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 \(c\).
\(|P_{f(4PIN_1)}(2) \setminus P_{f(1QMQ_1)}(2)|=54\),
\(|P_{f(1QMQ_1)}(2) \setminus P_{f(4PIN_1)}(2)|=104\).
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 1:10111011001110011011000101110100001110110011100110010011000100001011000010110111100110110100000011101100101100111101011110011111110011101011010100011011011001111110011010111010110011001011001111001100010110100011110111000111110001010101011001101000000101110100100101111010101111001100100010100111011011100001100111011101110
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 \((0.85)(0.8)(\frac{737
}{\log_{20}
737}-\frac{323}{\log_{20}323})=113.\)
Status
Protein1
Protein2
d
d1/2
Query variables
4PIN_1
1QMQ_1
145
126
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]