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Parikh vectors
1VTY_1 1XGB_1 3DOG_1 Letter Amino acid
0 1 17 E Glutamic acid
0 1 10 H Histidine
0 1 19 P Proline
0 1 15 S Serine
1 0 17 T Threonine
0 1 10 Y Tyrosine
0 0 23 V Valine
1 1 19 A Alanine
0 1 8 N Asparagine
0 0 16 D Aspartic acid
2 4 3 C Cysteine
0 0 9 Q Glutamine
0 0 16 I Isoleucine
0 0 39 L Leucine
0 0 16 F Phenylalanine
0 1 16 R Arginine
0 0 21 K Lycine
0 0 5 M Methionine
0 0 4 W Tryptophan
2 1 16 G Glycine 

1VTY_1|Chains A, B|DNA (5'-D(*CP*(NH2)AP*CP*GP*TP*G)-3')|synthetic construct (32630)
>1XGB_1|Chain A|ALPHA-CONOTOXIN GI|null
>3DOG_1|Chains A, C|Cell division protein kinase 2|Homo sapiens (9606)
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 \(w=f(c)\)
1VTY , Knot 5 6 0.49 8 5 4 
CACGTG
1XGB , Knot 11 14 0.69 22 13 12 
ECCNPACGRHYSCX
3DOG , Knot 135 299 0.85 40 191 285 
SMENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL

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 \(c\).

\(|P_{f(1VTY_1)}(2) \setminus P_{f(1XGB_1)}(2)|=3\), \(|P_{f(1XGB_1)}(2) \setminus P_{f(1VTY_1)}(2)|=11\). 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 1:010101
Pair \(Z_2\) Length of longest common subsequence
1VTY_1,1XGB_1 14 3
1VTY_1,3DOG_1 192 2
1XGB_1,3DOG_1 198 2

Newick tree

 
[
	3DOG_1:11.52,
	[
		1VTY_1:7,1XGB_1:7
	]: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 \((0.85)(0.8)(\frac{20 }{\log_{20} 20}-\frac{6}{\log_{20}6})=6.77\)
Status Protein1 Protein2 d d1/2
Query variables 1VTY_1 1XGB_1 8 5.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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