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Parikh vectors
4WVI_1 1ECS_1 8CDU_1 Letter Amino acid
50 11 383 A Alanine
45 12 491 G Glycine
29 6 0 I Isoleucine
9 4 0 M Methionine
29 7 0 P Proline
19 8 0 S Serine
10 8 0 R Arginine
13 2 0 H Histidine
26 6 0 T Threonine
27 3 0 Y Tyrosine
38 3 0 V Valine
1 3 365 C Cysteine
32 6 0 E Glutamic acid
52 1 0 K Lycine
34 2 0 N Asparagine
36 10 0 D Aspartic acid
14 6 0 Q Glutamine
39 17 0 L Leucine
21 8 0 F Phenylalanine
9 3 0 W Tryptophan 

4WVI_1|Chain A|Maltose-binding periplasmic protein,Signal peptidase IB|Escherichia coli K-12 (83333)
>1ECS_1|Chains A, B|BLEOMYCIN RESISTANCE PROTEIN|Klebsiella pneumoniae (573)
>8CDU_1|Chain A|16S rRNA|Bacillus subtilis subsp. subtilis str. 168 (224308)
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 \(w=f(c)\)
4WVI , Knot 215 533 0.84 40 237 487 
MSYYHHHHHHHMLVIWINGDKGYNGLAQVGKKFEKDTGIKVTVEHPYKLEEKFPQVAATGDGPDIIFWAHDRFGGYACSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDGELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDEALKDAQTNAGAIVTPYTIKGESMDPTLKDGERVAVNIVGYKTGGLEKGNVVVFHANKNDDYVKRVIGVPGDKVEYKNDTLYVNGKKQDEPYLNYNLKHKQGDYITGTFQVKDLPNANPKSNVIPKGKYLVLGDNREVSKDSRAFGLIDEDQIVGKVSFRFWSHPQFEK
1ECS , Knot 66 126 0.84 40 104 123 
MTDQATPNLPSRDFDSTAAFYERLGFGIVFRDAGWMILQRGDLMLEFFAHPGLDPLASWFSCCLRLDDLAEFYRQCKSVGIQETSSGYPRIHAPELQGWGGTMAALVDPDGTLLRLIQNELLAGIS
8CDU , Knot 288 1554 0.45 8 16 64 
UUUAUCGGAGAGUUUGAUCCUGGCUCAGGACGAACGCUGGCGGCGUGCCUAAUACAUGCAAGUCGAGCGGACAGAUGGGAGCUUGCUCCCUGAUGUUAGCGGCGGACGGGUGAGUAACACGUGGGUAACCUGCCUGUAAGACUGGGAUAACUCCGGGAAACCGGGGCUAAUACCGGAUGGUUGUUUGAACCGCAUGGUUCAAACAUAAAAGGUGGCUUCGGCUACCACUUACAGAUGGACCCGCGGCGCAUUAGCUAGUUGGUGAGGUAACGGCUCACCAAGGCGACGAUGCGUAGCCGACCUGAGAGGGUGAUCGGCCACACUGGGACUGAGACACGGCCCAGACUCCUACGGGAGGCAGCAGUAGGGAAUCUUCCGCAAUGGACGAAAGUCUGACGGAGCAACGCCGCGUGAGUGAUGAAGGUUUUCGGAUCGUAAAGCUCUGUUGUUAGGGAAGAACAAGUGCCGUUCGAAUAGGGCGGUACCUUGACGGUACCUAACCAGAAAGCCACGGCUAACUACGUGCCAGCAGCCGCGGUAAUACGUAGGUGGCAAGCGUUGUCCGGAAUUAUUGGGCGUAAAGGGCUCGCAGGCGGUUUCUUAAGUCUGAUGUGAAAGCCCCCGGCUCAACCGGGGAGGGUCAUUGGAAACUGGGGAACUUGAGUGCAGAAGAGGAGAGUGGAAUUCCACGUGUAGCGGUGAAAUGCGUAGAGAUGUGGAGGAACACCAGUGGCGAAGGCGACUCUCUGGUCUGUAACUGACGCUGAGGAGCGAAAGCGUGGGGAGCGAACAGGAUUAGAUACCCUGGUAGUCCACGCCGUAAACGAUGAGUGCUAAGUGUUAGGGGGUUUCCGCCCCUUAGUGCUGCAGCUAACGCAUUAAGCACUCCGCCUGGGGAGUACGGUCGCAAGACUGAAACUCAAAGGAAUUGACGGGGGCCCGCACAAGCGGUGGAGCAUGUGGUUUAAUUCGAAGCAACGCGAAGAACCUUACCAGGUCUUGACAUCCUCUGACAAUCCUAGAGAUAGGACGUCCCCUUCGGGGGCAGAGUGACAGGUGGUGCAUGGUUGUCGUCAGCUCGUGUCGUGAGAUGUUGGGUUAAGUCCCGCAACGAGCGCAACCCUUGAUCUUAGUUGCCAGCAUUCAGUUGGGCACUCUAAGGUGACUGCCGGUGACAAACCGGAGGAAGGUGGGGAUGACGUCAAAUCAUCAUGCCCCUUAUGACCUGGGCUACACACGUGCUACAAUGGACAGAACAAAGGGCAGCGAAACCGCGAGGUUAAGCCAAUCCCACAAAUCUGUUCUCAGUUCGGAUCGCAGUCUGCAACUCGACUGCGUGAAGCUGGAAUCGCUAGUAAUCGCGGAUCAGCAUGCCGCGGUGAAUACGUUCCCGGGCCUUGUACACACCGCCCGUCACACCACGAGAGUUUGUAACACCCGAAGUCGGUGAGGUAACCUUUUAGGAGCCAGCCGCCGAAGGUGGGACAGAUGAUUGGGGUGAAGUCGUAACAAGGUAGCCGUAUCGGAAGGUGCGGCUGGAUCACCUCCUUUCUA

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 \(c\).

\(|P_{f(4WVI_1)}(2) \setminus P_{f(1ECS_1)}(2)|=170\), \(|P_{f(1ECS_1)}(2) \setminus P_{f(4WVI_1)}(2)|=37\). 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 1:10000000000111111010010011101100100001101010010010001101110101101111100011101001111010100110001011010110001011101111011011000011101100100111101010101001111010010101111110110110000100010011100111011101110110000101000001101110010011010111110010000100110111010101001111110111011010001100110001100011011000011111110000001100101110100100101110110101110110011101101000100110010001111101001010010101001001110111000111001011110100000010011111100100000010101000001010001000010010101010011010100011101001111000010000011111000011101010110010100
Pair \(Z_2\) Length of longest common subsequence
4WVI_1,1ECS_1 207 3
4WVI_1,8CDU_1 243 3
1ECS_1,8CDU_1 112 2

Newick tree

 
[
	4WVI_1:12.24,
	[
		1ECS_1:56,8CDU_1:56
	]:70.24
]

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 \((0.85)(0.8)(\frac{659 }{\log_{20} 659}-\frac{126}{\log_{20}126})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 4WVI_1 1ECS_1 192 118.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]