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Parikh vectors
5VQX_1 2NCL_1 9QGF_1 Letter Amino acid
28 5 35 A Alanine
40 7 21 I Isoleucine
13 3 12 F Phenylalanine
39 7 26 T Threonine
20 1 18 N Asparagine
8 2 17 H Histidine
38 2 23 P Proline
20 3 17 S Serine
19 0 3 W Tryptophan
21 1 15 Y Tyrosine
40 5 25 V Valine
15 5 20 R Arginine
2 1 1 C Cysteine
7 3 2 M Methionine
24 1 18 D Aspartic acid
36 6 18 Q Glutamine
47 11 20 E Glutamic acid
34 5 40 G Glycine
48 4 35 L Leucine
58 9 9 K Lycine 

5VQX_1|Chain A|Reverse transcriptase/ribonuclease H|Human immunodeficiency virus type 1 group M subtype B (isolate BH10) (11678)
>2NCL_1|Chain A|BolA-like protein 3|Homo sapiens (9606)
>9QGF_1|Chains A, B|All1865 protein|Nostoc sp. PCC 7120 = FACHB-418 (103690)
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 \(w=f(c)\)
5VQX , Knot 221 557 0.83 40 253 515 
MVPISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKKKKSVTVLDVGDAYFSVPLDEDFRKYTAFTIPSINNETPGIRYQYNVLPQGWKGSPAIFQSSMTKILEPFAAQNPDIVICQYMDDLYVGSDLEIGQHRTKIEELRQHLLRWGLTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDIQKLVGKLNWASQIYPGIKVRQLSKLLRGTKALTEVIPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMRGAHTNDVKQLTEAVQKITTESIVIWGKTPKFKLPIQKETWETWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIVGAETFYVDGAANRETKLGKAGYVTNKGRQKVVPLTNTTNQKTELQAIYLALQDSGLEVNIVTDSQYALGIIQAQPDKSESELVNQIIEQLIKKEKVYLAWVPAHKGIGGNEQVDKLVSAG
2NCL , Knot 44 81 0.79 38 66 76 
ATQTEGELRVTQILKEKFPRATAIKVTDISGGCGAMYEIKIESEEFKEKRTVQQHQMVNQALKEEIKEMHGLRIFTSVPKR
9QGF , Knot 155 375 0.81 40 206 354 
MGHHHHHHGSGSTNINLFSSYQLGELELPNRIVMAPLTRSRAGEGNVPHQLNAIYYGQRASAGLIIAEATQVTPQGQGYPHTPGIHSPEQVAGWKLVTDTVHQQGGRIFLQLWHVGRISHPDLQPDGGLPVAPSAIAPKGEVLTYEGKKPYVTPRALDTSEIPAIVEQYRQGAANALAAGFDGVEIHAANGYLIDQFLRDGTNQRTDEYGGAIENRARLLLEVTEAITSVWDSQRVGVRLSPLTTLNGCVDSHPLETFGYVAQALNRFNLSYLHIFEAIDADIRHGGTVVPTSHLRDRFTGTLIVNGGYTREKGDTVIANKAADLVAFGTLFISNPDLPERLEVNAPLNQADPKTFYGGGEKGYTDYPFLAVANK

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 \(c\).

\(|P_{f(5VQX_1)}(2) \setminus P_{f(2NCL_1)}(2)|=202\), \(|P_{f(2NCL_1)}(2) \setminus P_{f(5VQX_1)}(2)|=15\). 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 1:11110110011101011101101001110000101110100010001010011100100011111000000010011010010000001101011110111100000101101101010111000100001101101000011100000111011010111100010011011110010111000100101100101100000100100011011100100000001111111001010010101111100001010010011101011001011101001001101001100111100010101100001100110110001000111010001010100010001100100100101011000010010011001000011111001010111000010011000101011101011001111011001000111110010101110000011011010001000111100000000010110111000110101100000111110101000000110011001100001011111100111100010011011
Pair \(Z_2\) Length of longest common subsequence
5VQX_1,2NCL_1 217 4
5VQX_1,9QGF_1 159 4
2NCL_1,9QGF_1 190 3

Newick tree

 
[
	2NCL_1:10.43,
	[
		5VQX_1:79.5,9QGF_1:79.5
	]:28.93
]

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 \((0.85)(0.8)(\frac{638 }{\log_{20} 638}-\frac{81}{\log_{20}81})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 5VQX_1 2NCL_1 204 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]