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Parikh vectors
1ECN_1 3POE_1 6OWC_1 Letter Amino acid
4 2 15 M Methionine
2 6 4 Y Tyrosine
0 4 3 C Cysteine
9 9 13 I Isoleucine
5 10 9 P Proline
5 4 9 N Asparagine
6 5 48 L Leucine
4 2 13 H Histidine
10 5 14 K Lycine
14 6 7 F Phenylalanine
9 13 21 S Serine
9 7 11 T Threonine
9 4 13 V Valine
9 7 13 D Aspartic acid
4 4 9 Q Glutamine
5 10 15 E Glutamic acid
11 8 11 G Glycine
1 2 3 W Tryptophan
17 2 15 A Alanine
3 5 12 R Arginine 

1ECN_1|Chain A|ERYTHROCRUORIN (CYANO MET)|Chironomus thummi thummi (7155)
>3POE_1|Chain A|Mannan-binding lectin serine protease 1|Rattus norvegicus (10116)
>6OWC_1|Chains A, B|Estrogen receptor|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 \(w=f(c)\)
1ECN , Knot 68 136 0.81 38 114 132 
LSADQISTVQASFDKVKGDPVGILYAVFKADPSIMAKFTQFAGKDLESIKGTAPFETHANRIVGFFSKIIGELPNIEADVNTFVASHKPRGVTHDQLNNFRAGFVSYMKAHTDFAGAEAAWGATLDTFFGMIFSKM
3POE , Knot 63 115 0.86 40 96 112 
MVECSGNLFTQRTGTITSPDYPNPYPKSSECSYTIDLEEGFMVTLQFEDIFDIEDHPEVPCPYDYIKIKAGSKVWGPFCGEKSPEPISTQSHSIQILFRSDNSGENRGWRLSYRA
6OWC , Knot 118 258 0.84 40 166 247 
GIKRSKKNSLALSLTADQMVSALLDAEPPILYSEYDPTRPFSEASMMGLLTNLADRELVHMINWAKRVPGFVDLTLHDQVHLLECAWLEILMIGLVWRSMEHPGKLLFAPNLLLDRNQGKSVEGMVEIFDMLLATSSRFRMMNLQGEEFVCLKSIILLNSGVYTFLSSTLKSLEEKDHIHRVLDKITDTLIHLMAKAGLTLQQQHQRLAQLLLILSHIRHMSNKGMEHLYSMKCKNVVPLSDLLLEMLDAHRLHAPTS

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 \(c\).

\(|P_{f(1ECN_1)}(2) \setminus P_{f(3POE_1)}(2)|=79\), \(|P_{f(3POE_1)}(2) \setminus P_{f(1ECN_1)}(2)|=61\). 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 1:1010010010101001010111110111010101110100111001001010111000100111110011101101010100111000101100001001011110010100011110111110100111111001
Pair \(Z_2\) Length of longest common subsequence
1ECN_1,3POE_1 140 3
1ECN_1,6OWC_1 162 3
3POE_1,6OWC_1 164 3

Newick tree

 
[
	6OWC_1:84.99,
	[
		1ECN_1:70,3POE_1:70
	]:14.99
]

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 \((0.85)(0.8)(\frac{251 }{\log_{20} 251}-\frac{115}{\log_{20}115})=43.1\)
Status Protein1 Protein2 d d1/2
Query variables 1ECN_1 3POE_1 53 49.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]