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Parikh vectors
9KJT_1 4JIS_1 4HFU_1 Letter Amino acid
60 11 10 A Alanine
51 14 28 G Glycine
15 10 9 H Histidine
54 14 24 S Serine
49 10 29 T Threonine
1 1 7 W Tryptophan
37 12 14 R Arginine
41 19 12 D Aspartic acid
35 11 7 Q Glutamine
49 16 24 E Glutamic acid
53 30 18 I Isoleucine
18 6 6 M Methionine
7 2 9 C Cysteine
48 18 22 K Lycine
36 8 19 P Proline
27 12 22 N Asparagine
59 19 30 L Leucine
24 9 8 F Phenylalanine
20 6 11 Y Tyrosine
67 19 18 V Valine 

9KJT_1|Chains A, B, C|Polyribonucleotide nucleotidyltransferase 1, mitochondrial|Homo sapiens (9606)
>4JIS_1|Chains A, B|ribitol-5-phosphate cytidylyltransferase|Bacillus subtilis subsp. spizizenii (655816)
>4HFU_1|Chain A|Hemagglutinin HA1|Influenza A virus (382813)
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 \(w=f(c)\)
9KJT , Knot 286 751 0.84 40 279 672 
MGAVAVDLGNRKLEISSGKLARFADGSAVVQSGDTAVMVTAVSKTKPSPSQFMPLVVDYRQKAAAAGRIPTNYLRREVGTSDKEILTSRIIDRSIRPLFPAGYFYDTQVLCNLLAVDGVNEPDVLAINGASVALSLSDIPWNGPVGAVRIGIIDGEYVVNPTRKEMSSSTLNLVVAGAPKSQIVMLEASAENILQQDFCHAIKVGVKYTQQIIQGIQQLVKETGVTKRTPQKLFTPSPEIVKYTHKLAMERLYAVFTDYEHDKVSRDEAVNKIRLDTEEQLKEKFPEADPYEIIESFNVVAKEVFRSIVLNEYKRCDGRDLTSLRNVSCEVDMFKTLHGSALFQRGQTQVLCTVTFDSLESGIKSDQVITAINGIKDKNFMLHYEFPPYATNEIGKVTGLNRRELGHGALAEKALYPVIPRDFPFTIRVTSEVLESNGSSSMASACGGSLALMDSGVPISSAVAGVAIGLVTKTDPEKGEIEDYRLLTDILGIEDYNGDMDFKIAGTNKGITALQADIKLPGIPIKIVMEAIQQASVAKKEILQIMNKTISKPRASRKENGPVVETVQVPLSKRAKFVGPGGYNLKKLQAETGVTISQVDEETFSVFAPTPSAMHEARDFITEICKDDQEQQLEFGAVYTATITEIRDTGVMVKLYPNMTAVLLHNTQLDQRKIKHPTALGLEVGQEIQVKYFGRDPADGRMRLSRKVLQSPATTVVRTLNDRSSIVMGEPISQSSSNSQAAALEHHHHHH
4JIS , Knot 111 247 0.82 40 170 238 
MGMIYAEILAGGKGSRMGNVNMPKQFLPLNKRPIIIHTVEKFLLNDRFDKILIVSPKEWINHTKDILKKFIGQDDRLVVVEGGSDRNESIMSGIRYIEKEFGIQDNDVIITHDSVRPFLTHRIIDENIDAVLQYGAVDTVISAIDTIIASEDQEFISDIPVRDNMYQGQTPQSFRISKLVELYNKLSDEQKAVLTDACKICSLAGEKVKLVRGEVFNIKVTTPYDLKVANAILQERISQLEHHHHHH
4HFU , Knot 144 327 0.85 40 201 316 
PGDQICIGYHANNSTEKVDTILERNVTVTHAKDILEKTHNGKLCKLNGIPPLELGDCSIAGWLLGNPECDRLLSVPEWSYIMEKENPRDGLCYPGSFNDYEELKYLLSSVKHFEKVKILPKDRWTQHTTTGGSRACAVSGNPSFFRNMVWLTKKGSDYPVAKGSYNNTSGEQMLIIWGVHHPNDETEQRTLYQNVGTYVSVGTSTLNKRSTPEIATRPKVNGLGSRMEFSWTLLDMWDTINFESTGNLIAPEYGFKISKRGSSGIMKTEGTLENCETKCQTPLGAINTTLPFHNVHPLTIGECPRYVKSEKLVLATGLRNVPQIESR

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 \(c\).

\(|P_{f(9KJT_1)}(2) \setminus P_{f(4JIS_1)}(2)|=133\), \(|P_{f(4JIS_1)}(2) \setminus P_{f(9KJT_1)}(2)|=24\). 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 1:1111110110001010010110110101110010011110110000101001111110000011111011000100011000001100011000101111110100001100111101100101111011011101001110111111011110100110100001000010111111100011110101001100010011011100000110110011000110000100110101011000001110010111000000010000110010100000100011010100110010111001100111000000010010010010001011001010111001000110010100100110000110110110000111000111010001101011000011011110011011110011101010001100010001101011011110011110011111111110000100101000011001111000010101011100011011010101111110111011001011000110110001001010000011110010111000101111110010010100110100100001011110101100100110010000000010111100101001000111101010101111000010000100101111011001010011001101010100011001100110010000011110110000000011110000000
Pair \(Z_2\) Length of longest common subsequence
9KJT_1,4JIS_1 157 8
9KJT_1,4HFU_1 166 4
4JIS_1,4HFU_1 163 4

Newick tree

 
[
	4HFU_1:83.46,
	[
		9KJT_1:78.5,4JIS_1:78.5
	]:4.96
]

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 \((0.85)(0.8)(\frac{998 }{\log_{20} 998}-\frac{247}{\log_{20}247})=203.\)
Status Protein1 Protein2 d d1/2
Query variables 9KJT_1 4JIS_1 253 166.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]