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Parikh vectors
1QVK_1 3BUT_1 4LUP_1 Letter Amino acid
0 3 1 M Methionine
2 3 1 W Tryptophan
0 6 3 N Asparagine
0 9 10 L Leucine
0 11 3 T Threonine
0 18 14 V Valine
0 8 5 G Glycine
0 5 3 P Proline
1 4 4 F Phenylalanine
0 10 7 S Serine
0 0 0 C Cysteine
0 12 4 E Glutamic acid
0 4 6 D Aspartic acid
0 1 7 Q Glutamine
0 8 7 H Histidine
0 8 2 I Isoleucine
0 9 5 K Lycine
0 3 6 Y Tyrosine
0 8 9 A Alanine
3 6 9 R Arginine 

1QVK_1|Chain A|c-RW|null
>3BUT_1|Chain A|Uncharacterized protein Af_0446|Archaeoglobus fulgidus DSM 4304 (224325)
>4LUP_1|Chains A, B[auth C]|RNA polymerase sigma factor|Escherichia coli (562)
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 \(w=f(c)\)
1QVK , Knot 4 6 0.39 6 5 4 
RRWWRF
3BUT , Knot 65 136 0.78 38 101 128 
MSLESVKAMWGVVTDSQTEIVALAKVRNEDVVPIVVSGYHYTIEMNGVKVADGYENSPVTVKPASATTLKFSLRLNNSFLREWWVTHIANGEKTKIRVAIKPTIEIGGRDVEVPVFLRESEFTTKLLSEGHHHHHH
4LUP , Knot 54 106 0.79 38 80 101 
MSEQLTDQVLVERVQKGDQKAFNLLVVRYQHKVASLVSRYVPSGDVPDVVQEAFIKAYRALDSFRGDSAFYTWLYRIAVNTAKNYLVAQGRRLELVPRGSHHHHHH

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 \(c\).

\(|P_{f(1QVK_1)}(2) \setminus P_{f(3BUT_1)}(2)|=4\), \(|P_{f(3BUT_1)}(2) \setminus P_{f(1QVK_1)}(2)|=100\). 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 1:001101
Pair \(Z_2\) Length of longest common subsequence
1QVK_1,3BUT_1 104 2
1QVK_1,4LUP_1 83 2
3BUT_1,4LUP_1 123 6

Newick tree

 
[
	3BUT_1:61.23,
	[
		1QVK_1:41.5,4LUP_1:41.5
	]:19.73
]

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 \((0.85)(0.8)(\frac{142 }{\log_{20} 142}-\frac{6}{\log_{20}6})=51.5\)
Status Protein1 Protein2 d d1/2
Query variables 1QVK_1 3BUT_1 64 33.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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