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Parikh vectors
8WPM_1 5UJQ_1 3UBT_1 Letter Amino acid
39 2 16 R Arginine
46 0 17 E Glutamic acid
26 1 6 M Methionine
53 3 29 K Lycine
37 2 7 T Threonine
11 0 4 W Tryptophan
39 1 24 N Asparagine
19 0 6 C Cysteine
31 4 25 G Glycine
105 2 24 L Leucine
27 0 17 P Proline
65 1 19 S Serine
30 1 15 Y Tyrosine
48 1 15 V Valine
41 1 18 D Aspartic acid
29 1 13 Q Glutamine
16 0 8 H Histidine
45 1 19 F Phenylalanine
45 4 19 A Alanine
45 4 30 I Isoleucine 

8WPM_1|Chain A|Short transient receptor potential channel 1|Homo sapiens (9606)
>5UJQ_1|Chain A|Bacteriocin|Carnobacterium maltaromaticum (2751)
>3UBT_1|Chains A, B, C[auth Y]|Modification methylase HaeIII|Haemophilus aegyptius (197575)
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 \(w=f(c)\)
8WPM , Knot 303 797 0.84 40 307 725 
GPVDMMAALYPSTDLSGASSSSLPSSPSSSSPNEVMALKDVREVKEENTLNEKLFLLACDKGDYYMVKKILEENSSGDLNINCVDVLGRNAVTITIENENLDILQLLLDYGCQSADALLVAIDSEVVGAVDILLNHRPKRSSRPTIVKLMERIQNPEYSTTMDVAPVILAAHRNNYEILTMLLKQDVSLPKPHAVGCECTLCSAKNKKDSLRHSRFRLDIYRCLASPALIMLTEEDPILRAFELSADLKELSLVEVEFRNDYEELARQCKMFAKDLLAQARNSRELEVILNHTSSDEPLDKRGLLEERMNLSRLKLAIKYNQKEFVSQSNCQQFLNTVWFGQMSGYRRKPTCKKIMTVLTVGIFWPVLSLCYLIAPKSQFGRIIHTPFMKFIIHGASYFTFLLLLNLYSLVYNEDKKNTMGPALERIDYLLILWIIGMIWSDIKRLWYEGLEDFLEESRNQLSFVMNSLYLATFALKVVAHNKFHDFADRKDWDAFHPTLVAEGLFAFANVLSYLRLFFMYTTSSILGPLQISMGQMLQDFGKFLGMFLLVLFSFTIGLTQLYDKGYTSKEQKDCVGIFCEQQSNDTFHSFIGTCFALFWYIFSLAHVAIFVTRFSYGEELQSFVGAVIVGTYNVVVVIVLTKLLVAMLHKSFQLIANHEDKEWKFARAKLWLSYFDDKCTLPPPFNIIPSPKTICYMISSLSKWICSHTSKGKVKRQNSLKEWRNLKQKRDENYQKVMCCLVHRYLTSMRQKMQSTDQATVENLNELRQDLSKFRNEIRDLLGFRTSKYAMFYPRN
5UJQ , Knot 18 29 0.69 30 26 27 
SAILAITLGIFATGYGMGVQKAINDRRKK
3UBT , Knot 148 331 0.86 40 216 324 
MNLISLFSGAGGLDLGFQKAGFRIICANEYDKSIWKTYESNHSAKLIKGDISKISSDEFPKCDGIIGGPPSQSWSEGGSLRGIDDPRGKLFYEYIRILKQKKPIFFLAENVKGMMAQRHNKAVQEFIQEFDNAGYDVHIILLNANDYGVAQDRKRVFYIGFRKELNINYLPPIPHLIKPTFKDVIWDLKDNPIPALDKNKTNGNKCIYPNHEYFIGSYSTIFMSRNRVRQWNEPAFTVQASGRQCQLHPQAPVMLKVSKNLNKFVEGKEHLYRRLTVRECARVQGFPDDFIFHYESLNDGYKMIGNAVPVNLAYEIAKTIKSALEICKGNN

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 \(c\).

\(|P_{f(8WPM_1)}(2) \setminus P_{f(5UJQ_1)}(2)|=285\), \(|P_{f(5UJQ_1)}(2) \setminus P_{f(8WPM_1)}(2)|=4\). 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 1:11101111101000101100001100100001001111001001000001000111110001000110011000001010100101110011010100001011011100100010111111000111110111000100000101101100100100000101111111100000011011100010110101110000100100000010000101010001101111110000111011010101001011010100000011000011100111010000010111000000011000111000101001011100000011000000011001111010100001000011011011111111010011110001101100111011101100101111101001100000000111110010011111111111001001100110011000000101110010110111011100010011000010110101110111111011001011110000011111010110110011011111111110101110010001000000000111100000000100111001111101101101111100100100100111111110001111111001111110001011100000010110101110010000011111011101001001100100110000001010000010010010000000000110011000100100010000010100100100010010001001111000001110100
Pair \(Z_2\) Length of longest common subsequence
8WPM_1,5UJQ_1 289 3
8WPM_1,3UBT_1 159 4
5UJQ_1,3UBT_1 204 3

Newick tree

 
[
	5UJQ_1:13.92,
	[
		8WPM_1:79.5,3UBT_1:79.5
	]:57.42
]

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 \((0.85)(0.8)(\frac{826 }{\log_{20} 826}-\frac{29}{\log_{20}29})=232.\)
Status Protein1 Protein2 d d1/2
Query variables 8WPM_1 5UJQ_1 298 153.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]