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Parikh vectors
1QYN_1 1PJT_1 6TPK_1 Letter Amino acid
9 18 32 F Phenylalanine
10 43 41 V Valine
4 34 29 R Arginine
14 24 10 Q Glutamine
1 5 12 W Tryptophan
5 6 10 Y Tyrosine
13 47 51 A Alanine
12 53 67 L Leucine
5 9 14 M Methionine
8 20 31 S Serine
8 21 36 I Isoleucine
7 27 18 D Aspartic acid
4 7 17 C Cysteine
13 27 25 E Glutamic acid
9 38 31 G Glycine
3 9 7 H Histidine
3 17 27 K Lycine
7 17 16 P Proline
7 14 10 N Asparagine
11 21 18 T Threonine 

1QYN_1|Chains A, B, C, D|Protein-export protein secB|Escherichia coli (562)
>1PJT_1|Chains A, B|Siroheme synthase|Salmonella typhimurium (90371)
>6TPK_1|Chain A|Oxytocin receptor|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 \(w=f(c)\)
1QYN , Knot 75 153 0.82 40 124 149 
MSEQNNTEMTFQIQRIYTKDISFEAPNAPHVFQKDWQPEVKLDLDTASSQLADDVYEVVLRVTVTASLGEETAFLCEVQQGGIFSIAGIEGTQMAHCLGAYCPNILFPYARECITSMVSRGTFPQLNLAPVNFDALFMNYLQQQAGEGTEEHQ
1PJT , Knot 184 457 0.82 40 215 418 
MDHLPIFCQLRDRDCLIVGGGDVAERKARLLLEAGARLTVNALTFIPQFTVWANEGMLTLVEGPFDETLLDSCWLAIAATDDDTVNQRVSDAAESRRIFCNVVDAPKAASFIMPSIIDRSPLMVAVSAGGTSPVLARLLREKLESLLPQHLGQVARYAGQLRARVKKQFATMGERRRFWEKFFVNDRLAQSLANADEKAVNATTERLFSEPLDHRGEVVLVGAGPGDAGLLTLKGLQQIQQADIVVYDRLVSDDIMNLVRRDADRVFVGKRAGYHCVPQEEINQILLREAQKGKRVVRLKGGDPFIFGRGGEELETLCHAGIPFSVVPGITAASGCSAYSGIPLTHRDYAQSVRLVTGHLKTGGELDWENLAAEKQTLVFYMGLNQAATIQEKLIAFGMQADMPVALVENGTSVKQRVVHGVLTQLGELAQQVESPALIIVGRVVALRDKLNWFSNH
6TPK , Knot 199 502 0.82 40 243 459 
NEALARVEVAVLCLILLLALSGNACVLLALHTTRQKHSRLFFFMKHLSIADLVVAVFQVLPQLLWDITFRFYGPDLLCRLVKYLQLVGMFASTYLLLLMSLDRCLAICQPLRSLRRRTARLAVLATWLGCLVVSAPQVHIFSLREVADGVFDCWAVFIRPWGPKAYITWITLAVYIVPVIVLATCYGLIAFKIWQNLRLKTAAAAAAEAPEGAAAGDGGRVALARVSSVKLISKAKIRTVKMTFIIVLAFIVCWTPFFFVQMWSVWDANAPKEASAFIIVMLLASLNCCCKPWIYMLFMGHLFHGIDCSFWCCNESYLTGSRDERKKSLLSKFGMDEGVTFMFIGRFDRGQKGVDVLLKAIEILSSKKEFQEMRFIIIGKGDPELEGWARSLEEKHGNVKVITEMLSREFVRELYGSVDFVIIPSYFEPFGLVALEAMCLGAIPIASAVGGLRDIITNETGILVKAGDPGELANAILKALELSRSDLSKFRENCKKRAMSFS

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 \(c\).

\(|P_{f(1QYN_1)}(2) \setminus P_{f(1PJT_1)}(2)|=46\), \(|P_{f(1PJT_1)}(2) \setminus P_{f(1QYN_1)}(2)|=137\). 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 1:100000001010100100001010110110110001010101010010001100100111010101011000111001001111011110100110011100101111010001001100101101011110101111001000110100000
Pair \(Z_2\) Length of longest common subsequence
1QYN_1,1PJT_1 183 4
1QYN_1,6TPK_1 211 4
1PJT_1,6TPK_1 144 4

Newick tree

 
[
	1QYN_1:10.17,
	[
		1PJT_1:72,6TPK_1:72
	]:34.17
]

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 \((0.85)(0.8)(\frac{610 }{\log_{20} 610}-\frac{153}{\log_{20}153})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 1QYN_1 1PJT_1 164 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]