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Parikh vectors
8RBL_1 3RBM_1 6GPP_1 Letter Amino acid
13 29 7 Y Tyrosine
29 20 16 A Alanine
12 8 11 Q Glutamine
11 40 20 I Isoleucine
14 12 19 T Threonine
18 15 10 F Phenylalanine
5 4 1 W Tryptophan
9 28 8 N Asparagine
15 19 18 G Glycine
7 15 11 H Histidine
7 9 8 M Methionine
23 11 9 R Arginine
25 30 27 E Glutamic acid
9 34 16 K Lycine
12 23 21 S Serine
14 21 12 V Valine
21 26 16 D Aspartic acid
3 8 0 C Cysteine
33 39 19 L Leucine
7 5 7 P Proline 

8RBL_1|Chains A, B|Mycolic acid methyltransferase MmaA1|Mycobacterium tuberculosis (1773)
>3RBM_1|Chains A, B, C, D|Farnesyl pyrophosphate synthase|Plasmodium vivax (126793)
>6GPP_1|Chain A|Heat shock protein HSP 90-alpha|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 \(w=f(c)\)
8RBL , Knot 125 287 0.82 40 177 279 
HMAKLRPYYEESQSAYDISDDFFALFLDPTWVYTCAYFERDDMTLEEAQLAKVDLALDKLNLEPGMTLLDVGCGWGGALVRAVEKYDVNVIGLTLSRNHYERSKDRLAAIGTQRRAEARLQGWEEFEENVDRIVSFEAFDAFKKERYLTFFERSYDILPDDGRMLLHSLFTYDRRWLHEQGIALTMSDLRFLKFLRESIFPGGELPSEPDIVDNAQAAGFTIEHVQLLQQHYARTLDAWAANLQAARERAIAVQSEEVYNNFMHYLTGCAERFRRGLINVAQFTMTK
3RBM , Knot 163 396 0.82 40 210 374 
MGSSHHHHHHSSGRENLYFQGMKETNSEEADSGLAFFRNMYDKYRDAFLSHLNEYSLEEEIKEHISKYYKLLFDYNCLGGKNNRGILVILIYEYVKNRDINSSEWEKAACLAWCIEILQAAFLVADDIMDKGEMRRNKYCWYLLKDVETKNAVNDVLLLYNSIYKLIEIYLRNESCYVDVIATFRDATLKTIIGQHLDTNIFSDKYSDAHREIDVNNINVPEQPVIDINMINFGVYKNIVIHKTAYYSFFLPIVCGMLLAGIAVDNLIYKKIEDISMLMGEYFQIHDDYLDIFGDSTKTGKVGSDIQNNKLTWPLIKTFELCSEPDKIKIVKNYGKNNLACVKVIDSLYEQYKIRKHYESYEKAQKAKILSAINELHHEGIEYVLKYLLEILFTGV
6GPP , Knot 112 256 0.80 38 163 244 
MGSSHHHHHHSSGLVPRGSHMPEETQTQDQPMEEEEVETFAFQAEIAQLMSLIINTFYSNKEIFLRELISNSSDALDKIRYESLTDPSKLDSGKELHINLIPNKQDRTLTIVDTGIGMTKADLINNLGTIAASGTKAFMEALQAGADISMIGQFGVGFYSAYLVAEKVTVITKHNDDEQYAWESSAGGSFTVRTDTGEPMGRGTKVILHLKEDQTEYLEERRIKEIVKKHSQFIGYPITLFVEKERDKEVSDDEAE

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 \(c\).

\(|P_{f(8RBL_1)}(2) \setminus P_{f(3RBM_1)}(2)|=65\), \(|P_{f(3RBM_1)}(2) \setminus P_{f(8RBL_1)}(2)|=98\). 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 1:01101010000000100100011111101011000101000010100101101011100101011101101101111111011000010111101000000000001111100001010101100100010011010110110000010110000011100101110011000001100011110100101101100011111011001011001011110100101100001001011110101100011110000100011001010100100111011010100
Pair \(Z_2\) Length of longest common subsequence
8RBL_1,3RBM_1 163 4
8RBL_1,6GPP_1 144 4
3RBM_1,6GPP_1 149 13

Newick tree

 
[
	3RBM_1:80.00,
	[
		8RBL_1:72,6GPP_1:72
	]:8.00
]

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 \((0.85)(0.8)(\frac{683 }{\log_{20} 683}-\frac{287}{\log_{20}287})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 8RBL_1 3RBM_1 138 120
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]