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Parikh vectors
1QXK_1 1JOV_1 6GHS_1 Letter Amino acid
9 7 13 Y Tyrosine
13 15 22 A Alanine
6 5 5 C Cysteine
11 16 12 N Asparagine
14 17 17 Q Glutamine
30 18 10 E Glutamic acid
14 15 34 G Glycine
18 17 16 I Isoleucine
29 29 20 L Leucine
11 5 2 M Methionine
19 7 19 R Arginine
18 7 23 D Aspartic acid
13 15 10 T Threonine
6 6 2 W Tryptophan
16 17 31 V Valine
14 11 5 F Phenylalanine
24 19 17 S Serine
22 11 20 P Proline
11 17 17 H Histidine
23 16 16 K Lycine 

1QXK_1|Chain A|Protein-tyrosine phosphatase, non-receptor type 1|Homo sapiens (9606)
>1JOV_1|Chain A|HI1317|Haemophilus influenzae (727)
>6GHS_1|Chain A|TagI restriction endonuclease|Thermocrispum agreste (37925)
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 \(w=f(c)\)
1QXK , Knot 141 321 0.84 40 207 309 
MEMEKEFEQIDKSGSWAAIYQDIRHEASDFPCRVAKLPKNKNRNRYRDVSPFDHSRIKLHQEDNDYINASLIKMEEAQRSYILTQGPLPNTCGHFWEMVWEQKSRGVVMLNRVMEKGSLKCAQYWPQKEEKEMIFEDTNLKLTLISEDIKSYYTVRQLELENLTTQETREILHFHYTTWPDFGVPESPASFLNFLFKVRESGSLSPEHGPVVVHCSAGIGRSGTFCLADTCLLLMDKRKDPSSVDIKKVLLEMRKFRMGLIQTADQLRFSYLAVIEGAKFIMGDSSVQDQWKELSHEDLEPPPEHIPPPPRPPKRILEPHN
1JOV , Knot 123 270 0.85 40 185 258 
MKTTLLKTLTPELHLVQHNDIPVLHLKHAVGTAKISLQGAQLISWKPQNAKQDVLWLSEVEPFKNGNAIRGGVPICYPWFGGVKQPAHGTARIRLWQLSHYYISVHKVRLEFELFSDLNIIEAKVSMVFTDKCHLTFTHYGEESAQAALHTYFNIGDINQVEVQGLPETCFNSLNQQQENVPSPRHISENVDCIYSAENMQNQILDKSFNRTIALHHHNASQFVLWNPWHKKTSGMSETGYQKMLCLETARIHHLLEFGESLSVEISLKG
6GHS , Knot 139 311 0.85 40 189 296 
MAYKRTFGHIPGHPEGSTYSNRRQVQKAGLHAHLQAGISGTAKQGADAIVLNGGYPDDRDYGDEIIYTGHGGQDPVTKKQIRDQDLDDPGNAGLVRSQLEGLPVRVIRGAGGEKPYSPSSGYRYDGLYKVVAHWFANHEDAPQFRVCQFQLVKIYDQVAAGVVVDNPDLSATAESTSVQGPAPHKKTTVSKAVRSAQVVKNVKGWHKHRCQVCGIVIEVDVGPYSQGAHIRPLGRKHGGPDVESNMLCLCPNDHVRFDNGALYITDDLKVVNALNGEVIGPLRVHPRHVIDLDHIRYHRSQLPNIPLEGSS

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 \(c\).

\(|P_{f(1QXK_1)}(2) \setminus P_{f(1JOV_1)}(2)|=102\), \(|P_{f(1JOV_1)}(2) \setminus P_{f(1QXK_1)}(2)|=80\). 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 1:101000100100010111100010001001100110110000000000101100001010000000101011010010000110011110001011011100000111110011001010010011000000111000010101100010000010010100100000001101000011011110011011011101000101010011111000111100101011000111100000100101001110100101111001001010011110110111100010001001000010111001111101100110100
Pair \(Z_2\) Length of longest common subsequence
1QXK_1,1JOV_1 182 3
1QXK_1,6GHS_1 184 3
1JOV_1,6GHS_1 158 4

Newick tree

 
[
	1QXK_1:95.30,
	[
		1JOV_1:79,6GHS_1:79
	]:16.30
]

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 \((0.85)(0.8)(\frac{591 }{\log_{20} 591}-\frac{270}{\log_{20}270})=90.4\)
Status Protein1 Protein2 d d1/2
Query variables 1QXK_1 1JOV_1 117 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]