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 \(c\).
\(|P_{f(5WTE_1)}(2) \setminus P_{f(1EQB_1)}(2)|=70\),
\(|P_{f(1EQB_1)}(2) \setminus P_{f(5WTE_1)}(2)|=103\).
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 1:11000111000100000110101110010010101001010101101111110010011110011001101011000000001010011100011001010000000011101000001101110010111011010011101011101100101111101111110011100001101000011111010000010101011100010110111011100000011110101100000000101000101000001011011100011100001100
Pair
\(Z_2\)
Length of longest common subsequence
5WTE_1,1EQB_1
173
4
5WTE_1,1GDV_1
166
3
1EQB_1,1GDV_1
185
3
Newick tree
[
1EQB_1:91.62,
[
5WTE_1:83,1GDV_1:83
]:8.62
]
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 \((0.85)(0.8)(\frac{695
}{\log_{20}
695}-\frac{278}{\log_{20}278})=115.\)
Status
Protein1
Protein2
d
d1/2
Query variables
5WTE_1
1EQB_1
147
122.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]