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Parikh vectors
1RZK_1 8JGG_1 5FPX_1 Letter Amino acid
20 6 8 E Glutamic acid
25 0 8 G Glycine
12 1 6 F Phenylalanine
19 3 8 V Valine
9 3 11 D Aspartic acid
5 1 6 H Histidine
26 3 9 I Isoleucine
19 3 5 S Serine
12 3 4 R Arginine
30 3 5 N Asparagine
14 1 1 C Cysteine
13 2 2 Q Glutamine
21 5 6 L Leucine
20 6 9 K Lycine
6 2 6 M Methionine
6 0 2 W Tryptophan
12 11 10 A Alanine
5 1 2 Y Tyrosine
25 2 3 T Threonine
14 4 2 P Proline 

1RZK_1|Chain A[auth G]|ENVELOPE GLYCOPROTEIN GP120|Human immunodeficiency virus 1 (11676)
>8JGG_1|Chain A[auth G]|Guanine nucleotide-binding protein G(I)/G(S)/G(O) subunit gamma-2|Homo sapiens (9606)
>5FPX_1|Chains A, B|PECTIN DEGRADATION PROTEIN|YERSINIA ENTEROCOLITICA (393305)
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 \(w=f(c)\)
1RZK , Knot 134 313 0.82 40 188 295 
GARSEVKLENVTENFNMWKNNMVEQMHEDIISLWDQSLKPCVKLTPLCVGAGSCNTSVITQACPKVSFEPIPIHYCAPAGFAILKCNDKKFNGTGPCTNVSTVQCTHGIRPVVSTQLLLNGSLAEEEIVIRSENFTNNAKTIIVQLNESVVINCTGAGHCNLSKTQWENTLEQIAIKLKEQFGNNKTIIFNPSSGGDPEIVTHSFNCGGEFFYCNSTQLFTWNDTRKLNNTGRNITLPCRIKQIINMWQEVGKAMYAPPIRGQIRCSSNITGLLLTRDGGKDTNGTEIFRPGGGDMRDNWRSELYKYKVVKIE
8JGG , Knot 36 60 0.82 36 56 58 
NTASIAQARKLVEQLKMEANIDRIKVSKAAADLMAYCEAHAKEDPLLTPVPASENPFREK
5FPX , Knot 62 113 0.86 40 102 111 
HMASKMFFINDETPWEELGNGIKRKVMTWSDDLMMVCVHFDKGAIGVAHKHDIHDQIAYVAAGSFEVEIEGQKRILKAGDAYRAVKNEMHGAVSLEDNSILIDTFNPKRDDFL

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 \(c\).

\(|P_{f(1RZK_1)}(2) \setminus P_{f(8JGG_1)}(2)|=156\), \(|P_{f(8JGG_1)}(2) \setminus P_{f(1RZK_1)}(2)|=24\). 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 1:1100010100100010110001100100011011000101010101101111000001100101010101111000111111110000001010110001001000011011100011101011000111000010001001110100011100011100010000100010011101000110000111010011010110001001101100000011010000010001001011001001101100110110111101010000010111100011000010011011110100010001000011010
Pair \(Z_2\) Length of longest common subsequence
1RZK_1,8JGG_1 180 3
1RZK_1,5FPX_1 180 3
8JGG_1,5FPX_1 112 3

Newick tree

 
[
	1RZK_1:98.76,
	[
		8JGG_1:56,5FPX_1:56
	]:42.76
]

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 \((0.85)(0.8)(\frac{373 }{\log_{20} 373}-\frac{60}{\log_{20}60})=98.4\)
Status Protein1 Protein2 d d1/2
Query variables 1RZK_1 8JGG_1 122 72
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]