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Parikh vectors
4XNK_1 3AIL_1 3LZX_1 Letter Amino acid
46 18 17 D Aspartic acid
33 25 16 S Serine
20 11 12 Q Glutamine
11 11 7 H Histidine
17 25 26 I Isoleucine
5 2 6 M Methionine
23 19 15 F Phenylalanine
34 12 14 T Threonine
44 20 27 A Alanine
26 16 31 E Glutamic acid
52 28 27 G Glycine
41 30 28 L Leucine
13 20 26 K Lycine
18 16 12 P Proline
15 2 2 W Tryptophan
15 16 11 Y Tyrosine
36 13 11 R Arginine
22 13 16 N Asparagine
0 3 2 C Cysteine
19 23 26 V Valine 

4XNK_1|Chain A|Alginate production protein AlgE|Pseudomonas aeruginosa PAO1 (208964)
>3AIL_1|Chains A, B, C, D|303aa long hypothetical esterase|Sulfolobus tokodaii (111955)
>3LZX_1|Chains A, B|Ferredoxin--NADP reductase 2|Bacillus subtilis (1423)
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 \(w=f(c)\)
4XNK , Knot 190 490 0.80 38 229 459 
MNSSRSVNPRPSFAPRALSLAIALLLGAPAFAANSGEAPKNFGLDVKITGESENDRDLGTAPGGTLNDIGIDLRPWAFGQWGDWSAYFMGQAVAATDTIETDTLQSDTDDGNNSRNDGREPDKSYLAAREFWVDYAGLTAYPGEHLRFGRQRLREDSGQWQDTNIEALNWSFETTLLNAHAGVAQRFSEYRTDLDELAPEDKDRTHVFGDISTQWAPHHRIGVRIHHADDSGHLRRPGEEVDNLDKTYTGQLTWLGIEATGDAYNYRSSMPLNYWASATWLTGDRDNLTTTTVDDRRIATGKQSGDVNAFGVDLGLRWNIDEQWKAGVGYARGSGGGKDGEEQFQQTGLESNRSNFTGTRSRVHRFGEAFRGELSNLQAATLFGSWQLREDYDASLVYHKFWRVDDDSDIGTSGINAALQPGEKDIGQELDLVVTKYFKQGLLPASMSQYVDEPSALIRFRGGLFKPGDAYGPGTDSTMHRAFVDFIWRF
3AIL , Knot 139 323 0.82 40 197 309 
MGSSHHHHHHSSGLVPRGSHMIDPKIKKLLESTIQLPIGKASVEEIRSLFKQFSSLTPREEVGKIEDITIPGSETNIKARVYYPKTQGPYGVLVYYHGGGFVLGDIESYDPLCRAITNSCQCVTISVDYRLAPENKFPAAVVDSFDALKWVYNNSEKFNGKYGIAVGGDSAGGNLAAVTAILSKKENIKLKYQVLIYPAVSFDLITKSLYDNGEGFFLTREHIDWFGQQYLRSFADLLDFRFSPILADLNDLPPALIITAEHDPLRDQGEAYANKLLQSGVQVTSVRFNNVIHGFVSFFPFIEQGRDAIGLIGYVLRKVFYGK
3LZX , Knot 144 332 0.84 40 197 321 
MREDTKVYDITIIGGGPVGLFTAFYGGMRQASVKIIESLPQLGGQLSALYPEKYIYDVAGFPKIRAQELINNLKEQMAKFDQTICLEQAVESVEKQADGVFKLVTNEETHYSKTVIITAGNGAFKPRKLELENAEQYEGKNLHYFVDDLQKFAGRRVAILGGGDSAVDWALMLEPIAKEVSIIHRRDKFRAHEHSVENLHASKVNVLTPFVPAELIGEDKIEQLVLEEVKGDRKEILEIDDLIVNYGFVSSLGPIKNWGLDIEKNSIVVKSTMETNIEGFFAAGDICTYEGKVNLIASGFGEAPTAVNNAKAYMDPKARVQPLHSTSLFENK

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 \(c\).

\(|P_{f(4XNK_1)}(2) \setminus P_{f(3AIL_1)}(2)|=91\), \(|P_{f(3AIL_1)}(2) \setminus P_{f(4XNK_1)}(2)|=59\). 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 1:1000001010101110110111111111111110010110011101010100000001101111010011101011111011010101110111100010000100000010000001001000011100111001110101100101100010000101000010110101000110101111001000000100111000000011101000111000111010010001010011001001000001010111101010100000011100110101101000010000100001101000101011110111010100010111101010111001000100011000000101000010011011010100101101110101000001011000110100000110011011101100011001011100010011111010001001011101011110110101110000100111011101
Pair \(Z_2\) Length of longest common subsequence
4XNK_1,3AIL_1 150 4
4XNK_1,3LZX_1 154 4
3AIL_1,3LZX_1 130 5

Newick tree

 
[
	4XNK_1:79.33,
	[
		3AIL_1:65,3LZX_1:65
	]:14.33
]

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 \((0.85)(0.8)(\frac{813 }{\log_{20} 813}-\frac{323}{\log_{20}323})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 4XNK_1 3AIL_1 166 138
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]