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Parikh vectors
5WEK_1 1CZU_1 5TBW_1 Letter Amino acid
79 11 860 A Alanine
37 11 0 N Asparagine
25 1 0 M Methionine
78 10 0 V Valine
55 8 0 F Phenylalanine
66 8 0 T Threonine
45 2 0 R Arginine
13 0 0 H Histidine
85 12 0 S Serine
42 8 0 Y Tyrosine
90 18 885 G Glycine
60 13 0 I Isoleucine
94 12 0 L Leucine
66 8 0 K Lycine
49 16 0 D Aspartic acid
20 1 609 C Cysteine
32 12 0 Q Glutamine
66 11 0 E Glutamic acid
33 3 0 P Proline
22 4 0 W Tryptophan 

5WEK_1|Chains A, B, C, D|Chimera of Glutamate receptor 2,Germ cell-specific gene 1-like protein|Rattus norvegicus (10116)
>1CZU_1|Chain A|FLAVODOXIN|Synechococcus elongatus (1140)
>5TBW_1|Chains A[auth 1], TA[auth AR]|25S ribosomal RNA|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
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 \(w=f(c)\)
5WEK , Knot 390 1057 0.85 40 333 957 
NSIQIGGLFPRGADQEYSAFRVGMVQFSTSEFRLTPHIDNLEVANSFAVTNAFCSQFSRGVYAIFGFYDKKSVNTITSFCGTLHVSFITPSFPTDGTHPFVIQMRPDLKGALLSLIEYYQWDKFAYLYDSDRGLSTLQAVLDSAAEKKWQVTAINVGNINNDKKDETYRSLFQDLELKKERRVILDCERDKVNDIVDQVITIGKHVKGYHYIIANLGFTDGDLLKIQFGGAEVSGFQIVDYDDSLVSKFIERWSTLEEKEYPGAHTATIKYTSALTYDAVQVMTEAFRNLRKQRIEISRRGNAGDCLANPAVPWGQGVEIERALKQVQVEGLSGNIKFDQNGKRINYTINIMELKTNGPRKIGYWSEVDKMVLTEDDTSGLEQKTVVVTTILESPYVMMKKNHEMLEGNERYEGYCVDLAAEIAKHCGFKYKLTIVGDGKYGARDADTKIWNGMVGELVYGKADIAIAPLTITLVREEVIDFSKPFMSLGISIMIKKPQKSKPGVFSFLDPLAYEIWMCIVFAYIGVSVVLFLVSRFSPYEWHTEEFEDGRETQSSESTNEFGIFNSLWFSLGAFMQQGCDISPRSLSGRIVGGVWWFFTLIIISSYTANLAAFLTVERMVSPIESAEDLSKQTEIAYGTLDSGSTKEFFRRSKIAVFDKMWTYMRSAEPSVFVRTTAEGVARVRKSKGKYAYLLESTMNEYIEQRKPCDTMKVGGNLDSKGYGIATPKGSSLGTPVNLAVLKLSEQGVLDKLKNKWWYDKGECGAKDSGSKEKTSALSLSNVAGVFYILVGGLGLAMLVALIEFCYKSRAEAKRMKGTGKTSRRGRALLAVALNLLALLFATTAFLTTYWCQGTQRVPKPGCGQGGGANCPNSGANATANSTAAPVAASPAGAPYSWEAGDERFQLRRFHTGIWYSCEEELGGPGEKCRSFIDLAPASEKGVLWLSVVSEVLYILLLVVGFSLMCLELLHSSSVIDGLKLNAFAAVFTVLSGLLGMVAHMMYTQVFQVTVSLGPEDWRPHSWDYGWSFCLAWGSFTCCMAASVTTLNSYTKTVIEF
1CZU , Knot 82 169 0.83 38 129 161 
AKIGLFYGTQTGVTQTIAESIQQEFGGESIVDLNDIANADASDLNAYDYLIIGCPTWNVGELQSDWEGIYDDLDSVNFQGKKVAYFGAGDQVGYSDNFQDAMGILEEKISSLGSQTVGYWPIEGYDFNESKAVRNNQFVGLAIDEDNQPDLTKNRIKTWVSQLKSEFGL
5TBW , Knot 533 3149 0.45 8 16 64 
UUGACCUCAAAUCAGGUAGGAGUACCCGCUGAACUUAAGCAUAUCAAUAAGCGGAGGAAAAGAAACCAACCGGGAUUGCCUUAGUAACGGCGAGUGAAGCGGCAAAAGCUCAAAUUUGAAAUCUGGUACCUUCGGUGCCCGAGUUGUAAUUUGGAGAGGGCAACUUUGGGGCCGUUCCUUGUCUAUGUUCCUUGGAACAGGACGUCAUAGAGGGUGAGAAUCCCGUGUGGCGAGGAGUGCGGUUCUUUGUAAAGUGCCUUCGAAGAGUCGAGUUGUUUGGGAAUGCAGCUCUAAGUGGGUGGUAAAUUCCAUCUAAAGCUAAAUAUUGGCGAGAGACCGAUAGCGAACAAGUACAGUGAUGGAAAGAUGAAAAGAACUUUGAAAAGAGAGUGAAAAAGUACGUGAAAUUGUUGAAAGGGAAGGGCAUUUGAUCAGACAGGCCAGCAUCAGUUUUGGUGGCAGGAUAAAUCCAUAGGAAUGUAGCUUGCCUCGGUAAGUAUUAUAGCCUGUGGGAAUACUGCCAGCUGGGACUGAGGACUGCGACGUAAGUCAAGGAUGCUGGCAUAAUGGUUAUAUGCCGCCCGUCUUGAAACACGGACCAAGGAGUCUAACGUCUAUGCGAGUGUUUGGGUGUAAAACCCAUACGCGUAAUGAAAGUGAACGUAGGUUGGGGCCUCGCAAGAGGUGCACAAUCGACCGAUCCUGAUGUCUUCGGAUGGAUUUGAGUAAGAGCAUAGCUGUUGGGACCCGAAAGAUGGUGAACUAUGCCUGAAUAGGGUGAAGCCAGAGGAAACUCUGGUGGAGGCUCGUAGCGGUUCUGACGUGCAAAUCGAUCGUCGAAUUUGGGUAUAGGGGCGAAAGACUAAUCGAACCAUCUAGUAGCUGGUUCCUGCCGAAGUUUCCCUCAGGAUAGCAGAAGCUCGUAUCAGUUUUAUGAGGUAAAGCGAAUGAUUAGAGGUUCCGGGGUCGAAAUGACCUUGACCUAUUCUCAAACUUUAAAUAUGUAAGAAGUCCUUGUUACUUAAUUGAACGUGGACAUUUGAAUGAAGAGCUUUUAGUGGGCCAUUUUUGGUAAGCAGAACUGGCGAUGCGGGAUGAACCGAACGUAGAGUUAAGGUGCCGGAAUACACGCUCAUCAGACACCACAAAAGGUGUUAGUUCAUCUAGACAGCCGGACGGUGGCCAUGGAAGUCGGAAUCCGCUAAGGAGUGUGUAACAACUCACCGGCCGAAUGAACUAGCCCUGAAAAUGGAUGGCGCUCAAGCGUGUUACCUAUACUCUACCGUCAGGGUUGAUAUGAUGCCCUGACGAGUAGGCAGGCGUGGAGGUCAGUGACGAAGCCUAGACCGUAAGGUCGGGUCGAACGGCCUCUAGUGCAGAUCUUGGUGGUAGUAGCAAAUAUUCAAAUGAGAACUUUGAAGACUGAAGUGGGGAAAGGUUCCACGUCAACAGCAGUUGGACGUGGGUUAGUCGAUCCUAAGAGAUGGGGAAGCUCCGUUUCAAAGGCCUGAUUUUAUGCAGGCCACCAUCGAAAGGGAAUCCGGUUAAGAUUCCGGAACCUGGAUAUGGAUUCUUCACGGUAACGUAACUGAAUGUGGAGACGUCGGCGCGAGCCCUGGGAGGAGUUAUCUUUUCUUCUUAACAGCUUAUCACCCCGGAAUUGGUUUAUCCGGAGAUGGGGUCUUAUGGCUGGAAGAGGCCAGCACCUUUGCUGGCUCCGGUGCGCUUGUGACGGCCCGUGAAAAUCCACAGGAAGGAAUAGUUUUCAUGCCAGGUCGUACUGAUAACCGCAGCAGGUCUCCAAGGUGAACAGCCUCUAGUUGAUAGAAUAAUGUAGAUAAGGGAAGUCGGCAAAAUAGAUCCGUAACUUCGGGAUAAGGAUUGGCUCUAAGGGUCGGGUACGAUCAACUUAGAACUGGUACGGACAAGGGGAAUCUGACUGUCUAAUUAAAACAUAGCAUUGCGAUGGUCAGAAAGUGAUGUUGACGCAAUGUGAUUUCUGCCCAGUGCUCUGAAUGUCAAAGUGAAGAAAUUCAACCAAGCGCGGGUAAACGGCGGGAGUAACUAUGACUCUCUUAAGGUAGCCAAAUGCCUCGUCAUCUAAUUAGUGACGCGCAUGAAUGGAUUAACGAGAUUCCCACUGUCCCUAUCUACUAUCUAGCGAAACCACAGCCAAGGGAACGGGCUUGGCAGAAUCAGCGGGGAAAGAAGACCCUGUUGAGCUUGACUCUAGUUUGACAUUGUGAAGAGACAUAGUUUCUUUACUUAUUCAAUGAAGCGGAGCUGGAAUUCAUUUUCCACGUUCUAGCAUUCAAGGUCCCAUUCGGGGCUGAUCCGGGUUGAAGACAUUGUCAGGUGGGGAGUUUGGCUGGGGCGGCACAUCUGUUAAACGAUAACGCAGAUGUCCUAAGGGGGGCUCAUGGAGAACAGAAAUCUCCAGUAGAACAAAAGGGUAAAAGCCCCCUUGAUUUUGAUUUUCAGUGUGAAUACAAACCAUGAAAGUGUGGCCUAUCGAUCCUUUAGUCCCUCGGAAUUUGAGGCUAGAGGUGCCAGAAAAGUUACCACAGGGAUAACUGGCUUGUGGCAGUCAAGCGUUCAUAGCGACAUUGCUUUUUGAUUCUUCGAUGUCGGCUCUUCCUAUCAUACCGAAGCAGAAUUCGGUAAGCGUUGGAUUGUUCACCCACUAAUAGGGAACGUGAGCUGGGUUUAGACCGUCGUGAGACAGGUUAGUUUUACCCUACUGAUGAAUGUUACCGCAAUAGUAAUUGAACUUAGUACGAGAGGAACAGUUCAUUCGGAUAAUUGGUUUUUGCGGCUGUCUGAUCAGGCAUUGCCGCGAAGCUACCAUCCGCUGGAUUAUGGCUGAACGCCUCUAAGUCAGAAUCCAUGCUAGAACGCGGUGAUUUCUUUGCUCCACACAAUAUAGAUGGAUACGAAUAAGGCGUCCUUGUGGCGUCGCUGAACCAUAGCAGGCUAGCAACGGUGCACUUGGCGGAAAGGCCUUGGGUGCUUGCUGGCGAAUUGCAAUGUCAUUUUGCGUGGGGAUAAAUCAUUUGUAUACGACUUAGAUGUACAACGGGGUAUUGUAAGCAGUAGAGUAGCCUUGUUGUUACGAUCUGCUGAGAUUAAGCCUUUGUUGUCUGAUUUGU

