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Parikh vectors
2XFY_1 1RZY_1 8VOR_1 Letter Amino acid
5 2 0 C Cysteine
14 7 4 H Histidine
22 12 7 I Isoleucine
40 9 3 L Leucine
25 2 2 Y Tyrosine
42 12 7 A Alanine
28 6 6 Q Glutamine
32 6 9 E Glutamic acid
23 4 5 F Phenylalanine
56 13 12 G Glycine
22 10 10 K Lycine
18 4 2 M Methionine
35 7 1 P Proline
21 2 5 T Threonine
10 1 1 W Tryptophan
36 8 14 V Valine
27 4 8 R Arginine
26 2 1 N Asparagine
30 9 3 D Aspartic acid
23 6 3 S Serine 

2XFY_1|Chain A|BETA-AMYLASE|HORDEUM VULGARE (4513)
>1RZY_1|Chain A|Histidine triad nucleotide-binding protein 1|Oryctolagus cuniculus (9986)
>8VOR_1|Chain A[auth 0]|Ribosomal protein L21|Escherichia coli (562)
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 \(w=f(c)\)
2XFY , Knot 211 535 0.82 40 260 491 
MEVNVKGNYVQVYVMLPLDAVSVNNRFEKGDELRAQLRKLVEAGVDGVMVDVWWGLVEGKGPKAYDWSAYKQLFELVQKAGLKLQAIMSFHQCGGNVGDAVNIPIPQWVRDVGTRDPDIFYTDGHGTRNIEYLTLGVDNQPLFHGRSAVQMYADYMTSFRENMKEFLDAGVIVDIEVGLGPAGEMRYPSYPQSHGWSFPGIGEFICYDKYLQADFKAAAAAVGHPEWEFPNDVGQYNDTPERTQFFRDNGTYLSEKGRFFLAWYSNNLIKHGDRILDEANKVFLGYKVQLAIKISGIHWWYKVPSHAAELTAGYYNLHDRDGYRTIARMLKRHRASINFTCAEMRDSEQSSQAMSAPEELVQQVLSAGWREGLNVACENALPRYDPTAYNTILRNARPHGINQSGPPEHKLFGFTYLRLSNQLVEGQNYANFKTFVDRMHANLPRDPYVDPMAPLPRSGPEISIEMILQAAQPKLQPFPFQEHTDLPVGPTGGMGGQAEGPTCGMGGQVKGPTGGMGGQAEDPTSGIGGELPATM
1RZY , Knot 65 126 0.83 40 102 122 
MADEIAKAQVARPGGDTIFGKIIRKEIPAKIIFEDDQCLAFHDISPQAPTHFLVIPKKHISQISAAEDADESLLGHLMIVGKKCAADLGLKKGYRMVVNEGSDGGQSVYHVHLHVLGGRQMNWPPG
8VOR , Knot 53 103 0.79 38 82 98 
MYAVFQSGGKQHRVSEGQTVRLEKLDIATGETVEFAEVLMIANGEEVKIGVPFVDGGVIKAEVVAHGRGEKVKIVKFRRRKHYRKQQGHRQWFTDVKITGISA

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 \(c\).

\(|P_{f(2XFY_1)}(2) \setminus P_{f(1RZY_1)}(2)|=188\), \(|P_{f(1RZY_1)}(2) \setminus P_{f(2XFY_1)}(2)|=30\). 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 1:1010101001010111110110100010010010101001101110111101111110101101001010001101100111010111010001101101101111011001100010110001010001001011100011101001101010010010001001101111101011111110100100100011011111011000001010101111111010101100110000010000110001001000101111100001100100110010011110010111010110110011001101011000100001000110110000101010010100000000110110011001101110011011000111000101000110010101100011100011110010100011010001010011001010110010101111110011010101110110101011110000011111011111010110011110101101111101001001111011101
Pair \(Z_2\) Length of longest common subsequence
2XFY_1,1RZY_1 218 3
2XFY_1,8VOR_1 214 3
1RZY_1,8VOR_1 132 3

Newick tree

 
[
	2XFY_1:11.74,
	[
		8VOR_1:66,1RZY_1:66
	]:52.74
]

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 \((0.85)(0.8)(\frac{661 }{\log_{20} 661}-\frac{126}{\log_{20}126})=154.\)
Status Protein1 Protein2 d d1/2
Query variables 2XFY_1 1RZY_1 196 120.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]