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Parikh vectors
8IAB_1 7YFC_1 7FPR_1 Letter Amino acid
13 7 2 C Cysteine
39 18 6 F Phenylalanine
12 4 5 W Tryptophan
22 15 4 Y Tyrosine
62 23 11 V Valine
53 29 13 A Alanine
46 18 6 N Asparagine
18 4 5 H Histidine
106 27 11 L Leucine
66 16 10 G Glycine
49 26 12 I Isoleucine
18 7 5 M Methionine
32 8 10 P Proline
45 20 9 S Serine
47 18 6 T Threonine
34 29 9 R Arginine
35 22 12 E Glutamic acid
31 24 6 K Lycine
32 28 13 D Aspartic acid
15 18 4 Q Glutamine 

8IAB_1|Chains A, B|Chloride channel protein CLC-a|Arabidopsis thaliana (3702)
>7YFC_1|Chain A|Engineered G-alpha-q|Homo sapiens (9606)
>7FPR_1|Chain A|Dihydrofolate reductase|Escherichia coli K-12 (83333)
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 \(w=f(c)\)
8IAB , Knot 293 775 0.83 40 296 700 
MDEDGNLQISNSNYNGEEEGEDPENNTLNQPLLKRHRTLSSTPLALVGAKVSHIESLDYEINENDLFKHDWRSRSKAQVFQYIFLKWTLACLVGLFTGLIATLINLAVENIAGYKLLAVGYYIAQDRFWTGLMVFTGANLGLTLVATVLVVYFAPTAAGPGIPEIKAYLNGIDTPNMFGFTTMMVKIVGSIGAVAAGLDLGKEGPLVHIGSCIASLLGQGGPDNHRIKWRWLRYFNNDRDRRDLITCGSASGVCAAFRSPVGGVLFALEEVATWWRSALLWRTFFSTAVVVVVLRAFIEICNSGKCGLFGSGGLIMFDVSHVEVRYHAADIIPVTLIGVFGGILGSLYNHLLHKVLRLYNLINQKGKIHKVLLSLGVSLFTSVCLFGLPFLAECKPCDPSIDEICPTNGRSGNFKQFNCPNGYYNDLSTLLLTTNDDAVRNIFSSNTPNEFGMVSLWIFFGLYCILGLITFGIATPSGLFLPIILMGSAYGRMLGTAMGSYTNIDQGLYAVLGAASLMAGSMRMTVSLCVIFLELTNNLLLLPITMFVLLIAKTVGDSFNLSIYEIILHLKGLPFLEANPEPWMRNLTVGELNDAKPPVVTLNGVEKVANIVDVLRNTTHNAFPVLDGADQNTGTELHGLILRAHLVKVLKKRWFLNEKRRTEEWEVREKFTPVELAEREDNFDDVAITSSEMQLYVDLHPLTNTTPYTVVQSMSVAKALVLFRSVGLRHLLVVPKIQASGMSPVIGILTRQDLRAYNILQAFPHLDKHKSGKAR
7YFC , Knot 158 361 0.86 40 216 345 
MGCTLSAEDKAAVERSKMIEKQLQKDKQVYRRTLRLLLLGADNSGKSTIVKQMRIYHVNGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDSARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTSGIFETKFQVDKVNFHMFDVGAQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNDFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRKEFVDISTASGDGRHICYPHFTCSVDTENARRIFNDCKDIILQMNLREYNLV
7FPR , Knot 81 159 0.86 40 124 153 
MISLIAALAVDRVIGMENAMPWNLPADLAWFKRNTLNKPVIMGRHTWESIGRPLPGRKNIILSSQPGTDDRVTWVKSVDEAIAACGDVPEIMVIGGGRVYEQFLPKAQKLYLTHIDAEVEGDTHFPDYEPDDWESVFSEFHDADAQNSHSYCFEILERR

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 \(c\).

\(|P_{f(8IAB_1)}(2) \setminus P_{f(7YFC_1)}(2)|=122\), \(|P_{f(7YFC_1)}(2) \setminus P_{f(8IAB_1)}(2)|=42\). 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 1:1000101010000001000100100001001110000010001111111010010010001000011000100000101100111010110111110111101101110011100111110011000110111110110111011101111011101111111010101011001011110011101110111111110110011110110011011101110000101011001000000001100101011011100111111111001101100111100110011111110111010001001111011111101001010001101111011111111110100011001101001100010100111011101100101111111100010010100101001001010010010100001001110000011001100001001111011111110011111011110101111111111010101110111000010011011111101111010101010111101000111111011111110011001010100111010111110101011100101101001011110101100110110110000001111101100001001011110101101100011100000000101000101101100000100111000010101010110000100110010110111110011100111110101011011111100001010011011101000001010
Pair \(Z_2\) Length of longest common subsequence
8IAB_1,7YFC_1 164 4
8IAB_1,7FPR_1 214 4
7YFC_1,7FPR_1 196 3

Newick tree

 
[
	7FPR_1:10.60,
	[
		8IAB_1:82,7YFC_1:82
	]:26.60
]

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 \((0.85)(0.8)(\frac{1136 }{\log_{20} 1136}-\frac{361}{\log_{20}361})=204.\)
Status Protein1 Protein2 d d1/2
Query variables 8IAB_1 7YFC_1 255 186
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]