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Parikh vectors
8JOG_1 7UMR_1 1KOU_1 Letter Amino acid
3 11 9 F Phenylalanine
4 12 7 T Threonine
5 10 9 A Alanine
6 15 2 R Arginine
3 4 1 C Cysteine
3 3 5 Q Glutamine
3 12 2 H Histidine
2 15 5 I Isoleucine
2 8 5 M Methionine
0 7 5 Y Tyrosine
7 9 9 V Valine
5 15 5 S Serine
1 2 1 W Tryptophan
5 6 6 N Asparagine
4 21 7 E Glutamic acid
6 10 13 G Glycine
12 9 7 L Leucine
7 14 11 K Lycine
4 5 4 P Proline
4 9 12 D Aspartic acid 

8JOG_1|Chain A|RAF proto-oncogene serine/threonine-protein kinase|Homo sapiens (9606)
>7UMR_1|Chain A|Protein PA-X|Influenza A virus (11320)
>1KOU_1|Chain A|PHOTOACTIVE YELLOW PROTEIN|Halorhodospira halophila (1053)
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 \(w=f(c)\)
8JOG , Knot 48 86 0.82 38 76 84 
GPLGSPSKTSNTIRVFLPNKQRTVVNVRNGMSLHDCLMKALKVRGLQPECCAVFRLLHEHKGKKARLDWNTDAASLIGEELQVDFL
7UMR , Knot 93 197 0.83 40 147 190 
MGSSHHHHHHSSGLVPRGSHMEDFVRQCFNPMIVELAEKAMKEYGEDPKIETNKFAAICTHLEVCFMYSDGGSKHRFEIIEGRDRIMAWTVVNSICNTTGVEKPKFLPDLYDYKENRFIEIGVTRREVHIYYLEKANKIKSEKTHIHIFSFTGEEMATKADYTLDEESRARIKTRLFTIRQEMASRSLWDSFRQSER
1KOU , Knot 65 125 0.83 40 102 120 
MEHVAFGSEDIENTLAKMDDGQLDGLAFGAIQLDGDGNILQYNAAEGDITGRDPKQVIGKNFFKDVAPCTDSPEFYGKFKEGVASGNLNTMFEYTFDYQMTPTKVKVHMKKALSGDSYWVFVKRV

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 \(c\).

\(|P_{f(8JOG_1)}(2) \setminus P_{f(7UMR_1)}(2)|=47\), \(|P_{f(7UMR_1)}(2) \setminus P_{f(8JOG_1)}(2)|=118\). 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 1:11110100000010111100000110100110100011011010110100011101100001001010100011011100101011
Pair \(Z_2\) Length of longest common subsequence
8JOG_1,7UMR_1 165 4
8JOG_1,1KOU_1 134 3
7UMR_1,1KOU_1 169 3

Newick tree

 
[
	7UMR_1:88.32,
	[
		8JOG_1:67,1KOU_1:67
	]:21.32
]

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 \((0.85)(0.8)(\frac{283 }{\log_{20} 283}-\frac{86}{\log_{20}86})=62.7\)
Status Protein1 Protein2 d d1/2
Query variables 8JOG_1 7UMR_1 82 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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