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Parikh vectors
5OAY_1 2IBJ_1 7JXS_1 Letter Amino acid
1 4 8 H Histidine
2 3 0 M Methionine
3 4 1 F Phenylalanine
3 1 2 W Tryptophan
5 7 6 N Asparagine
6 6 5 D Aspartic acid
3 2 10 Q Glutamine
9 3 9 R Arginine
6 12 12 E Glutamic acid
4 5 10 S Serine
6 8 7 V Valine
11 7 10 A Alanine
4 0 2 C Cysteine
4 7 13 K Lycine
4 1 3 P Proline
5 4 5 T Threonine
1 3 4 Y Tyrosine
9 5 10 G Glycine
1 3 6 I Isoleucine
7 3 14 L Leucine 

5OAY_1|Chain A|Transcriptional regulator WhiB1|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>2IBJ_1|Chain A|Cytochrome b5|Musca domestica (7370)
>7JXS_1|Chains A, B, C, D, E, F|Matrix protein|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
5OAY , Knot 53 94 0.85 40 80 91 
AGENLYFQGAMDWRHKAVCRDEDPELFFPVGNSGPALAQIADAKLVCNRCPVTTECLSWALNTGQDSGVWGGMSEDERRALKRRNARTKARTGV
2IBJ , Knot 47 88 0.79 38 72 85 
MSSEDVKYFTRAEVAKNNTKDKNWFIIHNNVYDVTAFLNEHPGGEEVLIEQAGKDATEHFEDVGHSSDAREMMKQYKVGELVAEERSN
7JXS , Knot 65 137 0.77 38 106 131 
GARASVLSGGELDKWEKIRLRPGGKKQYKLKHIVWASRELERFAVNPGLLETSEGCRQILGRLQPSLQTGSEELRSLYNTIAVLYCVHQRIDVKDTKEALDKIEEEQNKSKKKAQQAAADTGNNSQVSQNYHHHHHH

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 \(c\).

\(|P_{f(5OAY_1)}(2) \setminus P_{f(2IBJ_1)}(2)|=60\), \(|P_{f(2IBJ_1)}(2) \setminus P_{f(5OAY_1)}(2)|=52\). 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 1:1100101011101000110000010111111001111101101011000011000010111001000111111000000110000100010011
Pair \(Z_2\) Length of longest common subsequence
5OAY_1,2IBJ_1 112 3
5OAY_1,7JXS_1 130 2
2IBJ_1,7JXS_1 126 4

Newick tree

 
[
	7JXS_1:66.46,
	[
		5OAY_1:56,2IBJ_1:56
	]:10.46
]

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 \((0.85)(0.8)(\frac{182 }{\log_{20} 182}-\frac{88}{\log_{20}88})=31.2\)
Status Protein1 Protein2 d d1/2
Query variables 5OAY_1 2IBJ_1 39 37.5
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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