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 \(c\).
\(|P_{f(5EXG_1)}(2) \setminus P_{f(4PAL_1)}(2)|=77\),
\(|P_{f(4PAL_1)}(2) \setminus P_{f(5EXG_1)}(2)|=47\).
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 1:101110111111011011011110011010110101101101010001000000011011010011100100111111011000111111011100000101111001101101111110101111011011011110101101001010
Pair
\(Z_2\)
Length of longest common subsequence
5EXG_1,4PAL_1
124
3
5EXG_1,4KCF_1
180
6
4PAL_1,4KCF_1
196
4
Newick tree
[
4KCF_1:10.57,
[
5EXG_1:62,4PAL_1:62
]:40.57
]
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 \((0.85)(0.8)(\frac{258
}{\log_{20}
258}-\frac{108}{\log_{20}108})=47.6\)
Status
Protein1
Protein2
d
d1/2
Query variables
5EXG_1
4PAL_1
57
50
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]