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Parikh vectors
8HOJ_1 3VOH_1 2WKN_1 Letter Amino acid
38 18 19 E Glutamic acid
24 18 25 I Isoleucine
25 11 19 F Phenylalanine
10 19 11 Y Tyrosine
41 26 27 V Valine
21 26 33 D Aspartic acid
9 9 13 H Histidine
23 17 18 T Threonine
14 5 10 M Methionine
28 49 33 A Alanine
18 21 16 N Asparagine
47 26 31 L Leucine
33 28 46 G Glycine
27 17 22 K Lycine
25 21 29 P Proline
34 19 19 S Serine
5 8 8 W Tryptophan
17 20 12 R Arginine
6 4 5 C Cysteine
19 11 13 Q Glutamine 

8HOJ_1|Chains A, B|UGT71AP2|Scutellaria baicalensis (65409)
>3VOH_1|Chain A|Cellobiohydrolase|Coprinopsis cinerea (5346)
>2WKN_1|Chains A, B, C, D, E, F, G, H|FORMAMIDASE|DELFTIA ACIDOVORANS (80866)
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 \(w=f(c)\)
8HOJ , Knot 194 464 0.85 40 248 439 
GMTNVELVFIPSPGLSHLVSTVEAAKLLLHRDHRLSITVLITNLHKDTKVENYTLKTTSNPNTSPRLRFINLPTQEEDSIDSQIRQVREIVSVLIKNPDSKLGGLVLDMFCTSFIQVAEEFAIPSYVFFTSGASALGLLQHLVSLKLEHNQDLTQFKDSDVELSVPCFSVPVPAKVLPAVMVEGGPIEDTFMNYFKRIPETRGIMVNTFYELESFAIQSLLSDAKAPKVYPVGPILGFDQTQGPVGDDAIKKWLDDQPENSVVFLCFGTMGSFCEGQVKEIAMALEKSGNRFLWSLRKPEKGRVTEYENYEEVLPQGFLERTKGVGKVMGWAPQAAVLSHPAVGGFVSHCGWNSTLESLWFGVPMATFPLYAEQQMNAFLLVKEEGMAEMITLDYKIDFKGEKQPEIVGSDEIEAAIRRLMAEESGVRRKVKEMQNKARSALLEGGSSYDAQCLFVHDVINNIG
3VOH , Knot 156 373 0.82 40 211 351 
TPVPSTGNPFEGYDIYLSPYYAEEVEAAAAMIDDPVLKAKALKVKEIPTFIWFDVVRKTPDLGRYLADATAIQQRTGRKQLVQIVVYDLPDRDCAAAASNGEFSLADGGMEKYKDYVDRLASEIRKYPDVRIVAVIEPDSLANMVTNMNVAKCRGAEAAYKEGVIYALRQLSALGVYSYVDAGHAGWLGWNANLAPSARLFAQIYKDAGRSAFIRGLATNVSNYNALSATTRDPVTQGNDNYDELRFINALAPLLRNEGWDAKFIVDQGRSGVQNIRQEWGNWCNVYGAGFGMRPTLNTPSSAIDAIVWIKPGGEADGTSDTSAPRYDTHCGKSDSHKPAPEAGTWFQEYFVNLVKNANPPLAAALEHHHHHH
2WKN , Knot 177 409 0.86 40 224 383 
MAETLIKVDLNQSPYDNPQVHNRWHPDIPMAVWVEPGAEFKLETYDWTGGAIKNDDSAEDVRDVDLSTVHFLSGPVGVKGAEPGDLLVVDLLDIGARDDSLWGFNGFFSKQNGGGFLDEHFPLAQKSIWDFHGMFTKSRHIPGVNFAGLIHPGLIGCLPDPKMLASWNERETGLIATDPDRIPGLANPPNATTAHMGQMQGEARDKAAAEGARTVPPREHGGNCDIKDLSRGSRVFFPVYVDGAGLSVGDLHFSQGDGEITFCGAIEMAGWVHMKVSLIKGGMAKYGIKNPIFKPSPMTPNYKDYLIFEGISVDEKGKQHYLDVTVAYRQACLNAIEYLKKFGYSGAQAYSLLGTAPVQGHISGVVDVPNACATLWLPTEIFDFDINPTAEGPQKIITGGVDLPIAQDK

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 \(c\).

\(|P_{f(8HOJ_1)}(2) \setminus P_{f(3VOH_1)}(2)|=98\), \(|P_{f(3VOH_1)}(2) \setminus P_{f(8HOJ_1)}(2)|=61\). 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 1:11001011111011100110010110111000001010111001000001000010000010001010110110000001000100100110111001000111111011000110110011110011100110111110011010100000100100001010110101111101111111011110001100100110001111001001001110011001011010111111110000111100110011000100011110110110100101001111100010011101001001010000000011101110000111011111101111001111111000110001001111111101110100010111110001110110100010101000101110001011100111000110001001000100111011000010011100110011
Pair \(Z_2\) Length of longest common subsequence
8HOJ_1,3VOH_1 159 5
8HOJ_1,2WKN_1 158 4
3VOH_1,2WKN_1 173 4

Newick tree

 
[
	3VOH_1:84.38,
	[
		8HOJ_1:79,2WKN_1:79
	]:5.38
]

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 \((0.85)(0.8)(\frac{837 }{\log_{20} 837}-\frac{373}{\log_{20}373})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 8HOJ_1 3VOH_1 160 142
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]