Find similar proteins with similar tertiary structure? I'm a mathematician studying knotted curves and am trying to find some applications of my work in proteins. Using https://www.rcsb.org/, how do I find proteins that have similar  (but perhaps not the same) tertiary structure? Preferably they would have similar primary structure and function... 
Perhaps what I'm looking for is "similar proteins", if there is such a thing?

Also, is there a way in the database to select more "common proteins" (important or well studied, not just some rare exotic one (if there is such a thing)?
As a general question on the board, more detail is often good, because of the answers can be inaccurate due to misunderstanding of the biological context.

There are several ways to do this and at least 3 that I'm aware of. 


Blast - sequence similarity searching the PDB database
Contact map identity, I use this approach, but its complicated - and its precision or utility is context specific, i.e. it could be loads of work and miss the vital detail you need.
Direct mapping one protein structure against another, this in particular is true for specific domains, or sites within the protein and the result is angstroms. Its not difficult and I assume this can be scaled to the whole protein.


Point 1 "Blackboxes" 

By far the easiest is option one and I suspect I-Tasser (here) use this principle before instigating "threading". This package is considered "good" (others appear to consider it the best of its type), I believe it has done well in CASP tests (Critical Assessment of Techniques for Protein Structure Prediction). So the simple answer is dump your sequence in there an watch the output. Pyre is something similar. These are interesting tools but certainly not definitive, really there a starting point.

Point 1 "Formal Blasting PDB"

This is the easiest and most mature approach for your question, without knowing the details. I've run a test to ask the following question.

Which protein structure available on PBD is most similar to Japanese encephalitis virus (JEV) based on protein homology?


Step A
I've no specific interest in the virus BTW. The most robust Blast engine IMO is NCBI here . I dump a JEV sequence into the box, and click three important options (otherwise it doesn't work). 


Database = protein databank (PDB), i.e. structural data
Organism "Japanese encephalitis virus (taxid:11072)": the taxid will come up automatically (if not just proceed as normal)
Next to the Organism box I click "Exclude", otherwise this whole approach doesn't work
Obviously I select "blastp", otherwise it definitely will not work


The output is/was here (but is now inactive). Summarising what it presented, E is so high its "0" (>2e-139) protein ID "77.59%" and just states "envelope protein" Chain A. I hit the link and the protein that comes up has a name which ends in "6A0P_A". This is the PDB reference and Percentage ID and PDB reference are the only two bits of info you need. The PDB reference needs the "_A" removing, thus its 6A0P. The _A just refers to chain A of a dimer.

Step B
You then go back to the Web site you mentioned and input the PDB ID you stated into the search bar. This results in this quite interesting page here. The structure is Usutu virus, which surprised me. 

Point 2 . This is much harder and not only involves specific Python code, but a post unsupervised learning analysis and results in an average similarity of structure over the whole protein. There are a number of published algorithms for generating a contact map, or its easy to write your own. Its more advanced, its complicated and unless you've seen loads of them they not immediately understandable. Contact maps angle at a very specialist approach of predicting and understanding protein structure. It is not for the faint hearted and some biological details will be omitted, which may or may not be important for your purpose.

Point 3 . I don't have research grade expertise here, so I forget the exact button to click, but its quite easy via PyMol and other graphical viewers to perform spatial similarity for specific epitopes and this is something you need to learn within your sphere. Basically you will load two proteins into the image viewer and then they will be a "similarity aligning" function and you zoom in on your output to generate an answer in angstroms. You will need to learn this approach for the later parts of your project.

Summary, The orthodox way would be to perform point 1, select several targets and then examine them in detail via point 3. I definitely recommend direct Blast if you go down this route, because it is the most tractable and avoids potential unknown subsampling.  I personally would not recommend Point 2 in a general context because you have to consider an unsupervised learning approach towards processing a heap of contact maps. Ultimately, which ever approach you take you can never avoid Point 3 (which someone else does, in my case).

Note This may or may not be obvious to your but just because two sequence have the closest homology doesn't mean they will have the closest structural similarity.