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Parikh vectors
6ZOB_1 8JZE_1 2XNY_1 Letter Amino acid
45 10 1 P Proline
27 5 1 Y Tyrosine
97 9 9 V Valine
105 21 4 A Alanine
17 3 1 H Histidine
103 11 11 L Leucine
43 18 9 K Lycine
42 5 1 M Methionine
54 6 0 F Phenylalanine
67 7 0 T Threonine
9 8 0 W Tryptophan
48 5 8 Q Glutamine
51 11 7 E Glutamic acid
2 3 2 C Cysteine
72 13 7 I Isoleucine
38 7 10 R Arginine
47 10 8 D Aspartic acid
62 7 6 S Serine
39 10 1 N Asparagine
89 15 1 G Glycine 

6ZOB_1|Chains A, B, C|Multidrug efflux pump subunit AcrB|Escherichia coli K-12 (83333)
>8JZE_10|Chain J[auth f]|Photosystem I PsaF|Symbiodinium (2949)
>2XNY_1|Chains A, D|FIBRINOGEN ALPHA CHAIN|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 \(w=f(c)\)
6ZOB , Knot 377 1057 0.82 40 302 941 
MPNFFIDRPIFAWVIAIIIMLAGGLAILKLPVAQYPTIAPPAVTISASYPGADAKTVQDTVTQVIEQNMNGIDNLMYMSSNSDSTGTVQITLTFESGTDADIAQVQVQNKLQLAMPLLPQEVQQQGVSVEKSSSSFLMVVGVINTDGTMTQEDISDYVAANMKDAISRTSGVGDVQLFGSQYAMRIWMNPNELNKFQLTPVDVITAIKAQNAQVAAGQLGGTPPVKGQQLNASIIAQTRLTSTEEFGKILLKVNQDGSRVLLRDVAKIELGGENYDIIAEFNGQPASGLGIKLATGANALDTAAAIRAELAKMEPFFPSGLKIVYPYDTTPFVKISIHEVVKTLVEAIILVFLVMYLFLQNFRATLIPTIAVPVVLLGTFAVLAAFGFSINTLTMFGMVLAIGLLVDDAIVVVENVERVMAEEGLPPKEATRKSMGQIQGALVGIAMVLSAVFVPMAFFGGSTGAIYRQFSITIVSAMALSVLVALILTPALCATMLKPIAKGDHGEGKKGFFGWFNRMFEKSTHHYTDSVGGILRSTGRYLVLYLIIVVGMAYLFVRLPSSFLPDEDQGVFMTMVQLPAGATQERTQKVLNEVTHYYLTKEKNNVESVFAVNGFGFAGRGQNTGIAFVSLKDWADRPGEENKVEAITMRATRAFSQIKDAMVFAFNLPAIVELGTATGFDFELIDQAGLGHEKLTQARNQLLAEAAKHPDMLTSVRPNGLEDTPQFKIDIDQEKAQALGVSINDINTTLGAAWGGSYVNDFIDRGRVKKVYVMSEAKYRMLPDDIGDWYVRAADGQMVPFSAFSSSRWEYGSPRLERYNGLPSMEILGQAAPGKSTGEAMELMEQLASKLPTGVGYDWTGMSYQERLSGNQAPSLYAISLIVVFLCLAALYESWSIPFSVMLVVPLGVIGALLAATFRGLTNDVYFQVGLLTTIGLSAKNAILIVEFAKDLMDKEGKGLIEATLDAVRMRLRPILMTSLAFILGVMPLVISTGAGSGAQNAVGTGVMGGMVTATVLAIFFVPVFFVVVRRRFSRKNEDIEHSHTVDHHLEHHHHHH
8JZE , Knot 90 184 0.85 40 141 180 
AKPLEMFGGVDIDLEDTPAHWTIKASPVLEPCKDNKKFHKKLKDELYKVKKNQQKFAEGSAAYARFNKKIKMIELRETAYGDRLCGKKDGLPRTIATGEWNVRGSAMWPAAIFLYIAGWIGWAGRSYLIRTNDAAKEINIDVPLAVTCMASGFSWPVAAWQEIVNGEMAVPNDQIHNGGPWNQS
2XNY , Knot 43 87 0.73 34 62 84 
VSEDLRSRIEVLKRKVIEKVQHIQLLQKNVRAQLVDMKRLEVDIDIKIRSCRGSCSRALAREVDLKDYEDQQKQLEQVIAKDLLPSR

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 \(c\).

\(|P_{f(6ZOB_1)}(2) \setminus P_{f(8JZE_1)}(2)|=187\), \(|P_{f(8JZE_1)}(2) \setminus P_{f(6ZOB_1)}(2)|=26\). 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 1:1101110011111111111111111111011110010111111010100111010010001001100010110011010000000101010101001001011010100010111111100100011010000001111111100010100001000111010011000011101011100011011101001001010110110110100101111011101110100101011100010000011011101000100111001101011100001110101011011110110110110011110101101011110110110100001110101001100110111111111011100101011101111111110111111111010010111111111111001111100100111001111001000011010111111111101111111111100111000101011011110111111101110101101110100101001111110011000000000011111000100111011111111011101100111000011110110111110000000110010000100000010011110111111010001111101001100110000101101010011001001111110111110110101101011001111000100100011101100101100101011000101010100001011110100100011111110010011001010010110010001110011010101101011110110000100101010000111010111011110001011011001100110111001011000001010011010110111111011110001011101111111111111111010110001010111100111010011111011001100010111010101101010111100111111111111001110110011101111111010111111111111110001000000100000100010000000
Pair \(Z_2\) Length of longest common subsequence
6ZOB_1,8JZE_1 213 5
6ZOB_1,2XNY_1 256 4
8JZE_1,2XNY_1 141 4

Newick tree

 
[
	6ZOB_1:12.72,
	[
		8JZE_1:70.5,2XNY_1:70.5
	]:59.22
]

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 \((0.85)(0.8)(\frac{1241 }{\log_{20} 1241}-\frac{184}{\log_{20}184})=283.\)
Status Protein1 Protein2 d d1/2
Query variables 6ZOB_1 8JZE_1 356 208
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]