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 \(c\).
\(|P_{f(7VTF_1)}(2) \setminus P_{f(2QBR_1)}(2)|=85\),
\(|P_{f(2QBR_1)}(2) \setminus P_{f(7VTF_1)}(2)|=93\).
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 1:1000001111101010111000001001000110101011111001100111110011110111001010011000000110100011111000000101100010111110010100110100110000110010111100010001111110011011111011011001010001001001010010100101111010001101111100011001110111011000010100110111000110101110111111100110111110010110100011100000001010110110110010010
Pair
\(Z_2\)
Length of longest common subsequence
7VTF_1,2QBR_1
178
3
7VTF_1,2DPK_1
172
4
2QBR_1,2DPK_1
174
3
Newick tree
[
2QBR_1:88.66,
[
7VTF_1:86,2DPK_1:86
]:2.66
]
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 \((0.85)(0.8)(\frac{612
}{\log_{20}
612}-\frac{299}{\log_{20}299})=87.4\)
Status
Protein1
Protein2
d
d1/2
Query variables
7VTF_1
2QBR_1
110
108
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]