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Parikh vectors
8JIK_1 5ZZG_1 4XJU_1 Letter Amino acid
22 4 21 R Arginine
11 0 1 C Cysteine
9 5 25 Q Glutamine
10 2 9 M Methionine
6 2 8 W Tryptophan
15 4 35 Y Tyrosine
24 17 31 A Alanine
26 8 41 D Aspartic acid
29 9 46 I Isoleucine
22 5 29 F Phenylalanine
21 6 73 S Serine
23 1 45 N Asparagine
24 19 50 K Lycine
17 4 20 P Proline
21 5 42 T Threonine
34 11 52 G Glycine
14 11 10 H Histidine
39 18 51 L Leucine
24 8 35 V Valine
30 14 34 E Glutamic acid 

8JIK_1|Chain A|Serine decarboxylase|Camellia sinensis (4442)
>5ZZG_1|Chain A|Myoglobin|Physeter catodon (9755)
>4XJU_1|Chain A|Sialidase B|Streptococcus pneumoniae serotype 4 (strain ATCC BAA-334 / TIGR4) (170187)
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 \(w=f(c)\)
8JIK , Knot 175 421 0.83 40 243 400 
TTSLTITEPDVDDDSTSDMEAFMAGVLVRYRKTLIEKTKYHLGYPFNLDLDYGPLAELQHFAINNLGDPFIESNYGVHSRQFEVGVLDWFARLWEIEQKEYWGYITNGGTEGNLHGILIGREVFPDGIFYTSRESHYSIFKAARMYRMECVKVGTLINGEIDCADFKAKLLSNKDKPAIINLNIGTTVKGAVDDIDLVIQTLEECGFSHDRFYIHCDGALFGFMMPFLNRGPKITFKKPIGSVSVSGHKFMGCPTPCGVQITRLEHINALSRNVEYLASRDATITGSRNGHSPIILWYALNRKGFKGFQKEVQKCLRNAHYLKDRLREAGISAMLNELSSTVVFERPLDEEFVRRWQLACEGNMAHVIVMPNVTIEKLDEFLNELVQKRANWYNDGKAGPPCLAPDIGSENCDCDLHKLEH
5ZZG , Knot 73 153 0.80 38 111 148 
VLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKYDRFKHLKTEAEMKASEDLKKDGVTVLTALGAILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISEAIIHVLHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELGYQG
4XJU , Knot 256 658 0.84 40 261 595 
ISPIFQGGSYQLNNKSIDISSLLLDKLSGESQTVVMKFKADKPNSLQALFGLSNSKAGFKNNYFSIFMRDSGEIGVEIRDAQKGINYLFSRPASLWGKHKGQAVENTLVFVSDSKDKTYTMYVNGIEVFSETVDTFLPISNINGIDKATLGAVNREGKEHYLAKGSIDEISLFNKAISDQEVSTIPLSNPFQLIFQSGDSTQANYFRIPTLYTLSSGRVLSSIDARYGGTHDSKSKINIATSYSDDNGKTWSEPIFAMKFNDYEEQLVYWPRDNKLKNSQISGSASFIDSSIVEDKKSGKTILLADVMPAGIGNNNANKADSGFKEINGHYYLKLKKNGDNDFRYTVRENGVVYNETTNKPTNYTINDKYEVLEGGKSLTVEQYSVDFDSGSLRERHNGKQVPMNVFYKDSLFKVTPTNYIAMTTSQNRGESWEQFKLLPPFLGEKHNGTYLCPGQGLALKSSNRLIFATYTSGELTYLISDDSGQTWKKSSASIPFKNATAEAQMVELRDGVIRTFFRTTTGKIAYMTSRDSGETWSKVSYIDGIQQTSYGTQVSAIKYSQLIDGKEAVILSTPNSRSGRKGGQLVVGLVNKEDDSIDWKYHYGIDLPSYGYAYSAITELPNHHIGVLFEKYDSWSRNELHLSNVVQYIDLEINDLT

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 \(c\).

\(|P_{f(8JIK_1)}(2) \setminus P_{f(5ZZG_1)}(2)|=164\), \(|P_{f(5ZZG_1)}(2) \setminus P_{f(8JIK_1)}(2)|=32\). 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 1:0001010010100000001011111111000001100000011011010100111101001110011011100001100001011110111011010000011010011001010111110011101110000000001101101001001011011010100101010110000011110101100101110010111001000110000101000111111111110011010100111010101001110101011010010010110001001100010101000100111110110001101100010001001001000100111011100100011100110001100101100101101111101010010011001100010100010111101110110000000100100
Pair \(Z_2\) Length of longest common subsequence
8JIK_1,5ZZG_1 196 3
8JIK_1,4XJU_1 136 4
5ZZG_1,4XJU_1 194 4

Newick tree

 
[
	5ZZG_1:10.51,
	[
		8JIK_1:68,4XJU_1:68
	]:37.51
]

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 \((0.85)(0.8)(\frac{574 }{\log_{20} 574}-\frac{153}{\log_{20}153})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 8JIK_1 5ZZG_1 158 107
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]