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 \(c\).
\(|P_{f(1NCI_1)}(2) \setminus P_{f(5OML_1)}(2)|=74\),
\(|P_{f(5OML_1)}(2) \setminus P_{f(1NCI_1)}(2)|=46\).
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 1:10011111101100001111001101001000010100010111100110111110110101010011000111010101011010100100110111011010000101
Pair
\(Z_2\)
Length of longest common subsequence
1NCI_1,5OML_1
120
2
1NCI_1,6NBK_1
169
3
5OML_1,6NBK_1
167
4
Newick tree
[
6NBK_1:90.59,
[
1NCI_1:60,5OML_1:60
]:30.59
]
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 \((0.85)(0.8)(\frac{177
}{\log_{20}
177}-\frac{67}{\log_{20}67})=37.1\)
Status
Protein1
Protein2
d
d1/2
Query variables
1NCI_1
5OML_1
47
38
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]