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Parikh vectors
3HHT_1 6JCG_1 7PZS_1 Letter Amino acid
11 11 16 T Threonine
20 14 30 V Valine
3 3 0 C Cysteine
6 10 9 Q Glutamine
12 7 21 S Serine
20 8 27 L Leucine
5 4 19 F Phenylalanine
6 4 7 Y Tyrosine
12 14 23 A Alanine
5 5 11 N Asparagine
22 11 24 E Glutamic acid
8 6 11 H Histidine
5 2 6 M Methionine
10 15 18 I Isoleucine
13 13 16 K Lycine
17 5 10 P Proline
5 4 1 W Tryptophan
14 4 19 R Arginine
9 6 21 D Aspartic acid
13 17 23 G Glycine 

3HHT_1|Chain A|Nitrile hydratase alpha subunit|Geobacillus pallidus (33936)
>6JCG_1|Chain A|Integrase|Human immunodeficiency virus 1 (11676)
>7PZS_1|Chain A|UDP-3-O-acyl-N-acetylglucosamine deacetylase|Pseudomonas aeruginosa (strain ATCC 15692 / DSM 22644 / CIP 104116 / JCM 14847 / LMG 12228 / 1C / PRS 101 / PAO1) (208964)
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 \(w=f(c)\)
3HHT , Knot 102 216 0.84 40 149 207 
MTIGQKNTNIDPRFPHHHPRPQSFWEARAKALESLLIEKGHLSSDAIERVIKHYEHELGPMNGAKVVAKAWTDPAFKQRLLEDSETVLRELGYYGLQGEHIRVVENTDTVHNVVVCTLCSCYPWPLLGLPPSWYKEPAYRARVVKEPRQVLKEFGLDLPDSVEIRVWDSSSEIRFMVLPQRPEGTEGMTEEELAKLVTRDSMIGVAKIEPPKVTVG
6JCG , Knot 78 163 0.81 40 124 160 
GHGQVDCSPGIWQLDCTHLEGKVILVAVHVASGYIEAEVIPAETGQETAYFLLKLAGRWPVKTVHTDNGSNFTSTTVKAACWWAGIKQEFGIPYNPQSQGVIESMNKELKKIIGQVRDQAEHLKTAVQMAVFIHNKKRKGGIGGYSAGERIVDIIATDIQTKE
7PZS , Knot 133 312 0.81 38 185 295 
HHHHHHENLYFQSMIKQRTLKNIIRATGVGLHSGEKVYLTLKPAPVDTGIVFSRTDLDPVVEIPARAENVGETTMSTTLVKGDVKVDTVEHLLSAMAGLGIDNAYVELSASEVPIMDGSAGPFVFLIQSAGLQEQEAAKKFIRIKREVSVEEGDKRAVFVPFDGFKVSFEIDFDHPVFRGRTQQASVDFSSTSFVKEVSRARTFGFMRDIEYLRSQNLALGGSVENAIVVDENRVLNEDGLRYEDEFVKHKILDAIGDLYLLGNSLIGEFRGFKSGHALNNQLLRTLIADKDAWEVVTFEDARTAPISYMRP

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 \(c\).

\(|P_{f(3HHT_1)}(2) \setminus P_{f(6JCG_1)}(2)|=98\), \(|P_{f(6JCG_1)}(2) \setminus P_{f(3HHT_1)}(2)|=73\). 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 1:101100000101011000101001101010110011100101000110011000000111101101110110011100011000001100110011010010110000010011100100001111111110100011001011001001100111011001010110000010111110010100110000110110000111110101101011
Pair \(Z_2\) Length of longest common subsequence
3HHT_1,6JCG_1 171 3
3HHT_1,7PZS_1 168 3
6JCG_1,7PZS_1 165 3

Newick tree

 
[
	3HHT_1:85.49,
	[
		7PZS_1:82.5,6JCG_1:82.5
	]:2.99
]

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 \((0.85)(0.8)(\frac{379 }{\log_{20} 379}-\frac{163}{\log_{20}163})=64.8\)
Status Protein1 Protein2 d d1/2
Query variables 3HHT_1 6JCG_1 82 73
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]