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My question regards diffraction patterns and resolution limit of protein crystals. I understand that the inner symmetry (i.e. arrangement of the macromolecules) in a protein crystal determines how well it diffracts when hit by an x-ray beam. However, imagine that you collect a dataset (diffraction images) from a protein crystal and notice that the outer spots are located at a resolution of about 2.2Å, how can you explain that this crystal did not diffract to e.g. 1.5Å with more technical (specific) terms other than "it was not well ordered".

I am assume that it has something to do with Braggs law and the Ewald sphere construction? I am also not completely sure why the outer diffraction spots represents the higher resolution and the inner spots represents the lower resolution. I assume this is somehow explained when I understand what defines the resolution limit (?). I am not that strong in math/physics and find the general information on resolution limits (Braggs limit) to be very convoluted, and would therefore appreciate an intuitive explanation of how this works (i.e. what defines the resolution limit), if that is even possible.

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As you wrote in the question, the diffraction limit of the crystal is often explained by saying "it was not well ordered". I googled to find a more elaborate explanation; the closest thing I found was on Proteopedia, but it actually repeats the same:

What Limits Resolution?

In crystals of macromolecules, it is the degree of order in the crystal ("quality of the crystal") that limits the resolution of X-ray crystallography. Resolution is theoretically limited by the wave length of X-rays (on the order of 1 Å), but in practice, the quality of the available crystals determines resolution. More than half of the crystals obtained from various purified proteins are not of "atomic resolution", that is, the disorder it too great to permit determination of molecular structure. Perhaps the greatest challenge facing crystallographers is to obtain a well-ordered crystal that diffracts to "atomic resolution".

The resolution of a diffraction pattern depends on how ordered the crystal is. If it is highly ordered (atoms are in defined positions throughout the crystal and over time), the crystal will diffract to high resolution. The more disorder there is (atoms moving over time, or the content of one unit cell differing from the next), the lower the resolution of the diffraction image.

I think it's intuitive that the disorder limits the resolution. What you see from the diffraction experiment is an average over huge number of unit cell and over time. Disorder blurs the image and limits the resolution. And the resolution limit can mean that spots corresponding to d=2.2Å (lower resolution) are visible while spots corresponding to d=1.5Å (higher resolution) are not. The smaller the distance between aligned planes (in the Bragg's explanation), the more sensitive the reflection is to disorder. The same displacement of an atom causes bigger phase shift if the distance d between planes is smaller.

The question "why the outer diffraction spots represents the higher resolution and the inner spots represents the lower resolution?" is simpler. From the Bragg's law it follows that $\sin \theta \sim 1/d$. The angle increases when d decreases, i.e. when the resolution increases. The outer spots correspond to the higher resolution.

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    $\begingroup$ Thanks, this was clarifying. So intuitively, the outer spots are more "sensitive" to disorder because of the decreasing distances between miller planes. In connection I realized that the out spots are reciprocal to the electron density map, and thus the outer spots also contribute with more details in the final electron density (i.e. more important for high res). So, the outer spots with low "d" are more important for high resolution and also more sensitive to “disorder”, which is why the resolution limit is defined by “how far out” you can collect diffraction spots. $\endgroup$ Commented Feb 21, 2020 at 9:05
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I do not know much about protein crystallography, but if I understand the question correctly the "outer diffraction spots" are reflections at high order (large $n$ in Bragg's law at large angles: $2d \sin \theta = n \lambda$).

The wavelength $\lambda$ is accurately known, so the uncertainty in distances $d$ becomes smaller the larger the angle is.

(I am thinking of powder diffraction here. I am aware that single-crystal diffraction is much more involved, but maybe this way of reasoning gives some intuition.)

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  • $\begingroup$ Thanks!... So one could say that the resolution limit for one particular protein crystal is restricted to the highest order reflection that still gives a lattice plane distance (d) with a significant value? $\endgroup$ Commented Feb 4, 2020 at 15:22
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    $\begingroup$ @CuriousTree: Finding positions with the unit cell is probably more complicated. There is the Patterson function, a Fourier transform of the intensities, and also that can give an intuition that higher-order reflections contribute to increased resolution. But I have not worked with crystallography. $\endgroup$
    – user137289
    Commented Feb 4, 2020 at 15:35
  • $\begingroup$ Sorry, but this is wrong. The resolution (diffraction limit) depends on how well-ordered is the crystal. A protein crystal may diffract up to the resolution of 1Å, or 2Å, or 5Å. It is not about the uncertainty of d, but about the intensity of reflections. $\endgroup$
    – marcin
    Commented Feb 14, 2020 at 22:00

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