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Parikh vectors
2DBE_1 1URU_1 9IQR_1 Letter Amino acid
0 18 1 Q Glutamine
0 6 2 H Histidine
0 4 0 M Methionine
0 9 2 F Phenylalanine
0 12 1 S Serine
2 23 3 A Alanine
0 15 3 R Arginine
0 16 5 N Asparagine
0 2 1 W Tryptophan
0 8 3 Y Tyrosine
0 15 2 D Aspartic acid
2 14 9 T Threonine
0 10 2 V Valine
0 20 4 E Glutamic acid
4 10 3 G Glycine
0 20 4 K Lycine
0 4 4 P Proline
4 2 8 C Cysteine
0 11 6 I Isoleucine
0 25 2 L Leucine 

2DBE_1|Chains A, B|DNA (5'-D(*CP*GP*CP*GP*AP*AP*TP*TP*CP*GP*CP*G)-3')|
>1URU_1|Chain A|AMPHIPHYSIN|DROSOPHILA MELANOGASTER (7227)
>9IQR_1|Chain A|Muscarinic toxin 3|Dendroaspis angusticeps (8618)
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 \(w=f(c)\)
2DBE , Knot 6 12 0.41 8 7 8 
CGCGAATTCGCG
1URU , Knot 113 244 0.84 40 163 237 
MTENKGIMLAKSVQKHAGRAKEKILQNLGKVDRTADEIFDDHLNNFNRQQASANRLQKEFNNYIRCVRAAQAASKTLMDSVCEIYEPQWSGYDALQAQTGASESLWADFAHKLGDQVLIPLNTYTGQFPEMKKKVEKRNRKLIDYDGQRHSFQNLQANANKRKDDVKLTKGREQLEEARRTYEILNTELHDELPALYDSRILFLVTNLQTLFATEQVFHNETAKIYSELEAIVDKLATESQRGS
9IQR , Knot 39 65 0.83 38 58 63 
LTCVTKNTIFGITTENCPAGQNLCFKRWHYVIPRYTEITRGCAATCPIPENYDSIHCCKTDKCNE

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 \(c\).

\(|P_{f(2DBE_1)}(2) \setminus P_{f(1URU_1)}(2)|=4\), \(|P_{f(1URU_1)}(2) \setminus P_{f(2DBE_1)}(2)|=160\). 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 1:010111000101
Pair \(Z_2\) Length of longest common subsequence
2DBE_1,1URU_1 164 2
2DBE_1,9IQR_1 55 3
1URU_1,9IQR_1 165 3

Newick tree

 
[
	1URU_1:93.63,
	[
		2DBE_1:27.5,9IQR_1:27.5
	]:66.13
]

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 \((0.85)(0.8)(\frac{256 }{\log_{20} 256}-\frac{12}{\log_{20}12})=84.2\)
Status Protein1 Protein2 d d1/2
Query variables 2DBE_1 1URU_1 109 56.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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