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Parikh vectors
1XTZ_1 5HRQ_1 2ZZX_1 Letter Amino acid
21 2 18 I Isoleucine
21 0 18 K Lycine
1 0 10 W Tryptophan
22 1 29 V Valine
9 2 14 Q Glutamine
17 2 19 E Glutamic acid
11 0 6 H Histidine
10 0 20 P Proline
7 2 15 Y Tyrosine
11 0 27 R Arginine
20 0 17 D Aspartic acid
22 1 30 G Glycine
11 0 22 F Phenylalanine
7 1 14 T Threonine
21 0 40 A Alanine
12 2 8 N Asparagine
1 0 9 M Methionine
17 2 14 S Serine
2 4 0 C Cysteine
21 2 31 L Leucine 

1XTZ_1|Chain A|Ribose-5-phosphate isomerase|Saccharomyces cerevisiae (4932)
>5HRQ_1|Chains A, C, E, G, I, K|Insulin A-Chain|Homo sapiens (9606)
>2ZZX_1|Chains A, B, C, D|ABC transporter, solute-binding protein|Thermus thermophilus (300852)
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 \(w=f(c)\)
1XTZ , Knot 113 264 0.79 40 155 250 
MAAGVPKIDALESLGNPLEDAKRAAAYRAVDENLKFDDHKIIGIGSGSTVVYVAERIGQYLHDPKFYEVASKFICIPTGFQSRNLILDNKLQLGSIEQYPRIDIAFDGADEVDENLQLIKGGGACLFQEKLVSTSAKTFIVVADSRKKSPKHLGKNWRQGVPIEIVPSSYVRVKNDLLEQLHAEKVDIRQGGSAKAGPVVTDNNNFIIDADFGEISDPRKLHREIKLLVGVVETGLFIDNASKAYFGNSDGSVEVTEKHHHHHH
5HRQ , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN
2ZZX , Knot 153 361 0.83 38 206 346 
MKRVSRRAFLRRLGVGVAATAAFSPLAVAQARRYRWRIQTAWDAGTVGYSLFQKFTERVKELTDGQLEVQPFPAGAVVGTFDMFDAVKTGVLDGMNPFTLYWAGRMPVTAFLSSYALGLDRPDQWETWFYSLGGLDIARRAFAEQGLFYVGPVQHDLNIIHSKKPIRRFEDFKGVKLRVPGGMIAEVFAAAGASTVLLPGGEVYPALERGVIDAADFVGPAVNYNLGFHQVAKYIIMGPPETPAIHQPVDLMDFTINLNRWRSLPKPLQERFIAAVHEYSWIHYAGIQKANLEAWPKYRQAGVEVIRLSNEDVRKFRRLAIPIWFKWAKMDKYSREAFASQLEYMKGIGYVTDEELKGLSL

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 \(c\).

\(|P_{f(1XTZ_1)}(2) \setminus P_{f(5HRQ_1)}(2)|=144\), \(|P_{f(5HRQ_1)}(2) \setminus P_{f(1XTZ_1)}(2)|=9\). 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 1:111111010110011011001001110011000101000011111010011011001100100101001100110110110000111000101101000101011101100100010110111101100011000100111110000001001100100111101110001010001100101001010011010111110000011101011010010010001011111100111100100101100010101000000000
Pair \(Z_2\) Length of longest common subsequence
1XTZ_1,5HRQ_1 153 2
1XTZ_1,2ZZX_1 169 3
5HRQ_1,2ZZX_1 210 3

Newick tree

 
[
	2ZZX_1:10.79,
	[
		1XTZ_1:76.5,5HRQ_1:76.5
	]:24.29
]

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 \((0.85)(0.8)(\frac{285 }{\log_{20} 285}-\frac{21}{\log_{20}21})=88.6\)
Status Protein1 Protein2 d d1/2
Query variables 1XTZ_1 5HRQ_1 108 58
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]