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Parikh vectors
1FOL_1 9FCP_1 5FTE_1 Letter Amino acid
40 38 17 A Alanine
13 17 8 H Histidine
6 10 11 M Methionine
41 22 14 P Proline
23 18 27 T Threonine
13 17 18 Y Tyrosine
20 22 29 D Aspartic acid
10 4 6 C Cysteine
23 15 13 Q Glutamine
12 15 31 K Lycine
26 27 26 V Valine
30 24 19 R Arginine
26 20 33 E Glutamic acid
16 13 24 F Phenylalanine
12 4 3 W Tryptophan
16 18 21 N Asparagine
27 35 26 G Glycine
18 25 43 I Isoleucine
42 31 41 L Leucine
30 25 23 S Serine 

1FOL_1|Chains A, B|NITRIC-OXIDE SYNTHASE|Bos taurus (9913)
>9FCP_1|Chains A, B|12-oxophytodienoate reductase 3,Old yellow enzyme OYE4|Solanum lycopersicum (4081)
>5FTE_1|Chain A|TPR DOMAIN PROTEIN|BACTEROIDES (457390)
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 \(w=f(c)\)
1FOL , Knot 184 444 0.84 40 250 417 
SRAPAPATPHAPDHSPAPNSPTLTRPPEGPKFPRVKNWELGSITYDTLCAQSQQDGPCTPRRCLGSLVLPRKLQTRPSPGPPPAEQLLSQARDFINQYYSSIKRSGSQAHEERLQEVEAEVASTGTYHLRESELVFGAKQAWRNAPRCVGRIQWGKLQVFDARDCSSAQEMFTYICNHIKYATNRGNLRSAITVFPQRAPGRGDFRIWNSQLVRYAGYRQQDGSVRGDPANVEITELCIQHGWTPGNGRFDVLPLLLQAPDEAPELFVLPPELVLEVPLEHPTLEWFAALGLRWYALPAVSNMLLEIGGLEFSAAPFSGWYMSTEIGTRNLCDPHRYNILEDVAVCMDLDTRTTSSLWKDKAAVEINLAVLHSFQLAKVTIVDHHAATVSFMKHLDNEQKARGGCPADWAWIVPPISGSLTPVFHQEMVNYILSPAFRYQPDPW
9FCP , Knot 167 400 0.83 40 224 374 
MASHHHHHHMASSAQDGNNPLFSPYKMGKFNLSHRVVLAPMTRCRALNNIPQAALGEYYEQRATAGGFLITEGTMISPTSAGFPHVPGIFTKEQVREWKKIVDVVHAKGAVIFCQLWHVGRASHEVYQPAGAAPISSTEKPISNRWRILMPDGTHGIYPKPRAIGTYEISQVVEDYRRSALNAIEAGFDGIEIHGAHGYLIDQFLKDGINDRTDEYGGSLANRCKFITQVVQAVVSAIGADRVGVRVSPAIDHLDAMDSNPLSLGLAVVERLNKIQLHSGSKLAYLHVTAEADWDDAPDMPEEEEARLMRTLRNAYQGTFICSGGYTRELGIEAVAQGDADLVSYGRLFISNPDLVMRIKLNAPLNKYNRKTFYTQDPVVGYTDYPFLQGNGSNGPLSRL
5FTE , Knot 183 433 0.85 40 242 415 
MEDMILTEEMQKIMNLIQDDENNVFVTGKAGSGKTTFLKYLIEKSGKNCIVAAPTGIAAINAGGVTLHSLFGIPFGPITPYDRLENKFSEYKVELLLKMELLIIDEISMVRPDILDTIDRKLRWVYESDEPFGGVQVIMFGDLFQLPPVTKKQEREILSDFYDGFFFFNALVFKRTGFHIVELTKIFRQTEPEFINVLNNIRNYQVTSDELDLLSELKDRKISSSYDNEYIHICTHKADVEKINADKLGEQEIRNYDIVIKDKFPESSIPCDLHLKLRVGARVMSLVNDSLKGYYNGMLGIVTALEDNVITVRMDNGRTIKFERYTWSNTQYTLKDNEIVKEEIGSCTQFPLTLAWAITIHKSQGLTFDKIIIHVSHTFCPGQLYVALSRCRTLEGIVSDAFITKQMIIPEYALIDFERAYKSEGNYYGKRLD

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 \(c\).

\(|P_{f(1FOL_1)}(2) \setminus P_{f(9FCP_1)}(2)|=104\), \(|P_{f(9FCP_1)}(2) \setminus P_{f(1FOL_1)}(2)|=78\). 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 1:001111101011000111001010011011011010010110100001010000011001000110111100100010111111001100100110000001000100100001001010110010001000011111001100110011010110101101000001001100100010010001010011011100111010101100011001100000101010110101001010011011010101111110110011011111101110111001010111111101011111001110111101011110110100011000100100001100111010100000001100011101011110010110101100011010110010000010110110111111110101011100011001101110001011
Pair \(Z_2\) Length of longest common subsequence
1FOL_1,9FCP_1 182 4
1FOL_1,5FTE_1 176 3
9FCP_1,5FTE_1 158 4

Newick tree

 
[
	1FOL_1:92.75,
	[
		5FTE_1:79,9FCP_1:79
	]:13.75
]

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 \((0.85)(0.8)(\frac{844 }{\log_{20} 844}-\frac{400}{\log_{20}400})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 1FOL_1 9FCP_1 155 144
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]