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Parikh vectors
8DEO_1 8QMQ_1 6NXR_1 Letter Amino acid
19 28 13 I Isoleucine
40 19 18 K Lycine
3 15 3 M Methionine
19 17 9 F Phenylalanine
17 19 8 Q Glutamine
3 8 3 H Histidine
19 9 4 Y Tyrosine
37 22 12 D Aspartic acid
12 25 8 R Arginine
0 7 3 C Cysteine
54 41 23 G Glycine
38 19 9 P Proline
20 14 15 S Serine
26 27 34 V Valine
20 74 31 A Alanine
29 32 19 E Glutamic acid
18 38 16 L Leucine
57 22 11 T Threonine
2 7 6 W Tryptophan
34 19 12 N Asparagine 

8DEO_1|Chains A, B|Accumulation associated protein|Staphylococcus epidermidis RP62A (176279)
>8QMQ_1|Chains A, B, C, D|Succinate semialdehyde dehydrogenase [NAD(P)+] Sad|Escherichia coli K-12 (83333)
>6NXR_1|Chain A|Triosephosphate isomerase, cytosolic|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
8DEO , Knot 176 467 0.77 38 196 394 
GPAMIDIPPTTVKGRDNYDFYGRVDIESNPTDLNATNLTRYNYGQPPGTTTAGAVQFKNQVSFDKDFDFNIRVANNRQSNTTGADGWGFMFSKKDGDDFLKNGGILREKGTPSAAGFRIDTGYYNNDPLDKIQKQAGQGYRGYGTFVKNDSQGNTSKVGSGTPSTDFLNYADNTTNDLDGKFHGQKLNNVNLKYNASNQTFTATYAGKTWTATLSELGLSPTDSYNFLVTSSQYGNGNSGTYASGVMRADLDGATLTYTPKAVDGDPIISTKEIPFNKKREFDPNLAPGTEKVVQKGEPGIETTTTPTYVNPNTGEKVGEGEPTEKITKQPVDEIVHYGGEEIKPGHKDEFDPNAPKGSQTTQPGKPGVKNPDTGEVVTPPVDDVTKYGPVDGDPITSTEEIPFDKKREFNPDLKPGEERVKQKGEPGTKTITTPTTKNPLTGEKVGEGEPTEKITKQPVDEITEYG
8QMQ , Knot 187 462 0.82 40 225 425 
MTITPATHAISINPATGEQLSVLPWAGADDIENALQLAAAGFRDWRETNIDYRAEKLRDIGKALRARSEEMAQMITREMGKPINQARAEVAKSANLCDWYAEHGPAMLKAEPTLVENQQAVIEYRPLGTILAIMPWNFPLWQVMRGAVPIILAGNGYLLKHAPNVMGCAQLIAQVFKDAGIPQGVYGWLNADNDGVSQMIKDSRIAAVTVTGSVRAGAAIGAQAGAALKKCVLELGGSDPFIVLNDADLELAVKAAVAGRYQNTGQVCAAAKRFIIEEGIASAFTERFVAAAAALKMGDPRDEENALGPMARFDLRDELHHQVEKTLAQGARLLLGGEKMAGAGNYYPPTVLANVTPEMTAFREEMFGPVAAITIAKDAEHALELANDSEFGLSATIFTTDETQARQMAARLECGGVFINGYCASDARVAFGGVKKSGFGRELSHFGLHEFCNIQTVWKDRI
6NXR , Knot 114 257 0.82 40 160 238 
GPHMARKFFVGGNWKDNGTAEEVKKIVNTLNEAQVPSQDVVEVVVSPPYVFLPLVKSTLRSDFFVAAQNCWVKKGGAFTGEVSAEMLVNLDIPWVILGHSERRAILNESSEFVGDKVAYALAQGLKVIACVGETLEEREAGSTMDVVAAQTKAIADRVTNWSNVVIAYEPVWAIGTGKVASPAQAQEVHDELRKWLAKNVSADVAATTRIIYGGSVNGGNCKELGGQADVDGFLVGGASLKPEFIDIIKAAEVKKSA

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 \(c\).

\(|P_{f(8DEO_1)}(2) \setminus P_{f(8QMQ_1)}(2)|=63\), \(|P_{f(8QMQ_1)}(2) \setminus P_{f(8DEO_1)}(2)|=92\). 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 1:11111011100101000001010101000100101001000001011100011110100010100010101011000000001101111110000100110011110001010111101001000001100100011010010101100000100001101010001100100000010101010010010100010000101001100101010011101000001110000010100100101110101011010001011010111000011100000101011110001100101110000010010100100110101000100011001100110010110000101011010000011011100100101101110010001110101100000111000001010101100010001011000100100001101001101010001000110010001
Pair \(Z_2\) Length of longest common subsequence
8DEO_1,8QMQ_1 155 5
8DEO_1,6NXR_1 152 4
8QMQ_1,6NXR_1 167 4

Newick tree

 
[
	8QMQ_1:82.01,
	[
		8DEO_1:76,6NXR_1:76
	]:6.01
]

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 \((0.85)(0.8)(\frac{929 }{\log_{20} 929}-\frac{462}{\log_{20}462})=123.\)
Status Protein1 Protein2 d d1/2
Query variables 8DEO_1 8QMQ_1 155 149.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]