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Parikh vectors
4RLW_1 7BZN_1 2KYS_1 Letter Amino acid
15 12 5 E Glutamic acid
9 15 15 L Leucine
8 10 5 K Lycine
1 11 4 M Methionine
10 12 0 Y Tyrosine
12 13 5 D Aspartic acid
5 20 3 R Arginine
3 22 10 S Serine
1 4 1 W Tryptophan
21 25 5 A Alanine
7 14 1 F Phenylalanine
8 32 4 T Threonine
16 22 9 V Valine
4 5 2 H Histidine
2 3 3 C Cysteine
7 12 6 Q Glutamine
13 18 2 G Glycine
9 10 2 I Isoleucine
3 20 0 P Proline
4 18 3 N Asparagine 

4RLW_1|Chain A|(3R)-hydroxyacyl-ACP dehydratase subunit HadA|Mycobacterium tuberculosis H37Rv (83332)
>7BZN_1|Chain A|Capsid protein VP1|Coxsackievirus A10 (42769)
>2KYS_1|Chain A|Non-structural protein 7|SARS coronavirus (694009)
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 \(w=f(c)\)
4RLW , Knot 77 158 0.82 40 120 151 
VALSADIVGMHYRYPDHYEVEREKIREYAVAVQNDDAWYFEEDGAAELGYKGLLAPLTFICVFGYKAQAAFFKHANIATAEAQIVQVDQVLKFEKPIVAGDKLYCDVYVDSVREAHGTQIIVTKNIVTNEEGDLVQETYTTLAGRAGEDGEGFSDGAA
7BZN , Knot 129 298 0.82 40 190 288 
GDPVEDIIHDALGSTARRAISSVTNAESAANTTPSSHRLETGRVPALQAAETGATSNATDENMIETRCVVNRNGVLETTINHFFSRSGLVGVVNLTDGGTDTTGYATWDIDIMGFVQLRRKCEMFTYMRFNAEFTFVTTTKNGEARPYMLQYMYVPPGAPKPTGRDAFQWQTATNPSVFVKLTDPPAQVSVPFMSPASAYQWFYDGYPTFGQHPETSNTTYGLCPNNMMGTFAVRVVSREASQLKLQTRVYMKLKHVRAWVPRPIRSQPYLLKNFPNYDSSKITNSARDRSSIKQANM
2KYS , Knot 45 85 0.78 36 68 78 
GHSKMSDVKCTSVVLLSVLQQLRVESSSKLWAQCVQLHNDILLAKDTTEAFEKMVSLLSVLLSMQGAVDINRLCEEMLDNRATLQ

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 \(c\).

\(|P_{f(4RLW_1)}(2) \setminus P_{f(7BZN_1)}(2)|=55\), \(|P_{f(7BZN_1)}(2) \setminus P_{f(4RLW_1)}(2)|=125\). 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 1:11101011110000100001000010001111000011010001110110011111101101110010111100101101010110100110100111110010001010010010100111000110000101100000011101100101100111
Pair \(Z_2\) Length of longest common subsequence
4RLW_1,7BZN_1 180 3
4RLW_1,2KYS_1 136 3
7BZN_1,2KYS_1 184 3

Newick tree

 
[
	7BZN_1:97.47,
	[
		4RLW_1:68,2KYS_1:68
	]:29.47
]

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 \((0.85)(0.8)(\frac{456 }{\log_{20} 456}-\frac{158}{\log_{20}158})=88.1\)
Status Protein1 Protein2 d d1/2
Query variables 4RLW_1 7BZN_1 113 85
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]