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(This is in the context of macromolecular X-ray crystallography.)

From Lamb et al. 2015:

Model bias is the result of how maps are calculated: because the phase estimates [of the structure factor] for the calculation come from the model, maps will tend to show electron density for an atom in the model whether it is truly there or not. A simple omit map is normally a difference (Fo − Fc) map calculated after omitting specific atoms from a model, like a li- gand or a functionally important loop.

From this the purpose of an omit map is rather clear.

But what I don't understand is when omit maps are used w.r.t. the source of the phases. From what I can tell, if the phases have been determined experimentally (e.g., using MAD), there cannot be any extraneous electron densities in the atomic structure model. This drives me to the conclusion that omit maps are used when the source of the phases is molecular replacement (I presume this is what the "model" alludes to).

Is this correct? Are omit maps only used with molecular replacement?

Edit: I am also confused with regards to how an omit map actually works. If the phases are what determine the final electron density irrespective of the amplitudes used, how does deleting a part of the atomic model, getting the phases from that and recalculating a new map (with these phases and the experimental amplitudes) actually help?

Edit 2 (from comments): I got the impression that if we don't have the phases (say for a part of a model), it just wouldn't get built even if we have the amplitudes for that part, right? Or is it that in this case (of omitting a part), there would be an electron density (because the amplitudes are still there), just that it would be featureless?!

I would be very glad to receive some clarifications and/or pointers.

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    $\begingroup$ What kind of macromolecular system is this about? And I'm confused about your "phases". Phases of a signal/wavefunction, and/or amorphous/crystalline phases? $\endgroup$
    – Karl
    Oct 3 '20 at 10:36
  • $\begingroup$ Biomolecular (protein) crystals, I guess. And by phases I was referring to the relative phases of the the diffracted X-rays that together make up the diffraction pattern from the crystal (so I guess these are signal phases then?). I apologize if my terminology is confusing, I am new to this (+ information overload, so I am confused myself). $\endgroup$
    – Dunois
    Oct 3 '20 at 10:41
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[OP] ...if we don't have the phases (say for a part of a model), it just wouldn't get built even if we have the amplitudes for that part, right? Or is it that in this case (of omitting a part), there would be an electron density (because the amplitudes are still there), just that it would be featureless

For each reflection, the structure factor $F_{hkl}$ gives the magnitude and phase resulting from interference of all atoms in the unit cell (see e.g. this slide show). This is unlike other methods (like NMR) where signals originate from given parts of a structure, i.e. can be assigned to an atom or groups of atoms. In crystallography, every atom contributes to a given diffraction spot. On the other hand, a given atom contributes to every structure factor in the data set. In this respect, crystallography is a "holistic" method.

If you are familiar with the 3D fourier transform and the math behind it, you can see this immediately:

$$ F_\vec{h} = \int_{cell} \rho(\vec{r}) e^{2 \pi i \vec{r} \vec{h}} d \vec{r}$$

In the formula, $ F_\vec{h}$ refers to complex structure factors (so a combination of amplitudes and phase) in reciprocal space indexed by (hkl) or $\vec{h}$. On the other hand, $\rho(\vec{r})$ refers to the electron density at a location x,y,z or $\vec{r}$ in the unit cell, i.e. in direct space. So the electron density (and the coordinates) "live" in direct space, and the diffraction pattern and calculated phases of the structure factors "live" in reciprocal space (see e.g. this slide).

Omit map

The omit map will show features even in regions where no model has been built yet, or where the model is omitted from calculating phases (as long as the remainder of the model is decent and the diffraction data is decent). For a pictorial demonstration, see Kevin Cowtan's cat and Manx enter image description here

In this picture, a cat was used to calculate structure factor amplitudes, and a Manx (cat without tail) was used to calculate structure factor phases. Then, they were combined and Fourier-transformed. The resulting (2Fo-Fc) electron density shows the tail at lower signal strength than the remainder of the cat, and there is some noise. The difference between the perfect image of a cat and the observed image is caused by the phases lacking the information of the tail. (See this slide for the entire pictorial explanation.)

