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Parikh vectors
8BJY_1 4AZP_1 9CWV_1 Letter Amino acid
19 3 5 H Histidine
8 7 11 M Methionine
26 10 15 V Valine
14 5 11 N Asparagine
29 8 7 D Aspartic acid
9 6 2 C Cysteine
13 5 13 Q Glutamine
24 1 17 P Proline
21 6 11 S Serine
12 2 4 Y Tyrosine
22 12 18 E Glutamic acid
33 9 16 L Leucine
30 14 11 K Lycine
19 4 4 F Phenylalanine
42 8 23 A Alanine
12 2 5 W Tryptophan
20 5 11 R Arginine
37 11 16 G Glycine
22 5 15 I Isoleucine
20 15 16 T Threonine 

8BJY_1|Chain A|Fructosyl Peptide Oxidase mutant (X02B)|Parastagonospora nodorum SN15 (321614)
>4AZP_1|Chain A|FATTY ACID-BINDING PROTEIN, EPIDERMAL|MUS MUSCULUS (10090)
>9CWV_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R|Gag|Human immunodeficiency virus type 1 group M subtype B (isolate HXB2) (11706)
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 \(w=f(c)\)
8BJY , Knot 185 432 0.86 40 246 411 
MAPSRANTKVIVVGGGGTIGSSTALHLVRSGYTPSNVTVLDAYPIPSCQSAGNDLNKIMGVSADPAADAARQMWNEDELFKKFFHNTGRLDCAHGEKDIADLKKLYQNLVDWGLDATVEWLDSEDEILKRMPQLTRDQIKGWKAIFSKDGGWLAAAKAIKAIGEYLRDQGVRFGFYGAGSFKQPLLAEGVCIGVETVDGTRYYADKVVLAAGAWSPTLVELQEQCVSKAWVYGHIQLTPEEAARYKNSPVVYNGDVGFFFEPNEHGIIKVCDEFPGFTRFKMHQPFGAKAPKRISVPRSHCKHPTDTIPDASIVRIRRAIATFMPQFKNKPLFNQAMCWCTDTADGHLLICEHPEWKNFYLATGDSGHSFKLLPIIGKYVVELLEGTLADELAHKWRWRPGSGDALKSRREAPAKDLADMPGWNHDHHHHHH
4AZP , Knot 69 138 0.82 40 100 130 
GSHMASLKDLEGKWRLMESHGFEEYMKELGVGLALRKMAAMAKPDCIITCDGNNITVKTESTVKTTVFSCNLGEKFDETTADGRKTETVCTFQDGALVQHQQWDGKESTITRKLKDGKMIVECVMNNATCTRVYEKVQ
9CWV , Knot 108 231 0.84 40 159 225 
HQAISPRTLNAWVKVVEEKAFSPEVIPMFSALSEGATPQDLNTMLNTVGGHQAAMQMLKETINEEAAEWDRVHPVHAGPIAPGQMREPRGSDIAGTTSTLQEQIGWMTNNPPIPVGEIYKRWIILGLNKIVRMYSPTSILDIRQGPKEPFRDYVDRFYKTLRAEQASQEVKNWMTETLLVQNANPDCKTILKALGPAATLEEMMTACQGVGGPGHKARVLAEAMSQVINSA

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 \(c\).

\(|P_{f(8BJY_1)}(2) \setminus P_{f(4AZP_1)}(2)|=181\), \(|P_{f(4AZP_1)}(2) \setminus P_{f(8BJY_1)}(2)|=35\). 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 1:111001000111111110110001101100100100101101011100001100100111101011101100110000110011000101001010001101001000110111010101100000110011010000101101110001111111011011100100011011101110100111101101110010100001001111111101011010000100111010101010011000001110010111110100011101000111100101001111011001011000000100011010110100111011101000111001101000010101110001010010110100100101111110011011010110011001010110101100000111001101111000000000
Pair \(Z_2\) Length of longest common subsequence
8BJY_1,4AZP_1 216 4
8BJY_1,9CWV_1 185 4
4AZP_1,9CWV_1 169 3

Newick tree

 
[
	8BJY_1:10.35,
	[
		9CWV_1:84.5,4AZP_1:84.5
	]:20.85
]

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 \((0.85)(0.8)(\frac{570 }{\log_{20} 570}-\frac{138}{\log_{20}138})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 8BJY_1 4AZP_1 164 106
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]