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Parikh vectors
8PAI_1 8XTU_1 3QKB_1 Letter Amino acid
5 5 3 Q Glutamine
6 11 8 E Glutamic acid
13 11 10 G Glycine
8 3 7 H Histidine
7 1 1 P Proline
9 16 8 V Valine
2 0 0 C Cysteine
10 12 9 D Aspartic acid
9 13 10 L Leucine
9 10 11 K Lycine
6 16 4 S Serine
2 5 5 N Asparagine
4 9 5 F Phenylalanine
4 5 4 M Methionine
5 10 0 R Arginine
12 4 9 I Isoleucine
2 8 6 T Threonine
1 2 0 W Tryptophan
2 5 5 Y Tyrosine
10 22 6 A Alanine 

8PAI_1|Chains A, B|Histidine triad nucleotide-binding protein 1|Homo sapiens (9606)
>8XTU_1|Chain A|Rab5|Leishmania donovani (5661)
>3QKB_1|Chains A, B, C, D, E|Uncharacterized protein|Pediococcus pentosaceus (278197)
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 \(w=f(c)\)
8PAI , Knot 65 126 0.83 40 101 122 
MADEIAKAQVARPGGDTIFGKIIRKEIPAKIIFEDDRCLAFHDISPQAPTHFLVIPKKHISQISVAEDDDESLLGHLMIVGKKCAADLGLNKGYRMVVNEGSDGGQSVYHVHLHVLGGRQMHWPPG
8XTU , Knot 79 168 0.80 38 117 161 
TRKFKLVLLGESGVGKSSVVQRLMKDAFSEKLNSTVGASFFRYTSNVDDDTAVHFDIWDTAGLERFKSLASMYYRGAAAALVVFDIVSADTFEKARYWIRELQANSPETVVMLVGNKKDLESERQVSLADAQQSAGEMGAMYHETSARSGDGVRDAFHAVAAKLIEAN
3QKB , Knot 56 111 0.79 34 88 103 
MGSDKIHHHHHHENLYFQGMFITTEGINAGYTIKDVVEATSSLMLASEDIDKYNMFDQLFDEAKQKLKKKADLLEGDGIIGLKYNTEVVEVNGAPKFLVVHGYGTVILIDK

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 \(c\).

\(|P_{f(8PAI_1)}(2) \setminus P_{f(8XTU_1)}(2)|=57\), \(|P_{f(8XTU_1)}(2) \setminus P_{f(8PAI_1)}(2)|=73\). 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 1:110011010110111001110110001110111000001110010101100111110001001011000000111011111000110111001001110010011001001010111100101111
Pair \(Z_2\) Length of longest common subsequence
8PAI_1,8XTU_1 130 3
8PAI_1,3QKB_1 129 3
8XTU_1,3QKB_1 137 3

Newick tree

 
[
	8XTU_1:67.51,
	[
		8PAI_1:64.5,3QKB_1:64.5
	]:3.01
]

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 \((0.85)(0.8)(\frac{294 }{\log_{20} 294}-\frac{126}{\log_{20}126})=52.3\)
Status Protein1 Protein2 d d1/2
Query variables 8PAI_1 8XTU_1 65 57.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]