. The 2nd one is an antibody against phentobarbital. The 3rd/HIV one (1HZH = IgG1 b12) is not the one that's been (recently) approved for treatment as Ibalizumab.
The first of these papers (1997) notes some earlier attempts at full structure crystallography; mostly successful in special cases where there was reduced hinge mobility:
Other intact antibody structures have been studied by X-ray
crystallography (Silverton et al., 1977; Guddat et al., 1993;
Marquart et al., 1980). In two myeloma proteins, the flexible
hinge regions connecting Fab and Fc segments were deleted
(Silverton et al., 1977; Guddat et al., 1993). The molecules
were structurally restrained and, perhaps for this reason,
appeared as compact T-shapes, the angle between Fabs close
to 180°. A third antibody, Kol, had an intact hinge, but the
Fc was so disordered that it was not possible to orient it
with respect to the Fabs (Marquart et al., 1980). The two
Fabs and a portion of the hinge (upper and core) were,
The first two [fully successful] papers (1997-1998) came from the same lab/group. As I'm not expert in this area (or even in biochemsitry) it's hard for me say what was the breakthrough that enabled the full structures IgG crystal analysis in the late 1990s; the papers aren't incredibly explicit on that. There was certainly a heavy amount of computation involved on the X-ray diffraction data, conducted on a Cray C-90 supercomputer at that time; mentioned in these late 1990s papers. And the diffraction experiments were not conducted in the university where the authors were employed but at the Stanford Synchrotron Radiation
Laboratory (SSRL) and Brookhaven National Synchrotron Laboratory, respectively; almost certainly because they used the large synchrotrons available at these labs, as opposed to smaller X-ray sources. There are entire papers on the subtelties of cryportection needed in order to leverage high-energy X-ray sources.
The 2001 short paper in Science on IgG1 b12 is actually devoid of much method details, but the same (Scripps) group has published a longer paper in 2002 in JMB; there they say they (also) ran their diffraction at SSRL (beamline 7-1). No details on computational resources though.
The 2015 paper on IgG4 acknowledges the Canadian Light Source where the diffraction was conducted
on beamline 08ID-1 with synchrotron radiation. Interestingly, this paper also used TROSY (NMR) to confirm packing details and 1D NMR to compare hinge behavior in solution of a serine-to-proline replacement in the hinge (more on NMR below).
The 1997 and (to a lesser extent) the 2015 paper have most details on the computations involved (well, in this group of 4 papers). Basically prior Fc and Fab crystal diffraction "signatures" (ideally for the same antibody) are highly desirable and the software basically does a search trying to locate them in the larger "picture". But this can fail and manual hints based on parts of Fc/Fab (e.g. from similar but not the same antibody) may have to be entered to guide the search etc.
The 2002 paper has some hints why the automated search can fail (beyond the hinge problem) even when the Fab's structure has been previously analyzed separately; when the whole antibody is crystallized asymmetries are observed:
Importantly, all three of these intact antibodies
crystallize with one whole IgG molecule per asymmetric
unit, even though their respective space
groups and crystallization conditions vary. Thus,
although the two halves of the antibody are identical
in sequence, they are not identical in structure.
A (single) crystal structure will not--in itself--let you experimentally verify how much hinge freedom there is. Tomography techniques such as IPET are more sensible for the latter. Resolution is (much) lower, but if just want to capture the angles available in the hinge with something attached, it's far less time consuming. Feed the results into a targeted molecular dynamics (TMD) package together with the crystal structure of Fab & Fc, and you get something like this
As for [3D] NMR (which you also suggested), it cannot be easily used because of the too greater size of full-length antibodies (IgG: 145-160 kDa) vs 40 kDa typical limit for NMR. Basically with 3D NMR, the bigger the molecule, the "blurrier" the image your're going to get. While Wikipedia mentions a couple of proposals/experiments for larger molecules, it seems those didn't really take off in everyday practice.
There are actually some papers that have done spectroscopy NMR on full-length antibodies, but present the results with caveats like:
It should be noted that mAb2 used in this study [...] is an example of an intrinsically stable and soluble antibody. Despite this, at a very high mAb2 concentration the measurable NMR parameters registered quite significant differences as solvent conditions were varied, highlighting the inherent sensitivity of this NMR technique. It can be anticipated that other, less stable mAbs, which require more careful formulation to achieve satisfactory solubility and stability profile, would show even greater variation in NMR measurables.
There are also several papers e.g. Marino et al. (2017),
Brinson et al. (2019) on 2D NMR on MAb, but again for the purpose of determining the higher-order structure (in solution); although the resolution may be a bit better than IPET (but still below crystallography), it requires more complex math.