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Parikh vectors
5DBL_1 6RFG_1 5EFA_1 Letter Amino acid
20 11 14 E Glutamic acid
9 13 13 I Isoleucine
3 13 8 F Phenylalanine
4 7 9 S Serine
13 9 4 T Threonine
1 0 2 W Tryptophan
2 5 4 A Alanine
4 11 10 N Asparagine
7 9 9 D Aspartic acid
0 3 3 C Cysteine
16 9 13 K Lycine
17 4 2 P Proline
4 3 14 R Arginine
5 11 17 L Leucine
1 6 4 Y Tyrosine
8 15 7 V Valine
1 2 10 Q Glutamine
14 5 15 G Glycine
2 9 8 H Histidine
1 2 8 M Methionine 

5DBL_1|Chain A|Surface protein G|Staphylococcus aureus (93061)
>6RFG_1|Chains A, B, C, D|15 kDa core protein|Vaccinia virus (10245)
>5EFA_1|Chain A|Polymerase basic protein 2|Influenza B virus (518987)
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 \(w=f(c)\)
5DBL , Knot 61 132 0.75 38 91 119 
GPHMIAPGHRDEFDPKLPTGEKEEVPGKPGIKNPETGDVVRPPVDSVTKYGPVKGDSIVEKEEIPFEKERKFNPDLAPGTEKVTREGQKGEKTITTPTLKNPLTGEIISKGESKEEITKDPINELTEWGPET
6RFG , Knot 70 147 0.79 38 104 140 
HHHHHHIPTTENLYFQGAMELVNIFLETDAGRVKFAIKNTDDVCASELINKFVELLSEYIHIDQSEFYLVVKDKDIFYFKCDRGSISIVNNEFYVFDEPLLFVKDFTNVTGVEFIVTETMPCRIIPKNNHAVISVVTNHKFYNGLSL
5EFA , Knot 80 174 0.79 40 126 166 
MKIRQRQRFGRLELKRISGRGFKNDEEILIGNGTIQKIGIWDGEEEFHVRCGECRGILKKSQMRMEKLLINSAKKEDMKDLIILCMVFSQDTRMFQGVRGEINFLNRAGQLLSPMYQLQRYFLNRSNDLFDQWGYEESPKASELHGINELMNASDYTLKGVVVTKNVDHHHHHH

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 \(c\).

\(|P_{f(5DBL_1)}(2) \setminus P_{f(6RFG_1)}(2)|=63\), \(|P_{f(6RFG_1)}(2) \setminus P_{f(5DBL_1)}(2)|=76\). 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 1:110111110000101011010000111011100100101101110010001110100110000111000001010111100010001001000100101001101011001000001000110010011100
Pair \(Z_2\) Length of longest common subsequence
5DBL_1,6RFG_1 139 3
5DBL_1,5EFA_1 149 4
6RFG_1,5EFA_1 156 6

Newick tree

 
[
	5EFA_1:78.39,
	[
		5DBL_1:69.5,6RFG_1:69.5
	]:8.89
]

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 \((0.85)(0.8)(\frac{279 }{\log_{20} 279}-\frac{132}{\log_{20}132})=45.8\)
Status Protein1 Protein2 d d1/2
Query variables 5DBL_1 6RFG_1 57 54
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]