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 \(c\).
\(|P_{f(1ANK_1)}(2) \setminus P_{f(7UXQ_1)}(2)|=91\),
\(|P_{f(7UXQ_1)}(2) \setminus P_{f(1ANK_1)}(2)|=47\).
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 1:1011111111110100101110001110100101101110010011001001101101100011111100011000000111101110011010110011101001101011001110011100101101010010101101010001010010000000000100011000010111110000010110000101010011101010100111
Pair
\(Z_2\)
Length of longest common subsequence
1ANK_1,7UXQ_1
138
3
1ANK_1,4QPD_1
167
3
7UXQ_1,4QPD_1
167
7
Newick tree
[
4QPD_1:87.80,
[
1ANK_1:69,7UXQ_1:69
]:18.80
]
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 \((0.85)(0.8)(\frac{343
}{\log_{20}
343}-\frac{129}{\log_{20}129})=65.6\)
Status
Protein1
Protein2
d
d1/2
Query variables
1ANK_1
7UXQ_1
80
65
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]