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Parikh vectors
2VQP_1 3EYJ_1 8WPH_1 Letter Amino acid
18 24 28 V Valine
7 8 11 F Phenylalanine
5 9 3 C Cysteine
6 11 14 Q Glutamine
13 10 19 E Glutamic acid
5 28 23 G Glycine
22 22 28 I Isoleucine
2 5 1 W Tryptophan
13 19 15 D Aspartic acid
16 20 16 N Asparagine
25 24 35 L Leucine
7 2 8 M Methionine
12 17 22 P Proline
19 35 21 S Serine
27 24 22 T Threonine
17 13 33 A Alanine
3 7 13 H Histidine
23 19 25 K Lycine
10 9 10 Y Tyrosine
7 17 10 R Arginine 

2VQP_1|Chain A|MATRIX PROTEIN|HUMAN RESPIRATORY SYNCYTIAL VIRUS (11259)
>3EYJ_1|Chain A|Hemagglutinin HA1 chain|Influenza A virus (11320)
>8WPH_1|Chains A, B|McyI-NAD|Anabaena sp. 90 (46234)
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 \(w=f(c)\)
2VQP , Knot 117 257 0.84 40 169 247 
EMETYVNKLHEGSTYTAAVQYNVLEKDDDPASLTIWVPMFQSSMPADLLIKELANVNILVKQISTPKGPSLRVMINSRSAVLAQMPSKFTICANVSLDDRSKLAYDVTTPCEIKACSLTCLKSKNMLTTVKDLTMKTLNPTHDIIALCEFENIVTSKKVIIPTYLRSISVRNKDLNTLENITTTEFKNAITNAKIIPYSGLLLVITVTDNKGAFKYIKPQSQFIVDLGAYLEKESIYYVTTNWKHTATRFAIKPRED
3EYJ , Knot 139 323 0.82 40 192 312 
GNPIICLGHHAVENGTSVKTLTDNHVEVVSAKELVETKHTDELCPSPLKLVDGQDCDLINGALGSPGCDRLQDTTWDVFIERPTAVDTCYPFDVPDYQSLRSILASSGSLEFIAEQFTWNGVKVDGSSSACLRGGRNSFFSRLNWLTKATNGNYGPINVTKENTGSYVRLYLWGVHHPSSDNEQTDLYKVATGRVTVSTRSDQISIVPNIGSRPRVRNQSGRISIYWTLVNPGDSIIFNSIGNLIAPRGHYKISKSTKSTVLKSDKRIGSCTSPCLTDKGSIQSDKPFQNVSRIAIGNCPKYVKQGSLMLATGMRNIPGKQAK
8WPH , Knot 153 357 0.84 40 204 345 
MGSSHHHHHHSSGLVPRGSHMTIIYPPKNFPSKTKNHKVLLIGKMYDEIGEKLLAEYTNVEIIKEPKQHQIHEAIQDVSGVFVRYPTKLDAQAIGLAKNLKVISTSGFGTDAIDIAAATKRGIVVVNNPGLSTTAVTEHTLSMILALAKKLTFLNQCVKAGNYLIRNQVQPIQLEGKTLGIVGLGRIGSAVAKICSTALQMRVLAYDPYVPSGKADTVRATLVQDLDYLLTESDFVSLHPELTDETCEMFDLEAFKKMKPSAFLINTSRGKVVRQPDLVTAIREKLIAGAAIDVFEPEPPAINNPLYEFDNVIFSPHLAGVTPEAGMAAALSAANQILQVLQGEKPPYIINPKVWNS

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 \(c\).

\(|P_{f(2VQP_1)}(2) \setminus P_{f(3EYJ_1)}(2)|=69\), \(|P_{f(3EYJ_1)}(2) \setminus P_{f(2VQP_1)}(2)|=92\). 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 1:01000100100100001110001100000110101111110001110111001101011100100101101011100001111011001010101010000011001001001010010010000110010010100101000111100100110000111100100101000010010010000100110010111001111110100001110010100011101110100001001000100010011101000
Pair \(Z_2\) Length of longest common subsequence
2VQP_1,3EYJ_1 161 4
2VQP_1,8WPH_1 145 4
3EYJ_1,8WPH_1 164 4

Newick tree

 
[
	3EYJ_1:83.96,
	[
		2VQP_1:72.5,8WPH_1:72.5
	]:11.46
]

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 \((0.85)(0.8)(\frac{580 }{\log_{20} 580}-\frac{257}{\log_{20}257})=91.3\)
Status Protein1 Protein2 d d1/2
Query variables 2VQP_1 3EYJ_1 117 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]