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Parikh vectors
3WFR_1 1HSI_1 9BMK_1 Letter Amino acid
0 6 26 K Lycine
0 2 8 M Methionine
14 5 30 A Alanine
0 3 11 R Arginine
0 8 34 N Asparagine
0 6 22 E Glutamic acid
0 3 16 Y Tyrosine
0 4 20 D Aspartic acid
0 11 25 I Isoleucine
0 4 16 S Serine
0 7 18 T Threonine
0 3 17 F Phenylalanine
0 5 11 P Proline
0 9 22 V Valine
0 9 14 L Leucine
0 1 16 W Tryptophan
21 0 7 C Cysteine
0 2 13 Q Glutamine
24 11 30 G Glycine
0 0 10 H Histidine 

3WFR_1|Chains A, B|RNA (74-MER)|Thermotoga maritima MSB8 (243274)
>1HSI_1|Chains A, B|HIV-2 PROTEASE|Human immunodeficiency virus 2 (11709)
>9BMK_1|Chains A, B|Cellulase (Glycosyl hydrolase family 5)|Segatella copri DSM 18205 (537011)
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 \(w=f(c)\)
3WFR , Knot 23 74 0.44 8 16 44 
GGCCAGGUAGCUCAGUUGGUAGAGCACUGGACUGAAAAUCCAGGUGUCGGCGGUUCGAUUCCGCCCCUGGCCAC
1HSI , Knot 52 99 0.80 36 81 96 
PQFSLWKRPVVTAYIEGQPVEVLLDTGADDSIVAGIELGNNYSPKIVGGIGGFINTKEYKNVEIEVLNKKVRATIMTGDTPINIFGRNILTALGMSLNL
9BMK , Knot 158 366 0.85 40 225 353 
SNATAQQWNKDVVGWNLGNEFECSAPGQDGESMQIGNPDGSIHAETAWGNPVVTKKMIQAVKKAGFNAIRIPIRWQCHITNAQAMSIDKAWIARIKEVVGWCLDNGLKVIINVHHEKWLESRPTYQYKEENCQKLALLWMNIASEFANYDSRLAFAGTNEVHIRDNWGKPTAENLEVQNAYNQIFVDVVRATGGNNAKRHLILQTYVCNPWFGIENGDFIIPKDAEGNGNNYMSVEFHYYQPWSYAGDCTYDYWGDAYKDAGKIPADNEKTMTDFFDKAVNTWSNKGLGIVIGAWGVTDHYKSNSEKVHENMTYYCKFLTTEARKRGFSTFVWDNNHFGNGSEKYGIFDRFKSMKVNAPWILEGIF

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 \(c\).

\(|P_{f(3WFR_1)}(2) \setminus P_{f(1HSI_1)}(2)|=13\), \(|P_{f(1HSI_1)}(2) \setminus P_{f(3WFR_1)}(2)|=78\). 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 1:11001110110001100110111101001110011111000111010011011000110000100000110010
Pair \(Z_2\) Length of longest common subsequence
3WFR_1,1HSI_1 91 2
3WFR_1,9BMK_1 235 2
1HSI_1,9BMK_1 188 4

Newick tree

 
[
	9BMK_1:12.01,
	[
		3WFR_1:45.5,1HSI_1:45.5
	]:74.51
]

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 \((0.85)(0.8)(\frac{173 }{\log_{20} 173}-\frac{74}{\log_{20}74})=33.3\)
Status Protein1 Protein2 d d1/2
Query variables 3WFR_1 1HSI_1 49 35.5
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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