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Parikh vectors
6KPI_1 7UFX_1 2BYE_1 Letter Amino acid
9 0 3 N Asparagine
1 7 2 C Cysteine
29 3 8 G Glycine
4 0 2 H Histidine
23 0 8 I Isoleucine
10 0 3 M Methionine
10 5 9 T Threonine
12 0 5 R Arginine
29 0 7 L Leucine
29 0 9 K Lycine
11 0 6 F Phenylalanine
17 0 2 D Aspartic acid
10 0 10 Q Glutamine
30 0 14 E Glutamic acid
11 0 7 P Proline
14 0 3 S Serine
29 0 6 V Valine
32 7 2 A Alanine
5 0 1 W Tryptophan
18 0 3 Y Tyrosine 

6KPI_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Ketol-acid reductoisomerase|Saccharolobus solfataricus (2287)
>7UFX_1|Chain A|DNA (5'-D(A*AP*CP*CP*TP*GP*TP*GP*AP*CP*AP*AP*AP*TP*TP*GP*CP*CP*CP*TP*CP*A)-3')|Escherichia coli (562)
>2BYE_1|Chain A|PHOSPHOLIPASE C, EPSILON 1|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 \(w=f(c)\)
6KPI , Knot 141 333 0.82 40 191 309 
MDKTVLDANLDPLKGKTIGVIGYGNQGRVQATIMRENGLNVIVGNVKDKYYELAKKEGFEVYEIDEAVRRSDVALLLIPDEVMKEVYEKKIAPVLQGKKEFVLDFASGYNVAFGLIRPPKSVDTIMVAPRMVGEGIMDLHKQGKGYPVLLGVKQDASGKAWDYAKAIAKGIGAIPGGIAVISSFEEEALLDLMSEHTWVPILFGAIKACYDIAVKEYGVSPEAALLEFYASGELAEIARLIAEEGIFNQMVHHSTTSQYGTLTRMFKYYDVVRRIVENEAKYIWDGSFAKEWSLEQQAGYPVFYRLWELATQSEMAKAEKELYKLLGRKVKND
7UFX , Knot 10 22 0.46 8 12 19 
AACCTGTGACAAATTGCCCTCA
2BYE , Knot 58 110 0.82 40 89 105 
GEERKCLQTHRVTVHGVPGPEPFTVFTINGGTKAKQLLQQILTNEQDIKPVTTDYFLMEEKYFISKEKNECRKQPFQRAIGPEEEIMQILSSWFPEEGYMGRIVLKTQQE

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 \(c\).

\(|P_{f(6KPI_1)}(2) \setminus P_{f(7UFX_1)}(2)|=185\), \(|P_{f(7UFX_1)}(2) \setminus P_{f(6KPI_1)}(2)|=6\). 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 1:100011010101101001111101001010101100011011110100000011000110100100110000111111100110010000111110100011101101001111110110010011111011101110100010101111110001010110010111011111111111100100011101100001111111110100011100011010111101010101101101110011100110000000010100110000110011000100110101100101000110111001101100001101000100111001000
Pair \(Z_2\) Length of longest common subsequence
6KPI_1,7UFX_1 191 2
6KPI_1,2BYE_1 178 3
7UFX_1,2BYE_1 97 2

Newick tree

 
[
	6KPI_1:10.84,
	[
		2BYE_1:48.5,7UFX_1:48.5
	]:54.34
]

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 \((0.85)(0.8)(\frac{355 }{\log_{20} 355}-\frac{22}{\log_{20}22})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 6KPI_1 7UFX_1 139 73.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]