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 \(c\).
\(|P_{f(7FFR_1)}(2) \setminus P_{f(3KLL_1)}(2)|=57\),
\(|P_{f(3KLL_1)}(2) \setminus P_{f(7FFR_1)}(2)|=119\).
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 1:010000110010011000101111111100100011000101100100110001100010000001001110010011010000101100010001000001100000010110110101011001100000011000100110001010110111110000111000001100110111010010001010010001001010011001101111101000110101101001100100100000010110010001011001000000100010000001110000011010000111011011101100001000010100111111100010001000111110110000001000011101000010110010111001001100000110011000100111100000110100101101000110110000001010011010001011100101100001100010000000110001010110100001100101001010101001000000100001110110001010000101110011111000001000000110010011110011111
Pair
\(Z_2\)
Length of longest common subsequence
7FFR_1,3KLL_1
176
5
7FFR_1,1ADZ_1
216
3
3KLL_1,1ADZ_1
266
4
Newick tree
[
1ADZ_1:13.33,
[
7FFR_1:88,3KLL_1:88
]:42.33
]
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 \((0.85)(0.8)(\frac{1624
}{\log_{20}
1624}-\frac{585}{\log_{20}585})=260.\)
Status
Protein1
Protein2
d
d1/2
Query variables
7FFR_1
3KLL_1
335
260
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]