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Parikh vectors
2AUC_1 9BGL_1 3ZLR_1 Letter Amino acid
6 7 9 F Phenylalanine
11 13 14 S Serine
8 6 5 T Threonine
3 13 10 Q Glutamine
10 16 13 E Glutamic acid
5 12 13 G Glycine
3 9 2 H Histidine
6 10 4 I Isoleucine
1 0 5 W Tryptophan
1 10 3 P Proline
7 12 14 A Alanine
2 17 1 C Cysteine
14 27 14 L Leucine
10 12 5 K Lycine
5 5 4 M Methionine
11 16 7 N Asparagine
5 17 10 R Arginine
12 9 7 D Aspartic acid
5 4 6 Y Tyrosine
1 14 12 V Valine 

2AUC_1|Chains A, B, C|Myosin A Tail Interacting Protein|Plasmodium knowlesi (5850)
>9BGL_1|Chain A|Zinc finger CCCH-type antiviral protein 1|Homo sapiens (9606)
>3ZLR_1|Chains A, B|BCL-2-LIKE PROTEIN 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 \(w=f(c)\)
2AUC , Knot 63 126 0.80 40 97 123 
KDMFNTKSSNGKLRIEDASHNARKLGLAPSSTDEKKIRDLYGDSLTYEQYLEYLTMCVHDRDNMEELIKMFSHFDNNSSGFLTKNQMKNILTTWGDALTEQEANDALNAFSSEDRINYKLFCEDIL
9BGL , Knot 108 229 0.85 38 157 220 
SNAADPEVCCFITKILCAHGGRMALDALLQEIALSEPQLCEVLQVAGPDRFVVLETGGEAGITRSVVATTRARVCRRKYCQRPCDNLHLCKLNLLGRCNYSQSERNLCKYSHEVLSEENFKVLKNHELSGLNKEELAVLLLQSDPFFMPEICKSYKGEGRQQICNQQPPCSRLHICDHFTRGNCRFPNCLRSHNLMDRKVLAIMREHGLNPDVVQNIQDICNSKHMQKN
3ZLR , Knot 78 158 0.83 40 126 156 
GPLGSMSQSNRELVVDFLSYKLSQKGYSWSQMAAVKQALREAGDEFELRYRRAFSDLTSQLHITPGTAYQSFEQVVNELFRDGVNWGRIVAFFSFGGALCVESVDKEMQVLVSRIAAWMATYLNDHLEPWIQENGGWDTFVELYGNNAAAESRKGQER

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 \(c\).

\(|P_{f(2AUC_1)}(2) \setminus P_{f(9BGL_1)}(2)|=60\), \(|P_{f(9BGL_1)}(2) \setminus P_{f(2AUC_1)}(2)|=120\). 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 1:001100000010101001000100111110000000100101001000001001010100000100110110010000011100001001100110110000100110110000010001100011
Pair \(Z_2\) Length of longest common subsequence
2AUC_1,9BGL_1 180 3
2AUC_1,3ZLR_1 155 3
9BGL_1,3ZLR_1 173 3

Newick tree

 
[
	9BGL_1:91.57,
	[
		2AUC_1:77.5,3ZLR_1:77.5
	]:14.07
]

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 \((0.85)(0.8)(\frac{355 }{\log_{20} 355}-\frac{126}{\log_{20}126})=70.0\)
Status Protein1 Protein2 d d1/2
Query variables 2AUC_1 9BGL_1 93 72
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]