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Parikh vectors
6BOH_1 6BPO_1 7ZAF_1 Letter Amino acid
0 29 7 P Proline
0 52 5 D Aspartic acid
545 74 9 G Glycine
0 15 4 H Histidine
0 10 6 M Methionine
0 37 12 L Leucine
0 27 15 K Lycine
0 44 8 V Valine
310 62 7 A Alanine
0 32 4 R Arginine
0 23 10 E Glutamic acid
0 37 5 I Isoleucine
0 9 0 W Tryptophan
0 33 4 Y Tyrosine
0 54 5 N Asparagine
0 29 3 Q Glutamine
0 22 6 F Phenylalanine
0 81 2 T Threonine
422 2 1 C Cysteine
0 55 3 S Serine 

6BOH_1|Chains A, FB|16S ribosomal RNA|Thermus thermophilus (strain HB27 / ATCC BAA-163 / DSM 7039) (262724)
>6BPO_1|Chains A, B, C, D|Catecholate siderophore receptor Fiu|Escherichia coli (strain K12) (83333)
>7ZAF_1|Chain A[auth AAA]|KH domain-containing, RNA-binding, signal transduction-associated protein 1|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 \(w=f(c)\)
6BOH , Knot 271 1507 0.43 8 16 63 
UUGGAGAGUUUGAUCCUGGCUCAGGGUGAACGCUGGCGGCGUGCCUAAGACAUGCAAGUCGUGCGGGCCGCGGGGUUUUACUCCGUGGUCAGCGGCGGACGGGUGAGUAACGCGUGGGUGACCUACCCGGAAGAGGGGGACAACCCGGGGAAACUCGGGCUAAUCCCCCAUGUGGACCCGCCCCUUGGGGUGUGUCCAAAGGGCUUUGCCCGCUUCCGGAUGGGCCCGCGUCCCAUCAGCUAGUUGGUGGGGUAAUGGCCCACCAAGGCGACGACGGGUAGCCGGUCUGAGAGGAUGGCCGGCCACAGGGGCACUGAGACACGGGCCCCACUCCUACGGGAGGCAGCAGUUAGGAAUCUUCCGCAAUGGGCGCAAGCCUGACGGAGCGACGCCGCUUGGAGGAAGAAGCCCUUCGGGGUGUAAACUCCUGAACCCGGGACGAAACCCCCGAGGAGGGGACUGACGGUACCGGGGUAAUAGCGCCGGCCAACUCCGUGCCAGCAGCCGCGGUAAUACGGAGGGCGCGAGCGUUACCCGGAUUCACUGGGCGUAAAGGGCGUGUAGGCGGCCUGGGGCGUCCCAUGUGAAAGACCACGGCUCAACCGUGGGGGAGCGUGGGAUACGCUCAGGCUAGACGGUGGGAGAGGGUGGUGGAAUUCCCGGAGUAGCGGUGAAAUGCGCAGAUACCGGGAGGAACGCCGAUGGCGAAGGCAGCCACCUGGUCCACCCGUGACGCUGAGGCGCGAAAGCGUGGGGAGCAAACCGGAUUAGAUACCCGGGUAGUCCACGCCCUAAACGAUGCGCGCUAGGUCUCUGGGUCUCCUGGGGGCCGAAGCUAACGCGUUAAGCGCGCCGCCUGGGGAGUACGGCCGCAAGGCUGAAACUCAAAGGAAUUGACGGGGGCCCGCACAAGCGGUGGAGCAUGUGGUUUAAUUCGAAGCAACGCGAAGAACCUUACCAGGCCUUGACAUGCUAGGGAACCCGGGUGAAAGCCUGGGGUGCCCCGCGAGGGGAGCCCUAGCACAGGUGCUGCAUGGCCGUCGUCAGCUCGUGCCGUGAGGUGUUGGGUUAAGUCCCGCAACGAGCGCAACCCCCGCCGUUAGUUGCCAGCGGUUCGGCCGGGCACUCUAACGGGACUGCCCGCGAAAGCGGGAGGAAGGAGGGGACGACGUCUGGUCAGCAUGGCCCUUACGGCCUGGGCGACACACGUGCUACAAUGCCCACUACAAAGCGAUGCCACCCGGCAACGGGGAGCUAAUCGCAAAAAGGUGGGCCCAGUUCGGAUUGGGGUCUGCAACCCGACCCCAUGAAGCCGGAAUCGCUAGUAAUCGCGGAUCAGCCAUGCCGCGGUGAAUACGUUCCCGGGCCUUGUACACACCGCCCGUCACGCCAUGGGAGCGGGCUCUACCCGAAGUCGCCGGGAGCCUACGGGCAGGCGCCGAGGGUAGGGCCCGUGACUGGGGCGAAGUCGUAACAAGGUAGCUGUACCGGAAGGUGCGGCUGGAUC
6BPO , Knot 274 727 0.82 40 264 645 
AEGQTNADDTLVVEASTPSLYAPQQSADPKFSRPVADTTRTMTVISEQVIKDQGATNLTDALKNVPGVGAFFAGENGNSTTGDAIYMRGADTSNSIYIDGIRDIGSVSRDTFNTEQVEVIKGPSGTDYGRSAPTGSINMISKQPRNDSGIDASASIGSAWFRRGTLDVNQVIGDTTAVRLNVMGEKTHDAGRDKVKNERYGVAPSVAFGLGTANRLYLNYLHVTQHNTPDGGIPTIGLPGYSAPSAGTAALNHSGKVDTHNFYGTDSDYDDSTTDTATMRFEHDINDNTTIRNTTRWSRVKQDYLMTAIMGGASNITQPTSDVNSWTWSRTANTKDVSNKILTNQTNLTSTFYTGSIGHDVSTGVEFTRETQTNYGVNPVTLPAVNIYHPDSSIHPGGLTRNGANANGQTDTFAIYAFDTLQITRDFELNGGIRLDNYHTEYDSATACGGSGRGAITCPTGVAKGSPVTTVDTAKSGNLMNWKAGALYHLTENGNVYINYAVSQQPPGGNNFALAQSGSGNSANRTDFKPQKANTSEIGTKWQVLDKRLLLTAALFRTDIENEVEQNDDGTYSQYGKKRVEGYEISVAGNITPAWQVIGGYTQQKATIKNGKDVAQDGSSSLPYTPEHAFTLWSQYQATDDISVGAGARYIGSMHKGSDGAVGTPAFTEGYWVADAKLGYRVNRNLDFQLNVYNLFDTDYVASINKSGYRYHPGEPRTFLLTANMHF
7ZAF , Knot 61 116 0.83 38 95 114 
NYLDLFSHKNMKLKERVLIPVKQYPKFNFVGKILGPQGNTIKRLQEETGAKISVLGKGSMRDKAKEEELRKGGDPKYAHLNMDLHVFIEVFGPPCEAYALMAHAMEEVKKFLVPDM

