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 \(c\).
\(|P_{f(3VSH_1)}(2) \setminus P_{f(8GMK_1)}(2)|=63\),
\(|P_{f(8GMK_1)}(2) \setminus P_{f(3VSH_1)}(2)|=79\).
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 1:1011011111101110101011011011110001101111001011110000111110001100100011010001001101100101001110101000110110101100011110010101001111100111111100010001010001111110010000001111111110101100010100001100000010001101101101011001111010010101110010111111010101101110110010111110101
Pair
\(Z_2\)
Length of longest common subsequence
3VSH_1,8GMK_1
142
4
3VSH_1,7PTZ_1
153
4
8GMK_1,7PTZ_1
175
4
Newick tree
[
7PTZ_1:85.58,
[
3VSH_1:71,8GMK_1:71
]:14.58
]
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 \((0.85)(0.8)(\frac{650
}{\log_{20}
650}-\frac{271}{\log_{20}271})=105.\)
Status
Protein1
Protein2
d
d1/2
Query variables
3VSH_1
8GMK_1
126
110
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)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]