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Parikh vectors
4WEN_1 2DIR_1 5CZU_1 Letter Amino acid
14 2 12 D Aspartic acid
7 2 11 Q Glutamine
10 8 15 E Glutamic acid
7 1 6 H Histidine
13 5 4 I Isoleucine
9 4 8 F Phenylalanine
18 7 6 N Asparagine
12 9 9 K Lycine
19 11 10 S Serine
10 3 6 Y Tyrosine
26 5 15 A Alanine
22 6 28 L Leucine
7 4 2 P Proline
26 5 6 T Threonine
9 3 11 R Arginine
0 3 2 C Cysteine
37 7 11 G Glycine
4 1 3 M Methionine
4 1 1 W Tryptophan
21 11 8 V Valine 

4WEN_1|Chain A|K88 fimbrial protein AC|Escherichia coli (562)
>2DIR_1|Chain A|THUMP domain-containing protein 1|Homo sapiens (9606)
>5CZU_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 \(w=f(c)\)
4WEN , Knot 117 275 0.79 38 154 255 
WMTGHHHHHHDDYRQKWEWKVGTGLNGFGNVLNDLTNGGTKLTITVTGNKPILLGRTKEAFATPVTGGVDGIPHIAFTDYEGASVVLRNPDGETNKKGLAYFVLPMKNAEGTKVGSVKVNASYAGVLGRGGVTSADGELLSLFADGLSSIFYGGLPRGSELSAGSAAAARTKLFGSLSRNDILGQIQRVNANITSLVDVAGSYRENMEYTDGTVVSAAYALGIANGQTIEATFNQAVTTSTQWSAPLNVAITYYDNKQMTGDFNGSVDIGGSITA
2DIR , Knot 50 98 0.78 40 78 92 
GSSGSSGKAFLEDMKKYAETFLEPWFKAPNKGTFQIVYKSRNNSHVNREEVIRELAGIVCTLNSENKVDLTNPQYTVVVEIIKAVCCLSVVKSGPSSG
5CZU , Knot 83 174 0.82 40 125 170 
SSQIRQNYSTEVEAAVNRLVNLYLRASYTYLSLGFYFDRDDVALEGVCHFFRELAEEKREGAERLLKMQNQRGGRALFQDLQKPSQDEWGTTLDAMKAAIVLEKSLNQALLDLHALGSCQADPHLADFLESHFLDEEVKLIKKMGDHLTNIQRLVGSQAGLGEYLFERLTLKHD

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 \(c\).

\(|P_{f(4WEN_1)}(2) \setminus P_{f(2DIR_1)}(2)|=116\), \(|P_{f(2DIR_1)}(2) \setminus P_{f(4WEN_1)}(2)|=40\). 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 1:11010000000000001010110110111011001001100101010100111110000111011011101110111000011011100101000001110111110010100110101010011111011100101011011101100110111101001011011110001110100001110100101010011011100000100001011011011111010010101001100000101110111000000010101010101110101
Pair \(Z_2\) Length of longest common subsequence
4WEN_1,2DIR_1 156 4
4WEN_1,5CZU_1 145 3
2DIR_1,5CZU_1 131 4

Newick tree

 
[
	4WEN_1:78.29,
	[
		5CZU_1:65.5,2DIR_1:65.5
	]:12.79
]

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 \((0.85)(0.8)(\frac{373 }{\log_{20} 373}-\frac{98}{\log_{20}98})=84.7\)
Status Protein1 Protein2 d d1/2
Query variables 4WEN_1 2DIR_1 99 67
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]