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 \(c\).
\(|P_{f(4QMH_1)}(2) \setminus P_{f(4LMP_1)}(2)|=69\),
\(|P_{f(4LMP_1)}(2) \setminus P_{f(4QMH_1)}(2)|=102\).
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 1:1001110111010111010011100100001000001100101110010110101101111110011000101100011100011011111000010011111101110000110111100100110011000110001110110110111000111111001111110010000100111010101000010100010011111110111011001100001100001111100101011
Pair
\(Z_2\)
Length of longest common subsequence
4QMH_1,4LMP_1
171
4
4QMH_1,1CTJ_1
164
3
4LMP_1,1CTJ_1
187
4
Newick tree
[
4LMP_1:91.98,
[
4QMH_1:82,1CTJ_1:82
]:9.98
]
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 \((0.85)(0.8)(\frac{612
}{\log_{20}
612}-\frac{241}{\log_{20}241})=104.\)
Status
Protein1
Protein2
d
d1/2
Query variables
4QMH_1
4LMP_1
129
107
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]