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Parikh vectors
1ELY_1 7QZF_1 6ITZ_1 Letter Amino acid
11 5 6 Y Tyrosine
13 24 8 R Arginine
22 5 5 Q Glutamine
45 31 14 G Glycine
28 4 19 K Lycine
8 18 11 F Phenylalanine
3 2 7 W Tryptophan
21 20 6 S Serine
20 1 3 N Asparagine
28 26 15 D Aspartic acid
5 2 3 C Cysteine
24 24 22 E Glutamic acid
10 7 4 H Histidine
39 30 13 L Leucine
21 18 12 P Proline
64 35 16 A Alanine
16 10 17 I Isoleucine
8 8 4 M Methionine
40 20 9 T Threonine
23 26 22 V Valine 

1ELY_1|Chains A, B|ALKALINE PHOSPHATASE|Escherichia coli (562)
>7QZF_1|Chains A, B, C, D, E, F|Dyp-type peroxidase family|Streptomyces lividans (1916)
>6ITZ_1|Chains A, B|Peroxiredoxin|Thermococcus kodakarensis KOD1 (69014)
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 \(w=f(c)\)
1ELY , Knot 182 449 0.82 40 218 419 
TPEMPVLENRAAQGDITAPGGARRLTGDQTAALRDSLSDKPAKNIILLIGDGMGDSEITAARNYAEGAGGFFKGIDALPLTGQYTHYALNKKTGKPDYVTDCAASATAWSTGVKTYNGALGVDIHEKDHPTILEMAKAAGLATGNVSTAELQDATPAALVAHVTSRKCYGPSATSEKCPGNALEKGGKGSITEQLLNARADVTLGGGAKTFAETATAGEWQGKTLREQAQARGYQLVSDAASLNSVTEANQQKPLLGLFADGNMPVRWLGPKATYHGNIDKPAVTCTPNPQRNDSVPTLAQMTDKAIELLSKNEKGFFLQVEGASIDKQDHAANPCGQIGETVDLDEAVQRALEFAKKEGNTLVIVTADHAHASQIVAPDTKAPGLTQALNTKDGAVMVMSYGNSEEDSQEHTGSQLRIAAYGPHAANVVGLTDQTDLFYTMKAALGLK
7QZF , Knot 133 316 0.80 40 165 301 
MGGEVEEPEPQMVLSPLTSAAIFLVVTIDSGGEDTVRDLLSDVASLERAVGFRAQPDGRLSCVTGIGSEAWDRLFSGARPAGLHPFRELDGPVHRAVATPGDLLFHIRASRLDLCFALATEIMGRLRGAVTPQDEVHGFKYFDERDMLGFVAGTENPTGAAARRAVLVGAEDPAFAGGSYAVVQKYLHDIDAWEGLSVEAQERVIGRRKMTDVELSDDVKPADSHVALTSVTGPDGSDLEILRDAMPFGSVGREEFGTYFIGYARTPEVTETMLERMFLGTASAPHDRILDFSTAVTGSLFFTPAADFLEDLSARP
6ITZ , Knot 99 216 0.82 40 150 210 
MVVIGEKFPEVEVKTTHGVIKLPDYFAEQGKWFVLFSHPADFTPVCTTEFYAMQKRVDQFRELGVEPIGLSVDQVFSHIKWMEWIKENLGEEITFPVIADDRGELADKLGMIPSGATITARAVFIVDDKGIIRAIVYYPAEVGRDWDEILRLVKALKVSDEKGVALPHKWPNNELIGDKAIVPPASTVDEVKQREEAKAKGEIECYDWWFCYKKLE

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 \(c\).

\(|P_{f(1ELY_1)}(2) \setminus P_{f(7QZF_1)}(2)|=102\), \(|P_{f(7QZF_1)}(2) \setminus P_{f(1ELY_1)}(2)|=49\). 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 1:01011110001101010111110010100011100010001100111111011100010110001011111101101111010000011000010100100011010110011000011111010000010110110111110101001010010111111010000001101000001101100110101000110101010111110011001011010100100010101001100110100100100001111111010111011110100010100111000101000001101101000110110000011110101101000001101010110010100110011011000100111101001010011110001111001100001111110010000000000100101110110110111100000110010111110
Pair \(Z_2\) Length of longest common subsequence
1ELY_1,7QZF_1 151 4
1ELY_1,6ITZ_1 172 4
7QZF_1,6ITZ_1 157 4

Newick tree

 
[
	6ITZ_1:84.49,
	[
		1ELY_1:75.5,7QZF_1:75.5
	]:8.99
]

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 \((0.85)(0.8)(\frac{765 }{\log_{20} 765}-\frac{316}{\log_{20}316})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 1ELY_1 7QZF_1 157 132
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]