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Parikh vectors
8CGR_1 8FCV_1 1CKR_1 Letter Amino acid
388 16 7 A Alanine
0 0 14 L Leucine
0 0 3 M Methionine
0 0 0 W Tryptophan
0 0 6 R Arginine
0 0 8 Q Glutamine
486 15 11 G Glycine
0 0 3 F Phenylalanine
0 0 12 V Valine
0 0 14 E Glutamic acid
0 0 7 P Proline
0 14 16 T Threonine
0 0 3 Y Tyrosine
0 0 10 N Asparagine
0 0 11 D Aspartic acid
353 15 0 C Cysteine
0 0 0 H Histidine
0 0 11 I Isoleucine
0 0 14 K Lycine
0 0 9 S Serine 

8CGR_1|Chain A|16S rRNA|Escherichia coli BW25113 (679895)
>8FCV_1|Chain A[auth N]|DNA (60-MER)|synthetic construct (32630)
>1CKR_1|Chain A|HEAT SHOCK SUBSTRATE BINDING DOMAIN OF HSC-70|Rattus norvegicus (10116)
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 \(w=f(c)\)
8CGR , Knot 282 1540 0.44 8 16 64 
AAAUUGAAGAGUUUGAUCAUGGCUCAGAUUGAACGCUGGCGGCAGGCCUAACACAUGCAAGUCGAACGGUAACAGGAAGAAGCUUGCUUCUUUGCUGACGAGUGGCGGACGGGUGAGUAAUGUCUGGGAAACUGCCUGAUGGAGGGGGAUAACUACUGGAAACGGUAGCUAAUACCGCAUAACGUCGCAAGACCAAAGAGGGGGACCUUCGGGCCUCUUGCCAUCGGAUGUGCCCAGAUGGGAUUAGCUAGUAGGUGGGGUAACGGCUCACCUAGGCGACGAUCCCUAGCUGGUCUGAGAGGAUGACCAGCCACACUGGAACUGAGACACGGUCCAGACUCCUACGGGAGGCAGCAGUGGGGAAUAUUGCACAAUGGGCGCAAGCCUGAUGCAGCCAUGCCGCGUGUAUGAAGAAGCCCUUCGGGUUGUAAAGUACUUUCAGCGGGGAGGAAGGGAGUAAAGUUAAUACCUUUGCUCAUUGACGUUACCCGCAGAAGAAGCACCGGCUAACUCCGUGCCAGCAGCCGCGGUAAUACGGAGGGUGCAAGCGUUAAUCGGAAUUACUGGGCGUAAAGCGCACGCAGGCGGUUUGUUAAGUCAGAUGUGAAAUCCCCGGGCUCAACCUGGGAACUGCAUCUGAUACUGGCAAGCUUGAGUCUCGUAGAGGGGGGUAGAAUUCCAGGUGUAGCGGUGAAAUGCGUAGAGAUCUGGAGGAAUACCGGUGGCGAAGGCGGCCCCCUGGACGAAGACUGACGCUCAGGUGCGAAAGCGUGGGGAGCAAACAGGAUUAGAUACCCUGGUAGUCCACGCCGUAAACGAUGUCGACUUGGAGGUUGUGCCCUUGAGGCGUGGCUUCCGGAGCUAACGCGUUAAGUCGACCGCCUGGGGAGUACGGCCGCAAGGUUAAAACUCAAAUGAAUUGACGGGGGCCCGCACAAGCGGUGGAGCAUGUGGUUUAAUUCGAUGCAACGCGAAGAACCUUACCUGGUCUUGACAUCCACGGAAGUUUUCAGAGAUGAGAAUGUGCCUUCGGGAACCGUGAGACAGGUGCUGCAUGGCUGUCGUCAGCUCGUGUUGUGAAAUGUUGGGUUAAGUCCCGCAACGAGCGCAACCCUUAUCCUUUGUUGCCAGCGGUCCGGCCGGGAACUCAAAGGAGACUGCCAGUGAUAAACUGGAGGAAGGUGGGGAUGACGUCAAGUCAUCAUGGCCCUUACGACCAGGGCUACACACGUGCUACAAUGGCGCAUACAAAGAGAAGCGACCUCGCGAGAGCAAGCGGACCUCAUAAAGUGCGUCGUAGUCCGGAUUGGAGUCUGCAACUCGACUCCAUGAAGUCGGAAUCGCUAGUAAUCGUGGAUCAGAAUGCCACGGUGAAUACGUUCCCGGGCCUUGUACACACCGCCCGUCACACCAUGGGAGUGGGUUGCAAAAGAAGUAGGUAGCUUAACCUUCGGGAGGGCGCUUACCACUUUGUGAUUCAUGACUGGGGUGAAGUCGUAACAAGGUAACCGUAGGGGAACCUGCGGUUGGAUCACCUCCU
8FCV , Knot 17 60 0.38 8 14 27 
AAGACGTAGCCCAGCGCGTCGGCCGCTACGTATCGTAGATATATCTACGCGTAGATATAT
1CKR , Knot 76 159 0.80 34 118 155 
SENVQDLLLLDVTPLSLGIETAGGVMTVLIKRNTTIPTKQTQTFTTYSDNQPGVLIQVYEGERAMTKDNNLLGKFELTGIPPAPRGVPQIEVTFDIDANGILNVSAVDKSTGKENKITITNDKGRLSKEDIERMVQEAEKYKAEDEKQRDKVSSKNSLE

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 \(c\).

\(|P_{f(8CGR_1)}(2) \setminus P_{f(8FCV_1)}(2)|=7\), \(|P_{f(8FCV_1)}(2) \setminus P_{f(8CGR_1)}(2)|=5\). 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 1:1110011111100011001011000111001110100110110111000110101010111001110110110111111111000100000001001101110110111011101110110100011111100100011011111111101100100111110110110011010010101101001011110011111111111000001110000001001001110101000111011110011001101110111101101100010001110110110000011001100011111110110011001010011110011110101100011100000101111110110110111111010010101101110101110001101011001010010101010111111100000011100101111010000011011111111111111011110011010000010001001101001000101111111101001100110000101001101100101101101011111101011101001100111100100111010111101010101110110001001110011101011110000011100011000111110010100011010011011100011100001011111111101111000011101011011011110101011111000111111101001101101111101100000011101111100110100011101011111010111111011101111001110100001101100010100101110110100110001111100101000001111010110000011110011010100111001100100011111101011001011110011110001110111001101111100010101110110111101010110001100011010110101111110000100011000011010001011111000001111101111101010000011111001011110111010010101100100100110001010010111101001110011100001011011101011000001000000100100110110001100111110001111111100100110110111001111111110111110110100111001001011000001011001111001010101010010110110101010111111111011000010111110111011100001011110101001011000111001111000101100011000010111100111100100110110010111001111010010110111010100000111000010101010010001001010010111110111001011111111011101100011000001111111010001001000010110001011001111011110010110111101100101111111000101100111001000000
Pair \(Z_2\) Length of longest common subsequence
8CGR_1,8FCV_1 12 7
8CGR_1,1CKR_1 130 3
8FCV_1,1CKR_1 126 3

Newick tree

 
[
	1CKR_1:73.82,
	[
		8CGR_1:6,8FCV_1:6
	]:67.82
]

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 \((0.85)(0.8)(\frac{1600 }{\log_{20} 1600}-\frac{60}{\log_{20}60})=411.\)
Status Protein1 Protein2 d d1/2
Query variables 8CGR_1 8FCV_1 278 144.5
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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