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Parikh vectors
3ECV_1 8RCE_1 2DDH_1 Letter Amino acid
4 15 25 F Phenylalanine
10 17 44 E Glutamic acid
8 18 35 I Isoleucine
0 11 20 M Methionine
10 14 38 S Serine
9 11 45 T Threonine
1 0 8 W Tryptophan
0 7 28 Y Tyrosine
7 14 29 N Asparagine
11 16 25 D Aspartic acid
4 7 7 C Cysteine
5 10 32 P Proline
14 19 41 V Valine
10 23 46 A Alanine
4 10 33 R Arginine
8 15 22 H Histidine
11 21 39 K Lycine
3 10 31 Q Glutamine
25 13 44 G Glycine
9 34 69 L Leucine 

3ECV_1|Chains A, B, C, D|Superoxide dismutase [Cu-Zn]|Homo sapiens (9606)
>8RCE_1|Chain A|Peroxisome proliferator-activated receptor alpha|Homo sapiens (9606)
>2DDH_1|Chain A|Acyl-CoA oxidase|Rattus norvegicus (10116)
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 \(w=f(c)\)
3ECV , Knot 73 153 0.80 36 114 148 
ATKAVCVLKGDGPVQGIINFEQKESNGPVKVWGSIKGLTEGLHGFHVHEFGDNTAGCTSAGPHFNPLSRKHGGPKDEERHVGDLGNVTADKDGVADVSIEDSVISLSGDHCITGRTLVVHEKADDLGKGGNEESTKTGNAGSRLACGVIGIAQ
8RCE , Knot 122 285 0.80 38 182 269 
HHHHHHSSGLVPRGSHTADLKSLAKRIYEAYLKNFNMNKVKARVILSGKASNNPPFVIHDMETLCMAEKTLVAKLVANGIQNKEAEVRIFHCCQCTSVETVTELTEFAKAIPGFANLDLNDQVTLLKYGVYEAIFAMLSSVMNKDGMLVAYGNGFITREFLKSLRKPFCDIMEPKFDFAMKFNALELDDSDISLFVAAIICCGDRPGLLNVGHIEKMQEGIVHVLRLHLQSNHPDDIFLFPKLLQKMADLRQLVTEHAQLVQIIKKTESDAALHPLLQEIYRDMY
2DDH , Knot 262 661 0.85 40 285 614 
MNPDLRKERASATFNPELITHILDGSPENTRRRREIENLILNDPDFQHEDYNFLTRSQRYEVAVKKSATMVKKMREYGISDPEEIMWFKNSVHRGHPEPLDLHLGMFLPTLLHQATAEQQERFFMPAWNLEITGTYAQTEMGHGTHLRGLETTATYDPKTQEFILNSPTVTSIKWWPGGLGKTSNHAIVLAQLITQGECYGLHAFVVPIREIGTHKPLPGITVGDIGPKFGYEEMDNGYLKMDNYRIPRENMLMKYAQVKPDGTYVKPLSNKLTYGTMVFVRSFLVGNAAQSLSKACTIAIRYSAVRRQSEIKQSEPEPQILDFQTQQYKLFPLLATAYAFHFVGRYMKETYLRINESIGQGDLSELPELHALTAGLKAFTTWTANAGIEECRMACGGHGYSHSSGIPNIYVTFTPACTFEGENTVMMLQTARFLMKIYDQVRSGKLVGGMVSYLNDLPSQRIQPQQVAVWPTMVDINSLEGLTEAYKLRAARLVEIAAKNLQTHVSHRKSKEVAWNLTSVDLVRASEAHCHYVVVKVFSDKLPKIQDKAVQAVLRNLCLLYSLYGISQKGGDFLEGSIITGAQLSQVNARILELLTLIRPNAVALVDAFDFKDMTLGSVLGRYDGNVYENLFEWAKKSPLNKTEVHESYHKHLKPLQSKL

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 \(c\).

\(|P_{f(3ECV_1)}(2) \setminus P_{f(8RCE_1)}(2)|=48\), \(|P_{f(8RCE_1)}(2) \setminus P_{f(3ECV_1)}(2)|=116\). 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 1:100110110101110111010000001110111010110011011010011000110001110101100001110000001101101010001110101000110101000101001110001001101100000001011001101111110
Pair \(Z_2\) Length of longest common subsequence
3ECV_1,8RCE_1 164 3
3ECV_1,2DDH_1 215 4
8RCE_1,2DDH_1 169 4

Newick tree

 
[
	2DDH_1:10.10,
	[
		3ECV_1:82,8RCE_1:82
	]:19.10
]

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 \((0.85)(0.8)(\frac{438 }{\log_{20} 438}-\frac{153}{\log_{20}153})=84.7\)
Status Protein1 Protein2 d d1/2
Query variables 3ECV_1 8RCE_1 107 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]