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Parikh vectors
1REA_1 1UGW_1 2OBQ_1 Letter Amino acid
31 9 15 L Leucine
9 2 5 M Methionine
10 10 3 F Phenylalanine
10 6 12 P Proline
7 11 5 Y Tyrosine
2 1 9 H Histidine
27 9 10 I Isoleucine
20 12 16 S Serine
17 10 19 T Threonine
15 8 4 N Asparagine
13 2 10 Q Glutamine
3 0 7 C Cysteine
30 5 4 E Glutamic acid
27 7 4 K Lycine
2 1 2 W Tryptophan
14 2 13 R Arginine
20 6 7 D Aspartic acid
22 12 16 V Valine
38 3 16 A Alanine
35 17 23 G Glycine 

1REA_1|Chain A|REC A|Escherichia coli (562)
>1UGW_1|Chains A, G|Agglutinin alpha chain|Artocarpus integer (3490)
>2OBQ_1|Chains A, C|Hepatitis C virus|Hepatitis C virus (3052230)
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 \(w=f(c)\)
1REA , Knot 143 352 0.79 40 179 330 
AIDENKQKALAAALGQIEKQFGKGSIMRLGEDRSMDVETISTGSLSLDIALGAGGLPMGRIVEIYGPESSGKTTLTLQVIAAAQREGKTCAFIDAEHALDPIYARKLGVDIDNLLCSQPDTGEQALEICDALARSGAVDVIVVDSVAALTPKAEIEGEIGDSHMGLAARMMSQAMRKLAGNLKQSNTLLIFINQIRMKIGVMFGNPETTTGGNALKFYASVRLDIRRIGAVKEGENVVGSETRVKVVKNKIAAPFKQAEFQILYGEGINFYGELVDLGVKEKLIEKAGAWYSYKGEKIGQGKANATAWLKDNPETAKEIEKKVRELLLSNPNSTPDFSVDDSEGVAETNEDF
1UGW , Knot 65 133 0.79 38 102 129 
GKAFDDGAFTGIREINLSYNKETAIGDFQVVYDLNGSPYVGQNHVSFITGFTPVKISLDFPSEYIMEVSGYTGNVSGYVVVRSLTFKTNKKTYGPYGVTSGTPFNLPIENGLIVGFKGSIGYWLDYFSMYLSL
2OBQ , Knot 93 200 0.82 40 139 191 
MASMTGGQQMGAPITAYAQQTRGLLGCIITSLTGRDKNQVEGEVQIVSTATQTFLATCINGVCWTVYHGAGTRTIASPKGPVIQMYTNVDQDLVGWPAPQGSRSLTPCTCGSSDLYLVTRHADVIPVRRRGDSRGSLLSPRPISYLKGSSGGPLLCPAGHAVGLFRAAVCTRGVAKAVDFIPVENLETTMRSGSHHHHHH

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 \(c\).

\(|P_{f(1REA_1)}(2) \setminus P_{f(1UGW_1)}(2)|=115\), \(|P_{f(1UGW_1)}(2) \setminus P_{f(1REA_1)}(2)|=38\). 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 1:1100000011111110100011010110110000101001001010101111111111101101011000100010101111100010001110100110110100111010011000100100110100111001110111100111101010101011000111110110011001110100000111110010101111110100001101101010101010011110010011100001011000111110010101101011010101101110001100111100001001101010101110001001001000100111001000101010000111000001
Pair \(Z_2\) Length of longest common subsequence
1REA_1,1UGW_1 153 4
1REA_1,2OBQ_1 162 3
1UGW_1,2OBQ_1 145 4

Newick tree

 
[
	1REA_1:80.76,
	[
		1UGW_1:72.5,2OBQ_1:72.5
	]:8.26
]

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 \((0.85)(0.8)(\frac{485 }{\log_{20} 485}-\frac{133}{\log_{20}133})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 1REA_1 1UGW_1 129 88.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]