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Parikh vectors
2KOU_1 5BQK_1 1ZHN_1 Letter Amino acid
10 14 14 E Glutamic acid
11 27 25 L Leucine
11 16 26 S Serine
0 3 13 W Tryptophan
2 7 8 Y Tyrosine
3 17 20 V Valine
8 14 14 D Aspartic acid
3 7 8 N Asparagine
3 10 5 C Cysteine
5 18 19 G Glycine
8 15 13 P Proline
6 35 16 A Alanine
7 6 12 K Lycine
2 5 8 M Methionine
6 12 11 F Phenylalanine
3 8 8 H Histidine
2 7 20 Q Glutamine
7 14 7 I Isoleucine
3 13 14 T Threonine
2 23 12 R Arginine 

2KOU_1|Chain A|Dicer-like protein 4|Arabidopsis thaliana (3702)
>5BQK_1|Chains A, B, C|ICP27|Human herpesvirus 1 (10298)
>1ZHN_1|Chain A|CD1d1 antigen|Mus musculus (10090)
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 \(w=f(c)\)
2KOU , Knot 52 102 0.78 38 82 97 
ISGGSSISMMYKYCSRLPHDEFFQPKPEFQFKPVDEFGGTICRITLPANAPISEIESSLLPSTEAAKKDACLKAVHELHNLGVLNDFLLPDSKDEIEDELSD
5BQK , Knot 120 271 0.82 40 173 259 
ADTIDATTRLVLRSISERAAVDRISESFGRSAQVMHDPFGGQPFPAANSPWAPVLAGQGGPFDAETRRVSWETLVAHGPSLYRTFAGNPRAASTAKAMRDCVLRQENFIEALASADETLAWCKMCIHHNLPLRPQDPIIGTAAAVLDNLATRLRPFLQCYLKARGLCGLDELCSRRRLADIKDIASFVFVILARLANRVERGVAEIDYATLGVGVGEKMHFYLPGACMAGLIEILDTHRQECSSRVCELTASHIVAPPYVHGKYFYCNSLF
1ZHN , Knot 122 273 0.83 40 178 260 
NYTFRCLQMSSFANRSWSRTDSVVWLGDLQTHRWSNDSATISFTKPWSQGKLSNQQWEKLQHMFQVYRVSFTRDIQELVKMMSPKEDYPIEIQLSAGCEMYPGNASESFLHVAFQGKYVVRFWGTSWQTVPGAPSWLDLPIKVLNADQGTSATVQMLLNDTCPLFVRGLLEAGKSDLEKQEKPVAWLSSVPSSAHGHRQLVCHVSGFYPKPVWVMWMRGDQEQQGTHRGDFLPNADETWYLQATLDVEAGEEAGLACRVKHSSLGGQDIILYW

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 \(c\).

\(|P_{f(2KOU_1)}(2) \setminus P_{f(5BQK_1)}(2)|=34\), \(|P_{f(5BQK_1)}(2) \setminus P_{f(2KOU_1)}(2)|=125\). 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 1:101100101100000011000110101010101100111010010111011100100011100011000101011001001111001111000001000100
Pair \(Z_2\) Length of longest common subsequence
2KOU_1,5BQK_1 159 3
2KOU_1,1ZHN_1 172 3
5BQK_1,1ZHN_1 171 4

Newick tree

 
[
	1ZHN_1:87.73,
	[
		2KOU_1:79.5,5BQK_1:79.5
	]:8.23
]

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 \((0.85)(0.8)(\frac{373 }{\log_{20} 373}-\frac{102}{\log_{20}102})=83.3\)
Status Protein1 Protein2 d d1/2
Query variables 2KOU_1 5BQK_1 104 71
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]