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 \(c\).
\(|P_{f(2WEW_1)}(2) \setminus P_{f(5UIH_1)}(2)|=94\),
\(|P_{f(5UIH_1)}(2) \setminus P_{f(2WEW_1)}(2)|=86\).
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 1:1001000100001001110100110101101011111110000110101100111011110111010101010100110110011001001000100010101000110000111111000101000111000010110001001001000100011110100001001000
Pair
\(Z_2\)
Length of longest common subsequence
2WEW_1,5UIH_1
180
3
2WEW_1,7OFT_1
201
4
5UIH_1,7OFT_1
209
4
Newick tree
[
7OFT_1:10.36,
[
2WEW_1:90,5UIH_1:90
]:16.36
]
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 \((0.85)(0.8)(\frac{341
}{\log_{20}
341}-\frac{169}{\log_{20}169})=52.0\)
Status
Protein1
Protein2
d
d1/2
Query variables
2WEW_1
5UIH_1
71
70.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]