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Parikh vectors
7TUE_1 1RAW_1 8OMO_1 Letter Amino acid
18 14 57 G Glycine
23 0 48 T Threonine
26 0 37 R Arginine
5 10 13 C Cysteine
23 0 74 E Glutamic acid
13 0 55 S Serine
16 0 59 Q Glutamine
19 0 93 L Leucine
8 0 70 K Lycine
15 0 28 P Proline
13 0 67 V Valine
8 0 12 H Histidine
9 0 57 I Isoleucine
6 0 47 F Phenylalanine
3 0 26 M Methionine
8 0 3 W Tryptophan
16 0 25 Y Tyrosine
21 7 69 A Alanine
5 0 46 N Asparagine
19 0 48 D Aspartic acid 

7TUE_1|Chain A|HLA class I histocompatibility antigen, B alpha chain|Homo sapiens (9606)
>1RAW_1|Chain A|RNA APTAMER|
>8OMO_1|Chain A|DNA mismatch repair protein Msh2|Homo sapiens (9606)
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 \(w=f(c)\)
7TUE , Knot 122 274 0.83 40 181 266 
GSHSMRYFYTAMSRPGRGEPRFITVGYVDDTLFVRFDSDATSPRKEPRAPWIEQEGPEYWDRETQISKTNTQCYRENLRTALRYYNQSEAGSHIIQRMYGCDVGPDGRLLRGYDQYAYDGKDYIALNEDLSSWTAADTAAQITQRKWEAARVAEQDRAYLEGLCVESLRRYLENGKETLQRADPPKTHVTHHPISDHEVTLRCWALGFYPAEITLTWQRDGEDQTQDTELVETRPAGDRTFQKWAAVVVPSGEEQRYTCHVQHEGLPKPLTLRW
1RAW , Knot 12 36 0.39 8 12 22 
GGGAAGGGAAGAAACUGCGGCUUCGGCCGGCUUCCC
8OMO , Knot 349 934 0.85 40 298 849 
MAVQPKETLQLESAAEVGFVRFFQGMPEKPTTTVRLFDRGDFYTAHGEDALLAAREVFKTQGVIKYMGPAGAKNLQSVVLSKMNFESFVKDLLLVRQYRVEVYKNRAGNKASKENDWYLAYKASPGNLSQFEDILFGNNDMSASIGVVGVKMSAVDGQRQVGVGYVDSIQRKLGLCEFPDNDQFSNLEALLIQIGPKECVLPGGETAGDMGKLRQIIQRGGILITERKKADFSTKDIYQDLNRLLKGKKGEQMNSAVLPEMENQVAVSSLSAVIKFLELLSDDSNFGQFELTTFDFSQYMKLDIAAVRALNLFQGSVEDTTGSQSLAALLNKCKTPQGQRLVNQWIKQPLMDKNRIEERLNLVEAFVEDAELRQTLQEDLLRRFPDLNRLAKKFQRQAANLQDCYRLYQGINQLPNVIQALEKHEGKHQKLLLAVFVTPLTDLRSDFSKFQEMIETTLDMDQVENHEFLVKPSFDPNLSELREIMNDLEKKMQSTLISAARDLGLDPGKQIKLDSSAQFGYYFRVTCKEEKVLRNNKNFSTVDIQKNGVKFTNSKLTSLNEEYTKNKTEYEEAQDAIVKEIVNISSGYVEPMQTLNDVLAQLDAVVSFAHVSNGAPVPYVRPAILEKGQGRIILKASRHACVEVQDEIAFIPNDVYFEKDKQMFHIITGPNMGGKSTYIRQTGVIVLMAQIGCFVPCESAEVSIVDCILARVGAGDSQLKGVSTFMAEMLETASILRSATKDSLIIIDELGRGTSTYDGFGLAWAISEYIATKIGAFCMFATHFHELTALANQIPTVNNLHVTALTTEETLTMLYQVKKGVCDQSFGIHVAELANFPKHVIECAKQKALELEEFQYIGESQGYDIMEPAAKKCYLEREQGEKIIQEFLSKVKQMPFTEMSEENITIKLKQLKAEVIAKNNSFVNEIISRIKVTT

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 \(c\).

\(|P_{f(7TUE_1)}(2) \setminus P_{f(1RAW_1)}(2)|=178\), \(|P_{f(1RAW_1)}(2) \setminus P_{f(7TUE_1)}(2)|=9\). 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 1:1000100100110011010101101101000111010001001000101111000110010000010000000000010011000000011001100101001110101101000010010001110001001011001101000010110110000101011010010001001000100101100010001100001010011111011010101000100000000110001110001001111111010000000010001110110101
Pair \(Z_2\) Length of longest common subsequence
7TUE_1,1RAW_1 187 2
7TUE_1,8OMO_1 191 5
1RAW_1,8OMO_1 298 3

Newick tree

 
[
	8OMO_1:13.04,
	[
		7TUE_1:93.5,1RAW_1:93.5
	]:40.54
]

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 \((0.85)(0.8)(\frac{310 }{\log_{20} 310}-\frac{36}{\log_{20}36})=89.6\)
Status Protein1 Protein2 d d1/2
Query variables 7TUE_1 1RAW_1 120 66
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]