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Parikh vectors
2OCB_1 8CQF_1 3PDC_1 Letter Amino acid
17 25 30 L Leucine
9 18 13 Q Glutamine
13 46 22 G Glycine
4 20 12 I Isoleucine
15 27 12 D Aspartic acid
4 13 24 P Proline
3 13 10 W Tryptophan
17 33 24 V Valine
9 39 21 A Alanine
9 14 16 R Arginine
8 39 12 N Asparagine
9 39 29 S Serine
12 23 9 T Threonine
16 18 26 E Glutamic acid
2 13 16 H Histidine
10 14 16 K Lycine
6 19 12 Y Tyrosine
3 8 8 C Cysteine
2 9 17 M Methionine
12 20 15 F Phenylalanine

2OCB_1|Chain A|Ras-related protein Rab-9B|Homo sapiens (9606)
>8CQF_1|Chain A|Alpha-amylase|Pseudoalteromonas haloplanktis (228)
>3PDC_1|Chains A, B|Epoxide hydrolase 2|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 \(w=f(c)\)
2OCB , Knot 85 180 0.81 40 132 172
GSGKSLLLKVILLGDGGVGKSSLMNRYVTNKFDSQAFHTIGVEFLNRDLEVDGRFVTLQIWDTAGQERFKSLRTPFYRGADCCLLTFSVDDRQSFENLGNWQKEFIYYADVKDPEHFPFVVLGNKVDKEDRQVTTEEAQTWCMENGDYPYLETSAKDDTNVTVAFEEAVRQVLAVEEQLE
8CQF , Knot 184 450 0.83 40 234 428
HPTTFVHLFEWNWQDVAQECEQYLGPKGYAAVQVSPPNEHITGSQWWTRYQPVSYELQSRGGNRAQFIDMVNRCSAVGVDIYVDTLINHMAAGSGTGTAGNSFGNKSFPIYSPQDFHESCTINNSDYGNDRYRVQNCELVGLADLDTASNYVQNTIAAYINDLQAIGVKGFRFDASKHVAASDIQSLMAKVNGSPVVFQEVIDLGGEAIKSSEYLSTGLVTEFKYGAKLGTVFRNGSLAWLSNFGEGWGFMPSSSAVVFVDNHDNQRGHGAGGSSILTFEDGRLYDLANVFMLAYPYGYPRVMSSYDFHGNDWAGGPNVPVHNNGNLECFASNWKCEHRWSYIAGGVDFRNNTADNWAVTNWWDNTNNQISFGRGSSGHMAINKEDSTLTATVQTDMASGQYCNVLKGELSADAKSCSGEVITVNSDGTINLNIGAWDAMAIHKNAKLGA
3PDC , Knot 147 344 0.83 40 200 328
MGSSHHHHHHSSGLVPRGSHMAPLPTSCNPSDMSHGYVTVKPRVRLHFVELGSGPAVCLCHGFPESWYSWRYQIPALAQAGYRVLAMDMKGYGESSAPPEIEEYCMEVLCKEMVTFLDKLGLSQAVFIGHDWGGMLVWYMALFYPERVRAVASLNTPFIPANPNMSPLESIKANPVFDYQLYFQEPGVAEAELEQNLSRTFKSLFRASDESVLSMHKVCEAGGLFVNSPEEPSLSRMVTEEEIQFYVQQFKKSGFRGPLNWYRNMERNWKWACKSLGRKILIPALMVTAEKDFVLVPQMSQHMEDWIPHLKRGHIEDCGHWTQMDKPTEVNQILIKWLDSDARN

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 \(c\).

\(|P_{f(2OCB_1)}(2) \setminus P_{f(8CQF_1)}(2)|=41\), \(|P_{f(8CQF_1)}(2) \setminus P_{f(2OCB_1)}(2)|=143\). 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 1:101001110111110111100011000100010001100111011000101010110101100110001001001100110001101010000010011010001100101001001111111001000000100001001010010010100010000010111001100111100010
Pair \(Z_2\) Length of longest common subsequence
2OCB_1,8CQF_1 184 3
2OCB_1,3PDC_1 188 4
8CQF_1,3PDC_1 182 3

Newick tree

 
[
	2OCB_1:93.66,
	[
		8CQF_1:91,3PDC_1:91
	]:2.66
]

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 \((0.85)(0.8)(\frac{630 }{\log_{20} 630}-\frac{180}{\log_{20}180})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 2OCB_1 8CQF_1 160 111.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]