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 \(c\).

\(|P_{f(5WEK_1)}(2) \setminus P_{f(1CZU_1)}(2)|=214\), \(|P_{f(1CZU_1)}(2) \setminus P_{f(5WEK_1)}(2)|=10\). 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 1:0010111111011000001101111010000101010100101100111001100010011011111000001001001010101011010110010011110101010111101100001001101000001100101110011000101011011010000000000011001010000011100000010011001101100101000111011100101101011110101101100000110011001001000001110010100001100011011001100100001010001011001101111110110100110010101101010100010010001011010001100110100100111000000110000111001100101110000011010000010010111011000110001011101001100100011011110110101011111101011000110100111011101110010000111101101110011101111011101111110010100100001001000000000001111001110111110010010100101011111111101111000010111110100110110010010000011010100100001100001111001100100101011100010111010000100101100010001000010001011101000101110101001101101111010001110010001100010011000100000011010011111011111111111111101000001010010101000001011111110111111100111000100100011011010111100100110101000111111011111001011000101001001110000001111100000110111100011111011001101111111101101011000011011010111111011011111110110001101010111001010010011010111101000111010010000001101
Pair \(Z_2\) Length of longest common subsequence
5WEK_1,1CZU_1 224 4
5WEK_1,5TBW_1 335 4
1CZU_1,5TBW_1 137 3

Newick tree

 
[
	5WEK_1:15.69,
	[
		1CZU_1:68.5,5TBW_1:68.5
	]:91.19
]

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 \((0.85)(0.8)(\frac{1226 }{\log_{20} 1226}-\frac{169}{\log_{20}169})=284.\)
Status Protein1 Protein2 d d1/2
Query variables 5WEK_1 1CZU_1 365 209
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]