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Parikh vectors
8WGK_1 8EQL_1 4XTZ_1 Letter Amino acid
9 0 2 K Lycine
4 0 1 M Methionine
8 0 1 F Phenylalanine
7 0 13 S Serine
1 0 1 Y Tyrosine
5 0 22 D Aspartic acid
14 0 9 E Glutamic acid
2 0 4 H Histidine
8 0 13 P Proline
16 0 32 V Valine
1 2 1 C Cysteine
9 6 29 G Glycine
4 0 6 W Tryptophan
5 0 6 N Asparagine
10 0 12 Q Glutamine
10 0 13 I Isoleucine
23 0 32 L Leucine
14 2 10 T Threonine
25 6 36 A Alanine
10 0 27 R Arginine 

8WGK_1|Chain A|ABC-type uncharacterized transport system periplasmic component-like protein|Rhodothermus marinus DSM 4252 (518766)
>8EQL_1|Chain A[auth C]|DNA (5'-D(*AP*AP*TP*AP*AP*CP*CP*GP*GP*AP*AP*GP*TP*GP*GP*G)-3')|synthetic construct (32630)
>4XTZ_1|Chains A, B|Bifunctional ligase/repressor BirA|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
8WGK , Knot 85 185 0.80 40 126 177 
MHTLRLFTVLSLLLTSAFTLFPETETEVTPIQQLFLIKELKPGIARIGVIWDKNAANRDEVLPQLQRASAATGIKVVVAEVASLQEVAPQFRTLLRDHQVEALWVLEESGLLGQAAARSFLIKNATQAGMPVFAPSETWLKEGACVTWRKDAEGIRLVVNKAVAEAMGITIPAKYQDRTAFLAMN
8EQL , Knot 8 16 0.46 8 11 14 
AATAACCGGAAGTGGG
4XTZ , Knot 110 270 0.76 40 139 254 
GSHMVTDRDRLRPPLDERSLRDQLIGAGSGWRQLDVVAQTGSTNADLLARAASGADIDGVVLIAEHQTAGRGRHGRGWAATARAQIILSVGVRVVDVPVQAWGWLSLAAGLAVLDSVAPLIAVPPAETGLKWPNDVLARGGKLAGILAEVAQPFVVLGVGLNVTQAPEEVDPDATSLLDLGVAAPDRNRIASRLLRELEARIIQWRNANPQLAADYRARSLTIGSRVRVELPGGQDVVGIARDIDDQGRLCLDVGGRTVVVSAGDVVHLR

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 \(c\).

\(|P_{f(8WGK_1)}(2) \setminus P_{f(8EQL_1)}(2)|=119\), \(|P_{f(8EQL_1)}(2) \setminus P_{f(8WGK_1)}(2)|=4\). 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 1:10010110110111001101110000010110011110010111101111100011000011101001011011011110110100111010011000010111110001111011100111001001111111100011001101010001011011100111011110111000000111110
Pair \(Z_2\) Length of longest common subsequence
8WGK_1,8EQL_1 123 3
8WGK_1,4XTZ_1 129 4
8EQL_1,4XTZ_1 136 4

Newick tree

 
[
	4XTZ_1:67.78,
	[
		8WGK_1:61.5,8EQL_1:61.5
	]:6.28
]

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 \((0.85)(0.8)(\frac{201 }{\log_{20} 201}-\frac{16}{\log_{20}16})=65.4\)
Status Protein1 Protein2 d d1/2
Query variables 8WGK_1 8EQL_1 82 44
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]