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Parikh vectors
8COH_1 3NCL_1 4UPJ_1 Letter Amino acid
2 14 11 I Isoleucine
1 6 5 K Lycine
3 3 2 M Methionine
3 10 3 F Phenylalanine
2 14 5 P Proline
8 16 9 V Valine
10 13 3 R Arginine
5 10 6 E Glutamic acid
15 28 11 G Glycine
9 6 3 Y Tyrosine
12 16 5 A Alanine
7 15 4 D Aspartic acid
8 14 2 Q Glutamine
7 7 0 H Histidine
9 19 10 L Leucine
4 6 8 N Asparagine
16 18 4 S Serine
6 12 7 T Threonine
3 8 1 W Tryptophan
2 6 0 C Cysteine 

8COH_1|Chain A|Nanobody TPP-3444|Lama glama (9844)
>3NCL_1|Chain A|Suppressor of tumorigenicity 14 protein|Homo sapiens (9606)
>4UPJ_1|Chains A, B|HIV-2 PROTEASE|Human immunodeficiency virus 2 (11709)
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 \(w=f(c)\)
8COH , Knot 63 132 0.77 40 97 123 
EVQLVESGGGLVQPGGSLRLSCAASGRAHSDYAMAWFRQAPGQEREFVAGIGWSGGDTLYADSVRGRFTNSRDNSKNTLYLQMNSLRAEDTAVYYCAARQGQYIYSSMRSDSYDYWGQGTLVTVSSHHHHHH
3NCL , Knot 113 241 0.85 40 183 236 
VVGGTDADEGEWPWQVSLHALGQGHICGASLISPNWLVSAAHCYIDDRGFRYSDPTQWTAFLGLHDQSQRSAPGVQERRLKRIISHPFFNDFTFDYDIALLELEKPAEYSSMVRPISLPDASHVFPAGKAIWVTGWGHTQYGGTGALILQKGEIRVINQTTCENLLPQQITPRMMCVGFLSGGVDSCQGDSGGPLSSVEADGRIFQAGVVSWGDGCAQRNKPGVYTRLPLFRDWIKENTGV
4UPJ , Knot 52 99 0.80 36 81 96 
PQFSLWKRPVVTAYIEGQPVEVLLDTGADDSIVAGIELGNNYSPKIVGGIGGFINTLEYKNVEIEVLNKKVRATIMTGDTPINIFGRNILTALGMSLNL

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 \(c\).

\(|P_{f(8COH_1)}(2) \setminus P_{f(3NCL_1)}(2)|=45\), \(|P_{f(3NCL_1)}(2) \setminus P_{f(8COH_1)}(2)|=131\). 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 1:010110011111011101010011010100001111100111000011111110110010100101010000000000101010010100011000110010010001000000011010110100000000
Pair \(Z_2\) Length of longest common subsequence
8COH_1,3NCL_1 176 4
8COH_1,4UPJ_1 130 4
3NCL_1,4UPJ_1 182 3

Newick tree

 
[
	3NCL_1:96.30,
	[
		8COH_1:65,4UPJ_1:65
	]:31.30
]

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 \((0.85)(0.8)(\frac{373 }{\log_{20} 373}-\frac{132}{\log_{20}132})=73.2\)
Status Protein1 Protein2 d d1/2
Query variables 8COH_1 3NCL_1 97 71.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]