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Parikh vectors
8YBN_1 7QPP_1 2OUO_1 Letter Amino acid
20 7 2 N Asparagine
12 17 10 E Glutamic acid
6 14 9 P Proline
24 16 11 D Aspartic acid
16 34 7 L Leucine
33 14 14 K Lycine
9 9 8 M Methionine
13 10 1 T Threonine
16 15 8 V Valine
2 6 4 C Cysteine
9 12 3 G Glycine
14 25 9 S Serine
1 1 1 W Tryptophan
5 12 12 A Alanine
5 13 5 R Arginine
8 14 5 Q Glutamine
5 10 5 H Histidine
9 18 4 I Isoleucine
12 8 6 F Phenylalanine
21 8 6 Y Tyrosine 

8YBN_1|Chain A|Enterotoxin type B|Staphylococcus aureus (1280)
>7QPP_1|Chain A|Vitamin D3 receptor|Homo sapiens (9606)
>2OUO_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 \(w=f(c)\)
8YBN , Knot 110 240 0.83 40 158 231 
MESQPDPKPDELHKSSKFTGLMENMKVLYDDNHVSAINVKSIDQFLYFDLIYSIKDTKLGNYDNVRVEFKNKDLADKYKDKYVDVFGANYYYQCYFSKKTNDINSHQTDKRKTCMYGGVTEHNGNQLDKYRSITVRVFEDGKNLLSFDVQTNKKKVTAQELDYLTRHYLVKNKKLYEFNNSPYETGYIKFIENENSFWYDMMPAPGDKFDQSKYLMMYNDNKMVDSKDVKIEVYLTTKKK
7QPP , Knot 117 263 0.82 40 174 250 
GSHMDSLRPKLSEEQQRIIAILLDAHHKTYDPTYSDFCQFRPPVRVNDGGGSVTLELSQLSMLPHLADLVSYSIQKVIGFAKMIPGFRDLTSEDQIVLLKSSAIEVIMLRSNESFTMDDMSWTCGNQDYKYRVSDVTKAGHSLELIEPLIKFQVGLKKLNLHEEEHVLLMAICIVSPDRPGVQDAALIEAIQDRLSNTLQTYIRCRHPPPGSHLLYAKMIQKLADLRSLNEEHSKQYRCLSFQPECSMKLTPLVLEVFGNEIS
2OUO , Knot 65 130 0.81 40 110 127 
SMKDVPDSQQHPAPEKSSKVSEQLKCCSGILKEMFAKKHAAYAWPFYKPVDVEALGLHDYCDIIKHPMDMSTIKSKLEAREYRDAQEFGADVRLMFSNCYKYNPPDHEVVAMARKLQDVFEMRFAKMPDE

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 \(c\).

\(|P_{f(8YBN_1)}(2) \setminus P_{f(7QPP_1)}(2)|=78\), \(|P_{f(7QPP_1)}(2) \setminus P_{f(8YBN_1)}(2)|=94\). 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 1:100010101001000001011100101100000101101001001101011001000011000010101000011000000010111100000001000000100000000000101110000100100000101011001001101010000001010010010000110000100100010001010110000011001111110010000011100000110000101010100000
Pair \(Z_2\) Length of longest common subsequence
8YBN_1,7QPP_1 172 3
8YBN_1,2OUO_1 174 4
7QPP_1,2OUO_1 170 4

Newick tree

 
[
	8YBN_1:86.99,
	[
		7QPP_1:85,2OUO_1:85
	]:1.99
]

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 \((0.85)(0.8)(\frac{503 }{\log_{20} 503}-\frac{240}{\log_{20}240})=75.5\)
Status Protein1 Protein2 d d1/2
Query variables 8YBN_1 7QPP_1 96 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]