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Parikh vectors
5PQX_1 5IVW_1 9IYX_1 Letter Amino acid
5 39 14 I Isoleucine
5 42 19 T Threonine
5 30 6 Y Tyrosine
14 48 6 R Arginine
1 14 14 C Cysteine
7 40 15 G Glycine
10 18 1 H Histidine
6 33 16 F Phenylalanine
11 54 5 A Alanine
13 63 2 E Glutamic acid
6 61 2 K Lycine
7 22 7 M Methionine
7 57 25 S Serine
0 8 13 W Tryptophan
8 37 7 Q Glutamine
18 60 35 L Leucine
6 32 21 P Proline
10 54 10 V Valine
4 22 5 N Asparagine
13 48 3 D Aspartic acid 

5PQX_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>5IVW_1|Chain A[auth V]|TFIIH basal transcription factor complex helicase XPB subunit|Homo sapiens (9606)
>9IYX_1|Chains A, B|Large envelope protein|HBV genotype D3 (1219353)
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 \(w=f(c)\)
5PQX , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
5IVW , Knot 302 782 0.85 40 313 716 
MGKRDRADRDKKKSRKRHYEDEEDDEEDAPGNDPQEAVPSAAGKQVDESGTKVDEYGAKDYRLQMPLKDDHTSRPLWVAPDGHIFLEAFSPVYKYAQDFLVAIAEPVCRPTHVHEYKLTAYSLYAAVSVGLQTSDITEYLRKLSKTGVPDGIMQFIKLCTVSYGKVKLVLKHNRYFVESCHPDVIQHLLQDPVIRECRLRNSEGEATELITETFTSKSAISKTAESSGGPSTSRVTDPQGKSDIPMDLFDFYEQMDKDEEEEEETQTVSFEVKQEMIEELQKRCIHLEYPLLAEYDFRNDSVNPDINIDLKPTAVLRPYQEKSLRKMFGNGRARSGVIVLPCGAGKSLVGVTAACTVRKRCLVLGNSAVSVEQWKAQFKMWSTIDDSQICRFTSDAKDKPIGCSVAISTYSMLGHTTKRSWEAERVMEWLKTQEWGLMILDEVHTIPAKMFRRVLTIVQAHCKLGLTATLVREDDKIVDLNFLIGPKLYEANWMELQNNGYIAKVQCAEVWCPMSPEFYREYVAIKTKKRILLYTMNPNKFRACQFLIKFHERRNDKIIVFADNVFALKEYAIRLNKPYIYGPTSQGERMQILQNFKHNPKINTIFISKVGDTSFDLPEANVLIQISSHGGSRRQEAQRLGRVLRAKKGMVAEEYNAFFYSLVSQDTQEMAYSTKRQRFLVDQGYSFKVITKLAGMEEEDLAFSTKEEQQQLLQKVLAATDLDAEEEVVAGEFGSRSSQASRRFGTMSSMSGADDTVYMEYHSSRSKAPSKHVHPLFKRFRK
9IYX , Knot 102 226 0.81 40 138 212 
MENITSGFLGPLLVLQAGFFLLTRILTIPQSLDSWWTSLNFLGGTTVCLGQNSQSPTSNHSPTSCPPTCPGYRWMCLRRFIIFLFILLLCLIFLLVLLDYQGMLPVCPLIPGSSTTSTGPCRTCMTTAQGTSMYPSCCCTKPSDGNCTCIPIPSSWAFGKFLWEWASARFSWLSLLVPFVQWFVGLSPTVWLSVIWMMWYWGPSLYSILSPFLPLLPIFFCLWVYI

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 \(c\).

\(|P_{f(5PQX_1)}(2) \setminus P_{f(5IVW_1)}(2)|=16\), \(|P_{f(5IVW_1)}(2) \setminus P_{f(5PQX_1)}(2)|=209\). 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 1:100000000110110001010010011101010010011001100100001101110110100110010010011011010001010100010010001011100010001000110011101000111110010001001110010110110011
Pair \(Z_2\) Length of longest common subsequence
5PQX_1,5IVW_1 225 3
5PQX_1,9IYX_1 172 3
5IVW_1,9IYX_1 231 4

Newick tree

 
[
	5IVW_1:12.92,
	[
		5PQX_1:86,9IYX_1:86
	]:35.92
]

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 \((0.85)(0.8)(\frac{938 }{\log_{20} 938}-\frac{156}{\log_{20}156})=216.\)
Status Protein1 Protein2 d d1/2
Query variables 5PQX_1 5IVW_1 278 164.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]