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Parikh vectors
2OYF_1 8ETI_1 5LPT_1 Letter Amino acid
7 0 24 D Aspartic acid
5 0 17 P Proline
5 0 18 T Threonine
6 946 38 A Alanine
5 0 32 R Arginine
13 0 14 K Lycine
7 0 30 S Serine
1 0 4 W Tryptophan
9 0 8 Y Tyrosine
7 0 24 E Glutamic acid
5 0 22 I Isoleucine
11 967 33 G Glycine
2 0 17 M Methionine
4 0 20 F Phenylalanine
3 0 18 V Valine
14 678 5 C Cysteine
1 0 17 Q Glutamine
9 0 27 L Leucine
6 0 8 N Asparagine
1 0 10 H Histidine 

2OYF_1|Chain A|Phospholipase A2 VRV-PL-VIIIa|Daboia russellii pulchella (97228)
>8ETI_1|Chain A[auth 1]|RNA (1564-MER)|Schizosaccharomyces pombe (4896)
>5LPT_1|Chains A, B|Queuine tRNA-ribosyltransferase|Zymomonas mobilis subsp. mobilis ZM4 = ATCC 31821 (264203)
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 \(w=f(c)\)
2OYF , Knot 60 121 0.79 40 95 118 
SLLEFGKMILEETGKLAIPSYSSYGCYCGWGGKGTPKDATDRCCFVHDCCYGNLPDCNPKSDRYKYKRVNGAIVCEKGTSCENRICECDKAAAICFRQNLNTYSKKYMLYPDFLCKGELKC
8ETI , Knot 573 3497 0.44 8 16 64 
AUUUGACCUCAAAUCAGGUAGGACUACGCGCUGAACUUAAGCAUAUCAAUAAGCGCAGGAAAAGAAAAUAACCAUGAUUCCCUCAGUAACGGCGAGUGAAGCGGGAAAAGCUCAAAUUUGAAAUCUGGCAACAUUUCUUUUGUUGUCCGAGUUGUAAUUUCAAGAAGCUGCUUUGAGUGUAGACGAUCGGUCUAAGUUCCUUGGAACAGGACGUCAGAGAGGGUGAGAACCCCGUCUUUGGUCGAUUGGAUAUGCCAUAUAAAGCGCUUUCGAAGAGUCGAGUUGUUUGGGAAUGCAGCUCUAAAUGGGUGGUAAAUUUCAUCUAAAGCUAAAUAUUGGCGAGAGACCGAUAGCGAACAAGUAGAGUGAUCGAAAGAUGAAAAGAACUUUGAAAAGAGAGUUAAAUAGUACGUGAAAUUGCUGAAAGGGAAGCAUUGGAAAUCAGUCUUACCUGGGUGAGAUCAGUAGUCUCUUCGCGAGACUAUGCACUCUGAACCUGGGUUAGGUCAGCAUCAGUUUUCGGGGGCGGAAAAAGAAUAAGGGAAGGUGGCUUUCCGGGUUCUGCCUGGGGAGUGUUUAUAGCCCUUGUUGUAAUACGUCCACUGGGGACUGAGGACUGCGGCUUCGUGCCAAGGAUGCUGACAUAAUGGUUUUCAAUGGCCCGUCUUGAAACACGGACCAAGGAGUCUAGCAUCUAUGCGAGUGUUUGGGUGAUGAAAACCCAUCCGCGAAAUGAAAGUGAAUGCAGGUGGGAACGCCCUUGUGGCGUGCACCAUCGACCGACCCGGAAGUUUGUCAAUGGAAGGGUUUGAGUAAGAGCAUAGCUGUUGGGACCCGAAAGAUGGUGAACUAUGCCUGAAUAGGGUGAAGCCAGAGGAAACUCUGGUGGAGGCUCGUAGAGAUUCUGACGUGCAAAUCGAUCUUCAAAUUUGGGUAUAGGGGCGAAAGACUAAUCGAACCAUCUAGUAGCUGGUUCCUGCCGAAGUUUCCCUCAGGAUAGCAGAAACUCAGAUCAGUUUUAUGAGGUAAAGCGAAUGAUUAGAGGUCUUGGGGAAGGAAUUUCCUCAACCUAUUCUCAAACUUUAAAUAUGUAAGACGCCCUUGUCGCUUAAUUGGACGUGGGCCAUCGAAUGAGAGUUUCUAGUGGGCCAUUUUUGGUAAGCAGAACUGGCGAUGCGGGAUGAACCGAACGUGAGGUUAAGGUGCCGGAAUGUACGCUCAUCAGACACCAGAAAAGGUGUUAGUUCAUCUAGACAGCAGGACGGUGGCCAUGGAAGUCGGAAUCCGCUAAGGAGUGUGUAACAACUCACCUGCCGAAUGAACUAGCCCUGAAAAUGGAUGGCGCUUAAGCGUACUACCCAUACCUCACCGUCUGGGUUAGCUUUGAGAAGCUCAGACGAGUAGGCAGGCGUGGAGGUUUGUGACGAAGCCUUGGGCGUGAGCCUGGGUCGAACAGCCUCUAGUGCAGAUCUUGGUGGAAGUAGCAAAUAUUCAAAUGAGAACUUUGAAGACUGAAGUGGGGAAAGGUUCCAUGUGAACAGCAGUUGGACAUGGGUUAGUCGAUCCUAAGAGAUAGGGAAGCUCCGUAUGAAAGUUGCACGAUUUUUCGUGCCUCCUAUCGAAAGGGAAUCCGGUUAAUAUUCCGGAACCAGAAGGUGGAAUCAACACGGCAACGUAAAUGAAGUUGGAGACGUCGGCGGGAGCCCUGGGAAGAGUUCUCUUUUCUUUUUAACAAACCAUUGAACUACCCUGAAAUCGGUUUAUCCGGAGCUAGGGUAUGGUGUUUGGAAGAGUUCAGCGCCUCAUGCUGAAUCCGGUGCGCUCUCGACGGCCCUUGAAAAUCCAACGGAAGAAUGGACCUUCGGGUCCUUGUUUUCACAUCUGGUCGUACUCAUAACCGCAGCAGGUCUCCAAGGUGAACAGCCUCUAGUUGAUAGAACAAUGUAGAUAAGGGAAGUCGGCAAAAUGGAUCCGUAACUUCGGGAUAAGGAUUGGCUCUAAGGGUUGGGUACGUUGGGCCUUGGAACCUGAACGGUUGCUGGACUGAGCGUGGACCGAUGUCUUUUCUCGCCUUUCGGGGUGAGAAGGGAUGUUGGACCUGCUUGGACCUUGGCGGCCGGGAAGUCCUUGGUCGGGCUUUUCUCCUUCUCGGGGAUUAUGCUCUUACUGGCGUACGUUUAACAACCAACUUAGAACUGGUACGGACAAGGGGAAUCUGACUGUCUAAUUAAAACAUAGCAUUGCGAUGGCCAGAAAGUGGUGUUGACGCAAUGUGAUUUCUGCCCAGUGCUCUGAAUGUCAAAGUGAAGAAAUUCAACCAAGCGCGGGUAAACGGCGGGAGUAACUAUGACUCUCUUAAGGUAGCCAAAUGCCUCGUCAUCUAACUAGUGACGCGCAUGAAUGGAUUAACGAGAUUCCCACUGUCCCUAUCUACUAUCUAGCGAAACCACAGCCUGGGGAACGGGCCAGGCAAAAUCAGCGGGGAAAGAAGACCCUGUUGAGCUUGACUCUAGUUUGACAUUGUGAAGAGACAUAGAGGGUGUAGGAUAAGUGGGAGUAUGUUUCGGCAUACGCCGGUGAAAUACCACUACCUUUAUCGUUUCUUUACUUAAUCAAUGAAGCGGAAUUGGGAUUUAUUUCCCAUAUUCUAGCGUUAAAGUUUCUUCGCGAACUGAUCCGCGUUGAUGACAUUGUCAGGUGGGGAGUUUGGCUGGGGCGGCACAUCUGUUAAAAGAUAACGCAGGUGUCCUAAGGGGGACUCAUCGAGAACAGAAAUCUCGAGUAGAAUAAAAGGGUAAAAGUCCCCUUGAUUUUGAUUUUCAGUGUGAAUACAAACCAUGAAAGUGUGGCCUAUCGAUCCUUUGUUCCCUCGAAAUUUGAGGACAGAGGUGCCAGAAAAGUUACCACAGGGAUAACUGGCUUGUGGCAGUCAAGCGUUCAUAGCGACGUUGCUUUUUGAUUCUUCGAUGUCGGCUCUUCCUAUCAUACCGAAGCAGAAUUCGGUAAGCGUUGGAUUGUUCACCCACUAAUAGGGAACGUGAGCUGGGUUUAGACCGUCGUGAGACAGGUUAGUUUUACCCUACUGAUGAAGUGUCGUCGCAAUGGUAAUUCAACUUAGUACGAGAGGAACCGUUGAUUCAGAUCAUUGGUAUUUGCGGCUGCCUGACAAGGCAAUGCCGCGGAGCUAUCAUCUGCCGGAUAACGGCUGAACGCCUCUAAGCCAGAAUCCGUGCCAGAAAGCGACGAUUUUUUGGUCCGCAUGAUUUAUAUGUAUAAAAAUAGAGGUAGGACUUGUUCCUACUCUCCUGUAUCGUAGAAGAUGGGCGAUGGUUGAUGAAACGGAAGUGUUUUAUUGACUUGUCCAUGAAAUUCCAUUGAAAUCUUGUGCGGAAUCGAAUCCAUUGCAUACGACUUUAAUGUGGAACGGGGUAUUGUAAGCAGUAGAGUAGCCUUGUUGUUACGAUCUGCUGAGAUUAAGCCUUUGUUCCCAAGAUUUG
5LPT , Knot 161 386 0.82 40 215 372 
MVEATAQETDRPRFSFSIAAREGKARTGTIEMKRGVIRTPAFMPVGTAATVKALKPETVRATGADIILGNTYHLMLRPGAERIAKLGGLHSFMGWDRPILTDSGGYQVMSLSSLTKQSEEGVTFKSHLDGSRHMLSPERSIEIQHLLGSDIVMAFDECTPYPATPSRAASSMERSMRWAKRSRDAFDSRKEQAENAALFGIQQGSVFENLRQQSADALAEIGFDGYAVGGLAVGEGQDEMFRVLDFSVPMLPDDKPHYLMGVGKPDDIVGAVERGIDMFDCVLPTRSGRNGQAFTWDGPINIRNARFSEDLKPLDSECHCAVCQKWSRAYIHHLIRAGEILGAMLMTEHNIAFYQQLMQKIRDSISEGRFSQFAQDFRARYFARNS

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 \(c\).

\(|P_{f(2OYF_1)}(2) \setminus P_{f(8ETI_1)}(2)|=89\), \(|P_{f(8ETI_1)}(2) \setminus P_{f(2OYF_1)}(2)|=10\). 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 1:0110110111000101111000001000111101010010000011000001011000100000000010111100010000001000001111010001000000011010110010100
Pair \(Z_2\) Length of longest common subsequence
2OYF_1,8ETI_1 99 3
2OYF_1,5LPT_1 212 3
8ETI_1,5LPT_1 221 2

Newick tree

 
[
	5LPT_1:12.71,
	[
		2OYF_1:49.5,8ETI_1:49.5
	]:72.21
]

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 \((0.85)(0.8)(\frac{3618 }{\log_{20} 3618}-\frac{121}{\log_{20}121})=848.\)
Status Protein1 Protein2 d d1/2
Query variables 2OYF_1 8ETI_1 572 314
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]