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 \(c\).
\(|P_{f(6LZG_1)}(2) \setminus P_{f(7NQE_1)}(2)|=177\),
\(|P_{f(7NQE_1)}(2) \setminus P_{f(6LZG_1)}(2)|=37\).
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 1:00100010011001000100110000110100000100001001001100101110000011010110010010101010110001001100000001001100100100010100100100011101110011100100000111100100011001011000011100011010000001001010001011010000010110010001001011000101010101101010010111011101110111011001001011110010101001110011010011001001110111101001110001100110100110010110110101011100010100110100011010001101101111001100110011101101011010010011110101000000010111001101110111001100101111010110001100110100011111011100000001101101000001100000010010100110011000111000010000011001101101100011011100111100101011100101110110000000111100010101
Pair
\(Z_2\)
Length of longest common subsequence
6LZG_1,7NQE_1
214
4
6LZG_1,9IUK_1
110
5
7NQE_1,9IUK_1
212
4
Newick tree
[
7NQE_1:11.80,
[
6LZG_1:55,9IUK_1:55
]:63.80
]
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 \((0.85)(0.8)(\frac{818
}{\log_{20}
818}-\frac{222}{\log_{20}222})=164.\)
Status
Protein1
Protein2
d
d1/2
Query variables
6LZG_1
7NQE_1
214
144.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]