Use of omit maps

[OP] But what I don't understand is when omit maps are used w.r.t. the source of the phases.

To reiterate the other answer, omit maps may be used whenever the phases of the structure factor are based on an imperfect model of the structure. If parts of the electron density are difficult to interpret, the model in this part is left out, and a new density is calculated. This density is not biased by a potentially incorrect model in this area, so it might help to correctly build that part. (Here is a slide on model bias.)

With the computer power available nowadays, you can also omit every part of the structure in sequence and piece back together the entire electron density (if you are not sure which part of the model is incorrect). See e.g. https://www.phenix-online.org/documentation/reference/composite_omit_map.html

Omit maps can be calculated no matter where initial phases came from. In the final stages of model building and refinement, the phases are usually obtained exclusively from the model, even if you start with experimental phases.

Additional aspects raised in the chat

[OP in comments] is the reason why omit mapping works because of the fact that every atom contributes to a diffraction spot?

Yes, exactly. The phases of each structure factor are almost correct, and the amplitudes are experimental (so unbiased but they do have experimental error, e.g. noise). It is worse to have a bad part of the model compared to leaving that part out all together.

[OP in comments] So even if the phases from that particular part of the structure itself are missing/omitted, there are still phases from elsewhere that do contribute to the electron density at that location. So if the feature were a model artifact, these other contributive phases wouldn't exist, and the omit map would show zero density there. Is this correct?

No, the language you are using is incorrect. There is no one-to-one correspondence of spots in the diffraction pattern and locations in the unit cell. Intensities and phases are indexed by (hkl) or $\vec{h}$, i.e. which plane of reflection they originate from. Locations in the unit cell are designated by x, y, z or $\vec{r}$. Every atom contributes to each diffraction spot. Every diffraction spot contributes to the calculated electron density in a given location.

Here is a way to say it: Even if the phases lack the information from the omitted part of the model, they are still good approximations of the phases of the actual structure. What really carries the information is that the structure factor amplitudes are derived from the diffraction pattern, which of course reflects the real structure. The amplitudes carries less information than the phases, but as long as you don't introduce incorrect information via the phases, that weak signal from the intensities is good enough to build or correct the model

[OP in comments] So if the feature were a model artifact [...], the omit map would show zero density there. Is this correct? Or is it more that an omit map does not "delete" the density in question so much as it "un-defines" it?

If you put atoms into the model in a region that is "empty" (filled with solvent) in the real structure and calculate electron density based on that, there will be some electron density (with lower signal) looking like the structure of the added atoms. If you remove those atoms again, the artifact will disappear again. If you replace an correct part of the model with incorrect atomic positions, the density will be a mix of both. Depending on the quality of the diffraction data (especially the resolution limit), the density will reflect the correct or the incorrect model more. In the scenario of removing a spurious part of the model, you could say it deletes the density, in the scenario of removing an incorrect part of the model, you could say it corrects the density.

