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Parikh vectors
4PEI_1 3UVJ_1 8SBQ_1 Letter Amino acid
14 11 11 Q Glutamine
20 11 20 E Glutamic acid
25 18 23 G Glycine
33 17 24 L Leucine
28 17 16 S Serine
11 13 14 A Alanine
9 7 9 R Arginine
17 11 24 D Aspartic acid
9 9 7 M Methionine
15 13 12 F Phenylalanine
16 19 15 V Valine
6 9 0 C Cysteine
11 5 7 H Histidine
31 11 21 I Isoleucine
31 14 17 K Lycine
5 0 3 W Tryptophan
28 6 11 Y Tyrosine
18 10 10 N Asparagine
15 13 18 P Proline
14 11 17 T Threonine 

4PEI_1|Chains A, D[auth B], G[auth C], J[auth D], M[auth E]|RNA lariat debranching enzyme, putative|Entamoeba histolytica (5759)
>3UVJ_1|Chains A, C|Guanylate cyclase soluble subunit alpha-3|Homo sapiens (9606)
>8SBQ_1|Chains A, B|Fluorophosphonate-binding serine hydrolases E|Staphylococcus aureus USA100-CA-126 (1385546)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
4PEI , Knot 153 356 0.84 40 220 347 
GAMATEQIQHIAIVGSVHGKYREMYRQLSEYEKSTGKEISFVICTGDMQTLRYEADLVYLKVPPKYKQMGDFHLYYEGKEKAPYLTLFIGGNHESSNVLLHLYNGGFVCFNMYYLGVCSCININGLRIVGVSGIYKSFDEKKPYTYPPSPNDVVSLFHTRNYVIQMLSNLSQSSQIDISLSHDWPQGIVMKGNYKQLYRFQPGFKKDGASLGSPINKVILNTLKPKYWISGHMHCEYHAEEGPTHFIALGKIGYKNAISYLDLPLKQKTDLEYDKDWVCNLIMTWPAFSNKAQFPDLSYSISELLSKRTKELDKKIIELWEKYIGLKIIYDSDTFDIQFTSRRFYIEKIYNELNIN
3UVJ , Knot 103 225 0.82 38 166 219 
SMQGQVVQAKKFSNVTMLFSDIVGFTAICSQCSPLQVITMLNALYTRFDQQCGELDVYKVETIGDAYCVAGGLHKESDTHAVQIALMALKMMELSDEVMSPHGEPIKMRIGLHSGSVFAGVVGVKMPRYCLFGNNVTLANKFESCSVPRKINVSPTTYRLLKDCPGFVFTPRSREELPPNFPSEIPGICHFLDAYQQGTNSKPCFQKKDVEDGNANFLGKASGID
8SBQ , Knot 122 279 0.82 38 184 266 
GPGMETLELQGAKLRYHQVGQGPVLIFIPGANGTGDIFLPLAEQLKDHFTVVAVDRRDYGESELTEPLPDSASNPDSDYRVKRDAQDIAELAKSLSDEPVYILGSSSGSIVAMHVLKDYPEVVKKIAFHEPPINTFLPDSTYWKDKNDDIVHQILTEGLEKGMKTFGETLNIAPIDAKMMSQPADTEEGRIEQYKRTMFWLEFEIRQYTHSNITLDDFTKYSDKITLLNGTDSRGSFPQDVNFYINKETGIPIVDIPGGHLGYIQKPEGFADVLLNMWG

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(4PEI_1)}(2) \setminus P_{f(3UVJ_1)}(2)|=121\), \(|P_{f(3UVJ_1)}(2) \setminus P_{f(4PEI_1)}(2)|=67\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:11110001001111101010000100010000000100101110010100100010110101110000110101000100011010111110000001110100111101010011100010101101111011000100001000110100110110000011011001000001010100011011110100001001011100011011011001110010100110101000001001100111110110001100101110000010000011001110111100010110100010011000000100011011000111011000001010100001010010001010
Pair \(Z_2\) Length of longest common subsequence
4PEI_1,3UVJ_1 188 3
4PEI_1,8SBQ_1 154 3
3UVJ_1,8SBQ_1 176 4

Newick tree

 
[
	3UVJ_1:95.27,
	[
		4PEI_1:77,8SBQ_1:77
	]:18.27
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{581 }{\log_{20} 581}-\frac{225}{\log_{20}225})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 4PEI_1 3UVJ_1 133 107.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]