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 \(c\).
\(|P_{f(6MQS_1)}(2) \setminus P_{f(7ICM_1)}(2)|=140\),
\(|P_{f(7ICM_1)}(2) \setminus P_{f(6MQS_1)}(2)|=2\).
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 1:010100011111010001010011011010010010110001100101110111001100001010001010100000010101001011001100010011100100100110111101001000110111111000000000111101100011011010100101001100111110001100100110110001100001001000100001000101000
Pair
\(Z_2\)
Length of longest common subsequence
6MQS_1,7ICM_1
142
2
6MQS_1,3UIF_1
164
3
7ICM_1,3UIF_1
212
2
Newick tree
[
3UIF_1:10.45,
[
6MQS_1:71,7ICM_1:71
]:30.45
]
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 \((0.85)(0.8)(\frac{232
}{\log_{20}
232}-\frac{7}{\log_{20}7})=79.4\)
Status
Protein1
Protein2
d
d1/2
Query variables
6MQS_1
7ICM_1
97
49.5
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]