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Parikh vectors
2GBI_1 7BZU_1 6IKT_1 Letter Amino acid
33 12 5 Q Glutamine
40 12 14 E Glutamic acid
21 5 5 H Histidine
56 15 16 L Leucine
20 4 1 W Tryptophan
50 12 7 Y Tyrosine
30 20 14 R Arginine
44 13 10 D Aspartic acid
13 3 1 C Cysteine
44 10 10 I Isoleucine
13 11 6 M Methionine
41 25 18 A Alanine
33 18 4 N Asparagine
31 14 8 F Phenylalanine
28 20 11 P Proline
47 32 4 T Threonine
38 18 14 G Glycine
41 10 1 K Lycine
65 22 8 S Serine
42 22 12 V Valine 

2GBI_1|Chains A, B|Dipeptidyl peptidase 4|Rattus norvegicus (10116)
>7BZU_1|Chain A|Capsid protein VP1|Coxsackievirus A10 (42769)
>6IKT_1|Chain A|Peptide deformylase|Xanthomonas oryzae pv. oryzae (291331)
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 \(w=f(c)\)
2GBI , Knot 287 730 0.86 40 316 687 
RRTYTLADYLKNTFRVKSYSLRWVSDSEYLYKQENNILLFNAEHGNSSIFLENSTFEIFGDSISDYSVSPDRLFVLLEYNYVKQWRHSYTASYSIYDLNKRQLITEEKIPNNTQWITWSQEGHKLAYVWKNDIYVKIEPHLPSHRITSTGKENVIFNGINDWVYEEEIFGAYSALWWSPNGTFLAYAQFNDTGVPLIEYSFYSDESLQYPKTVWIPYPKAGAVNPTVKFFIVNTDSLSSTTTTIPMQITAPASVTTGDHYLCDVAWVSEDRISLQWLRRIQNYSVMAICDYDKTTLVWNCPTTQEHIETSATGWCGRFRPAEPHFTSDGSSFYKIVSDKDGYKHICQFQKDRKPEQVCTFITKGAWEVISIEALTSDYLYYISNEYKEMPGGRNLYKIQLTDHTNKKCLSCDLNPERCQYYSVSLSKEAKYYQLGCRGPGLPLYTLHRSTDQKELRVLEDNSALDKMLQDVQMPSKKLDFIVLNETRFWYQMILPPHFDKSKKYPLLIDVYAGPCSQKADAAFRLNWATYLASTENIIVASFDGRGSGYQGDKIMHAINKRLGTLEVEDQIEAARQFLKMGFVDSKRVAIWGWSYGGYVTSMVLGSGSGVFKCGIAVAPVSRWEYYDSVYTERYMGLPTPEDNLDHYRNSTVMSRAENFKQVEYLLIHGTADDNVHFQQSAQISKALVDAGVDFQAMWYTDEDHGIASSTAHQHIYSHMSHFLQQCFSLR
7BZU , Knot 129 298 0.82 40 190 288 
GDPVEDIIHDALGSTARRAISSVTNAESAANTTPSSHRLETGRVPALQAAETGATSNATDENMIETRCVVNRNGVLETTINHFFSRSGLVGVVNLTDGGTDTTGYATWDIDIMGFVQLRRKCEMFTYMRFNAEFTFVTTTKNGEARPYMLQYMYVPPGAPKPTGRDAFQWQTATNPSVFVKLTDPPAQVSVPFMSPASAYQWFYDGYPTFGQHPETSNTTYGLCPNNMMGTFAVRVVSREASQLKLQTRVYMKLKHVRAWVPRPIRSQPYLLKNFPNYDSSKITNSARDRSSIKQANM
6IKT , Knot 81 169 0.82 40 123 164 
MIRDIIRMGDKRLLRVAPQVTNLGSAELHALVSDMFETMGAAHGVGLAAPQIAVDLQLMVFGFEASERYPEAPAVPLTALANAQIEPLSDEMENGWEGCLSIPGLRAVIPRYRYIRYRGFAPDGSPIEREAEGFHARVVQHEYDHLVGRLYPSRIENFDTFGFDDVLSY

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 \(c\).

\(|P_{f(2GBI_1)}(2) \setminus P_{f(7BZU_1)}(2)|=162\), \(|P_{f(7BZU_1)}(2) \setminus P_{f(2GBI_1)}(2)|=36\). 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 1:0000011001000101000010110000010000001111010010001110000101110010000101001111100001001000001000100100001100001100001101000100110110001010101011000100010001110110011000011110011110101011101010001111100010000010010011110101111010101111000010000001110101110100100010011110000101011001000011110000000111001000001000101101010110101000100100110000100010010000010010011001110110101100001001000000111100100101000000001000101000000010100010000110011111100100000000101100001100110010110001011110000110011111010000001111010111000010111010110011000011110101010100100110110001101010001011001101111000011111100110100111101011100111111100100000100000111101000100000001100100100100111010100010100010100111011101011100000011100010001000100110001010
Pair \(Z_2\) Length of longest common subsequence
2GBI_1,7BZU_1 198 4
2GBI_1,6IKT_1 229 3
7BZU_1,6IKT_1 167 3

Newick tree

 
[
	2GBI_1:11.79,
	[
		7BZU_1:83.5,6IKT_1:83.5
	]:30.29
]

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 \((0.85)(0.8)(\frac{1028 }{\log_{20} 1028}-\frac{298}{\log_{20}298})=195.\)
Status Protein1 Protein2 d d1/2
Query variables 2GBI_1 7BZU_1 252 176
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]