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Parikh vectors
5JUO_1 6ZMB_1 4HBX_1 Letter Amino acid
0 6 7 Y Tyrosine
0 10 6 D Aspartic acid
0 18 8 E Glutamic acid
459 10 2 G Glycine
0 10 2 S Serine
0 10 7 I Isoleucine
0 9 12 P Proline
0 5 8 T Threonine
0 1 3 W Tryptophan
482 23 5 A Alanine
0 18 3 R Arginine
0 6 9 Q Glutamine
0 10 1 H Histidine
0 4 12 N Asparagine
0 4 5 M Methionine
0 12 5 V Valine
348 3 2 C Cysteine
0 28 13 L Leucine
0 7 12 K Lycine
0 6 5 F Phenylalanine 

5JUO_1|Chain A|18S ribosomal RNA|Saccharomyces cerevisiae (4932)
>6ZMB_1|Chains A[auth B], B[auth C]|tRNA hydroxylase|Pseudomonas putida KT2440 (160488)
>4HBX_1|Chain A|Bromodomain-containing protein 4|Homo sapiens (9606)
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 \(w=f(c)\)
5JUO , Knot 328 1798 0.45 8 16 64 
UAUCUGGUUGAUCCUGCCAGUAGUCAUAUGCUUGUCUCAAAGAUUAAGCCAUGCAUGUCUAAGUAUAAGCAAUUUAUACAGUGAAACUGCGAAUGGCUCAUUAAAUCAGUUAUCGUUUAUUUGAUAGUUCCUUUACUACAUGGUAUAACUGUGGUAAUUCUAGAGCUAAUACAUGCUUAAAAUCUCGACCCUUUGGAAGAGAUGUAUUUAUUAGAUAAAAAAUCAAUGUCUUCGGACUCUUUGAUGAUUCAUAAUAACUUUUCGAAUCGCAUGGCCUUGUGCUGGCGAUGGUUCAUUCAAAUUUCUGCCCUAUCAACUUUCGAUGGUAGGAUAGUGGCCUACCAUGGUUUCAACGGGUAACGGGGAAUAAGGGUUCGAUUCCGGAGAGGGAGCCUGAGAAACGGCUACCACAUCCAAGGAAGGCAGCAGGCGCGCAAAUUACCCAAUCCUAAUUCAGGGAGGUAGUGACAAUAAAUAACGAUACAGGGCCCAUUCGGGUCUUGUAAUUGGAAUGAGUACAAUGUAAAUACCUUAACGAGGAACAAUUGGAGGGCAAGUCUGGUGCCAGCAGCCGCGGUAAUUCCAGCUCCAAUAGCGUAUAUUAAAGUUGUUGCAGUUAAAAAGCUCGUAGUUGAACUUUGGGCCCGGUUGGCCGGUCCGAUUUUUUCGUGUACUGGAUUUCCAACGGGGCCUUUCCUUCUGGCUAACCUUGAGUCCUUGUGGCUCUUGGCGAACCAGGACUUUUACUUUGAAAAAAUUAGAGUGUUCAAAGCAGGCGUAUUGCUCGAAUAUAUUAGCAUGGAAUAAUAGAAUAGGACGUUUGGUUCUAUUUUGUUGGUUUCUAGGACCAUCGUAAUGAUUAAUAGGGACGGUCGGGGGCAUCAGUAUUCAAUUGUCAGAGGUGAAAUUCUUGGAUUUAUUGAAGACUAACUACUGCGAAAGCAUUUGCCAAGGACGUUUUCAUUAAUCAAGAACGAAAGUUAGGGGAUCGAAGAUGAUCAGAUACCGUCGUAGUCUUAACCAUAAACUAUGCCGACUAGGGAUCGGGUGGUGUUUUUUUAAUGACCCACUCGGCACCUUACGAGAAAUCAAAGUCUUUGGGUUCUGGGGGGAGUAUGGUCGCAAGGCUGAAACUUAAAGGAAUUGACGGAAGGGCACCACCAGGAGUGGAGCCUGCGGCUUAAUUUGACUCAACACGGGGAAACUCACCAGGUCCAGACACAAUAAGGAUUGACAGAUUGAGAGCUCUUUCUUGAUUUUGUGGGUGGUGGUGCAUGGCCGUUCUUAGUUGGUGGAGUGAUUUGUCUGCUUAAUUGCGAUAACGAACGAGACCUUAACCUACUAAAUAGUGGUGCUAGCAUUUGCUGGUUAUCCACUUCUUAGAGGGACUAUCGGUUUCAAGCCGAUGGAAGUUUGAGGCAAUAACAGGUCUGUGAUGCCCUUAGACGUUCUGGGCCGCACGCGCGCUACACUGACGGAGCCAGCGAGUCUAACCUUGGCCGAGAGGUCUUGGUAAUCUUGUGAAACUCCGUCGUGCUGGGGAUAGAGCAUUGUAAUUAUUGCUCUUCAACGAGGAAUUCCUAGUAAGCGCAAGUCAUCAGCUUGCGUUGAUUACGUCCCUGCCCUUUGUACACACCGCCCGUCGCUAGUACCGAUUGAAUGGCUUAGUGAGGCCUCAGGAUCUGCUUAGAGAAGGGGGCAACUCCAUCUCAGAGCGGAGAAUUUGGACAAACUUGGUCAUUUAGAGGAACUAAAAGUCGUAACAAGGUUUCCGUAGGUGAACCUGCGGAAGGAUCAU
6ZMB , Knot 91 200 0.80 40 133 188 
SLIPEIDAFLGCPTPDAWIEAALADQETLLIDHKNCEFKAASTALSLIAKYNTHLDLINMMSRLAREELVHHEQVLRLMKRRGVPLRPVSAGRYASGLRRLVRAHEPVKLVDTLVVGAFIEARSCERFAALVPHLDEELGRFYHGLLKSEARHYQGYLKLAHNYGDEADIARRVELVRAAEMELIQSPDQELRFHSGIPQ
4HBX , Knot 66 127 0.84 40 110 123 
SMNPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTEE

