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Parikh vectors
5AWO_1 8QUQ_1 4LKL_1 Letter Amino acid
33 25 21 S Serine
24 5 3 W Tryptophan
35 12 16 R Arginine
36 27 9 N Asparagine
14 19 7 Q Glutamine
66 45 10 G Glycine
22 9 11 I Isoleucine
18 16 11 K Lycine
14 2 5 M Methionine
30 3 7 P Proline
70 25 11 A Alanine
39 22 11 T Threonine
19 10 13 E Glutamic acid
13 2 6 H Histidine
42 24 31 L Leucine
38 21 10 V Valine
2 0 6 C Cysteine
22 18 7 F Phenylalanine
26 28 13 Y Tyrosine
47 34 14 D Aspartic acid 

5AWO_1|Chain A|Isomaltodextranase|Arthrobacter globiformis (1665)
>8QUQ_1|Chains A, B, C|OmpK36|Klebsiella pneumoniae (573)
>4LKL_1|Chain A|Serine/threonine-protein kinase PLK1|Homo sapiens (9606)
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 \(w=f(c)\)
5AWO , Knot 239 610 0.83 40 264 561 
GSHMATAVTARPGVPVTAAPPLRLASRNSVFTRSGAGPRYWNIYGYSFPHNAPIPENEWKVNIDWLAGNFADFGYDIACTDGWIEGSSRTTGNGYITSYNDSWQHDWAYWANYLAARKMKLGVYYNPLWVHRAAVEDASKTVLGRPDVKIADLVVPGDFFARDIGGNQLYWLDVTKSGAKEYVQGYVRYFKDLGVPYLRIDFLSWYEDGRDANIGQVNAPHGRANYELALSWINEAAGEDMEVSLVMPHMFQDGSAELANGDLVRINADADKGGWDRLSGMRQNWQDAWPNWANPFCGFTGWSHRNGRGQLILDGDFMRASTFASDEERKTMMNLMVAAGSPLAIADTYQQIGNNAWVYTNKEVLQLNADGLVGKPLYRSATPFSKDPGSRDTERWAGQLPDGSWGVALFNRSDTETVTKTIDFAKDLGLATGGNVRDLWEHRNLGMDSRATAALAPHASAIFRVTPPKMHGTTRYPAAFAAWGGGAGFNYNHPGYDGNGFVDGLQAGSGSADPLVTFAVQVPHRGSYAIRYRYANATGDTSTMTVTAEKADRSTVDGPVHVSFPGLATWDTWGVADGTITLDAGLNLVTIGRGATDKGAINLNWIELDM
8QUQ , Knot 147 347 0.82 38 175 321 
GAEIYNKDGNKLDLYGKIDGLHYFSDDKSVDGDQTYMRVGVKGETQINDQLTGYGQWEYNVQANNTESSSDQAWTRLAFAGLKFGDAGSFDYGRNYGVVYDVTSWTDVLPEFGGDGDTYGSDNFLQSRANGVATYRNSDFFGLVDGLNFALQYQGKNGSVSGEGATNNGRGWSKQNGDGFGTSLTYDIWDGISAGFAYSHSKRTDEQNSVPALGRGDNAETYTGGLKYDANNIYLASQYTQTYNATRAGSLGFANKAQNFEVVAQYQFDFGLRPSVAYLQSKGKDLERGYGDQDILKYVDVGATYYFNKNMSTYVDYKINLLDDNSFTRNAGISTDDVVALGLVYQF
4LKL , Knot 108 222 0.87 40 161 215 
CHLSDMLQQLHSVNASKPSERGLVRQEEAEDPACIPIFWVSKWVDYSDKYGLGYQLCDNSVGVLFNDSTRLILYNDGDSLQYIERDGTESYLTVSSHPNSLMKKITLLKYFRNYMSEHLLKAGANITPREGDELARLPYLRTWFRTRSAIILHLSNGSVQINFFQDHTKLILCPLMAAVTYIDEKRDFRTYRLSLLEEYGCCKELASRLRYARTMVDKLLSS

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 \(c\).

\(|P_{f(5AWO_1)}(2) \setminus P_{f(8QUQ_1)}(2)|=129\), \(|P_{f(8QUQ_1)}(2) \setminus P_{f(5AWO_1)}(2)|=40\). 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 1:1001101101011111011111011000011000111100101010011001111000101010111101101100110001110100000101010000001000110110011100101110001111001110010001110101011011111011100111001011010001100010101001001111010101101000100101101011010100011101100111001010111101100101011010110101010011100101100010011101101101101100001010111010110100110000000110111111011111000001100111000001101010111101100010110001100000011101101011111100000001000101100111101101001100001110001011111010111010110101000011111111111110000110010111011011010101110111011001001100001010100001010100100001011101011111010011110101010111011011011000111010110101
Pair \(Z_2\) Length of longest common subsequence
5AWO_1,8QUQ_1 169 4
5AWO_1,4LKL_1 175 4
8QUQ_1,4LKL_1 160 4

Newick tree

 
[
	5AWO_1:87.92,
	[
		8QUQ_1:80,4LKL_1:80
	]:7.92
]

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 \((0.85)(0.8)(\frac{957 }{\log_{20} 957}-\frac{347}{\log_{20}347})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 5AWO_1 8QUQ_1 209 161
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]