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Parikh vectors
2OLQ_1 4BJP_1 6HVB_1 Letter Amino acid
27 25 0 N Asparagine
5 0 2 C Cysteine
11 12 0 M Methionine
27 30 0 P Proline
45 47 0 A Alanine
23 34 0 R Arginine
38 23 1 D Aspartic acid
30 24 0 E Glutamic acid
28 12 1 F Phenylalanine
25 37 0 S Serine
45 28 0 T Threonine
17 21 1 Y Tyrosine
9 6 0 H Histidine
27 31 0 I Isoleucine
46 39 0 L Leucine
34 27 1 K Lycine
9 4 1 W Tryptophan
32 50 1 V Valine
15 13 0 Q Glutamine
47 48 0 G Glycine 

2OLQ_1|Chain A|Phosphoenolpyruvate carboxykinase|Escherichia coli (83333)
>4BJP_1|Chain A|PENICILLIN BINDING PROTEIN TRANSPEPTIDASE DOMAIN PROTEIN|ESCHERICHIA COLI (562)
>6HVB_1|Chain A|Urotensin-2|Homo sapiens (9606)
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 \(w=f(c)\)
2OLQ , Knot 213 540 0.82 40 250 503 
MRVNNGLTPQELEAYGISDVHDIVYNPSYDLLYQEELDPSLTGYERGVLTNLGAVAVDTGIFTGRSPKDKYIVRDDTTRDTFWWADKGKGKNDNKPLSPETWQHLKGLVTRQLSGKRLFVVDAFCGANPDTRLSVRFITEVAWQAHFVKNMFIRPSDEELAGFKPDFIVMNGAKCTNPQWKEQGLNSENFVAFNLTERMQLIGGTWYGGEMKKGMFSMMNYLLPLKGIASMHCSANVGEKGDVAVFFGLSGTGKTTLSTDPKRRLIGDDEHGWDDDGVFNFEGGCYAKTIKLSKEAEPEIYNAIRRDALLENVTVREDGTIDFDDGSKTENTRVSYPIYHIDNIVKPVSKAGHATKVIFLTADAFGVLPPVSRLTADQTQYHFLSGFTAKLAGTERGITEPTPTFSACFGAAFLSLHPTQYAEVLVKRMQAAGAQAYLVNTGWNGTGKRISIKDTRAIIDAILNGSLDNAETFTLPMFNLAIPTELPGVDTKILDPRNTYASPEQWQEKAETLAKLFIDNFDKYTDTPAGAALVAAGPKL
4BJP , Knot 206 511 0.83 38 237 481 
VSTSRGMITDRSGRPLAVSVPVKAIWADPKEVHDAGGISVGDRWKALANALNIPLDQLSARINANPKGRFIYLARQVNPDMADYIKKLKLPGIHLREESRRYYPSGEVTAHLIGFTNVDSQGIEGVEKSFDKWLTGQPGERIVRKDRYGRVIEDISSTDSQAAHNLALSIDERLQALVYRELNNAVAFNKAESGSAVLVDVNTGEVLAMANSPSYNPNNLSGTPKEAMRNRTITDVFEPGSTVKPMVVMTALQRGVVRENSVLNTIPYRINGHEIKDVARYSELTLTGVLQKSSNVGVSKLALAMPSSALVDTYSRFGLGKATNLGLVGERSGLYPQKQRWSDIERATFSFGYGLMVTPLQLARVYATIGSYGIYRPLSITKVDPPVPGERVFPESIVRTVVHMMESVALPGGGGVKAAIKGYRIAIKTGTAKKVGPDGRYINKYIAYTAGVAPASQPRFALVVVINDPQAGKYYGGAVSAPVFGAIMGGVLRTMNIEPDALTTGDKNEFV
6HVB , Knot 7 8 0.60 14 7 6 
DCFWKYCV

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 \(c\).

\(|P_{f(2OLQ_1)}(2) \setminus P_{f(4BJP_1)}(2)|=71\), \(|P_{f(4BJP_1)}(2) \setminus P_{f(2OLQ_1)}(2)|=58\). 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 1:101001101001010110010011001000110000101010100011100111111001110100100001100000000111100101000001101001001011100010100111101101101000101011001110101100111010000111101011110110000101000110000111101000101111010110100111011001111011101000101100101111111010100010001000111000011000111010110010010100010101001100011100101000101010010000000100110010011011001101001111010111111110010100000011011010111000110010101010111111010100010111001011110101100110101001010000111011101010010010111101111001111000110100001010010001001101110010000001111111111101
Pair \(Z_2\) Length of longest common subsequence
2OLQ_1,4BJP_1 129 4
2OLQ_1,6HVB_1 249 2
4BJP_1,6HVB_1 240 2

Newick tree

 
[
	6HVB_1:13.18,
	[
		2OLQ_1:64.5,4BJP_1:64.5
	]:71.68
]

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 \((0.85)(0.8)(\frac{1051 }{\log_{20} 1051}-\frac{511}{\log_{20}511})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 2OLQ_1 4BJP_1 176 171
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]