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Parikh vectors
1MNL_1 3NFW_1 7EML_1 Letter Amino acid
7 12 11 R Arginine
5 3 6 N Asparagine
9 12 15 E Glutamic acid
1 3 3 M Methionine
2 12 10 S Serine
4 15 8 V Valine
5 13 11 D Aspartic acid
1 5 2 C Cysteine
0 14 7 H Histidine
8 10 4 I Isoleucine
4 16 6 T Threonine
3 8 11 Q Glutamine
8 18 11 G Glycine
6 15 28 L Leucine
9 7 9 K Lycine
5 12 8 F Phenylalanine
1 4 1 W Tryptophan
7 2 6 Y Tyrosine
3 16 15 A Alanine
6 13 2 P Proline 

1MNL_1|Chain A|MONELLIN|Dioscoreophyllum cumminsii (3457)
>3NFW_1|Chains A, B, C, D|Flavin reductase-like, FMN-binding protein|Mycobacterium thermoresistibile (1078020)
>7EML_1|Chain A|Ferritin light chain|Equus caballus (9796)
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 \(w=f(c)\)
1MNL , Knot 53 94 0.85 38 81 92 
GEWEIIDIGPFTQNLGKFAVDEENKIGQYGRLTFNKVIRPCMKKTIYENEREIKGYEYQLYVYASDKLFRADISEDYKTRGRKLLRFNGPVPPP
3NFW , Knot 96 210 0.81 40 155 202 
MAHHHHHHMGTLEAQTQGPGSMVTAEAIDQRTFRRVLGQFCTGVTIITTVHEGNPVGFACQSFAALSLDPPLVLFCPTKVSRSWKAIEASGRFCVNILHEKQQHVSARFGSREPDKFAGIDWRPSDLGSPIIDGSLAHIDCTVHDVHDGGDHFVVFGKVHGLSEVPERKPRPLLFYRGEYTGIEPEKNTPAQWRDDLEAFLTATTADTWL
7EML , Knot 82 174 0.81 40 127 170 
SSQIRQNYSTEVEAAVNRLVNLYLRASYTYLSLGFYFHRDDVALEGVCHFFRELAEEKREGAERLLKMQNQRGGRALFQDLQKPSQDEWGTTLDAMKAAIVLEKSLNQALLDLHALGSAQADPHLCDFLESHFLDEEVKLIKKMGDHLTNIQRLVGSQAGLGEYLFERLTLKHD

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 \(c\).

\(|P_{f(1MNL_1)}(2) \setminus P_{f(3NFW_1)}(2)|=48\), \(|P_{f(3NFW_1)}(2) \setminus P_{f(1MNL_1)}(2)|=122\). 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 1:1010110111100011011100000110010101001101010001000000101000010101000110101000000010011010111111
Pair \(Z_2\) Length of longest common subsequence
1MNL_1,3NFW_1 170 3
1MNL_1,7EML_1 152 3
3NFW_1,7EML_1 162 3

Newick tree

 
[
	3NFW_1:85.23,
	[
		1MNL_1:76,7EML_1:76
	]:9.23
]

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 \((0.85)(0.8)(\frac{304 }{\log_{20} 304}-\frac{94}{\log_{20}94})=66.1\)
Status Protein1 Protein2 d d1/2
Query variables 1MNL_1 3NFW_1 83 59
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]