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Parikh vectors
2QQN_1 5ETK_1 3DIW_1 Letter Amino acid
4 5 5 M Methionine
6 6 2 F Phenylalanine
5 3 1 W Tryptophan
7 8 3 N Asparagine
2 4 2 H Histidine
11 9 8 I Isoleucine
7 22 9 L Leucine
7 4 2 Y Tyrosine
7 14 5 A Alanine
7 10 8 R Arginine
6 8 7 D Aspartic acid
13 7 9 S Serine
12 5 7 K Lycine
12 8 12 V Valine
7 12 5 P Proline
14 9 8 T Threonine
2 0 0 C Cysteine
5 7 12 Q Glutamine
12 12 7 E Glutamic acid
13 9 12 G Glycine 

2QQN_1|Chain A|Neuropilin-1|Homo sapiens (9606)
>5ETK_1|Chain A|2-amino-4-hydroxy-6-hydroxymethyldihydropteridine pyrophosphokinase|Escherichia coli (562)
>3DIW_1|Chains A, B|Tax1-binding protein 3|Mus musculus (10090)
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 \(w=f(c)\)
2QQN , Knot 80 159 0.85 40 129 155 
GSHMFKCMEALGMESGEIHSDQITASSQYSTNWSAERSRLNYPENGWTPGEDSYREWIQVDLGLLRFVTAVGTQGAISKETKKKYYVKTYKIDVSSNGEDWITIKEGNKPVLFQGNTNPTDVVVAVFPKPLITRFVRIKPATWETGISMRFEVYGCKIT
5ETK , Knot 79 162 0.82 38 127 157 
GSAMTVAYIAIGSNLASPLEQVNAALKALGDIPESHILTVSSFYRTPPLGPQDQPDYLNAAVALETSLAPEELLNHTQRIELQQGRVRKAERWGPRTLDLDIMLFGNEVINTERLTVPHYDMKNRGFMLWPLFEIAPELVFPDGEMLRQILHTRAFDKLNKW
3DIW , Knot 65 124 0.84 38 99 121 
MSYTPGQPVTAVVQRVEIHKLRQGENLILGFSIGGGIDQDPSQNPFSEDKTDKGIYVTRVSEGGPAEIAGLQIGDKIMQVNGWDMTMVTHDQARKRLTKRSEEVVRLLVTRQSLQKAVQQSMLS

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 \(c\).

\(|P_{f(2QQN_1)}(2) \setminus P_{f(5ETK_1)}(2)|=86\), \(|P_{f(5ETK_1)}(2) \setminus P_{f(2QQN_1)}(2)|=84\). 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 1:100110010111100101000010100000001010000100100110110000001101011110110111001110000000001000010100010011010010011110100010011111110111001101011010011010101010010
Pair \(Z_2\) Length of longest common subsequence
2QQN_1,5ETK_1 170 3
2QQN_1,3DIW_1 142 4
5ETK_1,3DIW_1 138 3

Newick tree

 
[
	2QQN_1:81.18,
	[
		3DIW_1:69,5ETK_1:69
	]:12.18
]

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 \((0.85)(0.8)(\frac{321 }{\log_{20} 321}-\frac{159}{\log_{20}159})=49.4\)
Status Protein1 Protein2 d d1/2
Query variables 2QQN_1 5ETK_1 64 63
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]