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Parikh vectors
1RPI_1 2OJM_1 3VSD_1 Letter Amino acid
10 3 43 V Valine
1 4 5 H Histidine
6 1 11 K Lycine
2 3 14 F Phenylalanine
8 2 10 T Threonine
1 0 13 Y Tyrosine
3 0 19 D Aspartic acid
1 0 8 M Methionine
10 1 42 L Leucine
7 0 22 P Proline
0 0 32 S Serine
2 0 3 W Tryptophan
3 0 40 A Alanine
6 0 8 Q Glutamine
2 0 2 C Cysteine
4 0 27 E Glutamic acid
13 3 37 G Glycine
11 3 14 I Isoleucine
4 2 28 R Arginine
5 0 11 N Asparagine 

1RPI_1|Chains A, B|protease|Human immunodeficiency virus 1 (11676)
>2OJM_1|Chain A|Moronecidin|null
>3VSD_1|Chains A, B|Protein CysO|Aeropyrum pernix (272557)
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 \(w=f(c)\)
1RPI , Knot 51 99 0.79 38 81 93 
PQITLWQRPIVTIKIGGQLKEALLNTGADDTVLEEVNLPGRWKPKLIGGIGGFVKVRQYDQVPIEICGHKVIGTVLVGPTPANVIGRNLMTQIGCTLNF
2OJM , Knot 13 22 0.60 18 21 20 
FFHHIFRGIVHVGKTIHRLVTG
3VSD , Knot 157 389 0.80 40 194 360 
MALADISGYLDVLDSVRGFSYLENAREVLRSGEARCLGNPRSEPEYVKALYVIGASRIPVGDGCSHTLEELGVFDISVPGEMVFPSPLDFFERGKPTPLVRSRLQLPNGVRVWLKLEWYNPFSLSVADRPAVEIISRLSRRVEKGSLVADATSSNFGVALSAVARLYGYRARVYLPGAAEEFGKLLPRLLGAQVIVDPEAPSTVHLLPRVMKDSKNEGFVHVNQFYNDANFEAHMRGTAREIFVQSRRGGLALRGVAGSLGTSGHMSAAAFYLQSVDPSIRAVLVQPAQGDSIPGIRRVETGMLWINMLDISYTLAEVTLEEAMEAVVEVARSDGLVIGPSGGAAVKALAKKAAEGDLEPGDYVVVVPDTGFKYLSLVQNALEGAGDSV

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 \(c\).

\(|P_{f(1RPI_1)}(2) \setminus P_{f(2OJM_1)}(2)|=74\), \(|P_{f(2OJM_1)}(2) \setminus P_{f(1RPI_1)}(2)|=14\). 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 1:101011001110101110100111001100011001011101010111111111010000011101010011101111101101110011001100101
Pair \(Z_2\) Length of longest common subsequence
1RPI_1,2OJM_1 88 2
1RPI_1,3VSD_1 183 3
2OJM_1,3VSD_1 189 3

Newick tree

 
[
	3VSD_1:10.35,
	[
		1RPI_1:44,2OJM_1:44
	]:60.35
]

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 \((0.85)(0.8)(\frac{121 }{\log_{20} 121}-\frac{22}{\log_{20}22})=36.8\)
Status Protein1 Protein2 d d1/2
Query variables 1RPI_1 2OJM_1 46 27.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]