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Parikh vectors
2GHY_1 5EFT_1 7OTA_1 Letter Amino acid
3 0 21 N Asparagine
2 0 2 C Cysteine
2 1 13 F Phenylalanine
3 4 20 S Serine
6 1 26 A Alanine
4 0 15 R Arginine
7 3 23 D Aspartic acid
7 0 47 E Glutamic acid
2 0 7 M Methionine
0 0 38 P Proline
0 0 19 W Tryptophan
3 0 35 Q Glutamine
2 0 33 G Glycine
10 2 48 L Leucine
3 2 60 K Lycine
4 0 39 T Threonine
1 0 22 Y Tyrosine
0 2 40 V Valine
2 0 8 H Histidine
2 1 40 I Isoleucine 

2GHY_1|Chains A, B|Regulatory protein rop|Escherichia coli (562)
>5EFT_1|Chains A, C, E, G|p9-1|Rice black-streaked dwarf virus 2 (566568)
>7OTA_1|Chains A, C|Reverse transcriptase/ribonuclease H|Human immunodeficiency virus type 1 group M subtype B (isolate BH10) (11678)
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 \(w=f(c)\)
2GHY , Knot 35 63 0.76 34 52 59 
MTKQEKTALNMARFIRSQTLTLLEKLNELDADEQADICESLHDHADELYRSCLARFGDDGENL
5EFT , Knot 12 16 0.69 16 15 14 
ADSVIKSDFSSLKLDV
7OTA , Knot 219 556 0.83 40 253 511 
MVPISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKKKKSVTVLDVGDAYFSVPLDEDFRKYTAFTIPSINNETPGIRYQYNVLPQGWKGSPAIFQSSMTKILEPFKKQNPDIVIYQYMDDLYVGSDLEIGQHRTKIEELRQHLLRWGLTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDICKLVGKLNWASQIYPGIKVRQLSKLLRGTKALTEVIPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMRGAHTNDVKQLTEAVQKITTESIVIWGKTPKFKLPIQKETWETWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIVGAETFYVDGAANRETKLGKAGYVTNKGRQKVVPLTNTTNQKTELQAIYLALQDSGLEVNIVTNSQYALGIIQAQPDKSESELVNQIIEQLIKKEKVYLAWVPAHKGIGGNEQVDKLVSA

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 \(c\).

\(|P_{f(2GHY_1)}(2) \setminus P_{f(5EFT_1)}(2)|=48\), \(|P_{f(5EFT_1)}(2) \setminus P_{f(2GHY_1)}(2)|=11\). 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 1:100000011011011000010110010010100010100010001001000011011001001
Pair \(Z_2\) Length of longest common subsequence
2GHY_1,5EFT_1 59 2
2GHY_1,7OTA_1 237 3
5EFT_1,7OTA_1 242 3

Newick tree

 
[
	7OTA_1:13.23,
	[
		2GHY_1:29.5,5EFT_1:29.5
	]:10.73
]

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 \((0.85)(0.8)(\frac{79 }{\log_{20} 79}-\frac{16}{\log_{20}16})=25.0\)
Status Protein1 Protein2 d d1/2
Query variables 2GHY_1 5EFT_1 32 19.5
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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