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 \(c\).
\(|P_{f(6VUM_1)}(2) \setminus P_{f(9GKB_1)}(2)|=52\),
\(|P_{f(9GKB_1)}(2) \setminus P_{f(6VUM_1)}(2)|=96\).
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 1:111000101000100000010000000011000100100101010011100101010100101111001100100110011000101001101100101101000000100101110010111100011001101001001001111111111100111001001011101011001110110111011111100101101110011010000001110011011111110111111110011110011110011111001011101001100011011001000000111101000101111101100000100101011011101101110110100110010111100100011010111101100111111111101111100110111011011001100011000000000001011100110001000111110001001111111110111000111101011011111111111111011100000011101111001111011110100001010000110010110010100100001100000000
Pair
\(Z_2\)
Length of longest common subsequence
6VUM_1,9GKB_1
148
4
6VUM_1,6SYB_1
183
4
9GKB_1,6SYB_1
193
3
Newick tree
[
6SYB_1:99.82,
[
6VUM_1:74,9GKB_1:74
]:25.82
]
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 \((0.85)(0.8)(\frac{1323
}{\log_{20}
1323}-\frac{558}{\log_{20}558})=195.\)
Status
Protein1
Protein2
d
d1/2
Query variables
6VUM_1
9GKB_1
250
213.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]