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Parikh vectors
1RCB_1 8HUZ_1 3VOF_1 Letter Amino acid
9 5 20 R Arginine
6 7 25 N Asparagine
10 13 26 E Glutamic acid
6 10 12 F Phenylalanine
8 12 50 A Alanine
4 11 26 D Aspartic acid
5 8 13 H Histidine
16 22 24 L Leucine
3 11 27 V Valine
15 8 22 T Threonine
2 2 15 Y Tyrosine
1 6 22 P Proline
8 6 21 S Serine
6 2 4 C Cysteine
7 5 15 Q Glutamine
3 14 25 G Glycine
6 7 25 I Isoleucine
12 13 12 K Lycine
1 6 2 M Methionine
1 1 9 W Tryptophan 

1RCB_1|Chain A|INTERLEUKIN-4|Homo sapiens (9606)
>8HUZ_1|Chains A, B, C, D, E, F|Prolyl-tRNA synthetase associated domain-containing protein 1|Mus musculus (10090)
>3VOF_1|Chain A|Cellobiohydrolase|Coprinopsis cinerea (5346)
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 \(w=f(c)\)
1RCB , Knot 67 129 0.84 40 101 125 
HKCDITLQEIIKTLNSLTEQKTLCTELTVTDIFAASKNTTEKETFCRAATVLRQFYSHHEKDTRCLGATAQQFHRHKQLIRFLKRLDRNLWGLAGLNSCPVKEANQSTLENFLERLKTIMREKYSKCSS
8HUZ , Knot 85 169 0.86 40 135 166 
MAGSELRAELEQRLGALAIRTEVVEHPEVFTIEEMMPHIQHLKGAHSKNLFLKDKKKKNYWLVTVLHDRQINLNDLGKQLGVGSGNLRFADETAMLEKLKVGQGCATPLSLFCDDGDVKFVLDSAFLEGGHEKVYFHPMTNAATMGLSPEDFLIFVKATGHDPIILNFD
3VOF , Knot 166 395 0.83 40 218 370 
APSASFERRQGSVNPYIGRSPLVIKSYAEKLEETIAYFEAQGDELNAARTRTVQGIPTFAWISDSATIDTIQPLIADAVAHQEASGEQVLVQLVIYNLPDRACAAKASDGEFHLDDDGANKYRAYVDRIVAELSTADADKLHFSIVLEPDSLGNMVTNMHVPKCQGAATAYKEGIAYTIASLQKPNIDLYIDAAHGGWLGWNDNLRPSAEIFKETLDLARQITPNATVRGLAINVSNYNPYKTRAREDYTEWNNAYDEWNYVKTLTPHLQAVGFPAQFIVDQGRSGREGIRTEWGQWCNIRNAGFGIRPTTDQAIVDSANVDAIVWVKPGGESDGTSDVNAVRFDENCRSPASHVPAPEAGEWFNEFVVNLVINANPPLEPTYAAAALEHHHHHH

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 \(c\).

\(|P_{f(1RCB_1)}(2) \setminus P_{f(8HUZ_1)}(2)|=55\), \(|P_{f(8HUZ_1)}(2) \setminus P_{f(1RCB_1)}(2)|=89\). 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 1:000010100110010010000010001010011110000000001001101100100000000001110100100000110110010001111111000110010000100110010011000000000
Pair \(Z_2\) Length of longest common subsequence
1RCB_1,8HUZ_1 144 3
1RCB_1,3VOF_1 189 3
8HUZ_1,3VOF_1 185 4

Newick tree

 
[
	3VOF_1:99.64,
	[
		1RCB_1:72,8HUZ_1:72
	]:27.64
]

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 \((0.85)(0.8)(\frac{298 }{\log_{20} 298}-\frac{129}{\log_{20}129})=52.4\)
Status Protein1 Protein2 d d1/2
Query variables 1RCB_1 8HUZ_1 65 56.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]