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Parikh vectors
5FQE_1 4TND_1 9NSZ_1 Letter Amino acid
54 27 19 N Asparagine
43 54 20 E Glutamic acid
55 25 18 I Isoleucine
19 36 7 R Arginine
37 29 8 D Aspartic acid
4 17 2 C Cysteine
11 24 18 Q Glutamine
34 41 18 G Glycine
19 15 4 H Histidine
41 53 22 K Lycine
14 13 11 M Methionine
25 29 12 F Phenylalanine
34 23 14 T Threonine
15 34 21 V Valine
27 24 16 A Alanine
45 57 27 L Leucine
25 29 9 P Proline
60 34 14 S Serine
18 5 6 W Tryptophan
30 21 7 Y Tyrosine 

5FQE_1|Chains A, B|BETA-N-ACETYLGALACTOSAMINIDASE|CLOSTRIDIUM PERFRINGENS (1502)
>4TND_1|Chain A|G protein-coupled receptor kinase 5|Homo sapiens (9606)
>9NSZ_1|Chain A|Beta-lactamase OXA-23|Acinetobacter baumannii (470)
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 \(w=f(c)\)
5FQE , Knot 238 610 0.83 40 273 563 
MGSSHHHHHHSSGLVPRGSHMASMKKDTTLGASIGSTDFHYLQKDYDEIKKLNLNTWNEVAWIGDELNSKIVMWTNSSPVNNVTLSSSDFINENGDLISSNNIKISWLKETLANIGRSNPSAPLEPFPDIIHNSGSLNIEKNKIASAWINIKIPRNAKPGIYNGSIEVTADELEKSYTFDYSFEVLNLVQPLPSETNTQIEFWQHPYTIARYYKICKEDLFTEKHFKYLRGNLKEYRNMGGRGVIATIVHEAWNHQSYDSDPSMIKWRKNSYGTFEFDYSHFDKWIQLNIDLGILDPEKGFGQIKCYSIVPWNNRIQYFNEATNKEEAINPTPGSDLWINIWTQFLTSFMSHLEEKGWFNITYISMDERSMDDLKACVDLIENITNNSYEHFKISSAMDYESGNDYSFLDRIDDISIGLSHINHNSDDMKNMATHRQELGLLTTIYTCTGDYPSSFTISDPSEGAFTIWYSLYQNTNGFLRWSWDGWVENPLENVSYKYWEPGDPFLIYPAEKDSIGKTFYSTPRLEKLKEGIRDINKAKYLMEKAPNLKNSIENLIYSLKRPNKGENAYGSAVAASKEDRDLTISEANRIKNGINNFAREFISLTMETL
4TND , Knot 231 590 0.83 40 267 543 
MELENIVANTVLLKAREGGGGKRKGKSKKWKEILKFPHISQCEDLRRTIDRDYCSLCDKQPIGRLLFRQFCETRPGLECYIQFLDSVAEYEVTPDEKLGEKGKKIMTKYLTPKSPVFIAQVGQDLVSQTEEKLLQKPCKELFSACAQSVHEYLRGEPFHEYLDSMFFDRFLQWKWLERQPVTKNTFRQYRVLGKGGFGEVCACQVRATGKMYACKRLEKKRIKKRKGESMALNEKQILEKVNSQFVVNLAYAYETKDALCLVLTIMNGGDLKFHIYNMGNPGFEEERALFYAAEILCGLEDLHHENTVYRDLKPENILLDDYGHIRISDLGLAVKIPEGDLIRGRVGTVGYMAPEVLNNQRYGLSPDYWGLGCLIYEMIEGQSPFRGRKEKVKREEVDRRVLETEEVYSHKFSEEAKSICKMLLTKDAKQRLGCQEEEAAEVKRHPFFRNMNFKRLEAGMLDPPFVPDPRAVYCKDVLDIEQFSTVKGVNLDHTDDDFYSKFSTGSVSIPWQNEMIETECFKELNVFGPNGTLPPDLNRNHPPEPPKKGLLQRLFKRQHQNNSKSSPSSKTSFNHHINSNHVSSNSTGSS
9NSZ , Knot 122 273 0.83 40 185 264 
MNKYFTCYVVASLFLSGCTVQHNLINETPSQIVQGHNQVIHQYFDEKNTSGVLVIQTDKKINLYGNALSRANTEYVPASTFKMLNALIGLENQKTDINEIFKWKGEKRSFTAWEKDMTLGEAMKLSAVPVYQELARRIGLDLMQKEVKRIGFGNAEIGQQVDNFWLVGPLKVTPIQEVEFVSQLAHTQLPFSEKVQANVKNMLLLEESNGYKIFGKTGWAMDIKPQVGWLTGWVEQPDGKIVAFALNMEMRSEMPASIRNELLMKSLKQLNII

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 \(c\).

\(|P_{f(5FQE_1)}(2) \setminus P_{f(4TND_1)}(2)|=83\), \(|P_{f(4TND_1)}(2) \setminus P_{f(5FQE_1)}(2)|=77\). 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 1:1100000000001111010011010000011101100010010000001001010010011111001000111100001100101000011000101100001010110001101100010111011101100010101000011011101011001011100101010100100000100010110110111000000101100100110000100001100001001010100000111011110110011000000001011010000010101000010011010101111010011101000011110001001001000001101011001110110011001100100011101001010000100101010110010000000101001100001000011001001011100100000010011000001111001000010010010100100111011001000001110101011100110010000101101111011000011001000101001001100100100110011010001001100100100100101011110000001010010010011001100110101001
Pair \(Z_2\) Length of longest common subsequence
5FQE_1,4TND_1 160 4
5FQE_1,9NSZ_1 172 4
4TND_1,9NSZ_1 182 4

Newick tree

 
[
	9NSZ_1:91.20,
	[
		5FQE_1:80,4TND_1:80
	]:11.20
]

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 \((0.85)(0.8)(\frac{1200 }{\log_{20} 1200}-\frac{590}{\log_{20}590})=156.\)
Status Protein1 Protein2 d d1/2
Query variables 5FQE_1 4TND_1 201 195.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]