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 \(c\).
\(|P_{f(3JPT_1)}(2) \setminus P_{f(5QDH_1)}(2)|=75\),
\(|P_{f(5QDH_1)}(2) \setminus P_{f(3JPT_1)}(2)|=81\).
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 1:10000110001011100110011010001001100001000110111001001001101001111100110010011101010010010000000010110010111101100110011001001000000100000111001101000110001101001110010010000110101010011000101011100101000000010110011001001011000100100011110011000000001000101011100000011101010011000101011001101000010111101111011110000011001010000100000
Pair
\(Z_2\)
Length of longest common subsequence
3JPT_1,5QDH_1
156
3
3JPT_1,7QWN_1
164
3
5QDH_1,7QWN_1
164
4
Newick tree
[
7QWN_1:83.29,
[
3JPT_1:78,5QDH_1:78
]:5.29
]
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 \((0.85)(0.8)(\frac{656
}{\log_{20}
656}-\frac{321}{\log_{20}321})=92.7\)
Status
Protein1
Protein2
d
d1/2
Query variables
3JPT_1
5QDH_1
116
112.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]