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Parikh vectors
7JWD_1 1MVA_1 6JUK_1 Letter Amino acid
9 7 38 L Leucine
13 8 26 T Threonine
11 5 22 E Glutamic acid
1 0 16 H Histidine
4 2 6 W Tryptophan
4 4 17 Y Tyrosine
13 4 27 D Aspartic acid
14 6 19 K Lycine
3 2 6 C Cysteine
7 6 7 Q Glutamine
4 8 18 I Isoleucine
3 2 8 M Methionine
1 6 25 P Proline
3 13 18 S Serine
4 15 40 A Alanine
8 4 23 R Arginine
7 4 9 F Phenylalanine
13 14 32 V Valine
7 10 12 N Asparagine
9 9 32 G Glycine 

7JWD_1|Chains A, B|Retinol-binding protein 2|Homo sapiens (9606)
>1MVA_1|Chains A, B, C|BACTERIOPHAGE MS2 CAPSID|Enterobacterio phage MS2 (12022)
>6JUK_1|Chains A, B|Formate dehydrogenase|Pseudomonas sp. 101 (33067)
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 \(w=f(c)\)
7JWD , Knot 68 138 0.81 40 106 133 
MTRDQNGTWEMESNENFEGYMKALDIDFATRKIAVRLTQTKVIDQDGDNFKTKTTSTFRNYDVDFTVGVEFDEYTKSLDNRHVKALVTWEGDVLVCVQKGEKENRGWKQWIEGDKLYLELTCGDQVCRQVFKKKLVPR
1MVA , Knot 64 129 0.80 38 98 125 
ASNFTQFVLVDNGGTGDVTVAPSNFANGVAEWISSNSRSQAYKVACSVRQSSAQNRKYTIKVEVPKVATQTVGGVELPVAAWRSYLNMELTIPIFATNSDCELIVKAMQGLLKDGNPIPSAIAANSGIY
6JUK , Knot 170 401 0.84 40 229 386 
MAKVLCVLYDDPVDGYPKTYARDDLPKIDHYPGGQTLPTPKAIDFTPGQLLGSVSGELGLRKYLESNGHTLVVTSDKDGPDSVFERELVDADVVISQPFWPAYLTPERIAKAKNLKLALTAGIGSDHVDLQSAIDRNVTVAEVTYCNSISVAEHVVMMILSLVRNYLPSHEWARKGGWNIADCVSHAYDLEAMHVGTVAAGRIGLAVLRRLAPFDVHLHYTDRHRLPESVEKELNLTWHATREDMYPVCDVVTLNIPLHPPTEHMINDETLKLFKRGAYIVNTARGKLCDRDAVARALESGRLAGYAGDVWFPQPAPKDHPWRTMPYNGMTPHISGTTLTAQARYAAGTREILECFFEGRPIRDEYLIVQGGALAGTGAHIYSKGNATGGSEEAAKFKKAV

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 \(c\).

\(|P_{f(7JWD_1)}(2) \setminus P_{f(1MVA_1)}(2)|=69\), \(|P_{f(1MVA_1)}(2) \setminus P_{f(7JWD_1)}(2)|=61\). 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 1:100000101010000010101011010110001110100001100010010000000100001010111010000001000010111010101110100100000110011010010101001001000110001110
Pair \(Z_2\) Length of longest common subsequence
7JWD_1,1MVA_1 130 3
7JWD_1,6JUK_1 199 3
1MVA_1,6JUK_1 195 4

Newick tree

 
[
	6JUK_1:10.37,
	[
		7JWD_1:65,1MVA_1:65
	]:42.37
]

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 \((0.85)(0.8)(\frac{267 }{\log_{20} 267}-\frac{129}{\log_{20}129})=43.2\)
Status Protein1 Protein2 d d1/2
Query variables 7JWD_1 1MVA_1 56 53
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]