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 \(c\).
\(|P_{f(8PRW_1)}(2) \setminus P_{f(8XBH_1)}(2)|=154\),
\(|P_{f(8XBH_1)}(2) \setminus P_{f(8PRW_1)}(2)|=17\).
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 1:101010001101110011100110110110000111001000011011101011111000100000000111010001100000100100010100111000110001111011101111111111111111111111011001101011101111001000100111000100111100010001110110011001001000110011001000101111000001100110001111101011000100011110100011111110001110110010101110011000101111010011010111111111111110111100100000111000101110010101001000110000011010101001010110111011100100001001000101100011010101101110010001100110110000011101100010000000100001100110011000001101011000110101100110001010000010001001000100111111100000101000100100110101000010000010100100011011000001010000011110100110000000110110000100010001100011110100001110100000100111011001110110100001110111010111011011101110111000010001000000101001101001111110010000101110110000001111101011111111100110100100000110011100110111010000010110001101111101001011101100000101001100100001100101011111100101110100111011001110010000111011111010110100001111010111011101001010100011000010011010011000110100101101010101010101011010100010011110101110100111101110111110100010101110101010101111111100001010101001110000001100001010000011000110110101101001000011001110001011010000100100001001011011001000101101101011011010011110110110100011000110010110111110110111101100100100010100110001011111011011100010001100011000110010111011110001110011110100100101110011010101011110001000101011010100001001001001101001100000111010111101110101110111110111111101000110011111011100100000010010101010000001100010100110001011010100110000001110000010001000101100011001000010111101111001101001111010100001000000101001100110001011111100110101011111111011101100111110001001001100100110100010001101101001111001101111010010111000000001101010000100110011100011100001100001000101011101000000101010000100000010100100101100100001001111101011001010000110001010010000101010001110101001110011100111111100101101000111101010100110011100101010000101111110000
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 \((0.85)(0.8)(\frac{2241
}{\log_{20}
2241}-\frac{354}{\log_{20}354})=468.\)
Status
Protein1
Protein2
d
d1/2
Query variables
8PRW_1
8XBH_1
596
350.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]