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 \(c\).
\(|P_{f(4KFQ_1)}(2) \setminus P_{f(1GAV_1)}(2)|=137\),
\(|P_{f(1GAV_1)}(2) \setminus P_{f(4KFQ_1)}(2)|=32\).
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 1:1100010110100011101010100100000101010110011001100001101000110000110101110110010100010111010110000100000001011110110101011111101000010010100110001101110010010110010100100011010100001010100010100100010000000110110110000101111001110101000001100101110011111100001100010101100000111001000110000000
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 \((0.85)(0.8)(\frac{421
}{\log_{20}
421}-\frac{129}{\log_{20}129})=87.8\)
Status
Protein1
Protein2
d
d1/2
Query variables
4KFQ_1
1GAV_1
114
79
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]