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Parikh vectors
4LSK_1 2DLW_1 2XFP_1 Letter Amino acid
0 8 52 L Leucine
0 3 31 K Lycine
0 1 16 M Methionine
0 5 29 P Proline
0 2 20 D Aspartic acid
428 2 9 C Cysteine
0 3 13 Q Glutamine
545 15 42 G Glycine
0 3 23 Y Tyrosine
0 11 31 R Arginine
0 8 14 F Phenylalanine
0 6 35 T Threonine
0 3 40 V Valine
0 3 17 N Asparagine
0 13 42 E Glutamic acid
0 2 13 H Histidine
0 13 20 S Serine
311 8 34 A Alanine
0 2 28 I Isoleucine
0 2 11 W Tryptophan 

4LSK_1|Chains A[auth QA], EB[auth XA]|16S rRNA|Thermus thermophilus (300852)
>2DLW_1|Chain A|Docking protein 2, isoform a|Homo sapiens (9606)
>2XFP_1|Chains A, B|Amine oxidase [flavin-containing] B|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 \(w=f(c)\)
4LSK , Knot 274 1522 0.44 8 16 63 
UUUGUUGGAGAGUUUGAUCCUGGCUCAGGGUGAACGCUGGCGGCGUGCCUAAGACAUGCAAGUCGUGCGGGCCGCGGGGUUUUACUCCGUGGUCAGCGGCGGACGGGUGAGUAACGCGUGGGUGACCUACCCGGAAGAGGGGGACAACCCGGGGAAACUCGGGCUAAUCCCCCAUGUGGACCCGCCCCUUGGGGUGUGUCCAAAGGGCUUUGCCCGCUUCCGGAUGGGCCCGCGUCCCAUCAGCUAGUUGGUGGGGUAAUGGCCCACCAAGGCGACGACGGGUAGCCGGUCUGAGAGGAUGGCCGGCCACAGGGGCACUGAGACACGGGCCCCACUCCUACGGGAGGCAGCAGUUAGGAAUCUUCCGCAAUGGGCGCAAGCCUGACGGAGCGACGCCGCUUGGAGGAAGAAGCCCUUCGGGGUGUAAACUCCUGAACCCGGGACGAAACCCCCGACGAGGGGACUGACGGUACCGGGGUAAUAGCGCCGGCCAACUCCGUGCCAGCAGCCGCGGUAAUACGGAGGGCGCGAGCGUUACCCGGAUUCACUGGGCGUAAAGGGCGUGUAGGCGGCCUGGGGCGUCCCAUGUGAAAGACCACGGCUCAACCGUGGGGGAGCGUGGGAUACGCUCAGGCUAGACGGUGGGAGAGGGUGGUGGAAUUCCCGGAGUAGCGGUGAAAUGCGCAGAUACCGGGAGGAACGCCGAUGGCGAAGGCAGCCACCUGGUCCACCCGUGACGCUGAGGCGCGAAAGCGUGGGGAGCAAACCGGAUUAGAUACCCGGGUAGUCCACGCCCUAAACGAUGCGCGCUAGGUCUCUGGGUCUCCUGGGGGCCGAAGCUAACGCGUUAAGCGCGCCGCCUGGGGAGUACGGCCGCAAGGCUGAAACUCAAAGGAAUUGACGGGGGCCCGCACAAGCGGUGGAGCAUGUGGUUUAAUUCGAAGCAACGCGAAGAACCUUACCAGGCCUUGACAUGCUAGGGAACCCGGGUGAAAGCCUGGGGUGCCCCGCGAGGGGAGCCCUAGCACAGGUGCUGCAUGGCCGUCGUCAGCUCGUGCCGUGAGGUGUUGGGUUAAGUCCCGCAACGAGCGCAACCCCCGCCGUUAGUUGCCAGCGGUUCGGCCGGGCACUCUAACGGGACUGCCCGCGAAAGCGGGAGGAAGGAGGGGACGACGUCUGGUCAGCAUGGCCCUUACGGCCUGGGCGACACACGUGCUACAAUGCCCACUACAAAGCGAUGCCACCCGGCAACGGGGAGCUAAUCGCAAAAAGGUGGGCCCAGUUCGGAUUGGGGUCUGCAACCCGACCCCAUGAAGCCGGAAUCGCUAGUAAUCGCGGAUCAGCCAUGCCGCGGUGAAUACGUUCCCGGGCCUUGUACACACCGCCCGUCACGCCAUGGGAGCGGGCUCUACCCGAAGUCGCCGGGAGCCUACGGGCAGGCGCCGAGGGUAGGGCCCGUGACUGGGGCGAAGUCGUAACAAGGUAGCUGUACCGGAAGGUGCGGCUGGAUCACCUCCUUUCU
2DLW , Knot 58 113 0.80 40 84 107 
GSSGSSGHKEFAVTMRPTEASERCHLRGSYTLRAGESALELWGGPEPGTQLYDWPYRFLRRFGRDKVTFSFEAGRRCVSGEGNFEFETRQGNEIFLALEEAISAQKNSGPSSG
2XFP , Knot 215 520 0.86 40 259 494 
MSNKCDVVVVGGGISGMAAAKLLHDSGLNVVVLEARDRVGGRTYTLRNQKVKYVDLGGSYVGPTQNRILRLAKELGLETYKVNEVERLIHHVKGKSYPFRGPFPPVWNPITYLDHNNFWRTMDDMGREIPSDAPWKAPLAEEWDNMTMKELLDKLCWTESAKQLATLFVNLCVTAETHEVSALWFLWYVKQCGGTTRIISTTNGGQERKFVGGSGQVSERIMDLLGDRVKLERPVIYIDQTRENVLVETLNHEMYEAKYVISAIPPTLGMKIHFNPPLPMMRNQMITRVPLGSVIKCIVYYKEPFWRKKDYCGTMIIDGEEAPVAYTLDDTKPEGNYAAIMGFILAHKARKLARLTKEERLKKLCELYAKVLGSLEALEPVHYEEKNWCEEQYSGGCYTTYFPPGILTQYGRVLRQPVDRIYFAGTETATHWSGYMEGAVEAGERAAREILHAMGKIPEDEIWQSEPESVDVPAQPITTTFLERHLPSVPGLLRLIGLTTIFSATALGFLAHKRGLLVRV

