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 \(c\).
\(|P_{f(8ZQD_1)}(2) \setminus P_{f(3FTS_1)}(2)|=59\),
\(|P_{f(3FTS_1)}(2) \setminus P_{f(8ZQD_1)}(2)|=67\).
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 1:11000001100111011011000010010010011101010011110110110100110101111000010110111111111000010001011110001100111100000110100101001110100100000011000100110110010011001001011001110100010000011001100100010001000000000000110101000011011000101100101010000010000000011010101101101100011110011010011100111110111101111011001000111110011100110001110101100110100010001110001011110000101101100001011011001001101111110011100011000010011111011000010010101010001001100001101100011010010000100111000111111100111101100011001010010010101101101101001011010101111011001101100100100110110111011101111101000100011100010110010000010000001011110000100110001001100101101000011011101111101110111010010100
Pair
\(Z_2\)
Length of longest common subsequence
8ZQD_1,3FTS_1
126
4
8ZQD_1,2YIC_1
120
4
3FTS_1,2YIC_1
116
4
Newick tree
[
8ZQD_1:62.64,
[
2YIC_1:58,3FTS_1:58
]:4.64
]
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 \((0.85)(0.8)(\frac{1285
}{\log_{20}
1285}-\frac{611}{\log_{20}611})=171.\)
Status
Protein1
Protein2
d
d1/2
Query variables
8ZQD_1
3FTS_1
215
204
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]