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  • $\begingroup$ Thank you for this very detailed answer. I am still trying to wrap my head around this, so I have a question for you: is the reason why omit mapping works because of the fact that every atom contributes to a diffraction spot? I'll go read all the links you've included. I think I am getting it, but I still haven't had that "a-ha" moment yet. $\endgroup$
    – Dunois
    Oct 3 '20 at 11:40
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    $\begingroup$ @Dunois Yes, exactly. The phases of each structure factor are almost correct, and the amplitudes are experimental (so unbiased but they do have experimental error, e.g. noise). It is worse to have a bad part of the model compared to leaving that part out all together. $\endgroup$ Oct 3 '20 at 11:45
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    $\begingroup$ No, the language you are using is incorrect. There is no one-to-one correspondence of spots in the diffraction pattern and locations in the unit cell. Intensities and phases are index by (hkl) or $\vec{h}$, i.e. which plane of reflection they originate from. Locations in the unit cell are designated by x, y, z or $\vec{r}$. Every atom contributes to each diffraction spot. Every diffraction spot contributes to the calculated electron density in a given location. $\endgroup$ Oct 3 '20 at 11:55
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    $\begingroup$ Here is a way to say it: Even if the phases lack the information from the omitted part of the model, they are still good approximations of the phases of the actual structure. What really carries the information is that the structure factor amplitudes are derived from the diffraction pattern, which of course reflects the real structure. The amplitudes carries less information than the phases, but as long as you don't introduce incorrect information via the phases, that weak signal from the intensities is good enough to build or correct the model. $\endgroup$ Oct 3 '20 at 11:59
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    $\begingroup$ Thank you for the explanation and also pointing out where I am mistaken. I think I was trying to formulate the essence of what you've stated, albeit with very poor language (as you pointed out). But is my reasoning regarding the model artifact aspect of this nevertheless correct? Or is it more that an omit map does not "delete" the density in question so much as it "un-defines" it? (So that we can correct/rebuild it.) $\endgroup$
    – Dunois
    Oct 3 '20 at 12:07
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Experimental phasing and molecular replacement provide initial phases used to start model building. Later on (refinement, validation) phases are calculated from the model.

If the phases are what determine the final electron density irrespective of the amplitudes used

Both amplitudes and phases are used to determine the electron density. If you calculate the phases omitting part of the model, the electron density in this place won't be biased by that part of the model. (It may be biased by the absence of the model in this place).

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  • $\begingroup$ I apologize if this is a dumb question (and if I am running in circles), but I got the impression that if we don't have the phases (say for a part of a model), it just wouldn't get built even if we have the amplitudes for that part, right? Or is it that in this case (of omitting a part), there would be an electron density (because the amplitudes are still there), just that it would be featureless?! $\endgroup$
    – Dunois
    Oct 3 '20 at 10:46
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    $\begingroup$ Leaving out part of the model results in lower intensity of the electron density in that region, and in noise throughout the unit cell. You might get some bias with the omitted region has a peculiar shape (like leaving out atoms whose x-coordinate is between 0 and 0.1 in fractional coordinates - but who would do that?). $\endgroup$ Oct 3 '20 at 11:49
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The following doesn't fit into a comments box.

  • From the perspective of small molecule crystallography (say, ShelX), once you solved your model completely or completely enough to engage refining, you may recognize reflections leaving the straight line in the Fobs vs. Fcalc map. Within reason, it is possible to omit these outlier reflections from the .hkl file and still obtain a refinement which is both chemically reasonable, and converges. Not necessarily designed for macromolecules, but one of tutorials for Olex2 showcases this for a small complex.

    Independent to the intent to remove a few outliers, it is possible to constrain structure solution and refinement to reflections within a certain range of resolution, even if there are recorded data passing beyond this interval. This equally falls into the category of omission of data.

  • The second perspective is when experimentally recorded electron density can't be nicely attributed using the «standard» approach to solve a structure. Orientationally disordered solvent molecules may be an example for this to happen, or long flexible alkyl chains lowering crystallinity of a sample. The old way is to use Platon's squeeze instruction (again, small molecule perspective, solution and refinement with ShelX, reference) to keep pockets of electron density in Fobs for those you do not attempt (any more) allocation to specific atoms at specific positions. Eventually, you document a model lacking a few atoms, including a statement that such a map was used and how many electrons are in these pockets (application in a paper, a small molecule tutorial). The visualization of the structure model highlighting these pockets and writing a .cif file including the reduced .hkl help others to understand what you did encounter.

    For small molecule samples, there are libraries like DSR with preset geometries about molecules known to possibly yield such problems. If there is enough experimental evidence, their input, complementary to squeeze's maps, may be very helpful.

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    $\begingroup$ To the first bullet: You are talking about omitting reflections. In the omit map, part of the model are omitted from phase calculation in preparation for calculating maps. Omitting reflections is done in macromolecular crystallography as well, to assess bias via calculation of R$_\mathrm{free}$. This is part of assessing model bias, but only distantly related to the question the OP posed. $\endgroup$ Oct 3 '20 at 13:03

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