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Parikh vectors
2KJI_1 2RVA_1 4PDZ_1 Letter Amino acid
3 7 4 I Isoleucine
2 4 8 K Lycine
0 3 7 F Phenylalanine
2 7 7 V Valine
4 20 5 A Alanine
3 4 1 R Arginine
2 8 2 N Asparagine
1 9 1 Y Tyrosine
4 7 7 D Aspartic acid
0 7 5 H Histidine
2 2 0 P Proline
3 13 5 S Serine
8 0 2 C Cysteine
2 3 3 Q Glutamine
3 5 16 E Glutamic acid
5 10 4 G Glycine
3 5 8 L Leucine
1 0 4 M Methionine
2 18 3 T Threonine
0 5 0 W Tryptophan 

2KJI_1|Chain A|Probable insulin-like peptide beta-type 5|Caenorhabditis elegans (6239)
>2RVA_1|Chain A|Glucanase|Paenibacillus fukuinensis (170835)
>4PDZ_1|Chains A, B|Protein S100-B|Bos taurus (9913)
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 \(w=f(c)\)
2KJI , Knot 31 50 0.80 34 45 48 
GETRACGRKLISLVMAVCGDLCNPQEGKDIATECCGNQCSDDYIRSACCP
2RVA , Knot 60 137 0.71 36 91 125 
HHHHHHNLALNKATATSSIETAGHEGDKAVDGNAATRWASAYGASPQWIYINLGSTQSISRVKLNWEDAYATAYSIQVSNDSGSTPTNWTTVYSTTTGDGAIDDITFAATNAKFVRVYATTRATAYGYSLWEFEVYG
4PDZ , Knot 50 92 0.82 36 75 89 
MSELEKAVVALIDVFHQYSGREGDKHKLKKSELKELINNELSHFLEEIKEQEVVDKVMETLDSDGDGECDFQEFMAFVAMITTACHEFFEHE

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 \(c\).

\(|P_{f(2KJI_1)}(2) \setminus P_{f(2RVA_1)}(2)|=29\), \(|P_{f(2RVA_1)}(2) \setminus P_{f(2KJI_1)}(2)|=75\). 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 1:10001010011011111010100100100110000100000001001001
Pair \(Z_2\) Length of longest common subsequence
2KJI_1,2RVA_1 104 3
2KJI_1,4PDZ_1 92 2
2RVA_1,4PDZ_1 120 4

Newick tree

 
[
	2RVA_1:59.13,
	[
		2KJI_1:46,4PDZ_1:46
	]:13.13
]

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 \((0.85)(0.8)(\frac{187 }{\log_{20} 187}-\frac{50}{\log_{20}50})=46.7\)
Status Protein1 Protein2 d d1/2
Query variables 2KJI_1 2RVA_1 57 38.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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