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Parikh vectors
6COQ_1 8OIN_1 3BCI_1 Letter Amino acid
0 3 2 C Cysteine
0 12 5 G Glycine
2 24 16 L Leucine
1 29 12 A Alanine
1 16 14 E Glutamic acid
0 4 11 H Histidine
1 4 3 M Methionine
0 1 2 W Tryptophan
0 1 10 Y Tyrosine
3 9 2 R Arginine
1 10 9 I Isoleucine
1 8 10 S Serine
0 6 8 T Threonine
0 4 8 N Asparagine
1 6 15 D Aspartic acid
1 9 6 Q Glutamine
2 19 32 K Lycine
3 3 6 F Phenylalanine
0 16 6 P Proline
0 14 9 V Valine 

6COQ_1|Chain A|Competence-stimulating peptide type 1|Streptococcus pneumoniae (1313)
>8OIN_1|Chains A[auth B1], B[auth B2], C[auth B3], D[auth B4], E[auth B5], F[auth B6]|Mitochondrial ribosomal protein L12|Sus scrofa (9823)
>3BCI_1|Chain A|Disulfide bond protein A|Staphylococcus aureus (1280)
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 \(w=f(c)\)
6COQ , Knot 13 17 0.72 22 16 15 
EMRLSAFFRDFILQRKK
8OIN , Knot 89 198 0.79 40 125 185 
MLPAAASSLWGPCFGLRAAALRVARHQGPRLCGVRLMRCSSHRKGEALAGAPLDNAPKEYPPKIQQLVQDIASLTLLEISDLNELLKKTLKIQDVGFMPMGAVAPGAPPAAAAPEAAEEDLPKRKEQTHFTVRLTEAKPVDKVKLIKEIKSHIQGINLVQAKKLVESLPQEIKANVPKAEAEKIKAALEAVGGTVVLE
3BCI , Knot 82 186 0.76 40 124 170 
MASATTSSKNGKPLVVVYGDYKCPYCKELDEKVMPKLRKNYIDNHKVEYQFVNLAFLGKDSIVGSRASHAVLMYAPKSFLDFQKQLFAAQQDENKEWLTKELLDKHIKQLHLDKETENKIIKDYKTKDSKSWKAAEKDKKIAKDNHIKTTPTAFINGEKVEDPYDYESYEKLLKDKIKLEHHHHHH

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 \(c\).

\(|P_{f(6COQ_1)}(2) \setminus P_{f(8OIN_1)}(2)|=12\), \(|P_{f(8OIN_1)}(2) \setminus P_{f(6COQ_1)}(2)|=121\). 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 1:01010111001110000
Pair \(Z_2\) Length of longest common subsequence
6COQ_1,8OIN_1 133 2
6COQ_1,3BCI_1 128 2
8OIN_1,3BCI_1 145 3

Newick tree

 
[
	8OIN_1:71.32,
	[
		6COQ_1:64,3BCI_1:64
	]:7.32
]

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 \((0.85)(0.8)(\frac{215 }{\log_{20} 215}-\frac{17}{\log_{20}17})=69.3\)
Status Protein1 Protein2 d d1/2
Query variables 6COQ_1 8OIN_1 85 46
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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