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Parikh vectors
7KRC_1 4PTS_1 7OOF_1 Letter Amino acid
36 12 10 Q Glutamine
8 10 9 H Histidine
58 11 7 K Lycine
7 2 3 M Methionine
13 14 8 F Phenylalanine
20 14 24 S Serine
40 24 17 V Valine
15 24 6 R Arginine
20 10 8 N Asparagine
48 27 14 L Leucine
28 25 21 A Alanine
24 27 19 D Aspartic acid
1 4 2 C Cysteine
47 20 16 E Glutamic acid
34 29 10 G Glycine
38 27 5 P Proline
39 28 27 T Threonine
40 17 13 I Isoleucine
19 6 3 W Tryptophan
22 17 9 Y Tyrosine 

7KRC_1|Chain A|HIV-1 REVERSE TRANSCRIPTASE, P66 SUBUNIT|Human immunodeficiency virus type 1 group M subtype B (11678)
>4PTS_1|Chains A, B|glutathione S-transferase|Gordonia bronchialis (526226)
>7OOF_1|Chain A|Alginate lyase (PL7)|Paradendryphiella salina (179392)
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 \(w=f(c)\)
7KRC , Knot 221 557 0.83 40 252 513 
MVPISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKKKKSVTVLDVGDAYFSVPLDEDFRKYTAFTIPSINNETPGIRYQYNVLPQGWKGSPAIFQSSMTKILEPFAAQNPDIVIYQYMDDLYVGSDLEIGQHRTKIEELRQHLLRWGLTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDIQKLVGKLNWASQIYPGIKVRQLSKLLRGTKALTEVIPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMRGAHTNDVKQLTEAVQKITTESIVIWGKTPKFKLPIQKETWETWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIVGAETFYVDGAANRETKLGKAGYVTNKGRQKVVPLTNTTNQKTELQAIYLALQDSGLEVNIVTDSQYALGIIQAQPDKSESELVNQIIEQLIKKEKVYLAWVPAHKGIGGNEQVDKLVSAG
4PTS , Knot 146 348 0.81 40 204 329 
MAPDTHSSDSEQGSYVTTGQEFVRDTKYIETRITADGRDGYPVEPGRYRLIAARACPWANRTLIVRRLLGLENVLSLGLCGPTHDKNSWTFDLDPGGVDPVLGIPRLKDAYEKRFPGYPKGITVPAVVEIATGEVVTNDFPQITIDFSLEWTAYHRDGAPQLYPEHLRAEIDEVSKGIYTEVNNGVYRCGFAGSQDAYDAAYDRLFTKLDELSERLAQQRYLVGDTITEADVRLFTTLARFDAVYHGHFKCNRSKLSEMPVLWAYARDLFQTPGFGDTTDFTQIKQHYYIVHSDINPTQIVPKGPDLRNWLTPHHREELGGRPFGDGTPPGPPIAAEVVPAQNWVPTS
7OOF , Knot 104 231 0.81 40 146 222 
EFYTAPSTESKFTEVLSKAKLQYPTSTTVAFADDLLDGYAASYFYLTSDLYMQFQVAGSSQRSELREMETSGDEAAWDCTGSTAHVASAQIAIPVQEDGIEEVTILQVHDSDVTPVLRISWVSSITIDGVTSEDVVLATIRNGIDDSTATKTVLQAHTTSRTEFNINVQNSKLSITVDGTTELDEADISQFDGSTCYFKAGAYNNNPTDTSANARIKMYELEWVDHHHHHH

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 \(c\).

\(|P_{f(7KRC_1)}(2) \setminus P_{f(4PTS_1)}(2)|=103\), \(|P_{f(4PTS_1)}(2) \setminus P_{f(7KRC_1)}(2)|=55\). 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
7KRC_1,4PTS_1 158 4
7KRC_1,7OOF_1 168 3
4PTS_1,7OOF_1 152 4

Newick tree

 
[
	7KRC_1:83.30,
	[
		4PTS_1:76,7OOF_1:76
	]:7.30
]

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 \((0.85)(0.8)(\frac{905 }{\log_{20} 905}-\frac{348}{\log_{20}348})=149.\)
Status Protein1 Protein2 d d1/2
Query variables 7KRC_1 4PTS_1 190 152.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]