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Parikh vectors
9BEO_1 3MED_1 8PDL_1 Letter Amino acid
1 37 11 P Proline
5 25 15 D Aspartic acid
3 47 29 E Glutamic acid
5 57 24 K Lycine
9 34 28 G Glycine
1 8 8 H Histidine
9 41 26 I Isoleucine
2 40 21 T Threonine
1 22 16 Y Tyrosine
6 26 30 A Alanine
0 2 3 C Cysteine
4 37 18 Q Glutamine
9 19 40 S Serine
1 19 2 W Tryptophan
8 40 24 V Valine
2 18 19 R Arginine
5 49 44 L Leucine
3 6 9 M Methionine
3 20 17 N Asparagine
1 13 10 F Phenylalanine

9BEO_1|Chains A, B, C, D, E, F|Molybdenum-pterin binding domain-containing protein|Eubacterium limosum (1736)
>3MED_1|Chain A|p66 Reverse transcriptase|HIV-1 M:B_HXB2R (11706)
>8PDL_1|Chain A|Nucleoprotein|Human metapneumovirus (strain CAN97-83) (694067)
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 \(w=f(c)\)
9BEO , Knot 44 78 0.82 38 68 76
MKLSARNQLAGKVVSIKEGAVNGIVVLDIGGGNQISSTISMDSIRELGLQVGSDAYAVIKATSVMIGIDDWSHPQFEK
3MED , Knot 224 560 0.84 40 255 519
PISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKKNKSVTVLDVGDAYFSVPLDEDFRKYTAFTIPSINNETPGIRYQYNVLPQGWKGSPAIFQSSMTKILEPFRKQNPDIVIYQYMDDLYVGSDLEIGQHRTKIEELRQHLLRWGLTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDIQKLVGKLNWASQIYPGIKVRQLCKLLRGTKALTEVIPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMRGAHTNDVKQLTEAVQKITTESIVIWGKTPKFKLPIQKETWETWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIVGAETFYVDGAANRETKLGKAGYVTNRGRQKVVTLTDTTNQKTELQAIYLALQDSGLEVNIVTDSQYALGIIQAQPDQSESELVNQIIEQLIKKEKVYLAWVPAHKGIGGNEQVDKLVSAGIRKVL
8PDL , Knot 168 394 0.85 40 223 366
MSLQGIHLSDLSYKHAILKESQYTIKRDVGTTTAVTPSSLQQEITLLCGEILYAKHADYKYAAEIGIQYISTALGSERVQQILRNSGSEVQVVLTRTYSLGKIKNNKGEDLQMLDIHGVEKSWVEEIDKEARKTMATLLKESSGNIPQNQRPSAPDTPIILLCVGALIFTKLASTIEVGLETTVRRANRVLSDALKRYPRMDIPKIARSFYDLFEQKVYHRSLFIEYGKALGSSSTGSKAESLFVNIFMQAYGAGQTMLRWGVIARSSNNIMLGHVSVQAELKQVTEVYDLVREMGPESGLLHLRQSPKAGLLSLANCPNFASVVLGNASGLGIIGMYRGRVPNTELFSAAESYAKSLKESNKINFSSLGLTDEEKEAAEHFLNVSDDSQNDYE

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 \(c\).

\(|P_{f(9BEO_1)}(2) \setminus P_{f(3MED_1)}(2)|=13\), \(|P_{f(3MED_1)}(2) \setminus P_{f(9BEO_1)}(2)|=200\). 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 1:101010001110110100111011111011110010001010010011101100101110100111110010010100
Pair \(Z_2\) Length of longest common subsequence
9BEO_1,3MED_1 213 4
9BEO_1,8PDL_1 189 3
3MED_1,8PDL_1 144 4

Newick tree

 
[
	9BEO_1:10.56,
	[
		8PDL_1:72,3MED_1:72
	]:36.56
]

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 \((0.85)(0.8)(\frac{638 }{\log_{20} 638}-\frac{78}{\log_{20}78})=164.\)
Status Protein1 Protein2 d d1/2
Query variables 9BEO_1 3MED_1 203 114
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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