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Parikh vectors
8QBV_1 1EII_1 4HAN_1 Letter Amino acid
12 6 17 R Arginine
3 7 19 F Phenylalanine
11 14 11 T Threonine
7 4 24 I Isoleucine
1 0 15 P Proline
13 2 20 S Serine
9 11 24 V Valine
10 3 7 M Methionine
1 4 3 W Tryptophan
2 3 7 Y Tyrosine
10 9 20 N Asparagine
0 3 3 C Cysteine
23 7 6 Q Glutamine
22 11 17 E Glutamic acid
25 9 24 L Leucine
28 3 13 A Alanine
12 11 14 D Aspartic acid
9 11 21 G Glycine
0 1 10 H Histidine
19 15 18 K Lycine 

8QBV_1|Chain A|Membrane-associated protein Vipp1|Nostoc punctiforme (272131)
>1EII_1|Chain A|CELLULAR RETINOL-BINDING PROTEIN II|Rattus norvegicus (10116)
>4HAN_1|Chains A, B|Galectin-8|Homo sapiens (9606)
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 \(w=f(c)\)
8QBV , Knot 96 217 0.79 36 138 202 
MGLFDRIKRVVSSNLNDLVNKAEDPEKMLEQAILEMQEDLVQLRQGVAQAIAAQKRSEKQYNDAQNEINKWQRNAQLALQKGDENLARQALERKKTYTDTSAALKASLDTQSTQVETLKRNLIQLESKISEAKTKKEMLKARITTAKAQEQLQGMVRGMNTSSAMSAFERMEEKVLMQESRAQALGELAGADLETQFAQLEGGSDVDDELAALKAQM
1EII , Knot 67 134 0.81 38 103 130 
MTKDQNGTWEMESNENFEGYMKALDIDFATRKIAVRLTQTKIIVQDGDNFKTKTNSTFRNYDLDFTVGVEFDEHTKGLDGRNVKTLVTWEGNTLVCVQKGEKENRGWKQWVEGDKLYLELTCGDQVCRQVFKKK
4HAN , Knot 126 293 0.81 40 173 275 
GSMMLSLNNLQNIIYNPVIPFVGTIPDQLDPGTLIVIRGHVPSDADRFQVDLQNGSSVKPRADVAFHFNPRFKRAGCIVCNTLINEKWGREEITYDTPFKREKSFEIVIMVLKDKFQVAVNGKHTLLYGHRIGPEKIDTLGIYGKVNIHSIGFSFSSHMRLPFAARLNTPMGPGRTVVVKGEVNANAKSFNVDLLAGKSKDIALHLNPRLNIKAFVRNSFLQESWGEEERNITSFPFSPGMYFEMIIYCDVREFKVAVNGVHSLEYKHRFKELSSIDTLEINGDIHLLEVRSW

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 \(c\).

\(|P_{f(8QBV_1)}(2) \setminus P_{f(1EII_1)}(2)|=93\), \(|P_{f(1EII_1)}(2) \setminus P_{f(8QBV_1)}(2)|=58\). 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 1:1111001001100010011001001001100111010001101001110111100000000001000100100010111001000110011000000000011101010000001001000110100010010000011010100101000101110110000110110010001110000101110111101000110101100100011110101
Pair \(Z_2\) Length of longest common subsequence
8QBV_1,1EII_1 151 3
8QBV_1,4HAN_1 157 3
1EII_1,4HAN_1 178 3

Newick tree

 
[
	4HAN_1:86.53,
	[
		8QBV_1:75.5,1EII_1:75.5
	]:11.03
]

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 \((0.85)(0.8)(\frac{351 }{\log_{20} 351}-\frac{134}{\log_{20}134})=66.2\)
Status Protein1 Protein2 d d1/2
Query variables 8QBV_1 1EII_1 84 67
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]