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 \(c\).
\(|P_{f(1QIV_1)}(2) \setminus P_{f(4WUN_1)}(2)|=39\),
\(|P_{f(4WUN_1)}(2) \setminus P_{f(1QIV_1)}(2)|=142\).
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 1:1001000011010011011000101010000110110011001001010011001010101010110110111001000000001001101100010101011010011001100100001001100101010101000011011010
Pair
\(Z_2\)
Length of longest common subsequence
1QIV_1,4WUN_1
181
3
1QIV_1,6EYB_1
150
3
4WUN_1,6EYB_1
165
3
Newick tree
[
4WUN_1:90.12,
[
1QIV_1:75,6EYB_1:75
]:15.12
]
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 \((0.85)(0.8)(\frac{459
}{\log_{20}
459}-\frac{148}{\log_{20}148})=92.2\)
Status
Protein1
Protein2
d
d1/2
Query variables
1QIV_1
4WUN_1
121
86
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]