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Parikh vectors
1EKB_1 7FZN_1 2KEN_1 Letter Amino acid
0 2 7 H Histidine
1 5 12 L Leucine
0 6 2 R Arginine
1 2 7 Q Glutamine
0 8 9 I Isoleucine
0 6 2 F Phenylalanine
1 10 7 S Serine
0 2 2 Y Tyrosine
2 17 10 V Valine
0 6 3 A Alanine
1 9 13 E Glutamic acid
3 14 7 K Lycine
0 5 1 M Methionine
0 2 3 W Tryptophan
0 11 3 D Aspartic acid
1 2 0 C Cysteine
1 11 7 G Glycine
1 1 3 P Proline
1 12 4 T Threonine
0 4 7 N Asparagine 

1EKB_1|Chain A|ENTEROPEPTIDASE|Bos taurus (9913)
>7FZN_1|Chain A|Fatty acid-binding protein, adipocyte|Homo sapiens (9606)
>2KEN_1|Chain A|Conserved protein|Methanosarcina mazei (2209)
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 \(w=f(c)\)
1EKB , Knot 11 13 0.72 20 12 11 
CGKKLVTQEVSPK
7FZN , Knot 69 135 0.83 40 104 129 
GSHMCDAFVGTWKLVSSENFDDYMKEVGVGFATRKVAGMAKPNMIISVNGDVITIKSESTFKNTEISFILGQEFDEVTADDRKVKSTITLDGGVLVHVQKWDGKSTTIKRKREDDKLVVECVMKGVTSTRVYERA
2KEN , Knot 56 109 0.80 38 86 103 
MEPQLTKIVDIVENGQWANLKAKVIQLWENTHESISQVGLLGDETGIIKFTIWKNAELPLLEQGESYLLRSVVVGEYNDRFQVQVNKNSSIEKLSEPIEVGLEHHHHHH

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 \(c\).

\(|P_{f(1EKB_1)}(2) \setminus P_{f(7FZN_1)}(2)|=5\), \(|P_{f(7FZN_1)}(2) \setminus P_{f(1EKB_1)}(2)|=97\). 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 1:0100110001010
Pair \(Z_2\) Length of longest common subsequence
1EKB_1,7FZN_1 102 3
1EKB_1,2KEN_1 94 2
7FZN_1,2KEN_1 134 3

Newick tree

 
[
	7FZN_1:63.16,
	[
		1EKB_1:47,2KEN_1:47
	]:16.16
]

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 \((0.85)(0.8)(\frac{148 }{\log_{20} 148}-\frac{13}{\log_{20}13})=50.0\)
Status Protein1 Protein2 d d1/2
Query variables 1EKB_1 7FZN_1 61 33.5
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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