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Parikh vectors
4TWU_1 2DNO_1 2DAU_1 Letter Amino acid
7 8 2 T Threonine
12 7 0 E Glutamic acid
17 6 0 L Leucine
22 7 0 K Lycine
2 0 0 Y Tyrosine
15 7 2 A Alanine
11 2 0 H Histidine
9 2 0 I Isoleucine
3 2 0 M Methionine
7 6 0 F Phenylalanine
4 4 0 P Proline
6 4 0 Q Glutamine
2 1 0 N Asparagine
6 6 0 D Aspartic acid
0 2 4 C Cysteine
15 13 4 G Glycine
5 14 0 S Serine
2 0 0 W Tryptophan
7 6 0 V Valine
2 5 0 R Arginine 

4TWU_1|Chain A|Myoglobin|Equus caballus (9796)
>2DNO_1|Chain A|trinucleotide repeat containing 4 variant|Homo sapiens (9606)
>2DAU_1|Chains A, B|DNA (5'-D(*CP*GP*CP*GP*AP*AP*TP*TP*CP*GP*CP*G)-3')|
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 \(w=f(c)\)
4TWU , Knot 75 154 0.81 38 113 148 
MGLSDGEWQQVLNVWGKVEADIAGHGQEVLIRLFTGHPETLEKFKKFKHLKTEAEMKASEKLKKHGTVVLTALGGILKKKGHHEAKLKPLAQSHATKHKIPIKYLEFISDAIIHVLHSKHPGDFGADAQGAMTKALELFRNDIAAKYKELGFQG
2DNO , Knot 49 102 0.74 36 79 95 
GSSGSSGSRGEDRKLFVGMLGKQQTDEDVRKMFEPFGTIDECTVLRGPDGTSKGCAFVKFQTHAEAQAAINTLHSSRTLPGASSSLVVKFADTEKESGPSSG
2DAU , Knot 6 12 0.41 8 7 8 
CGCGAATTCGCG

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 \(c\).

\(|P_{f(4TWU_1)}(2) \setminus P_{f(2DNO_1)}(2)|=81\), \(|P_{f(2DNO_1)}(2) \setminus P_{f(4TWU_1)}(2)|=47\). 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 1:1110010100110111010101110100111011010100100100100100010101000100010111011111100010001010111000100001110010110011101100001101110101110011011000111000011101
Pair \(Z_2\) Length of longest common subsequence
4TWU_1,2DNO_1 128 3
4TWU_1,2DAU_1 114 2
2DNO_1,2DAU_1 80 2

Newick tree

 
[
	4TWU_1:66.05,
	[
		2DAU_1:40,2DNO_1:40
	]:26.05
]

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 \((0.85)(0.8)(\frac{256 }{\log_{20} 256}-\frac{102}{\log_{20}102})=49.1\)
Status Protein1 Protein2 d d1/2
Query variables 4TWU_1 2DNO_1 62 49
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]