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 \(c\).

\(|P_{f(6BOH_1)}(2) \setminus P_{f(6BPO_1)}(2)|=10\), \(|P_{f(6BPO_1)}(2) \setminus P_{f(6BOH_1)}(2)|=258\). 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 1:0011111100011000011000111101110100110110101000111101010111001010111001011110000100001011001101101110111011101101010111011000100011111111111101100011111110001110011000000101011100010000001111010100011111100001000100000111011100010100001001100110011011110110110001001111011011011101100110001111111011001100101111101001111010111000010000010111111011011001111100000010110111010111000110111101101001000111111111110000001111010111000001110001111011110000011111111110011011010011110110110100110011000010100110110010110110101111110101110100100011100010011101011111101010111011000111101000010101111110010110001100101111111010111101010001110011101101111111110110111100000111101101101111010101110100111111110100110110111110110010001100010001011010011110101111101011111101110011100111010001110110001010000111011010101001110000011100000011111001111001101010011101010010001111110101100101111001111000111111100110111110001010111011011110101011000110001111011010111111000010011100001101010011111100011101111100011110100001011111111000011010111010010101100100100110001010010111101001110011100001011011101011000001001001100100110110001100111010000110111100100010111110111111111111111101101000110011010110000010110001110110101010100101101000100101111011010010001101101111110011001011111110111000110001110011110001011000110000101111001111001001101100101110011001010010110111010100000111000010101010010001001010010111110111000010001111001001111100010111011101001111101111000101100111101111001011011110110010100111111010110011100
Pair \(Z_2\) Length of longest common subsequence
6BOH_1,6BPO_1 268 4
6BOH_1,7ZAF_1 107 2
6BPO_1,7ZAF_1 215 3

Newick tree

 
[
	6BPO_1:13.82,
	[
		6BOH_1:53.5,7ZAF_1:53.5
	]:83.32
]

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 \((0.85)(0.8)(\frac{2234 }{\log_{20} 2234}-\frac{727}{\log_{20}727})=365.\)
Status Protein1 Protein2 d d1/2
Query variables 6BOH_1 6BPO_1 271 270.5
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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