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Parikh vectors
6ICP_1 8XYX_1 4LGB_1 Letter Amino acid
13 21 10 G Glycine
25 13 9 P Proline
8 5 1 W Tryptophan
15 27 6 N Asparagine
14 30 5 Q Glutamine
25 28 13 E Glutamic acid
18 27 7 I Isoleucine
14 44 10 K Lycine
20 12 7 F Phenylalanine
32 29 14 S Serine
12 7 16 T Threonine
26 27 12 D Aspartic acid
28 28 13 L Leucine
6 8 2 M Methionine
14 11 5 Y Tyrosine
24 17 22 V Valine
9 3 2 C Cysteine
20 14 11 R Arginine
18 6 6 H Histidine
28 21 6 A Alanine 

6ICP_1|Chains A, B, C|Methylxanthine N1-demethylase NdmA|Pseudomonas putida (303)
>8XYX_1|Chain A|MT-a70 family protein|Tetrahymena thermophila SB210 (312017)
>4LGB_1|Chain A|Abscisic acid receptor PYL2|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
6ICP , Knot 160 369 0.85 40 223 350 
MGSSHHHHHHENLYFQGSMEQAIINDEREYLRHFWHPVCTVTELEKAHPSSLGPLAVKLLNEQLVVAKLGDEYVAMRDRCAHRSAKLSLGTVSGNRLQCPYHGWQYDTHGACQLVPACPNSPIPNKAKVDRFDCEERYGLIWIRLDSSFDCTEIPYFSAANDPRLRIVIQEPYWWDATAERRWENFTDFSHFAFIHPGTLFDPNNAEPPIVPMDRFNGQFRFVYDTPEDMAVPNQAPIGSFSYTCSMPFAINLEVSKYSSSSLHVLFNVSCPVDSHTTKNFLIFAREQSDDSDYLHIAFQDLVLAEDKPVIESQWPKDAPADEVSVVADKVSIQYRKWLRELKEAHKEGSQAFRSALLDPVIESDRSYI
8XYX , Knot 160 378 0.83 40 207 363 
GPEFKLMSKAVNKKGLRPRKSDSILDHIKNKLDQEFLEDNENGEQSDEDYDQKSLNKAKKPYKKRQTQNGSELVISQQKTKAKASANNKKSAKNSQKLDEEEKIVEEEDLSPQKNGAVSEDDQQQEASTQEDDYLDRLPKSKKGLQGLLQDIEKRILHYKQLFFKEQNEIANGKRSMVPDNSIPICSDVTKLNFQALIDAQMRHAGKMFDVIMMAPPWQLSSSQPSRGVAIAYDSLSDEKIQNMPIQSLQQDGFIFVWAINAKYRVTIKMIENWGYKLVDEITWVKKTVNGKIAKGHGFYLQHAKESCLIGVKGDVDNGRFKKNIASDVIFSERRGQSQKPEEIYQYINQLCPNGNYLEIFARRNNLHDNWVSIGNEL
4LGB , Knot 81 177 0.79 40 125 170 
GSEQKTLEPVIKTYHQFEPDPTTCTSLITQRIHAPASVVWPLIRRFDNPERYKHFVKRCRLISGDGDVGSVREVTVISGLPASTSTERLEFVDDDHRVLSFRVVGGEHRLKNYKSVTSVNEFLNQDSGKVYTVVLESYTVDIPEGNTEEDTKMFVDTVVKLNLQKLGVAATSAPMHD

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 \(c\).

\(|P_{f(6ICP_1)}(2) \setminus P_{f(8XYX_1)}(2)|=98\), \(|P_{f(8XYX_1)}(2) \setminus P_{f(6ICP_1)}(2)|=82\). 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 1:110000000000101010100111000000100110110010010010100111111011000111101100011100001000101011010100100100110000011001111010011100101001000000111110100010000110101100101011100101101010001001001001111011011010010111111001010101100010011110011110100000111110101000000010111010011000000011111000000000101110011110001110001100111001011100101000011001001000100110011101110000001
Pair \(Z_2\) Length of longest common subsequence
6ICP_1,8XYX_1 180 4
6ICP_1,4LGB_1 190 3
8XYX_1,4LGB_1 188 3

Newick tree

 
[
	4LGB_1:95.95,
	[
		6ICP_1:90,8XYX_1:90
	]:5.95
]

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 \((0.85)(0.8)(\frac{747 }{\log_{20} 747}-\frac{369}{\log_{20}369})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 6ICP_1 8XYX_1 132 131.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]