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Parikh vectors
1EYF_1 1ETX_1 8QNW_1 Letter Amino acid
4 10 27 V Valine
6 0 4 C Cysteine
5 2 13 F Phenylalanine
6 6 19 D Aspartic acid
4 9 16 Q Glutamine
3 0 17 H Histidine
4 4 26 S Serine
5 6 18 T Threonine
1 0 3 W Tryptophan
13 6 35 A Alanine
12 6 25 R Arginine
3 5 38 G Glycine
2 1 25 I Isoleucine
5 12 31 L Leucine
5 7 15 K Lycine
1 6 9 M Methionine
5 2 21 P Proline
4 8 18 N Asparagine
3 4 19 E Glutamic acid
1 4 17 Y Tyrosine 

1EYF_1|Chain A|ADA REGULATORY PROTEIN|Escherichia coli (562)
>1ETX_1|Chains A, B|FACTOR FOR INVERSION STIMULATION|Escherichia coli (562)
>8QNW_1|Chain A|12-oxophytodienoate reductase 3|Solanum lycopersicum (4081)
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 \(w=f(c)\)
1EYF , Knot 52 92 0.85 40 80 90 
MKKATCLTDDQRWQSVLARDPNADGEFVFAVRTTGIFCRPSCRARHALRENVSFYANASEALAAGFRPCKRCQPDKANPRQHRLDKITHACR
1ETX , Knot 52 98 0.81 34 82 95 
MFEQRVNSDVLTVSTVNSQDQVTQKPLRDSVKQALKNYFAQLNGQDVNDLYELVLAEVEQPLLDMVMQYTRGNATRAALMMGINRGTLRKKLKKYGMN
8QNW , Knot 166 396 0.83 40 220 370 
MASHHHHHHMASSAQDGNNPLFSPYKMGKFNLSHRVVLAPMTRCRALNNIPQAALGEYYEQRATAGGFLITEGTMISPTSAGFPHVPGIFTKEQVREWKKIVDVVHAKGAVIFCQLWHVGRASHEVYQPAGAAPISSTEKPISNRWRILMPDGTHGIYPKPRAIGTYEISQVVEDYRRSALNAIEAGFDGIEIHGAHGYLIDQFLKDGINDRTDEYGGSLANRCKFITQVVQAVVSAIGADRVGVRVSPAIDHLDAMDSNPLSLGLAVVERLNKIQLHSGSKLAYLHVTQPGDGRGSEEEEARLMRTLRNAYQGTFICSGGYTRELGIEAVAQGDADLVSYGRLFISNPDLVMRIKLNAPLNKYNRKTFYTQDPVVGYTDYPFLQGNGSNGPLSRL

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 \(c\).

\(|P_{f(1EYF_1)}(2) \setminus P_{f(1ETX_1)}(2)|=61\), \(|P_{f(1ETX_1)}(2) \setminus P_{f(1EYF_1)}(2)|=63\). 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 1:10010010000010011100101010111110001110010001001100010101010011111101000001001010000100100100
Pair \(Z_2\) Length of longest common subsequence
1EYF_1,1ETX_1 124 3
1EYF_1,8QNW_1 204 3
1ETX_1,8QNW_1 192 3

Newick tree

 
[
	8QNW_1:10.62,
	[
		1EYF_1:62,1ETX_1:62
	]:46.62
]

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 \((0.85)(0.8)(\frac{190 }{\log_{20} 190}-\frac{92}{\log_{20}92})=32.3\)
Status Protein1 Protein2 d d1/2
Query variables 1EYF_1 1ETX_1 40 39.5
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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