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Parikh vectors
6NCJ_1 3CVT_1 9FYW_1 Letter Amino acid
3 9 3 F Phenylalanine
4 9 0 W Tryptophan
7 13 5 D Aspartic acid
2 4 8 C Cysteine
10 14 8 Q Glutamine
15 12 3 I Isoleucine
11 8 11 K Lycine
6 20 5 P Proline
10 13 10 T Threonine
14 40 13 A Alanine
5 20 5 R Arginine
5 6 10 N Asparagine
19 20 4 G Glycine
14 4 4 H Histidine
4 8 5 M Methionine
13 9 18 S Serine
15 15 14 V Valine
10 13 5 E Glutamic acid
9 33 13 L Leucine
4 12 6 Y Tyrosine 

6NCJ_1|Chain A|Integrase|Human immunodeficiency virus 1 (11676)
>3CVT_1|Chains A, B, C, D|DNA-3-methyladenine glycosylase 2|Escherichia coli (562)
>9FYW_1|Chains A, B|Ribonuclease pancreatic|Bos taurus (9913)
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 \(w=f(c)\)
6NCJ , Knot 81 180 0.78 40 134 174 
MGSSHHHHHHSSGLVPRGSHMHGQVDSSPGIWQLDCTHLEGKVILVAVHVASGYIEAEVIPAETGQETAYFLLKLAGRWPVKTVHTDNGSNFTSTTVKAACWWAGIKQEDGIPYNPQSQGVIESMNKELKKIIGQVRDQAEHLKTAVQMAVFIHNHKRKGGIGGYSAGERIVDIIATDIQ
3CVT , Knot 125 282 0.83 40 177 273 
MYTLNWQPPYDWSWMLGFLAARAVSSVETVADSYYARSLAVGEYRGVVTAIPDIARHTLHINLSAGLEPVAAECLAKMSRLFDLQCNPQIVNGALGRLGAARPGLRLPGCVDAFEQGVRAILGQLVSVAMAAKLTARVAQLYGERLDDFPEYICFPTPQRLAAADPQALKALGMPLKRAEALIHLANAALEGTLPMTIPGDVEQAMKTLQTFPGIGRWTANYFALRGWQAKDVFLPDDYLIKQRFPGMTPAQIRRYAERWKPWRSYALLHIWYTEGWQPDEA
9FYW , Knot 72 150 0.80 38 120 143 
MALKSLVLLSLLVLVLLLVRVQPSLGKETAAAKFERQHMDSSTSAASSSNYCNQMMKSRNLTKDRCKPVNTFVHESLADVQAVCSQKNVACKNGQTNCYQSYSTMSITDCRETGSSKYPNCAYKTTQANKHIIVACEGNPYVPVHFDASV

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 \(c\).

\(|P_{f(6NCJ_1)}(2) \setminus P_{f(3CVT_1)}(2)|=72\), \(|P_{f(3CVT_1)}(2) \setminus P_{f(6NCJ_1)}(2)|=115\). 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 1:110000000000111101001010100011110100001010111111011010101011110010001011101110111001000010010000101101111100001110010001110010001001110100010010011011111000000111110011001101110010
Pair \(Z_2\) Length of longest common subsequence
6NCJ_1,3CVT_1 187 3
6NCJ_1,9FYW_1 158 3
3CVT_1,9FYW_1 193 3

Newick tree

 
[
	3CVT_1:99.77,
	[
		6NCJ_1:79,9FYW_1:79
	]:20.77
]

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 \((0.85)(0.8)(\frac{462 }{\log_{20} 462}-\frac{180}{\log_{20}180})=82.7\)
Status Protein1 Protein2 d d1/2
Query variables 6NCJ_1 3CVT_1 108 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]