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Parikh vectors
8ASR_1 4RDC_1 7EZO_1 Letter Amino acid
2 2 2 C Cysteine
6 29 14 Q Glutamine
5 25 19 I Isoleucine
5 32 22 K Lycine
2 1 10 M Methionine
13 35 14 T Threonine
4 6 17 R Arginine
6 30 26 N Asparagine
4 0 8 W Tryptophan
11 34 20 V Valine
2 18 22 E Glutamic acid
7 18 14 F Phenylalanine
12 22 17 S Serine
14 40 28 A Alanine
8 19 25 D Aspartic acid
21 29 17 G Glycine
5 27 23 L Leucine
6 10 19 Y Tyrosine
3 0 10 H Histidine
5 18 11 P Proline 

8ASR_1|Chains A, B, C, D|Agroavidin|Rhizobium sp. AAP43 (1523420)
>4RDC_1|Chain A|Amino acid/amide ABC transporter substrate-binding protein, HAAT family|Anabaena variabilis (240292)
>7EZO_1|Chain A|endoxylanase/arabinofuranosidase|Bacillus sp. (in: Bacteria) (1409)
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 \(w=f(c)\)
8ASR , Knot 70 141 0.82 40 108 136 
MAHGPFNAQGFTDFDSLSGTSTTWVNELGSVMTIDVDRKGGVTGYYVNNAPGTGCRGLPYDLSGHAHGSTIAFSVVWSNGIADCRSATSWAGYARKTFGGGVQIVTQWSLAFVGKAGGKIETGQNVFTYQAYDAQPSPADK
4RDC , Knot 157 395 0.79 36 189 368 
SNATNTDTNSTNNSPNNTTNTTTNVTTTSDKNTIPIGIALAQTSNVALLGQEQVAGAKIAEKYFNDKGGVNGTPIKLIFQDTAGDEAGTINAFQTLINKDKVVGIVGPTLSQQAFSANPIAERAKVPVVGPSNTAKGIPEIGDYVARVSAPVSVVAPNSVKAALKQNPNIKKVAVFFAQNDAFSKSETEIFQQTVKDQGLELVTVQKFQTTDTDFQSQATNAINLKPDLVIISGLAADGGNLVRQLRELGYQGAIIGGDGLNTSNVFAVCKALCDGVLIAQAYSPEYTGEINKAFRQAYVDQYKKEPPQFSAQAFAAVQVYVESLKALDTKNKVSKIQLPELRTELNKQLLTGKYNTPLGEISFTPIGEVVQKDFYVAQIKMEKDGSQGKFTFLK
7EZO , Knot 150 338 0.86 40 222 322 
NNLKDAYADYFTIGCAVNMANFNSSQQIALITSNFNSITAENDMKPQPTQPAEGKWNWENADKIANFARAHKIGLRGHCLVWHAQTGDWMFHDEKGDLVSKEVLFERMRTHIHTIVNRYKDVVYAWDVVNEAMTDDAKAEIPYRQSLYYKIAGDEFIKKAFEYAHEADPKALLFYNDYNETNPAKRDRIYNMVKSMKAEGIPISGIGMQGHYNVLSPTEDEFRKALELYSQVVDNIHITELDVRINTREQGGQLSVNQEGKKLELTPEADAAQVAQYDMLFRVMRDYKHVISNVTFWNGYDGDSWLDRRWGNRQRNYPLLFDENLLPKSSFSSYYKVL

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 \(c\).

\(|P_{f(8ASR_1)}(2) \setminus P_{f(4RDC_1)}(2)|=41\), \(|P_{f(4RDC_1)}(2) \setminus P_{f(8ASR_1)}(2)|=122\). 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 1:110111010110010010100001100110110101000111010010011101001110010101010011101110011100001001110100011111011001011111011101001001100010010101100
Pair \(Z_2\) Length of longest common subsequence
8ASR_1,4RDC_1 163 4
8ASR_1,7EZO_1 190 3
4RDC_1,7EZO_1 153 4

Newick tree

 
[
	8ASR_1:92.16,
	[
		4RDC_1:76.5,7EZO_1:76.5
	]:15.66
]

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 \((0.85)(0.8)(\frac{536 }{\log_{20} 536}-\frac{141}{\log_{20}141})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 8ASR_1 4RDC_1 142 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]