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Parikh vectors
2DPI_1 7OEL_1 8BKN_1 Letter Amino acid
3 5 0 C Cysteine
0 11 6 Q Glutamine
2 24 14 G Glycine
0 17 9 I Isoleucine
0 17 2 R Arginine
0 12 13 E Glutamic acid
0 6 2 M Methionine
0 14 4 P Proline
0 7 2 W Tryptophan
0 11 2 N Asparagine
0 18 8 D Aspartic acid
0 13 11 H Histidine
0 25 17 L Leucine
0 6 2 Y Tyrosine
2 7 15 A Alanine
0 9 19 K Lycine
0 8 7 F Phenylalanine
0 10 5 S Serine
0 10 7 T Threonine
0 16 7 V Valine

2DPI_1|Chain A[auth P]|5'-D(*AP*GP*GP*AP*CP*CP*(DOC))-3'|
>7OEL_1|Chain A|N6-adenosine-methyltransferase catalytic subunit|Homo sapiens (9606)
>8BKN_1|Chain A|Myoglobin|Equus caballus (9796)
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 \(w=f(c)\)
2DPI , Knot 5 7 0.46 6 5 5
AGGACCC
7OEL , Knot 113 246 0.84 40 167 234
MGHHHHHHSSGRENLYFQGALTQSVGGDSSADRLFPPQWICCDIRYLDVSILGKFAVVMADPPWDIHMELPYGTLTDDEMRRLNIPVLQDDGFLFLWVTGRAMELGRECLNLWGYERVDEIIWVKTNQLQRIIRTGRTGHWLNHGKEHCLVGVKGNPQGFNQGLDCDVIVAEVRSTSHKPDEIYGMIERLSPGTRKIELFGRPHNVQPNWITLGNQLDGIHLLDPDVVARFKQRYPDGIISKPKNL
8BKN , Knot 74 152 0.81 38 115 147
GLSDGEWQQVLNVWGKVEADIAGHGQEVLIRLFTGHPETLEKFDKFKHLKTEAEMKASEDLKKHGTVVLTALGGILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISDAIIHVLHSKHPGDFGADAQGAMTKALELFRNDIAAKYKELGFQ

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 \(c\).

\(|P_{f(2DPI_1)}(2) \setminus P_{f(7OEL_1)}(2)|=2\), \(|P_{f(7OEL_1)}(2) \setminus P_{f(2DPI_1)}(2)|=164\). 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 1:1111000
Pair \(Z_2\) Length of longest common subsequence
2DPI_1,7OEL_1 166 2
2DPI_1,8BKN_1 114 2
7OEL_1,8BKN_1 170 3

Newick tree

 
[
	7OEL_1:91.24,
	[
		2DPI_1:57,8BKN_1:57
	]:34.24
]

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 \((0.85)(0.8)(\frac{253 }{\log_{20} 253}-\frac{7}{\log_{20}7})=85.8\)
Status Protein1 Protein2 d d1/2
Query variables 2DPI_1 7OEL_1 109 55.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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