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Parikh vectors
5XZG_1 5PCA_1 5NRG_1 Letter Amino acid
43 14 0 K Lycine
5 6 0 M Methionine
5 2 0 W Tryptophan
16 6 0 P Proline
14 6 0 N Asparagine
8 2 597 C Cysteine
11 2 0 Q Glutamine
7 8 0 H Histidine
21 4 0 I Isoleucine
40 13 0 L Leucine
22 8 0 F Phenylalanine
14 8 0 T Threonine
11 4 0 Y Tyrosine
16 12 0 D Aspartic acid
38 9 0 E Glutamic acid
18 8 0 V Valine
12 4 811 A Alanine
20 5 0 R Arginine
15 6 863 G Glycine
26 11 0 S Serine 

5XZG_1|Chain A|Cyclic GMP-AMP synthase|Mus musculus (10090)
>5PCA_1|Chain A|Bromodomain adjacent to zinc finger domain protein 2B|Homo sapiens (9606)
>5NRG_1|Chain A[auth X]|23S ribosomal RNA|Staphylococcus aureus subsp. aureus NCTC 8325 (93061)
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 \(w=f(c)\)
5XZG , Knot 154 362 0.83 40 197 342 
SPDKLKKVLDKLRLKRKDISEAAETVNKVVERLLRRMQKRESEFKGVEQLNTGSYYEHVKISAPNEFDVMFKLEVPRIELQEYYETGAFYLVKFKRIPRGNPLSHFLEGEVLSATKMLSKFRKIIKEEVKEIKDIDVSVEKEKPGSPAVTLLIRNPEEISVDIILALESKGSWPISTKEGLPIQGWLGTKVRTNLRREPFYLVPKNAKDGNSFQGETWRLSFSHTEKYILNNHGIEKTCCESSGAKCCRKECLKLMKYLLEQLKKEFQELDAFCSYHVKTAIFHMWTQDPQDSQWDPRNLSSCFDKLLAFFLECLRTEKLDHYFIPKFNLFSQELIDRKSKEFLSKKIEYERNNGFPIFDKL
5PCA , Knot 66 138 0.78 40 109 129 
MHHHHHHSSGVDLGTENLYFQSMSVKKPKRDDSKDLALCSMILTEMETHEDAWPFLLPVNLKLVPGYKKVIKKPMDFSTIREKLSSGQYPNLETFALDVRLVFDNCETFNEDDSDIGRAGHNMRKYFEKKWTDTFKVS
5NRG , Knot 501 2923 0.45 8 16 64 
GAUUAAGUUAUUAAGGGCGCACGGUGGAUGCCUUGGCACUAGAAGCCAAUGAAGGACGUUACUAACGACGAUAUGCUUUGGGGAGCUGUAAGUAAGCUUUGAUCCAGAGAUUUCCGAAUGGGGAAACCCAGCAUGAGUUAUGUCAUGUUAUCGAUAUGUGAAUACAUAGCAUAUCAGAAGGCACACCCGGAGAACUGAAACAUCUUAGUACCCGGAGGAAGAGAAAGAAAAUUCGAUUCCCUUAGUAGCGGCGAGCGAAACGGGAAGAGCCCAAACCAACAAGCUUGCUUGUUGGGGUUGUAGGACACUCUAUACGGAGUUACAAAGGACGACAUUAGACGAAUCAUCUGGAAAGAUGAAUCAAAGAAGGUAAUAAUCCUGUAGUCGAAAAUGUUGUCUCUCUUGAGUGGAUCCUGAGUACGACGGAGCACGUGAAAUUCCGUCGGAAUCUGGGAGGACCAUCUCCUAAGGCUAAAUACUCUCUAGUGACCGAUAGUGAACCAGUACCGUGAGGGAAAGGUGAAAAGCACCCCGGAAGGGGAGUGAAAUAGAACCUGAAACCGUGUGCUUACAAGUAGUCAGAGCCCGUUAAUGGGUGAUGGCGUGCCUUUUGUAGAAUGAACCGGCGAGUUACGAUUUGAUGCAAGGUUAAGCAGUAAAUGUGGAGCCGUAGCGAAAGCGAGUCUGAAUAGGGCGUUUAGUAUUUGGUCGUAGACCCGAAACCAGGUGAUCUACCCUUGGUCAGGUUGAAGUUCAGGUAACACUGAAUGGAGGACCGAACCGACUUACGUUGAAAAGUGAGCGGAUGAACUGAGGGUAGCGGAGAAAUUCCAAUCGAACCUGGAGAUAGCUGGUUCUCUCCGAAAUAGCUUUAGGGCUAGCCUCAAGUGAUGAUUAUUGGAGGUAGAGCACUGUUUGGACGAGGGGCCCCUCUCGGGUUACCGAAUUCAGACAAACUCCGAAUGCCAAUUAAUUUAACUUGGGAGUCAGAACAUGGGUGAUAAGGUCCGUGUUCGAAAGGGAAACAGCCCAGACCACCAGCUAAGGUCCCAAAAUAUAUGUUAAGUGGAAAAGGAUGUGGCGUUGCCCAGACAACUAGGAUGUUGGCUUAGAAGCAGCCAUCAUUUAAAGAGUGCGUAAUAGCUCACUAGUCGAGUGACACUGCGCCGAAAAUGUACCGGGGCUAAACAUAUUACCGAAGCUGUGGAUUGUCCUUUGGACAAUGGUAGGAGAGCGUUCUAAGGGCGUUGAAGCAUGAUCGUAAGGACAUGUGGAGCGCUUAGAAGUGAGAAUGCCGGUGUGAGUAGCGAAAGACGGGUGAGAAUCCCGUCCACCGAUUGACUAAGGUUUCCAGAGGAAGGCUCGUCCGCUCUGGGUUAGUCGGGUCCUAAGCUGAGGCCGACAGGCGUAGGCGAUGGAUAACAGGUUGAUAUUCCUGUACCACCUAUAAUCGUUUUAAUCGAUGGGGGGACGCAGUAGGAUAGGCGAAGCGUGCGAUUGGAUUGCACGUCUAAGCAGUAAGGCUGAGUAUUAGGCAAAUCCGGUACUCGUUAAGGCUGAGCUGUGAUGGGGAGAAGACAUUGAGUCUUCGAGUCGUUGAUUUCACACUGCCGAGAAAAGCCUCUAGAUAGAAAAUAGGUGCCCGUACCGCAAACCGACACAGGUAGUCAAGAUGAGAAUUCUAAGGUGAGCGAGCGAACUCUCGUUAAGGAACUCGGCAAAAUGACCCCGUAACUUCGGGAGAAGGGGUGCUCUUUAGGGUUAACGCCCAGAAGAGCCGCAGUGAAUAGGCCCAAGCGACUGUUUAUCAAAAACACAGGUCUCUGCUAAACCGUAAGGUGAUGUAUAGGGGCUGACGCCUGCCCGGUGCUGGAAGGUUAAGAGGAGUGGUUAGCUUCUGCGAAGCUACGAAUCGAAGCCCCAGUAAACGGCGGCCGUAACUAUAACGGUCCUAAGGUAGCGAAAUUCCUUGUCGGGUAAGUUCCGACCCGCACGAAAGGCGUAACGAUUUGGGCACUGUCUCAACGAGAGACUCGGUGAAAUCAUAGUACCUGUGAAGAUGCAGGUUACCCGCGACAGGACGGAAAGACCCCGUGGAGCUUUACUGUAGCCUGAUAUUGAAAUUCGGCACAGCUUGUACAGGAUAGGUAGGAGCCUUUGAAACGUGAGCGCUAGCUUACGUGGAGGCGCUGGUGGGAUACUACCCUAGCUGUGUUGGCUUUCUAACCCGCACCACUUAUCGUGGUGGGAGACAGUGUCAAGCGGGCAGUUUGACUGGGGCGGUCGCCUCCUAAAAGGUAACGGAGGCGCUCAAAGGUUCCCUCAGAAUGGUUGGAAAUCAUUCAUAGAGUGUAAAGGCAUAAGGGAGCUUGACUGCGAGACCUACAAGUCGAGCAGGGUCGAAAGACGGACUUAGUGAUCCGGUGGUUCCGCAUGGAAGGGCCAUCGCUCAACGGAUAAAAGCUACCCCGGGGAUAACAGGCUUAUCUCCCCCAAGAGUUCACAUCGACGGGGAGGUUUGGCACCUCGAUGUCGGCUCAUCGCAUCCUGGGGCUGUAGUCGGUCCCAAGGGUUGGGCUGUUCGCCCAUUAAAGCGGUACGCGAGCUGGGUUCAGAACGUCGUGAGACAGUUCGGUCCCUAUCCGUCGUGGGCGUAGGAAAUUUGAGAGGAGCUGUCCUUAGUACGAGAGGACCGGGAUGGACAUACCUCUGGUGUACCAGUUGUCGUGCCAACGGCAUAGCUGGGUAGCUAUGUGUGGACGGGAUAAGUGCUGAAAGCAUCUAAGCAUGAAGCCCCCCUCAAGAUGAGAUUUCCCAACUUCGGUUAUAAGAUCCCUCAAAGAUGAUGAGGUUAAUAGGUUCGAGGUGGAAGCAUGGUGACAUGUGGAGCUGACGAAUACUAAUCGAUCGAAGACUUAAUCAA

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 \(c\).

\(|P_{f(5XZG_1)}(2) \setminus P_{f(5PCA_1)}(2)|=137\), \(|P_{f(5PCA_1)}(2) \setminus P_{f(5XZG_1)}(2)|=49\). 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 1:01001001100101000010011001001100110010000001011001001000001010110010111010110101000000111011010011010110011010110100110010011000100100101010000110111011100100101011111000101110000111101111001000100011011100100100101001010100000011000110000000011000000010110011001000100101100001001110110001000010100100010011111100100001000111010110001100000011000100000011111001
Pair \(Z_2\) Length of longest common subsequence
5XZG_1,5PCA_1 186 3
5XZG_1,5NRG_1 207 2
5PCA_1,5NRG_1 123 2

Newick tree

 
[
	5XZG_1:10.92,
	[
		5PCA_1:61.5,5NRG_1:61.5
	]:46.42
]

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 \((0.85)(0.8)(\frac{500 }{\log_{20} 500}-\frac{138}{\log_{20}138})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 5XZG_1 5PCA_1 138 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]