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 \(c\).
\(|P_{f(6RHW_1)}(2) \setminus P_{f(7GPW_1)}(2)|=118\),
\(|P_{f(7GPW_1)}(2) \setminus P_{f(6RHW_1)}(2)|=61\).
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 1:010001001000010100010000100000000100010101100100000011101010110110110101010001011100010100100000001001110000000010000100011010100111010100000000010000100001100000001111010101100110000001000000000001101000101110001010001110100110101111100000001000111000001001010100011110101000100000010110010100001011011000000
Pair
\(Z_2\)
Length of longest common subsequence
6RHW_1,7GPW_1
179
3
6RHW_1,6IJK_1
193
3
7GPW_1,6IJK_1
172
4
Newick tree
[
6RHW_1:95.30,
[
7GPW_1:86,6IJK_1:86
]:9.30
]
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 \((0.85)(0.8)(\frac{491
}{\log_{20}
491}-\frac{182}{\log_{20}182})=90.1\)
Status
Protein1
Protein2
d
d1/2
Query variables
6RHW_1
7GPW_1
112
90.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]