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Parikh vectors
7BOA_1 8EKZ_1 2XKY_1 Letter Amino acid
0 0 19 I Isoleucine
0 0 23 L Leucine
0 0 7 M Methionine
5 0 11 Y Tyrosine
8 8 16 A Alanine
2 0 9 Q Glutamine
4 0 29 E Glutamic acid
2 6 19 G Glycine
0 0 18 D Aspartic acid
5 0 16 K Lycine
4 0 10 F Phenylalanine
0 0 24 V Valine
0 0 20 R Arginine
0 0 17 N Asparagine
0 0 6 C Cysteine
0 0 2 W Tryptophan
0 0 10 H Histidine
0 0 12 P Proline
0 0 27 S Serine
0 2 14 T Threonine 

7BOA_1|Chains A, B, C, D, E, F, G, H|CC-Type2-(YaFd)4-W19(BrPhe)|synthetic construct (32630)
>8EKZ_1|Chain A[auth C]|DNA (5'-D(*AP*AP*TP*AP*AP*GP*GP*AP*GP*AP*AP*GP*TP*AP*GP*G)-3')|synthetic construct (32630)
>2XKY_1|Chains A[auth I], B[auth J], C[auth K], D[auth L]|INWARD RECTIFIER POTASSIUM CHANNEL 2|MUS MUSCULUS (10090)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
7BOA , Knot 13 32 0.46 16 14 17 
XGEFAQAYKEFAKAYKEFAYAYKEFAQAYKGX
8EKZ , Knot 7 16 0.40 6 7 12 
AATAAGGAGAAGTAGG
2XKY , Knot 136 309 0.84 40 198 296 
MGSVRTNRYSIVSSEEDGMKLATMAVANGFGNGKSKVHTRQQCRSRFVKKDGHCNVQFINVGEKGQRGGRNETLVFSHNAVIAMRDGKLCLMWRVGNLRKSHLVEAHVRAQLLKSRITSEGEYIPLDQIDINVGFDSGIDRIFLVSPITIVHEIDEDSPLYDLSKQDIDNADFEIVVILEGMVEATAMTTQCRSSYLANEILWGHRYEPVLFEEKHYYKVDYSRFHKTYEVPNTPLCSARDLAEKKYILSNANSFCYENEVALTSKEEEEDSENGVPESTSTDSPPGIDLHNQASVPLEPRPLRRESEI

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 \(c\).

\(|P_{f(7BOA_1)}(2) \setminus P_{f(8EKZ_1)}(2)|=14\), \(|P_{f(8EKZ_1)}(2) \setminus P_{f(7BOA_1)}(2)|=7\). 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 1:01011010001101000110100011010010
Pair \(Z_2\) Length of longest common subsequence
7BOA_1,8EKZ_1 21 1
7BOA_1,2XKY_1 200 2
8EKZ_1,2XKY_1 199 3

Newick tree

 
[
	2XKY_1:11.02,
	[
		7BOA_1:10.5,8EKZ_1:10.5
	]:10.52
]

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 \((0.85)(0.8)(\frac{48 }{\log_{20} 48}-\frac{16}{\log_{20}16})=13.5\)
Status Protein1 Protein2 d d1/2
Query variables 7BOA_1 8EKZ_1 10 8
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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