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 \(c\).
\(|P_{f(3GRN_1)}(2) \setminus P_{f(3CPL_1)}(2)|=53\),
\(|P_{f(3CPL_1)}(2) \setminus P_{f(3GRN_1)}(2)|=117\).
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 1:101001011010111000010111100000000011010111101010001001110011000110111101110101010000111111011011101010000100011010011110011101001100100000010011101000000
Pair
\(Z_2\)
Length of longest common subsequence
3GRN_1,3CPL_1
170
3
3GRN_1,8OPG_1
172
3
3CPL_1,8OPG_1
184
4
Newick tree
[
8OPG_1:90.36,
[
3GRN_1:85,3CPL_1:85
]:5.36
]
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 \((0.85)(0.8)(\frac{428
}{\log_{20}
428}-\frac{153}{\log_{20}153})=81.9\)
Status
Protein1
Protein2
d
d1/2
Query variables
3GRN_1
3CPL_1
106
80.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]