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Parikh vectors
6URU_1 1RKW_1 8TYX_1 Letter Amino acid
31 23 9 N Asparagine
19 6 5 Q Glutamine
19 9 2 H Histidine
31 17 4 I Isoleucine
12 3 1 M Methionine
12 2 2 R Arginine
44 3 7 D Aspartic acid
39 20 5 E Glutamic acid
46 8 11 G Glycine
3 3 1 W Tryptophan
41 7 10 V Valine
2 0 0 C Cysteine
24 9 4 F Phenylalanine
18 10 4 S Serine
35 11 5 T Threonine
24 13 0 Y Tyrosine
30 9 8 A Alanine
50 17 7 L Leucine
52 23 8 K Lycine
28 1 5 P Proline 

6URU_1|Chains A, B|iAChSnFR precursor|Escherichia coli (562)
>1RKW_1|Chains A, B, C[auth D], D[auth E]|Transcriptional regulator qacR|Staphylococcus aureus (1280)
>8TYX_1|Chain A|Ubl(BilA)|Ensifer aridi (1708715)
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 \(w=f(c)\)
6URU , Knot 221 560 0.83 40 249 522 
MHHHHHHGYPYDVPDYAGAQPARSANDTVVVGSINFTEQIIVANMVAEMIEAHTDLKVVRKLNLGGTNVNFEAIKRGGANNGIDIYVEYTGTGLVDILGFPEPNVYITADKQKNGIKANFKIRHNVEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSTQSVLSKDPNEKRDHMVLLEFVTAAGITLGMDELYKGGTGGSMSKGEELFTGVVPILVELDGDVNGHKFSVRGEGEGDATNGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKQHDFFKSAMPEGYVQERTISFKDDGTYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNFPPPPTTDPEKAYETVKKEYKDKWNIVWLKPLGFNNTYTLAVKDELAKQYNLKTFSDLAKISDKLILGATMEFLEKPDGYPGLQKVYNFKFKHTKSMDMGIRYTAIDNNEVQVIDAFATDGLLVSHKLKILEDDKHFFPPYYAAPIIRQDVLDKHPELKDVLNKLANQISDEEMQKLNYKVDGEGQDPAKVAKEFLKEKGLILQVDEQKLISEEDLN
1RKW , Knot 85 194 0.77 38 118 180 
MNLKDKILGVAKELFIKNGYNATTTGEIVKLSESSKGNLYYHFKTKENLFLEILNIEESKWQEQWKKEQIKAKTNREKFYLYNELSLTTEYYYPLQNAIIEFYTEYYKTNSINEKMNKLENKYIDAYHVIFKEGNLNGEWSINDVNAVSKIAANAVNGIVTFTHEQNINERIKLMNKFSQIFLNGLSKHHHHHH
8TYX , Knot 50 98 0.78 36 80 96 
MSKDSRKGDNHGGGSGKIEIIVVVNGQPTQVEANPNQPLHVVRTKALENTQNVAQPPDNWEFKDEAGNLLDVDKKIGDFGFANTVTLFLSLKAGVAGA

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 \(c\).

\(|P_{f(6URU_1)}(2) \setminus P_{f(1RKW_1)}(2)|=156\), \(|P_{f(1RKW_1)}(2) \setminus P_{f(6URU_1)}(2)|=25\). 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 1:10000001010011001110110010001111010100011110111011010001011001011100101011001110011010100010111011111010101010000011010101000100101011000000011101111110000100001100010000001111011011110111001001101101001001101111111010101010010101010101001010101100010111111011001001100100010010000110011101010000101000100000101010100110010101101000101110010001111100010010001000000010111101111000001110001100001001001101000111110101100101011100100101000001011100011000010110111001111000101100000111100111110001100010100110011001000010010001010100110110011000111101000011000010
Pair \(Z_2\) Length of longest common subsequence
6URU_1,1RKW_1 181 6
6URU_1,8TYX_1 193 3
1RKW_1,8TYX_1 132 3

Newick tree

 
[
	6URU_1:10.07,
	[
		1RKW_1:66,8TYX_1:66
	]:35.07
]

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 \((0.85)(0.8)(\frac{754 }{\log_{20} 754}-\frac{194}{\log_{20}194})=156.\)
Status Protein1 Protein2 d d1/2
Query variables 6URU_1 1RKW_1 195 128.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]