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Parikh vectors
6LSN_1 6ETE_1 7JKX_1 Letter Amino acid
21 18 3 R Arginine
36 25 2 G Glycine
20 18 12 P Proline
29 19 8 T Threonine
18 18 7 Y Tyrosine
10 10 4 M Methionine
4 8 3 W Tryptophan
16 19 12 N Asparagine
16 14 9 Q Glutamine
37 25 9 E Glutamic acid
13 12 1 H Histidine
27 20 7 I Isoleucine
32 22 14 L Leucine
27 13 6 D Aspartic acid
19 23 12 K Lycine
20 18 5 F Phenylalanine
23 15 1 S Serine
36 25 5 A Alanine
12 7 2 C Cysteine
34 13 5 V Valine 

6LSN_1|Chains A, C|Tubulin alpha-1B chain|Sus scrofa (9823)
>6ETE_1|Chain A|Lysine-specific demethylase 4D|Homo sapiens (9606)
>7JKX_1|Chain A|Bromodomain-containing protein 4|Homo sapiens (9606)
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 \(w=f(c)\)
6LSN , Knot 192 450 0.87 40 257 425 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFLVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLISQIVSSITASLRFDGALNVDLTEFQTNLVPYPRIHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRSIQFVDWCPTGFKVGINYQPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEE
6ETE , Knot 153 342 0.87 40 223 330 
METMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR
7JKX , Knot 66 127 0.84 40 108 122 
LENPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTEE

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 \(c\).

\(|P_{f(6LSN_1)}(2) \setminus P_{f(6ETE_1)}(2)|=107\), \(|P_{f(6ETE_1)}(2) \setminus P_{f(6LSN_1)}(2)|=73\). 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 1:100010101101110110101010010011010101100001111000100110001110011011110101011001001000011010011010001100010100011001101110010011000011011111001111010110011100101001000010101011101001110100011000001000001111000110010000101001000010011001100101010101110101001000111010101111001111010010000101101001010110011000100100110011001011100101111010000010110101011011100011011111011010011011000011101110100010110100111010110110010100100011110000001110010101000100
Pair \(Z_2\) Length of longest common subsequence
6LSN_1,6ETE_1 180 4
6LSN_1,7JKX_1 221 4
6ETE_1,7JKX_1 189 3

Newick tree

 
[
	7JKX_1:10.74,
	[
		6LSN_1:90,6ETE_1:90
	]:16.74
]

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 \((0.85)(0.8)(\frac{792 }{\log_{20} 792}-\frac{342}{\log_{20}342})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 6LSN_1 6ETE_1 157 139
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]