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 \(c\).

\(|P_{f(5JUO_1)}(2) \setminus P_{f(6ZMB_1)}(2)|=12\), \(|P_{f(6ZMB_1)}(2) \setminus P_{f(5JUO_1)}(2)|=129\). 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 1:0100011001100001001101100101010001000011111001110010101010001110101110110001010110111100101110110001001110011001001000100011011000000010010101101011001011011000011110011010101000111100001100000011111111010100010011101111110011010000011100000011011000101101100000011100101011000010100110110110001000111000001000010011000001101101111011011000100101100001101110110111111011111000110000111111111100011111101100100101000111111110110111010101110010001100001100011111110110110110111011011010111100010001110000101100111101110101101011101000011011111101100111111011100011010011011001011011000011000011011010101001111001001011001111110001011001110000111000110011001100011000000010101001110000011011110000000000011001100001110000010110000011011100111100000100001111111001111010001111011101010010001110101001101011110110111101111010001100001000010011000001111001001011011001101111101100111110100110100011001001111101111000001110001001111100110010010111110100010011111010000010011001111101111100111111001111101100111010010010110000110010111001010011001111100111011010000000110110001000110100001011111100111100000111000011111111010110010111100111100011111110011011111110100100111110111100010110001100011000110101111111000100111000111010110111110011011100111110000000001100001011101101101010110010000011001101111011000100010001100101101101110111100001100010011101101101001101000100110010001000000111111100100110000111001101111100011110110110111000101101000001110100001110010101010100101001101111001101110001100001100111111000011011000010111100001001010011111011110100101100100100000011011111100000110111010111001001100010100110010100000100000010101010010001001001101001100111011000110111100001111000100011111111111011000010000111101111110001110111000110010001111111001111100101101111000001011101110001011111110010
Pair \(Z_2\) Length of longest common subsequence
5JUO_1,6ZMB_1 141 2
5JUO_1,4HBX_1 126 1
6ZMB_1,4HBX_1 167 3

Newick tree

 
[
	6ZMB_1:81.47,
	[
		5JUO_1:63,4HBX_1:63
	]:18.47
]

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 \((0.85)(0.8)(\frac{1998 }{\log_{20} 1998}-\frac{200}{\log_{20}200})=458.\)
Status Protein1 Protein2 d d1/2
Query variables 5JUO_1 6ZMB_1 327 208
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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