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Parikh vectors
7HBF_1 3HET_1 4QAH_1 Letter Amino acid
9 0 14 Q Glutamine
19 4 28 L Leucine
17 5 23 K Lycine
17 0 12 T Threonine
7 1 9 Y Tyrosine
8 3 17 R Arginine
8 1 10 N Asparagine
16 1 18 D Aspartic acid
26 5 27 E Glutamic acid
20 3 16 I Isoleucine
7 1 11 M Methionine
15 1 13 A Alanine
10 0 14 F Phenylalanine
16 1 24 S Serine
1 0 6 W Tryptophan
0 0 6 C Cysteine
15 0 14 G Glycine
10 1 9 H Histidine
5 0 12 P Proline
11 0 16 V Valine 

7HBF_1|Chain A|Heat shock protein HSP 90-alpha|Homo sapiens (9606)
>3HET_1|Chains A, B|alpha/beta-peptide based on the GCN4-pLI side chain sequence with an (alpha-alpha-beta) backbone and a cyclic beta-residue at position 10|null
>4QAH_1|Chain A|Tyrosine-protein phosphatase non-receptor type 1|Homo sapiens (9606)
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 \(w=f(c)\)
7HBF , Knot 105 237 0.80 38 157 225 
MDQPMEEEEVETFAFQAEIAQLMSLIINTFYSNKEIFLRELISNSSDALDKIRYESLTDPSKLDSGKELHINLIPNKQDRTLTIVDTGIGMTKADLINNLGTIAKSGTKAFMEALQAGADISMIGQFGVGFYSAYLVAEKVTVITKHNDDEQYAWESSAGGSFTVRTDTGEPMGRGTKVILHLKEDQTEYLEERRIKEIVKKHSQFIGYPITLFVEKERDKEVSDDEAELEHHHHHH
3HET , Knot 20 34 0.69 26 28 31 
XRMKXIEDKLXEIXSKXYHXENELARIKKLLXER
4QAH , Knot 132 299 0.84 40 198 291 
MEMEKEFEQIDKSGSWAAIYQDIRHEASDFPCRVAKLPKNKNRNRYRDVSPFDHSRIKLHQEDNDYINASLIKMEEAQRSYILTQGPLPNTCGHFWEMVWEQKSRGVVMLNRVMEKGSLKCAQYWPQKEEKEMIFEDTNLKLTLISEDIKSYYTVRQLELENLTTQETREILHFHYTTWPDFGVPESPASFLNFLFKVRESGSLSPEHGPVVVHCSAGIGRSGTFCLADTCLLLMDKRKDPSSVDIKKVLLEMRKFRMGLIQKADQLRFSYLAVIEGAKFIMGDSSVQDQWKELSHEDL

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 \(c\).

\(|P_{f(7HBF_1)}(2) \setminus P_{f(3HET_1)}(2)|=147\), \(|P_{f(3HET_1)}(2) \setminus P_{f(7HBF_1)}(2)|=18\). 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 1:100110000100111010110110111001000001110011000001100100001001001001001010111000000101100111100101100110110010011101101110101110111110010111001011000000000110001110101000010111010011101000000010000100110000011101101110000000100001010000000
Pair \(Z_2\) Length of longest common subsequence
7HBF_1,3HET_1 165 3
7HBF_1,4QAH_1 183 3
3HET_1,4QAH_1 200 3

Newick tree

 
[
	4QAH_1:99.89,
	[
		7HBF_1:82.5,3HET_1:82.5
	]:17.39
]

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 \((0.85)(0.8)(\frac{271 }{\log_{20} 271}-\frac{34}{\log_{20}34})=78.9\)
Status Protein1 Protein2 d d1/2
Query variables 7HBF_1 3HET_1 100 56.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]