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 \(c\).
\(|P_{f(6QYA_1)}(2) \setminus P_{f(7UWH_1)}(2)|=206\),
\(|P_{f(7UWH_1)}(2) \setminus P_{f(6QYA_1)}(2)|=12\).
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 1:100101111001100101000001010001110010101101111100001111110001111101000100110000011001110001001000110100110011001111000000000100111001011000101101011010010000101100110110110001000011101101001101111100011010100101000100001011111110110011100111000110110110011010100001001000100010011011101001011010100101010010101011111
Pair
\(Z_2\)
Length of longest common subsequence
6QYA_1,7UWH_1
218
2
6QYA_1,6JVY_1
182
3
7UWH_1,6JVY_1
102
3
Newick tree
[
6QYA_1:11.13,
[
6JVY_1:51,7UWH_1:51
]:61.13
]
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 \((0.85)(0.8)(\frac{374
}{\log_{20}
374}-\frac{59}{\log_{20}59})=99.1\)
Status
Protein1
Protein2
d
d1/2
Query variables
6QYA_1
7UWH_1
139
79.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]