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Parikh vectors
1CLU_1 3HXQ_1 2VAI_1 Letter Amino acid
11 16 0 R Arginine
10 7 4 G Glycine
3 4 0 H Histidine
8 19 0 S Serine
11 3 6 T Threonine
0 1 0 W Tryptophan
9 7 0 Y Tyrosine
3 2 4 C Cysteine
13 17 0 E Glutamic acid
11 13 0 I Isoleucine
4 12 0 P Proline
4 3 0 N Asparagine
14 11 0 D Aspartic acid
5 8 0 F Phenylalanine
15 19 0 V Valine
11 15 3 A Alanine
11 11 0 Q Glutamine
11 23 0 L Leucine
8 14 0 K Lycine
4 4 0 M Methionine 

1CLU_1|Chain A|TRANSFORMING PROTEIN P21/H-RAS-1|Homo sapiens (9606)
>3HXQ_1|Chain A|von Willebrand Factor|Homo sapiens (9606)
>2VAI_1|Chain A|5'-D(*AP*GP*GP*AP*TP*CP*CP*TP*UP*TP *TP*GP*GP*AP*TP*CP*CP*T)-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 \(w=f(c)\)
1CLU , Knot 80 166 0.82 38 131 163 
MTEYKLVVVGAPGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHQYREQIKRVKDSDDVPMVLVGNKCDLAARTVESRQAQDLARSYGIPYIETSAKTRQGVEDAFYTLVREIRQH
3HXQ , Knot 91 209 0.77 40 137 193 
EDISEPPLHDFYCSRLLDLVFLLDGSSRLSEAEFEVLKAFVVDMMERLRISQKWVRVAVVEYHDGSHAYIGLKDRKRPSELRRIASQVKYAGSQVASTSEVLKYTLFQIFSKIDRPEASRIALLLMASQEPQRMSRNFVRYVQGLKKKKVIVIPVGIGPHANLKQIRLIEKQAPENKAFVLSSVDELEQQRDEIVSYLCDLAPEAPPPT
2VAI , Knot 9 18 0.48 10 11 11 
AGGATCCTUTTGGATCCT

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 \(c\).

\(|P_{f(1CLU_1)}(2) \setminus P_{f(3HXQ_1)}(2)|=75\), \(|P_{f(3HXQ_1)}(2) \setminus P_{f(1CLU_1)}(2)|=81\). 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 1:1000011111111110011010110001100001010000000111010001101100110000011000010010111011110000010010000001001000001111111000011100100001001100011101000100001100110011001000
Pair \(Z_2\) Length of longest common subsequence
1CLU_1,3HXQ_1 156 4
1CLU_1,2VAI_1 134 2
3HXQ_1,2VAI_1 146 2

Newick tree

 
[
	3HXQ_1:78.18,
	[
		1CLU_1:67,2VAI_1:67
	]:11.18
]

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 \((0.85)(0.8)(\frac{375 }{\log_{20} 375}-\frac{166}{\log_{20}166})=62.7\)
Status Protein1 Protein2 d d1/2
Query variables 1CLU_1 3HXQ_1 78 70
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]