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Parikh vectors
5LUI_1 7THY_1 6ERI_1 Letter Amino acid
26 13 749 A Alanine
15 12 0 R Arginine
14 8 0 N Asparagine
6 4 0 H Histidine
13 13 0 V Valine
12 14 0 D Aspartic acid
2 1 655 C Cysteine
4 9 0 Q Glutamine
13 8 0 E Glutamic acid
22 10 882 G Glycine
7 19 0 K Lycine
9 7 0 F Phenylalanine
20 11 0 T Threonine
10 5 0 Y Tyrosine
22 20 0 L Leucine
4 7 0 M Methionine
26 12 0 S Serine
17 12 0 I Isoleucine
19 6 0 P Proline
4 3 0 W Tryptophan 

5LUI_1|Chain A|Cutinase 1|Thermobifida cellulosilytica (144786)
>7THY_1|Chain A|Leucine-rich repeat serine/threonine-protein kinase 2|Homo sapiens (9606)
>6ERI_1|Chain A[auth AA]|23S ribosomal RNA|Spinacia oleracea (3562)
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 \(w=f(c)\)
5LUI , Knot 115 265 0.80 40 169 252 
MANPYERGPNPTDALLEASSGPFSVSEENVSRLSASGFGGGTIYYPRENNTYGAVAISPGYTGTEASIAWLGERIASHGFVVITIDTITTLDQPDSRAEQLNAALNHMINRASSTVRSRIDSSRLAVMGHSMGGGGTLRLASQRPDLKAAIPLTPWHLNKNWSSVTVPTLIIGADLDTIAPVATHAKPFYNSLPSSISKAYLELDGATHFAPNIPNKIIGKYSVAWLKRFVDNDTRYTQFLCPGPRDGLFGEVEEYRSTCPFALE
7THY , Knot 94 194 0.85 40 147 190 
YNRMKLMIVGNTGSGKTTLLQQLMKTKKSDLGMQSATVGIDVKDWPIQIRDKRKRDLVLNVWDFAGREEFYSTHPHFMTQRALYLAVYDLSKGQAEVDAMKPWLFNIKARASSSPVILVGTHLDVSDEKQRKACMSKITKELLNKRGFPAIRDYHFVNATEESDALAKLRKTIINESLNFKIRDQLVVGQLIPD
6ERI , Knot 477 2810 0.44 8 16 64 
UUCAAACGAGGAAAGGCUUACGGUGGAUACCUAGGCACCCAGAGACGAGGAAGGGCGUAUUAAUCGACGAAAUGCUUCGGGGAGUUGAAAAUAAGCAGAGAUCCGGAGAUUCCCGAAUAGGUCAACCUUUCGAACUUCUGCUGAAUCCAUGGGCAGGCAAGAGACAACCUGGCGAACUGAAACAUCUUAGUAGCCAGAGGAAAAGAAAGCAAAAGCGAUUCCCGUAGUAGCGGCGAGCGAAAUGGGAGCAGCCUAAACCGUGAAAACGGGGUUGUGGGAGAGCAAUACAAGCGUCGUGCUGCUAGGCGAAUCAGUGGAGUGCGGAACCCUAGAUGGUGAAAGUCCAGUAGCCGAAAGCAUCACUAGCUUAUGCUCUGACCCGAGUAGCAUGGGGCACGUGGAAUCCCGUGUGAAUCAGCAAGGACCACCUUGCAAGGCUAAAUACUCCUGGGUGACCGAUAGCGAAGUAGUACCGUGAGGGAAGGGUGAAAAGAACCCCCAUCGGGGAGUGAAAUAGAACAUGAAACCGUAAGCUCUCAAGCAGUGGGAGGGGGACCAGACCCUGACCGCGUGCCUGUUGAAGAAUGAGCCGGCGACUCAUAGGCAGUGGCUUGGUUAAGGGAACCCACCGGAGCCGUAGCGAAAGCGAGUCUUCAUAGGGCAAUUGUCACUGCUUAUGGACCCGAACCUGGGUGAUCUAUCCAUGACCAGGAUGAAGCUUGGGUGAAACUAAGUGGAGGUCCGAACCGACUGAUGUUGAAGAAUCAGCGGAUGAGUUGUGGUUAGGGGUGAAAUGCCACUCGAACCCAGAGCUAGCUGGUUCUCCCCGAAAUGCGUUGAGGCGCAGCAGUUGACUGGACAUCUAGGGGUAAAGCACUGUUUCGGUGCGGGCCGCGAGAGCGGUACCAAAUCGAGGCAAACUCUGAAUACUAGAUAUGACCUCCAAAUAACAGGGGUCAAGGUCGGCCAGUGAGACGAUGGGGGAUAAGCUUCAUCGUCGAGAGGGAAACAGCCCGGAUCACCAGCUAAGGCCCCUAAAUGACCGCUCAGUGAUAAAGGAGGUAGGGGUGCAGAGACAGCCAGGAGGUUUGCCUAGAAGCAGCCACCCUUGAAAGAGUGCGUAAUAGCUCACUGAUCGAGCGCUCUUGCGCCGAAGAUGAACGGGGCUAAGCGGUCUGCCGAAGCUGUGGGAUGUAAAAAAACAUCGGUAGGGGAGCGUUCCGUGUUAGGGAGAAACGCGUGCGUGAGCCGCGUUGGACGAAGCGGAAGCGAGAAUGUCGGCUUGAGUAACGCAAACAUUGGUGAGAAUCCAAUGCCCCGAAAACCUAAGGGUUCCUCCGCAAGGUUCGUCCACGGAGGGUGAGUCAGGGCCUAAGAUCAGGCCGAAAGGCGUAGUCGAUGGACAACAGGUGAAUAUUCCUGUACUACCCCUUGUUGGUCCCGAGGGACGGAGGAGGCUAGGUUAGCCGAAAGAUGGUUAUCGGUUCAAGGACGCAAGGUGACCCUGUUUUUCAGGGUAAGAAGGGGUAGAGAAAAUGCCUCGAGCCAAUGUUCGAGUACCAGGCGCUACGGCGCUGAAGUAACCGAUGCCAUACUCCCAGGAAAAGCUCGAACGACCUUCAACAAAAGGGUACCUGUACCCGAAACCGACACAGGUAGGUAGGUAGAGAAUACCUAGGGGCGCGAGACAACUCUCUCUAAGGAACUCGGCAAAAUAGCCCCGUAACUUCGGGAGAAGGGGUGCCCCCUCACAAAGGGGGUCGAAGUGACCAGGCCCGGGCGACUGUUUACCAAAAACACAGGUCUCCGCAAAGUCGUAAGACCAUGUAUGGGGGCUGACGCCUGCCCAGUGCCGGAAGGUCAAGGAAGUUGGUGACCUGAUGACAGGGGAGCCGGCGACCGAAGCCCCGGUGAACGGCGGCCGUAACUAUAACGGUCCUAAGGUAGCGAAAUUCCUUGUCGGGUAAGUUCCGACCCGCACGAAAGGCGUAACGAUCUGGGCACUGUCUCGGAGAGAGGCUCGGUGAAAUAGACAUGUCUGUGAAGAUGCGGACUACCUGCACCUGGACAGAAAGACCCUAUGAAGCUUUACUGUUCCCUGGGAUUGGCUUUGGGCUUUUCCUGCGCAGCUUAGGUGGAAGGCGAAGAAGGCCCCCUUCCGGGGGGGCCCGAGCCAUCAGUGAGAUACCACUCUGGAAGAGCUAGAAUUCUAACCUUGUGUCAGGACCUACGGGCCAAGGGACAUUCUCAGGUAGACAGUUUCUAUGGGGCGUAGGCCUCCCAAAAGGUAACGGAGGCGUGCAAAGGUUUCCUCGGGCCGGACGGAGAUUGGCCCUCGAGUGCAAAGGCAGAAGGGAGCUUGACUGCAAGACCCACCCGUCGAGCAGGGACGAAAGUCGGCCUUAGUGAUCCGACGGUGCCGAGUGGAAGGGCCGUCGCUCAACGGAUAAAAGUUACUCUAGGGAUAACAGGCUGAUCUUCCCCAAGAGUUCACAUCGACGGGAAGGUUUGGCACCUCGAUGUCGGCUCUUCGCCACCUGGGGCUGUAGUAUGUUCCAAGGGUUGGGCUGUUCGCCCAUUAAAGCGGUACGUGAGCUGGGUUCAGAACGUCGUGAGACAGUUCGGUCCAUAUCCGGUGUGGGCGUUAGAGCAUUGAGAGGACCUUUCCCUAGUACGAGAGGACCGGGAAGGACGCACCUCUGGUGUACCAGUUAUCGUGCCCACGGUAAACGCUGGGUAGCCAAGUGCGGAGCGGAUAACUGCUGAAAGCAUCUAAGUAGUAAGCCCACCCCAAGAUGAGUGCUCUCCUA

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 \(c\).

\(|P_{f(5LUI_1)}(2) \setminus P_{f(7THY_1)}(2)|=91\), \(|P_{f(7THY_1)}(2) \setminus P_{f(5LUI_1)}(2)|=69\). 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 1:1101000110100111010011101000010010101111101001000000111110110010010111110011001111101001001001000100101110011001000100010000111110011111010110001010111110110100010010110111110100111110010110001100100101010110011101100111000111100110000000011011100111101000000011110
Pair \(Z_2\) Length of longest common subsequence
5LUI_1,7THY_1 160 4
5LUI_1,6ERI_1 179 4
7THY_1,6ERI_1 159 2

Newick tree

 
[
	5LUI_1:86.60,
	[
		7THY_1:79.5,6ERI_1:79.5
	]:7.10
]

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 \((0.85)(0.8)(\frac{459 }{\log_{20} 459}-\frac{194}{\log_{20}194})=77.5\)
Status Protein1 Protein2 d d1/2
Query variables 5LUI_1 7THY_1 99 86
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]