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Parikh vectors
3OFW_1 1FAH_1 5XKT_1 Letter Amino acid
3 19 6 N Asparagine
6 3 2 C Cysteine
4 21 10 R Arginine
5 37 13 E Glutamic acid
3 23 12 I Isoleucine
4 38 11 K Lycine
5 25 6 F Phenylalanine
0 5 2 W Tryptophan
3 35 18 A Alanine
1 33 14 D Aspartic acid
1 23 4 Q Glutamine
1 13 2 H Histidine
2 50 21 L Leucine
3 22 16 T Threonine
3 12 3 Y Tyrosine
1 25 14 V Valine
9 25 22 G Glycine
0 14 5 M Methionine
3 28 9 P Proline
3 20 10 S Serine 

3OFW_1|Chain A|Kunitz-type proteinase inhibitor SHPI-1|Stichodactyla helianthus (6123)
>1FAH_1|Chains A, B|CYTOCHROME P450 BM-3|Bacillus megaterium (1404)
>5XKT_1|Chains A, B|Urease accessory protein UreG|Klebsiella pneumoniae subsp. rhinoscleromatis SB3432 (861365)
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 \(w=f(c)\)
3OFW , Knot 34 60 0.77 36 51 58 
EAEASICSEPKKVGRCKGYFPRFYFDSETGKCTPFIYGGCGGNGNNFETLHQCRAICRLG
1FAH , Knot 194 471 0.84 40 243 449 
TIKEMPQPKTFGELKNLPLLNTDKPVQALMKIADELGEIFKFEAPGRVTRYLSSQRLIKEACDESRFDKNLSQALKFVRDFAGDGLFTSWTHEKNWKKAHNILLPSFSQQAMKGYHAMMVDIAVQLVQKWERLNADEHIEVPEDMTRLTLDTIGLCGFNYRFNSFYRDQPHPFITSMVRALDEAMNKLQRANPDDPAYDENKRQFQEDIKVMNDLVDKIIADRKASGEQSDDLLTHMLNGKDPETGEPLDDENIRYQIITFLIAGHEATSGLLSFALYFLVKNPHVLQKAAEEAARVLVDPVPSYKQVKQLKYVGMVLNEALRLWPTAPAFSLYAKEDTVLGGEYPLEKGDELMVLIPQLHRDKTIWGDDVEEFRPERFENPSAIPQHAFKPFGNGQRACIGQQFALHEATLVLGMMLKHFDFEDHTNYELDIKETLTLKPEGFVVKAKSKKIPLGGIPSPSTEQSAKKVR
5XKT , Knot 96 200 0.84 40 144 194 
KHPLRVGVGGPVGSGKTALLEALCKAMRDTWQLAVVTNDIYTKEDQRILTEAGTLAPERIVGVETGGCPHTAIREDASMNLAAVEALSEKFGNLDLIFVESGGDNLSATFSPELADLTIYVIDVAEGEKIPRKGGPGITRSDFLVINKTDLAPYVGASLKVMASDTQRMRGDRPWTFTNLKQGDGLSTIIAFLEDKGMLG

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 \(c\).

\(|P_{f(3OFW_1)}(2) \setminus P_{f(1FAH_1)}(2)|=20\), \(|P_{f(1FAH_1)}(2) \setminus P_{f(3OFW_1)}(2)|=212\). 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 1:010101000100110001011010100001000111011011010010010000110011
Pair \(Z_2\) Length of longest common subsequence
3OFW_1,1FAH_1 232 3
3OFW_1,5XKT_1 157 3
1FAH_1,5XKT_1 167 4

Newick tree

 
[
	1FAH_1:10.53,
	[
		3OFW_1:78.5,5XKT_1:78.5
	]:29.03
]

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 \((0.85)(0.8)(\frac{531 }{\log_{20} 531}-\frac{60}{\log_{20}60})=142.\)
Status Protein1 Protein2 d d1/2
Query variables 3OFW_1 1FAH_1 182 101.5
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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