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Parikh vectors
1PUL_1 7AUZ_1 6IHA_1 Letter Amino acid
4 45 1 S Serine
19 38 7 A Alanine
6 41 0 E Glutamic acid
17 40 7 K Lycine
2 26 1 Q Glutamine
4 31 2 I Isoleucine
7 27 1 F Phenylalanine
3 21 2 N Asparagine
11 34 0 D Aspartic acid
0 11 0 C Cysteine
6 69 1 L Leucine
4 10 0 M Methionine
0 22 0 Y Tyrosine
3 23 1 R Arginine
9 32 2 G Glycine
7 16 0 H Histidine
6 38 4 V Valine
3 36 1 P Proline
11 40 0 T Threonine
3 13 0 W Tryptophan 

1PUL_1|Chain A|Hypothetical protein C32E8.3 in chromosome I|Caenorhabditis elegans (6239)
>7AUZ_1|Chain A|Leukotriene A-4 hydrolase|Homo sapiens (9606)
>6IHA_1|Chain A|SibaCec-A|Simulium bannaense (1619335)
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 \(w=f(c)\)
1PUL , Knot 57 125 0.73 36 87 117 
MGHHHHHHSHMAAAAGFNWDDADVKKRWDAFTKFGAATATEMTGKNFDKWLKDAGVLDNKAITGTMTGIAFSKVTGPKKKATFDETKKVLAFVAEDRARQSKKPIQDELDAITEKLAKLEAPSVG
7AUZ , Knot 246 613 0.85 40 285 572 
GPGPEIVDTCSLASPASVCRTKHLHLRCSVDFTRRTLTGTAALTVQSQEDNLRSLVLDTKDLTIEKVVINGQEVKYALGERQSYKGSPMEISLPIALSKNQEIVIEISFETSPKSSALQWLTPEQTSGKEHPYLFSQCQAIHCRAILPCQDTPSVKLTYTAEVSVPKELVALMSAIRDGETPDPEDPSRKIYKFIQKVPIPCYLIALVVGALESRQIGPRTLVWSEKEQVEKSAYEFSETESMLKIAEDLGGPYVWGQYDLLVLPPSFPYGGMENPCLTFVTPTLLAGDKSLSNVIAHEISHSWTGNLVTNKTWDHFWLNEGHTVYLERHICGRLFGEKFRHFNALGGWGELQNSVKTFGETHPFTKLVVDLTDIDPDVAYSSVPYEKGFALLFYLEQLLGGPEIFLGFLKAYVEKFSYKSITTDDWKDFLYSYFKDKVDVLNQVDWNAWLYSPGLPPIKPNYDMTLTNACIALSQRWITAKEDDLNSFNATDLKDLSSHQLNEFLAQTLQRAPLPLGHIKRMQEVYNFNAINNSEIRFRWLRLCIQSKWEDAIPLALKMATEQGRMKFTRPLFKDLAAFDKSHDQAVRTYQEHKASMHPVTAMLVGKDLKVD
6IHA , Knot 17 30 0.64 24 26 28 
KINKQKIKNGAKKALGVASKVAPVVAAFAR

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 \(c\).

\(|P_{f(1PUL_1)}(2) \setminus P_{f(7AUZ_1)}(2)|=14\), \(|P_{f(7AUZ_1)}(2) \setminus P_{f(1PUL_1)}(2)|=212\). 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 1:11000000001111111010010100010110011110100101001001100111100011010101111001011000101000001111110001000001100010110001101011011
Pair \(Z_2\) Length of longest common subsequence
1PUL_1,7AUZ_1 226 3
1PUL_1,6IHA_1 85 3
7AUZ_1,6IHA_1 267 3

Newick tree

 
[
	7AUZ_1:14.68,
	[
		1PUL_1:42.5,6IHA_1:42.5
	]:98.18
]

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 \((0.85)(0.8)(\frac{738 }{\log_{20} 738}-\frac{125}{\log_{20}125})=174.\)
Status Protein1 Protein2 d d1/2
Query variables 1PUL_1 7AUZ_1 221 131
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]