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Parikh vectors
4BIE_1 4JYA_1 9RID_1 Letter Amino acid
26 543 26 G Glycine
23 0 11 I Isoleucine
12 0 17 F Phenylalanine
2 0 3 W Tryptophan
16 0 27 V Valine
19 0 17 D Aspartic acid
10 0 14 Q Glutamine
27 0 9 E Glutamic acid
30 0 29 L Leucine
11 0 11 R Arginine
7 0 7 H Histidine
14 0 11 Y Tyrosine
13 0 24 T Threonine
18 312 17 A Alanine
11 0 21 N Asparagine
5 428 12 C Cysteine
27 0 11 K Lycine
6 0 10 M Methionine
17 0 13 P Proline
24 0 16 S Serine 

4BIE_1|Chains A, B|MITOGEN-ACTIVATED PROTEIN KINASE KINASE KINASE 5|HOMO SAPIENS (9606)
>4JYA_1|Chain A|16S ribosomal RNA|Thermus thermophilus (300852)
>9RID_1|Chains A, B|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
4BIE , Knot 138 318 0.83 40 195 302 
STEEGDCESDLLEYDYEYDENGDRVVLGKGTYGIVYAGRDLSNQVRIAIKEIPERDSRYSQPLHEEIALHKHLKHKNIVQYLGSFSENGFIKIFMEQVPGGSLSALLRSKWGPLKDNEQTIGFYTKQILEGLKYLHDNQIVHRDIKGDNVLINTYSGVLKISDFGTSKRLAGINPCTEEFTGTLQYMAPEIIDKGPRGYGKAADIWSLGCTIIEMATGKPPFYELGEPQAAMFKVGMFKVHPEIPESMSAEAKAFILKCFEPDPDKRACANDLLVDEFLKVSSKKKKTQPKLSALSAGSNEYLRSISLPVPVLVEDTS
4JYA , Knot 274 1516 0.44 8 16 63 
UGGAGAGUUUGAUCCUGGCUCAGGGUGAACGCUGGCGGCGUGCCUAAGACAUGCAAGUCGUGCGGGCCGCGGGAUUUUACUCCGUGGUCAGCGGCGGACGGGUGAGUAACGCGUGGGUGACCUACCCGGAAGAGGGGGACAACCCGGGGAAACUCGGGCUAAUCCCCCAUGUGGACCCGCCCCUUGGGGUGUGUCCAAAGGGCUUUGCCCGCUUCCGGAUGGGCCCGCGUCCCAUCAGCUAGUUGGUGGGGUAAUGGCCCACCAAGGCGACGACGGGUAGCCGGUCUGAGAGGAUGGCCGGCCACAGGGGCACUGAGACACGGGCCCCACUCCUACGGGAGGCAGCAGUUAGGAAUCUUCCGCAAUGGGCGCAAGCCUGACGGAGCGACGCCGCUUGGAGGAAGAAGCCCUUCGGGGUGUAAACUCCUGAACCCGGGACGAAACCCCCGACGAGGGGACUGACGGUACCGGGGUAAUAGCGCCGGCCAACUCCGUGCCAGCAGCCGCGGUAAUACGGAGGGCGCGAGCGUUACCCGGAUUCACUGGGCGUAAAGGGCGUGUAGGCGGCCUGGGGCGUCCCAUGUGAAAGACCACGGCUCAACCGUGGGGGAGCGUGGGAUACGCUCAGGCUAGACGGUGGGAGAGGGUGGUGGAAUUCCCGGAGUAGCGGUGAAAUGCGCAGAUACCGGGAGGAACGCCGAUGGCGAAGGCAGCCACCUGGUCCACCCGUGACGCUGAGGCGCGAAAGCGUGGGGAGCAAACCGGAUUAGAUACCCGGGUAGUCCACGCCCUAAACGAUGCGCGCUAGGUCUCUGGGUCUCCUGGGGGCCGAAGCUAACGCGUUAAGCGCGCCGCCUGGGGAGUACGGCCGCAAGGCUGAAACUCAAAGGAAUUGACGGGGGCCCGCACAAGCGGUGGAGCAUGUGGUUUAAUUCGAAGCAACGCGAAGAACCUUACCAGGCCUUGACAUGCUAGGGAACCCGGGUGAAAGCCUGGGGUGCCCCGCGAGGGGAGCCCUAGCACAGGUGCUGCAUGGCCGUCGUCAGCUCGUGCCGUGAGGUGUUGGGUUAAGUCCCGCAACGAGCGCAACCCCCGCCGUUAGUUGCCAGCGGUUCGGCCGGGCACUCUAACGGGACUGCCCGCGAAAGCGGGAGGAAGGAGGGGACGACGUCUGGUCAGCAUGGCCCUUACGGCCUGGGCGACACACGUGCUACAAUGCCCACUACAAAGCGAUGCCACCCGGCAACGGGGAGCUAAUCGCAAAAAGGUGGGCCCAGUUCGGAUUGGGGUCUGCAACCCGACCCCAUGAAGCCGGAAUCGCUAGUAAUCGCGGAUCAGCCAUGCCGCGGUGAAUACGUUCCCGGGCCUUGUACACACCGCCCGUCACGCCAUGGGAGCGGGCUCUACCCGAAGUCGCCGGGAGCCUACGGGCAGGCGCCGAGGGUAGGGCCCGUGACUGGGGCGAAGUCGUAACAAGGUAGCUGUACCGGAAGGUGCGGCUGGAUCACCUCCUUUC
9RID , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ

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 \(c\).

\(|P_{f(4BIE_1)}(2) \setminus P_{f(4JYA_1)}(2)|=189\), \(|P_{f(4JYA_1)}(2) \setminus P_{f(4BIE_1)}(2)|=10\). 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 1:000010000011000000000100111101001110110010001011100110000000011000111000100001100110100011101110011110101110001111000000111000011011001000011000101001110000111010011000011110100001010100111011001101010110110110011011010111001101011110111101010110010101011110010101000101001110011010000000010101101100001001011111110000
Pair \(Z_2\) Length of longest common subsequence
4BIE_1,4JYA_1 199 3
4BIE_1,9RID_1 179 4
4JYA_1,9RID_1 200 3

Newick tree

 
[
	4JYA_1:10.94,
	[
		4BIE_1:89.5,9RID_1:89.5
	]:13.44
]

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 \((0.85)(0.8)(\frac{1834 }{\log_{20} 1834}-\frac{318}{\log_{20}318})=384.\)
Status Protein1 Protein2 d d1/2
Query variables 4BIE_1 4JYA_1 271 202.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]