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Parikh vectors
7WUN_1 3RCZ_1 7YPX_1 Letter Amino acid
0 3 7 F Phenylalanine
0 1 6 Y Tyrosine
0 8 10 D Aspartic acid
0 7 6 H Histidine
0 5 4 K Lycine
0 1 3 M Methionine
0 2 14 P Proline
0 4 23 T Threonine
0 7 21 V Valine
0 3 9 Q Glutamine
0 9 6 E Glutamic acid
0 5 11 I Isoleucine
0 0 27 A Alanine
0 11 19 L Leucine
1 1 64 G Glycine
1 7 24 S Serine
0 1 6 W Tryptophan
1 5 7 R Arginine
0 1 21 N Asparagine
0 1 1 C Cysteine 

7WUN_1|Chain A[auth X]|3-mer peptide|Oryza sativa (4530)
>3RCZ_1|Chain A|DNA repair protein rad60|Schizosaccharomyces pombe (4896)
>7YPX_1|Chains A, B, C|Pam3 tail fiber proreins|uncultured cyanophage (215796)
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 \(w=f(c)\)
7WUN , Knot 3 3 0.36 6 2 1 
RSG
3RCZ , Knot 41 82 0.73 38 64 76 
MHHHHHHKLITLLLRSSKSEDLRLSIPVDFTVKDLIKRYCTEVKISFHERIRLEFEGEWLDPNDQVQSTELEDEDQVSVVLD
7YPX , Knot 117 289 0.76 40 150 244 
HHHHHHSSGMASINLPFSLSGSKRIPTSEELADGYQCGPLDVELDNWLMWWLTGQVDGVIEGAGLTTDDTDLARLYKAIQSMTSGNLRTVVLTAASGNLPIPSDVSVLNWVRAVGGGGAGGNSNTGNSKASGGGGGAGFDRFNVAVTPGSNVPYTVGAAGAVNGLGAGYNGGAGGSTAILGTTAGGGAGGLGVNNNATAVQVNGGTTSGTTPEISYPGGLGTEGIVGTGGGSVLSQPTQRAFTNAGNNNPANSWGGGGPGGSDFGGAWQPGGVGKQGIIIVQYFSRFAP

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 \(c\).

\(|P_{f(7WUN_1)}(2) \setminus P_{f(3RCZ_1)}(2)|=1\), \(|P_{f(3RCZ_1)}(2) \setminus P_{f(7WUN_1)}(2)|=63\). 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 1:001
Pair \(Z_2\) Length of longest common subsequence
7WUN_1,3RCZ_1 64 2
7WUN_1,7YPX_1 150 2
3RCZ_1,7YPX_1 154 6

Newick tree

 
[
	7YPX_1:85.79,
	[
		7WUN_1:32,3RCZ_1:32
	]:53.79
]

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 \((0.85)(0.8)(\frac{85 }{\log_{20} 85}-\frac{3}{\log_{20}3})=33.4\)
Status Protein1 Protein2 d d1/2
Query variables 7WUN_1 3RCZ_1 41 21
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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