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Parikh vectors
6RIN_1 2CHL_1 6JXV_1 Letter Amino acid
0 13 2 N Asparagine
9 26 8 G Glycine
0 13 4 M Methionine
0 17 3 F Phenylalanine
0 2 0 W Tryptophan
0 15 6 V Valine
0 22 3 R Arginine
0 21 6 D Aspartic acid
0 21 13 E Glutamic acid
0 17 5 I Isoleucine
0 26 11 K Lycine
0 17 3 P Proline
0 16 7 S Serine
11 14 2 A Alanine
12 4 1 C Cysteine
0 38 7 L Leucine
7 15 6 T Threonine
0 22 3 Y Tyrosine
0 17 7 Q Glutamine
0 15 4 H Histidine 

6RIN_1|Chain A[auth N]|Non-template DNA|Escherichia coli (562)
>2CHL_1|Chain A|CASEIN KINASE I ISOFORM GAMMA-3|HOMO SAPIENS (9606)
>6JXV_1|Chain A|Small ubiquitin-related modifier|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 \(w=f(c)\)
6RIN , Knot 14 39 0.43 8 14 26 
GCACATCACCCATTCAGAAGCTAAGGCATGGCTAGCTGC
2CHL , Knot 148 351 0.82 40 216 334 
MHHHHHHSSGVDLGTENLYFQSMGVLMVGPNFRVGKKIGCGNFGELRLGKNLYTNEYVAIKLEPMKSRAPQLHLEYRFYKQLGSGDGIPQVYYFGPCGKYNAMVLELLGPSLEDLFDLCDRTFSLKTVLMIAIQLISRMEYVHSKNLIYRDVKPENFLIGRPGNKTQQVIHIIDFALAKEYIDPETKKHIPYREHKSLTGTARYMSINTHLGKEQSRRDDLEALGHMFMYFLRGSLPWQGLKADTLKERYQKIGDTKRATPIEVLCENFPEMATYLRYVRRLDFFEKPDYDYLRKLFTDLFDRKGYMFDYEYDWIGKQLPTPVGAVQQDPALSSNREAHQHRDKMQQSKNQ
6JXV , Knot 54 101 0.82 38 84 98 
MSDQEAKPSTEDLGDKKEGEYIKLKVIGQDSSEIHFKVKMTTHLKKLKESYCQRQGVPMNSLRFLFEGQRIADNHTPKELGMEEEDVIEVYQEQTGGHSTV

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 \(c\).

\(|P_{f(6RIN_1)}(2) \setminus P_{f(2CHL_1)}(2)|=9\), \(|P_{f(2CHL_1)}(2) \setminus P_{f(6RIN_1)}(2)|=211\). 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 1:101010010001000111110011110101100110010
Pair \(Z_2\) Length of longest common subsequence
6RIN_1,2CHL_1 220 2
6RIN_1,6JXV_1 92 3
2CHL_1,6JXV_1 196 3

Newick tree

 
[
	2CHL_1:11.32,
	[
		6RIN_1:46,6JXV_1:46
	]:71.32
]

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 \((0.85)(0.8)(\frac{390 }{\log_{20} 390}-\frac{39}{\log_{20}39})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 6RIN_1 2CHL_1 144 78
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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