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Parikh vectors
2UXN_1 9DDN_1 6WAP_1 Letter Amino acid
37 8 15 Q Glutamine
73 25 18 L Leucine
31 22 15 I Isoleucine
39 15 9 S Serine
55 28 19 A Alanine
27 12 10 N Asparagine
15 5 6 H Histidine
45 15 18 G Glycine
11 9 11 M Methionine
27 14 4 F Phenylalanine
9 3 5 W Tryptophan
18 5 4 Y Tyrosine
32 13 11 R Arginine
30 5 7 D Aspartic acid
43 17 17 E Glutamic acid
34 10 16 T Threonine
53 11 15 V Valine
9 0 2 C Cysteine
41 8 11 K Lycine
37 5 18 P Proline 

2UXN_1|Chain A|LYSINE-SPECIFIC HISTONE DEMETHYLASE 1|HOMO SAPIENS (9606)
>9DDN_1|Chains A, B, C, D, E|Tol-Pal system protein TolQ|Escherichia coli (562)
>6WAP_1|Chain A|HIV-1 capsid protein|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
2UXN , Knot 262 666 0.85 40 278 616 
PSGVEGAAFQSRLPHDRMTSQEAACFPDIISGPQQTQKVFLFIRNRTLQLWLDNPKIQLTFEATLQQLEAPYNSDTVLVHRVHSYLERHGLINFGIYKRIKPLPTKKTGKVIIIGSGVSGLAAARQLQSFGMDVTLLEARDRVGGRVATFRKGNYVADLGAMVVTGLGGNPMAVVSKQVNMELAKIKQKCPLYEANGQAVPKEKDEMVEQEFNRLLEATSYLSHQLDFNVLNNKPVSLGQALEVVIQLQEKHVKDEQIEHWKKIVKTQEELKELLNKMVNLKEKIKELHQQYKEASEVKPPRDITAEFLVKSKHRDLTALCKEYDELAETQGKLEEKLQELEANPPSDVYLSSRDRQILDWHFANLEFANATPLSTLSLKHWDQDDDFEFTGSHLTVRNGYSCVPVALAEGLDIKLNTAVRQVRYTASGCEVIAVNTRSTSQTFIYKCDAVLCTLPLGVLKQQPPAVQFVPPLPEWKTSAVQRMGFGNLNKVVLCFDRVFWDPSVNLFGHVGSTTASRGELFLFWNLYKAPILLALVAGEAAGIMENISDDVIVGRCLAILKGIFGSSAVPQPKETVVSRWRADPWARGSYSYVAAGSSGNDYDLMAQPITPGPSIPGAPQPIPRLFFAGEHTIRNYPATVHGALLSGLREAGRIADQFLGAMYTL
9DDN , Knot 104 230 0.82 38 152 222 
MTDMNILDLFLKASLLVKLIMLILIGFSIASWAIIIQRTRILNAAAREAEAFEDKFWSGIELSRLYQESQGKRDNLTGSEQIFYSGFKEFVRLHRANSHAPEAVVEGASRAMRISMNRELENLETHIPFLGTVGSISPYIGLFGTVWGIMHAFIALGAVKQATLQMVAPGIAEALIATAIGLFAAIPAVMAYNRLNQRVNKLELNYDNFMEEFTAILHRQAFTVSESNKG
6WAP , Knot 103 231 0.81 40 158 221 
PIVQNLQGQMVHQAISPRTLNAWVKVVEEKAFSPEVIPMFSALSEGATPQDLNTMLNTVGGHQAAMQMLKETINEEAAEWDRLHPVHAGPIAPGQMREPRGSDIAGTTSTLQEQIGWMTHNPPIPVGEIYKRWIILGLNKIVRMYSPTSILDIRQGPKEPFRDYVDRFYKTLRAEQASQEVKNWMTETLLVQNANPDCKTILKALGPGATLEEMMTACQGVGGPGHKARVL

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 \(c\).

\(|P_{f(2UXN_1)}(2) \setminus P_{f(9DDN_1)}(2)|=158\), \(|P_{f(9DDN_1)}(2) \setminus P_{f(2UXN_1)}(2)|=32\). 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 1:101101111000110001000011011011011000001111100001011100101010101010010110000011100100010001110111000101110000101111101101111100100111010110100011101101001001101111110111101111100010101101000011001010111000001100010011010001000101011000110110110111010000100001001001100000100110011010001001000000100101100101011100000010110000001100010100010010101100101000000110101101011010110010100100000101010010100100011111101101010011001000101001111000000001100001110011111100011110111111010001100111101001110100111010101110110001001011111010011111111110111110010001111001111011110011101000110010101110100001111001000011101101110111110111011111000100011010111101100110110011111001
Pair \(Z_2\) Length of longest common subsequence
2UXN_1,9DDN_1 190 4
2UXN_1,6WAP_1 186 4
9DDN_1,6WAP_1 152 3

Newick tree

 
[
	2UXN_1:99.28,
	[
		6WAP_1:76,9DDN_1:76
	]:23.28
]

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 \((0.85)(0.8)(\frac{896 }{\log_{20} 896}-\frac{230}{\log_{20}230})=182.\)
Status Protein1 Protein2 d d1/2
Query variables 2UXN_1 9DDN_1 231 154
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]