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Parikh vectors
1DZW_1 7LEI_1 6CFJ_1 Letter Amino acid
6 4 0 D Aspartic acid
6 5 0 Q Glutamine
9 7 0 K Lycine
9 6 0 P Proline
10 0 0 S Serine
10 4 0 R Arginine
5 2 0 M Methionine
3 2 0 F Phenylalanine
24 3 636 A Alanine
4 2 795 C Cysteine
13 13 1065 G Glycine
15 13 0 I Isoleucine
10 4 0 N Asparagine
18 4 0 E Glutamic acid
11 1 0 H Histidine
24 12 0 L Leucine
14 8 0 T Threonine
2 2 0 W Tryptophan
7 1 0 Y Tyrosine
15 6 0 V Valine 

1DZW_1|Chain A[auth P]|L-fuculose phosphate aldolase|Escherichia coli (562)
>7LEI_1|Chains A, B|Protease|Human immunodeficiency virus 1 (11676)
>6CFJ_1|Chains A[auth 1A], EB[auth 2A]|23S Ribosomal RNA|Thermus thermophilus (strain HB8 / ATCC 27634 / DSM 579) (300852)
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 \(w=f(c)\)
1DZW , Knot 104 215 0.86 40 154 209 
MERNKLARQIIDTCLEMTRLGLNQGTAGNVSVRYQDGMLITPTGIPYEKLTESHIVFIDGNGKHEEGKLPSSEWRFHMAAYQSRPDANAVVHNHAVHCTAVSILNRSIPAIHYMIAAAGGNSIPCAPYATAGTRELSEHVALALKNRKATLLQHHGLIACEVNLEKALWLAHEVEVLAQLYLTTLAITDPVPVLSDEEIAVVLEKFKTYGLRIEE
7LEI , Knot 52 99 0.80 38 79 94 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVLEEMNLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF
6CFJ , Knot 480 2915 0.43 8 16 64 
GUCAAGAUGGUAAGGGCCCACGGUGGAUGCCUCGGCACCCGAGCCGAUGAAGGACGUGGCUACCUGCGAUAAGCCAGGGGGAGCCGGUAGCGGGCGUGGAUCCCUGGAUGUCCGAAUGGGGGAACCCGGCCGGCGGGAACGCCGGUCACCGCGCUUUUGCGCGGGGGGAACCUGGGGAACUGAAACAUCUCAGUACCCAGAGGAGAGGAAAGAGAAAUCGACUCCCUGAGUAGCGGCGAGCGAAAGGGGACCAGCCUAAACCGUCCGGCUUGUCCGGGCGGGGUCGUGGGGCCCUCGGACACCGAAUCCCCAGCCUAGCCGAAGCUGUUGGGAAGCAGCGCCAGAGAGGGUGAAAGCCCCGUAGGCGAAAGGUGGGGGGAUAGGUGAGGGUACCCGAGUACCCCGUGGUUCGUGGAGCCAUGGGGGAAUCUGGGCGGACCACCGCCUAAGGCUAAGUACUCCGGGUGACCGAUAGCGCACCAGUACCGUGAGGGAAAGGUGAAAAGAACCCCGGGAGGGGAGUGAAAUAGAGCCUGAAACCGUGGGCUUACAAGCAGUCACGGCCCCGCAAGGGGUUGUGGCGUGCCUAUUGAAGCAUGAGCCGGCGACUCACGGUCGUGGGCGAGCUUAAGCCGUUGAGGCGGAGGCGUAGGGAAACCGAGUCCGAACAGGGCGCAAGCGGGCCGCACGCGGCCCGCAAAGUCCGCGGCCGUGGACCCGAAACCGGGCGAGCUAGCCCUGGCCAGGGUGAAGCUGGGGUGAGACCCAGUGGAGGCCCGAACCGGUGGGGGAUGCAAACCCCUCGGAUGAGCUGGGGCUAGGAGUGAAAAGCUAACCGAGCCCGGAGAUAGCUGGUUCUCCCCGAAAUGACUUUAGGGUCAGCCUCAGGCGCUGACUGGGGCCUGUAGAGCACUGAUAGGGCUAGGGGGCCCACCAGCCUACCAAACCCUGUCAAACUCCGAAGGGUCCCAGGUGGAGCCUGGGAGUGAGGGCGCGAGCGAUAACGUCCGCGUCCGAGCGCGGGAACAACCGAGACCGCCAGCUAAGGCCCCCAAGUCUGGGCUAAGUGGUAAAGGAUGUGGCGCCGCGAAGACAGCCAGGAGGUUGGCUUAGAAGCAGCCAUCCUUUAAAGAGUGCGUAAUAGCUCACUGGUCGAGUGGCGCCGCGCCGAAAAUGAUCGGGGCUUAAGCCCAGCGCCGAAGCUGCGGGUCUGGGGGAUGACCCCAGGCGGUAGGGGAGCGUUCCCGAUGCCGAUGAAGGCCGACCCGCGAGGGCGGCUGGAGGUAAGGGAAGUGCGAAUGCCGGCAUGAGUAACGAUAAAGAGGGUGAGAAUCCCUCUCGCCGUAAGCCCAAGGGUUCCUACGCAAUGGUCGUCAGCGUAGGGUUAGGCGGGACCUAAGGUGAAGCCGAAAGGCGUAGCCGAAGGGCAGCCGGUUAAUAUUCCGGCCCUUCCCGCAGGUGCGAUGGGGGGACGCUCUAGGCUAGGGGGACCGGAGCCAUGGACGAGCCCGGCCAGAAGCGCAGGGUGGGAGGUAGGCAAAUCCGCCUCCCAACAAGCUCUGCGUGGUGGGGAAGCCCGUACGGGUGACAACCCCCCGAAGCCAGGGAGCCAAGAAAAGCCUCUAAGCACAACCUGCGGGAACCCGUACCGCAAACCGACACAGGUGGGCGGGUGCAAGAGCACUCAGGCGCGCGGGAGAACCCUCGCCAAGGAACUCUGCAAGUUGGCCCCGUAACUUCGGGAGAAGGGGUGCUCCCUGGGGUGAUGAGCCCCGGGGAGCCGCAGUGAACAGGCUCUGGCGACUGUUUACCAAAAACACAGCUCUCUGCGAACUCGUAAGAGGAGGUAUAGGGAGCGACGCUUGCCCGGUGCCGGAAGGUCAAGGGGAGGGGUGCAAGCCCCGAACCGAAGCCCCGGUGAACGGCGGCCGUAACUAUAACGGUCCUAAGGUAGCGAAAUUCCUUGUCGGGUAAGUUCCGACCUGCACGAAAAGCGUAACGACCGGAGCGCUGUCUCGGCGAGGGACCCGGUGAAAUUGAACUGGCCGUGAAGAUGCGGCCUACCCGUGGCAGGACGAAAAGACCCCGUGGAGCUUUACUGCAGCCUGGUGUUGGCUCUUGGUCGCGCCUGCGUAGGAUAGGUGGGAGCCUGUGAACCCCCGCCUCCGGGUGGGGGGGAGGCGCCGGUGAAAUACCACCCUGGCGCGGCUGGGGGCCUAACCCUCGGAUGGGGGGACAGCGCUUGGCGGGCAGUUUGACUGGGGCGGUCGCCUCCUAAAAGGUAACGGAGGCGCCCAAAGGUCCCCUCAGGCGGGACGGAAAUCCGCCGGAGAGCGCAAGGGUAGAAGGGGGCCUGACUGCGAGGCCUGCAAGCCGAGCAGGGGCGAAAGCCGGGCCUAGUGAACCGGUGGUCCCGUGUGGAAGGGCCAUCGAUCAACGGAUAAAAGUUACCCCGGGGAUAACAGGCUGAUCUCCCCCGAGCGUCCACAGCGGCGGGGAGGUUUGGCACCUCGAUGUCGGCUCGUCGCAUCCUGGGGCUGAAGAAGGUCCCAAGGGUUGGGCUGUUCGCCCAUUAAAGCGGCACGCGAGCUGGGUUCAGAACGUCGUGAGACAGUUCGGUCUCUAUCCGCCACGGGCGCAGGAGGCUUGAGGGGGGCUCUUCCUAGUACGAGAGGACCGGAAGGGACGCACCUCUGGUUUCCCAGCUGUCCCUCCAGGGGCAUAAGCUGGGUAGCCAUGUGCGGAAGGGAUAACCGCUGAAAGCAUCUAAGCGGGAAGCCCGCCCCAAGAUGAGGCCUCCCACGGCGUCAAGCCGGUAAGGACCCGGGAAGACCACCCGGUGGAUGGGCCGGGGGUGUAAGCGCCGCGAGGCGUUGAGCCGACCGGUCCCAAUCGUCCGAGGUCUUGACCCCUCC

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 \(c\).

\(|P_{f(1DZW_1)}(2) \setminus P_{f(7LEI_1)}(2)|=115\), \(|P_{f(7LEI_1)}(2) \setminus P_{f(1DZW_1)}(2)|=40\). 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 1:10000110011000101001110010110101000011110101110001000011110101000010110001010111000010101110001100011011000111100111111100110110101100010001111100001011000111100101001111100101110101001110011111000011111001000110100
Pair \(Z_2\) Length of longest common subsequence
1DZW_1,7LEI_1 155 3
1DZW_1,6CFJ_1 160 5
7LEI_1,6CFJ_1 87 2

Newick tree

 
[
	1DZW_1:87.40,
	[
		7LEI_1:43.5,6CFJ_1:43.5
	]:43.90
]

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 \((0.85)(0.8)(\frac{314 }{\log_{20} 314}-\frac{99}{\log_{20}99})=67.3\)
Status Protein1 Protein2 d d1/2
Query variables 1DZW_1 7LEI_1 87 62
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]