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Parikh vectors
9AVU_1 4DMK_1 3VQG_1 Letter Amino acid
14 9 0 M Methionine
30 14 8 S Serine
13 12 4 Q Glutamine
42 9 5 I Isoleucine
13 18 6 H Histidine
37 28 10 L Leucine
27 11 0 T Threonine
15 12 0 Y Tyrosine
16 29 5 A Alanine
17 8 5 N Asparagine
25 11 2 K Lycine
23 19 4 F Phenylalanine
10 5 0 W Tryptophan
44 19 10 V Valine
23 12 7 D Aspartic acid
19 24 9 G Glycine
25 23 7 E Glutamic acid
26 19 4 P Proline
12 17 8 R Arginine
6 2 0 C Cysteine 

9AVU_1|Chains A, C|Acetylcholine receptor subunit alpha|Bos taurus (9913)
>4DMK_1|Chains A, B, C, D|Putative hydrolase|Pseudomonas aeruginosa (208963)
>3VQG_1|Chain A|E3 ubiquitin-protein ligase LNX|Mus musculus (10090)
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 \(w=f(c)\)
9AVU , Knot 182 437 0.84 40 238 412 
SEHETRLVAKLFEDYNSVVRPVEDHRQAVEVTVGLQLIQLINVDEVNQIVTTNVRLKQQWVDYNLKWNPDDYGGVKKIHIPSEKIWRPDLVLYNNADGDFAIVKFTKVLLDYTGHITWTPPAIFKSYCEIIVTHFPFDEQNCSMKLGTWTYDGSVVVINPESDQPDLSNFMESGEWVIKESRGWKHWVFYACCPSTPYLDITYHFVMQRLPLYFIVNVIIPCLLFSFLTGLVFYLPTDSGEKMTLSISVLLSLTVFLLVIVELIPSTSSAVPLIGKYMLFTMVFVIASIIITVIVINTHHRSPSTHVMPEWVRKVFIDTIPNIMFFSTMKRPSREKQDKKIFTEDIDISDISGKPGPPPMGFHSPLIKHPEVKSAIEGIKYIAETMKSDQESNNAAEEWKYVAMVMDHILLAVFMLVCIIGTLAVFAGRLIELNQQG
4DMK , Knot 137 301 0.86 40 203 292 
AEEFPVPNGFESAYREVDGVKLHYVKGGQGPLVMLVHGFGQTWYEWHQLMPELAKRFTVIAPDLPGLGQSEPPKTGYSGEQVAVYLHKLARQFSPDRPFDLVAHDIGIWNTYPMVVKNQADIARLVYMEAPIPDARIYRFPAFTAQGESLVWHFSFFAADDRLAETLIAGKERFFLEHFIKSHASNTEVFSERLLDLYARSYAKPHSLNASFEYFRALNESVRQNAELAKTRLQMPTMTLAGGGHGGMGTFQLEQMKAYAEDVEGHVLPGCGHWLPEECAAPMNRLVIDFLSRGRHHHHHH
3VQG , Knot 47 94 0.75 30 76 90 
GPLGSDHDDSFHVILNKSSPEEQLGIKLVRRVDEPGVFIFNVLNGGVADRHGQLEENDRVLAINGHDLRFGSPESAAHLIQASERRVHLVVSRQ

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 \(c\).

\(|P_{f(9AVU_1)}(2) \setminus P_{f(4DMK_1)}(2)|=103\), \(|P_{f(4DMK_1)}(2) \setminus P_{f(9AVU_1)}(2)|=68\). 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 1:00000011101100000110110000011010111011011010010011000101000110001010100011100101100011010111000101011110100111000101010111110000011100111000000101101000101111010000101001100101110000110011101001001010100011100111011101111011101101111011000100101010111010111111101110000111111001110111111011101111000000100011101100111001101111001001000000001100010100101011111111001110010100110110011001000000001100100111110011111111101110111111011010001
Pair \(Z_2\) Length of longest common subsequence
9AVU_1,4DMK_1 171 3
9AVU_1,3VQG_1 206 3
4DMK_1,3VQG_1 171 3

Newick tree

 
[
	3VQG_1:97.51,
	[
		9AVU_1:85.5,4DMK_1:85.5
	]:12.01
]

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 \((0.85)(0.8)(\frac{738 }{\log_{20} 738}-\frac{301}{\log_{20}301})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 9AVU_1 4DMK_1 152 128.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]