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Parikh vectors
2OBZ_1 5DOF_1 5EGT_1 Letter Amino acid
0 5 2 H Histidine
0 16 12 L Leucine
0 3 4 P Proline
0 9 7 Y Tyrosine
0 7 7 V Valine
0 4 5 R Arginine
0 12 7 N Asparagine
0 11 8 D Aspartic acid
3 3 10 G Glycine
0 11 4 F Phenylalanine
0 10 4 S Serine
0 4 9 T Threonine
0 13 5 I Isoleucine
0 1 4 M Methionine
0 0 1 W Tryptophan
0 8 15 A Alanine
2 1 0 C Cysteine
0 21 6 Q Glutamine
0 8 13 E Glutamic acid
0 16 20 K Lycine 

2OBZ_1|Chains A, B|5'-D(*CP*GP*CP*GP*(BRU)P*G)-3'|
>5DOF_1|Chains A, B, C, D|Telomerase-associated protein 19|Tetrahymena thermophila (5911)
>5EGT_1|Chain A|Thermonuclease|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 \(w=f(c)\)
2OBZ , Knot 4 6 0.39 6 4 4 
CGCGUG
5DOF , Knot 78 163 0.81 38 118 158 
QQPKRNFDLYKLITDKQIDFQVADLIQDEQSSFVSVRIYGQFKCFVPKSTIQEQLDKIKNLSSKELAKNKIFKFLSEYNKNNQKQDELSHDYYGYFKVQQHQFILNLENAQREASLAVDDFYFINGRIYKTNHDILILQAHHVYQMQKPTLQLLQAASEINQN
5EGT , Knot 67 143 0.77 38 106 135 
ATSTKKLHKEPATLIKAIDGDTVKLMYKGQPMTFRLLLVDTPEFNEKYGPEASAFTKKMEENAKKIEVEFDKGQRTDKYGRGLAYIYADGKMVNEALVRQGLAKVAYVYKGNNTHEQLLRKAEAQAKKEKLNIWSEDNADSGQ

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 \(c\).

\(|P_{f(2OBZ_1)}(2) \setminus P_{f(5DOF_1)}(2)|=4\), \(|P_{f(5DOF_1)}(2) \setminus P_{f(2OBZ_1)}(2)|=118\). 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
2OBZ_1,5DOF_1 122 1
2OBZ_1,5EGT_1 110 1
5DOF_1,5EGT_1 148 3

Newick tree

 
[
	5DOF_1:71.57,
	[
		2OBZ_1:55,5EGT_1:55
	]:16.57
]

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 \((0.85)(0.8)(\frac{169 }{\log_{20} 169}-\frac{6}{\log_{20}6})=60.2\)
Status Protein1 Protein2 d d1/2
Query variables 2OBZ_1 5DOF_1 76 39.5
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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