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Parikh vectors
7RQU_1 2JCP_1 3DWO_1 Letter Amino acid
49 6 23 F Phenylalanine
18 6 0 C Cysteine
48 9 53 G Glycine
20 3 11 M Methionine
35 8 19 Y Tyrosine
48 10 34 D Aspartic acid
52 8 7 E Glutamic acid
22 5 14 H Histidine
29 5 15 Q Glutamine
40 10 17 I Isoleucine
41 7 25 P Proline
49 15 33 T Threonine
9 0 10 W Tryptophan
50 11 45 A Alanine
47 8 22 R Arginine
39 10 23 N Asparagine
56 10 22 V Valine
97 7 32 L Leucine
51 5 15 K Lycine
68 11 31 S Serine 

7RQU_1|Chains A, B, C, D|Transient receptor potential cation channel subfamily V member 1|Rattus norvegicus (10116)
>2JCP_1|Chain A|CD44 ANTIGEN|MUS MUSCULUS (10090)
>3DWO_1|Chain A[auth X]|Probable outer membrane protein|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
7RQU , Knot 325 868 0.84 40 308 787 
MEQRASLDSEESESPPQENSCLDPPDRDPNCKPPPVKPHIFTTRSRTRLFGKGDSEEASPLDCPYEEGGLASCPIITVSSVLTIQRPGDGPASVRPSSQDSVSAGEKPPRLYDRRSIFDAVAQSNCQELESLLPFLQRSKKRLTDSEFKDPETGKTCLLKAMLNLHNGQNDTIALLLDVARKTDSLKQFVNASYTDSYYKGQTALHIAIERRNMTLVTLLVENGADVQAAANGDFFKKTKGRPGFYFGELPLSLAACTNQLAIVKFLLQNSWQPADISARDSVGNTVLHALVEVADNTVDNTKFVTSMYNEILILGAKLHPTLKLEEITNRKGLTPLALAASSGKIGVLAYILQREIHEPECRHLSRKFTEWAYGPVHSSLYDLSCIDTCEKNSVLEVIAYSSSETPNRHDMLLVEPLNRLLQDKWDRFVKRIFYFNFFVYCLYMIIFTAAAYYRPVEGLPPYKLKNTVGDYFRVTGEILSVSGGVYFFFRGIQYFLQRRPSLKSLFVDSYSEILFFVQSLFMLVSVVLYFSQRKEYVASMVFSLAMGWTNMLYYTRGFQQMGIYAVMIEKMILRDLCRFMFVYLVFLFGFSTAVVTLIEDGKNNSLPMESTPHKCRGSACKPGNSYNSLYSTCLELFKFTIGMGDLEFTENYDFKAVFIILLLAYVILTYILLLNMLIALMGETVNKIAQESKNIWKLQRAITILDTEKSFLKCMRKAFRSGKLLQVGFTPDGKDDYRWCFRVDEVNWTTWNTNVGIINEDPGNCEGVKRTLSFSLRSGRVSGRNWKNFALVPLLRDASTRDRHATQQEEVQLKHYTGSLKPEDAEVFKDSMVPGEKENSLEVLFQGPDYKDDDDKAHHHHHHHHHH
2JCP , Knot 77 154 0.84 38 123 149 
MQQIDLNVTCRYAGVFHVEKNGRYSISRTEAADLCQAFNSTLPTMDQMKLALSKGFETCRYGFIEGNVVIPRIHPNAICAANHTGVYILVTSNTSHYDTYCFNASAPPEEDCTSVTDLPNSFDGPVTITIVNRDGTRYSKKGEYRTHQEDIDAS
3DWO , Knot 182 451 0.82 38 222 410 
AGFMVPTTNTAGWGRAMAGGSLFPNDPSAAFNNPAAMAFIDKRIAQLTVNYADIDIKYNGDAYDYQGNPMTGGYQDGPGTPELGTNDGGQAGFGAWLPTGFLVVPINDRFAFGLSQVVPMGMRSTWDPNWKGRDFAVDTKIETIGLTGSLSFKVNDNFSLGAGVIIQRTSGFVSQNLDLYASAANSPGMGGIPFPASNSSALMRVKVDNTSPGFFAGAVWKPTDRDTLGFAYHAKIRNKLKGHYNLYDHDGGLTEGAIEGGTPGLAYPGLDLRMGASASARLDIPAYASLDWVHQFNDRLSLGASATWTEWSSFQDLTLKSHGNTIVSIPYTYRNTWTLAVGGDYKVTDQWTMRAGVAYDQTPTHNATRDPRIPDGDRYFASLGAGYRFQSMPELSIDAAYSRQFVKEVPLKTVNQDRLGGGRLDGRATSKGQVFSLSATYDFHHHHHHHH

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 \(c\).

\(|P_{f(7RQU_1)}(2) \setminus P_{f(2JCP_1)}(2)|=205\), \(|P_{f(2JCP_1)}(2) \setminus P_{f(7RQU_1)}(2)|=20\). 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 1:1000101000000011000001011000100011110101100000001110100001011001000111100111010011010011011101010000010110011010000011011100000010011111000000100001001001000110111010010000111110110000010011010000000010011011100001011011100110101110101100001011101101110111000011110111000101101010001100110111011000100001100100011111101010101001000011011111100101111101100010010000100010011011100010010010000000110111000000100001111011001100010011001101011100101111011100011011110010001100101010110101110111011001100010100111000001111100111110111010000001101110111110011000011001110111100111001001111011111110011101100100001110001000010100110000010000101101011110101000001011111111101110011110111111100100110000011010011011000001100100110010110111010100000101010010100100011110001100011000101010010101001001111111001000000100000101000010101001011000111100000101110110000000010000000000
Pair \(Z_2\) Length of longest common subsequence
7RQU_1,2JCP_1 225 3
7RQU_1,3DWO_1 154 8
2JCP_1,3DWO_1 191 3

Newick tree

 
[
	2JCP_1:11.98,
	[
		7RQU_1:77,3DWO_1:77
	]:34.98
]

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 \((0.85)(0.8)(\frac{1022 }{\log_{20} 1022}-\frac{154}{\log_{20}154})=238.\)
Status Protein1 Protein2 d d1/2
Query variables 7RQU_1 2JCP_1 302 176.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]