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 \(c\).
\(|P_{f(1IPD_1)}(2) \setminus P_{f(5AEF_1)}(2)|=147\),
\(|P_{f(5AEF_1)}(2) \setminus P_{f(1IPD_1)}(2)|=9\).
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 1:101111110111101001110110110010111110011111111101110111010001100101111101111010111001010011101000001110101101111100101100011011011110010111011010110010110000000101001101110110000001101001011011011000100110101011100001011110110011010111010111011001101111011111010110101110110101101110111010111101111100111110110010011101110011101110110011010110011
Pair
\(Z_2\)
Length of longest common subsequence
1IPD_1,5AEF_1
156
3
1IPD_1,2WZQ_1
163
4
5AEF_1,2WZQ_1
253
3
Newick tree
[
2WZQ_1:11.31,
[
1IPD_1:78,5AEF_1:78
]:36.31
]
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 \((0.85)(0.8)(\frac{373
}{\log_{20}
373}-\frac{28}{\log_{20}28})=111.\)
Status
Protein1
Protein2
d
d1/2
Query variables
1IPD_1
5AEF_1
132
71.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]