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 \(c\).

\(|P_{f(4LSK_1)}(2) \setminus P_{f(2DLW_1)}(2)|=14\), \(|P_{f(2DLW_1)}(2) \setminus P_{f(4LSK_1)}(2)|=82\). 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 1:0001001111110001100001100011110111010011011010100011110101011100101011100101111000010000101100110110111011101110110101011101100010001111111111110110001111111000111001100000010101110001000000111101010001111110000100010000011101110001010000100110011001101111011011000100111101101101110110011000111111101100110010111110100111101011100001000001011111101101100111110000001011011101011100011011110110100100011111111111000000111101011100000111000111101111000001101111111001101101001111011011010011001100001010011011001011011010111111010111010010001110001001110101111110101011101100011110100001010111111001011000110010111111101011110101000111001110110111111111011011110000011110110110111101010111010011111111010011011011111011001000110001000101101001111010111110101111110111001110011101000111011000101000011101101010100111000001110000001111100111100110101001110101001000111111010110010111100111100011111110011011111000101011101101111010101100011000111101101011111100001001110000110101001111110001110111110001111010000101111111100001101011101001010110010010011000101001011110100111001110000101101110101100000100100110010011011000110011101000011011110010001011111011111111111111110110100011001101011000001011000111011010101010010110100010010111101101001000110110111111001100101111111011100011000111001111000101100011000010111100111100100110110010111001100101001011011101010000011100001010101001000100101001011111011100001000111100100111110001011101110100111110111100010110011110111100101101111011001010011111101011001110010000000000
Pair \(Z_2\) Length of longest common subsequence
4LSK_1,2DLW_1 96 2
4LSK_1,2XFP_1 263 3
2DLW_1,2XFP_1 213 4

Newick tree

 
[
	2XFP_1:13.35,
	[
		4LSK_1:48,2DLW_1:48
	]:87.35
]

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 \((0.85)(0.8)(\frac{1635 }{\log_{20} 1635}-\frac{113}{\log_{20}113})=401.\)
Status Protein1 Protein2 d d1/2
Query variables 4LSK_1 2DLW_1 272 163.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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