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Parikh vectors
6JVG_1 4INS_1 1FSF_1 Letter Amino acid
5 2 14 I Isoleucine
16 2 24 L Leucine
9 0 15 K Lycine
5 0 10 M Methionine
4 0 22 A Alanine
12 0 12 D Aspartic acid
12 2 18 E Glutamic acid
3 0 11 H Histidine
4 2 8 Y Tyrosine
16 1 17 G Glycine
7 0 13 P Proline
4 0 2 W Tryptophan
13 1 21 V Valine
8 0 12 R Arginine
1 2 16 N Asparagine
2 4 4 C Cysteine
11 2 7 Q Glutamine
11 0 12 F Phenylalanine
5 2 11 S Serine
8 1 17 T Threonine 

6JVG_1|Chains A, B|7,8-dihydro-8-oxoguanine triphosphatase|Homo sapiens (9606)
>4INS_1|Chains A, C|INSULIN (CHAIN A)|Sus scrofa (9823)
>1FSF_1|Chain A|GLUCOSAMINE-6-PHOSPHATE DEAMINASE|Escherichia coli (562)
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 \(w=f(c)\)
6JVG , Knot 74 156 0.79 40 119 151 
MGASRLYTLVLVLQPQRVLLGMKKRGFGAGRWNGFGGKVQEGETIEDGARRELQEESGLTVDALHKVGQIVFEFVGEPELMDVHVFCTDSIQGTPVESDEMRPCWFQLDQIPFKDMWPDDSYWFPLLLQKKKFHGYFKFQGQDTILDYTLREVDTV
4INS , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN
1FSF , Knot 122 266 0.85 40 188 254 
MRLIPLTTAEQVGKWAARHIVNRINAFKPTADRPFVLGLPTGGTPMTTYKALVEMHKAGQVSFKHVVTFNMDEYVGLPKEHPESYYSFMHRNFFDHVDIPAENINLLNGNAPDIDAECRQYEEKIRSYGKIHLFMGGVGNDGHIAFNEPASSLASRTRIKTLTHDTRVANSRFFDNDVNQVPKYALTVGVGTLLDAEEVMILVLGSQKALALQAAVEGCVNHMWTISCLQLHPKAIMVCDEPSTMELKVKTLRYFNELEAENIKGL

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 \(c\).

\(|P_{f(6JVG_1)}(2) \setminus P_{f(4INS_1)}(2)|=113\), \(|P_{f(4INS_1)}(2) \setminus P_{f(6JVG_1)}(2)|=14\). 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 1:111001001111101001111100011111010111101001001001100010000110101100110111011101011010110000101011000010101101001110011100001111110000101010101000110001001001
Pair \(Z_2\) Length of longest common subsequence
6JVG_1,4INS_1 127 2
6JVG_1,1FSF_1 183 3
4INS_1,1FSF_1 194 2

Newick tree

 
[
	1FSF_1:10.51,
	[
		6JVG_1:63.5,4INS_1:63.5
	]:39.01
]

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 \((0.85)(0.8)(\frac{177 }{\log_{20} 177}-\frac{21}{\log_{20}21})=55.6\)
Status Protein1 Protein2 d d1/2
Query variables 6JVG_1 4INS_1 68 39
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]