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Parikh vectors
9EIU_1 1AQA_1 8JUV_1 Letter Amino acid
10 12 13 E Glutamic acid
15 8 35 L Leucine
1 0 10 M Methionine
4 7 17 T Threonine
8 4 43 A Alanine
5 1 9 N Asparagine
1 6 13 H Histidine
11 10 12 K Lycine
8 6 28 S Serine
0 4 8 Y Tyrosine
7 3 19 R Arginine
0 0 4 C Cysteine
11 2 7 Q Glutamine
1 2 12 F Phenylalanine
1 3 15 P Proline
0 1 3 W Tryptophan
12 5 36 V Valine
5 9 21 D Aspartic acid
9 6 30 G Glycine
7 5 4 I Isoleucine 

9EIU_1|Chain A|Caveolae-associated protein 1|Homo sapiens (9606)
>1AQA_1|Chain A|CYTOCHROME B5|Rattus norvegicus (10116)
>8JUV_1|Chain A|LRR receptor-like serine/threonine-protein kinase FLS2|Oryza sativa subsp. japonica (39947)
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 \(w=f(c)\)
9EIU , Knot 55 116 0.75 34 81 109 
GSPGIQLIKSDQVNGVLVLSLLDKIIGAVDQIQLTQAQLEERQAEMEGAVQSIQGELSKLGKAHATDDNTVSKLLEKVRKVSVNVKTVRGSLERQAGQIKKLEVNEAELLRRRNFK
1AQA , Knot 47 94 0.75 36 76 91 
DKDVKYYTLEEIQKHKDSKSTWVILHHKVYDLTKFLEEHPGGEEVLREQAGGDATENFEDVGHSTDARELSKTYIIGELHPDDRSKIAKPSETL
8JUV , Knot 141 339 0.80 40 186 314 
MGSSHHHHHHSQGSPELRRFSYGQLAAATNSFDQGNVIGSSNLSTVYKGVLAGDADGGMVVAVKRLNLEQFPSKSDKCFLTELATLSRLRHKNLARVVGYAWEAGKIKALVLDYMVNGDLDGAIHGGAAAPPPAPSRWTVRERLRVCVSVAHGLVYLHSGYDFPVVHCAVKPSNVLLDGDWEARVSDFGTARMLGVHLPAAANAAAQSTATSSAFRGTVGYMAPEFAYMRTVSTKVDVFSFGVLAMELFTGRRPTGTIEEDGVPLTLQQLVDNAVSRGLDGVHAVLDPRMKVATEADLSTAADVLAVALSCAAFEPADRPDMGAVLSSLLKMSKLVGED

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 \(c\).

\(|P_{f(9EIU_1)}(2) \setminus P_{f(1AQA_1)}(2)|=55\), \(|P_{f(1AQA_1)}(2) \setminus P_{f(9EIU_1)}(2)|=50\). 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 1:10111011000010111110110011111001010010100001010111001010100110101000001001100100101010010101000110100101001011000010
Pair \(Z_2\) Length of longest common subsequence
9EIU_1,1AQA_1 105 4
9EIU_1,8JUV_1 159 3
1AQA_1,8JUV_1 174 3

Newick tree

 
[
	8JUV_1:91.32,
	[
		9EIU_1:52.5,1AQA_1:52.5
	]:38.82
]

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 \((0.85)(0.8)(\frac{210 }{\log_{20} 210}-\frac{94}{\log_{20}94})=37.8\)
Status Protein1 Protein2 d d1/2
Query variables 9EIU_1 1AQA_1 45 41
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]