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 \(c\).
\(|P_{f(9IJV_1)}(2) \setminus P_{f(5YCG_1)}(2)|=146\),
\(|P_{f(5YCG_1)}(2) \setminus P_{f(9IJV_1)}(2)|=50\).
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 1:000010010011011111000001010111011100010100010000011110100100100000100001000111001111101100010111000000010111010101000010101100101100110100011000000010101000111001010101000111100101100001010100100000010011100100001010001110101000011000100111100100000000000101000011011011110010011110010000011000101101100100010000011100100010011101010001000101011010000001110000111101001001010110110101111
Pair
\(Z_2\)
Length of longest common subsequence
9IJV_1,5YCG_1
196
3
9IJV_1,4WYC_1
176
4
5YCG_1,4WYC_1
186
4
Newick tree
[
5YCG_1:97.91,
[
9IJV_1:88,4WYC_1:88
]:9.91
]
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 \((0.85)(0.8)(\frac{544
}{\log_{20}
544}-\frac{157}{\log_{20}157})=112.\)
Status
Protein1
Protein2
d
d1/2
Query variables
9IJV_1
5YCG_1
142
99
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]