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 \(c\).
\(|P_{f(8FTP_1)}(2) \setminus P_{f(3GXG_1)}(2)|=97\),
\(|P_{f(3GXG_1)}(2) \setminus P_{f(8FTP_1)}(2)|=49\).
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 1:111101011011110010011111011010000100110110001000011000101111001100011111001101000110010001111110111011101010001011101011111000111001110010010000100000100010010100010010011001000100110111110100001101001001000100000010100001011010000001100100110010100
Pair
\(Z_2\)
Length of longest common subsequence
8FTP_1,3GXG_1
146
3
8FTP_1,5LZH_1
170
3
3GXG_1,5LZH_1
146
3
Newick tree
[
5LZH_1:81.19,
[
8FTP_1:73,3GXG_1:73
]:8.19
]
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 \((0.85)(0.8)(\frac{406
}{\log_{20}
406}-\frac{157}{\log_{20}157})=74.4\)
Status
Protein1
Protein2
d
d1/2
Query variables
8FTP_1
3GXG_1
93
75
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]