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Parikh vectors
2KCW_1 1ANC_1 1JRX_1 Letter Amino acid
16 19 37 L Leucine
4 10 43 K Lycine
5 18 29 N Asparagine
9 12 38 D Aspartic acid
2 12 8 C Cysteine
14 8 31 E Glutamic acid
11 23 61 G Glycine
4 4 18 H Histidine
7 12 17 P Proline
2 4 4 W Tryptophan
4 11 13 Q Glutamine
8 16 36 S Serine
5 8 34 T Threonine
4 8 16 Y Tyrosine
4 13 26 I Isoleucine
4 2 17 M Methionine
6 4 12 F Phenylalanine
17 20 48 V Valine
11 15 66 A Alanine
10 4 17 R Arginine 

2KCW_1|Chain A|Uncharacterized acetyltransferase yjaB|Escherichia coli (83333)
>1ANC_1|Chain A|ANIONIC TRYPSIN|Rattus rattus (10117)
>1JRX_1|Chains A, B|FLAVOCYTOCHROME C|Shewanella frigidimarina (56812)
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 \(w=f(c)\)
2KCW , Knot 71 147 0.80 40 112 141 
MVISIRRSRHEEGEELVAIWCRSVDATHDFLSAEYRTELEDLVRSFLPEAPLWVAVNERDQPVGFMLLSGQHMDALFIDPDVRGCGVGRVLVEHALSMAPELTTNVNEQNEQAVGFYKKVGFKVTGRSEVDDLGKPYPLLNLAYVGA
1ANC , Knot 104 223 0.84 40 158 216 
IVGGYTCQENSVPYQVSLNSGYHFCGGSLINDQWVVSAAHCYKSRIQVRLGEHNINVLEGNEQFVNAAKIIKHPNFDRKTLNNDIMLIKLSSPVKLNARVATVALPSSCAPAGTQCLISGWGNTLSSGVNEPDLLQCLDAPLLPQADCEASYPGKITDNMVCVGFLEGGKDSCQGDSGGPVVCNGELQGIVKWGYGCALPDNPGVYTKVCNYVDWIQDTIAAN
1JRX , Knot 224 571 0.83 40 255 514 
ADNLAEFHVQNQECDSCHTPDGELSNDSLTYENTQCVSCHGTLAEVAETTKHEHYNAHASHFPGEVACTSCHSAHEKSMVYCDSCHSFDFNMPYAKKWLRDEPTIAELAKDKSERQAALASAPHDTVDVVVVGSGGAGFSAAISATDSGAKVILIEKEPVIGGNAKLAAGGMNAAWTDQQKAKKITDSPELMFEDTMKGGQNINDPALVKVLSSHSKDSVDWMTAMGADLTDVGMMGGASVNRAHRPTGGAGVGAHVVQVLYDNAVKRNIDLRMNTRGIEVLKDDKGTVKGILVKGMYKGYYWVKADAVILATGGFAKNNERVAKLDPSLKGFISTNQPGAVGDGLDVAENAGGALKDMQYIQAHPTLSVKGGVMVTEAVRGNGAILVNREGKRFVNEITTADKASAAILAQTGKSAYLIFDDSVRKSLSKIDKYIGLGVAPTADSLVKLGKMEGIDGKALTETVARYNSLVSSGKDTDFERPNLPRALNEGNYYAIEVTPGVHHTMGGVMIDTKAEVMNAKKQVIPGLYGAGEVTGGVHGANRLGGNAISDIITFGRLAGEEAAKYSKKN

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 \(c\).

\(|P_{f(2KCW_1)}(2) \setminus P_{f(1ANC_1)}(2)|=62\), \(|P_{f(1ANC_1)}(2) \setminus P_{f(2KCW_1)}(2)|=108\). 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 1:111010000000100111110001010001101000001001100111011111110000011111110100101111010101011101110011011101000100000011110001110101000100110101110110111
Pair \(Z_2\) Length of longest common subsequence
2KCW_1,1ANC_1 170 3
2KCW_1,1JRX_1 213 3
1ANC_1,1JRX_1 191 4

Newick tree

 
[
	1JRX_1:10.98,
	[
		2KCW_1:85,1ANC_1:85
	]:20.98
]

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 \((0.85)(0.8)(\frac{370 }{\log_{20} 370}-\frac{147}{\log_{20}147})=67.4\)
Status Protein1 Protein2 d d1/2
Query variables 2KCW_1 1ANC_1 85 70.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]