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Parikh vectors
8SDE_1 3RJV_1 7HGO_1 Letter Amino acid
0 2 0 C Cysteine
12 11 12 K Lycine
4 6 2 T Threonine
8 10 2 V Valine
5 9 2 R Arginine
2 4 5 N Asparagine
9 19 1 Q Glutamine
1 20 9 G Glycine
2 5 8 F Phenylalanine
1 4 2 W Tryptophan
0 10 5 Y Tyrosine
7 15 7 D Aspartic acid
0 3 4 H Histidine
6 6 4 I Isoleucine
1 5 12 P Proline
2 15 3 S Serine
6 33 7 A Alanine
18 15 3 E Glutamic acid
8 17 4 L Leucine
0 3 2 M Methionine 

8SDE_1|Chain A|Tripartite motif-containing protein 29|Mus musculus (10090)
>3RJV_1|Chain A|Putative Sel1 repeat protein|Klebsiella pneumoniae subsp. pneumoniae (272620)
>7HGO_1|Chains A, B, C|Hepatoma-derived growth factor-related protein 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 \(w=f(c)\)
8SDE , Knot 44 92 0.72 32 67 87 
GPTVEEAKAEKETELSLQKEQLQLKIIEIEDDVEKWQKEKDRIKSFTTNEKAILEQNFRDLVRELEKQKEEVRAALEQREQDAVDQVKVIVD
3RJV , Knot 95 212 0.80 40 131 201 
GATEPGSQYQQQAEAGDRRAQYYLADTWVSSGDYQKAEYWAQKAAAQGDGDALALLAQLKIRNPQQADYPQARQLAEKAVEAGSKSGEIVLARVLVNRQAGATDVAHAITLLQDAARDSESDAAVDAQMLLGLIYASGVHGPEDDVKASEYFKGSSSLSRTGYAEYWAGMMFQQGEKGFIEPNKQKALHWLNVSCLEGFDTGCEEFDRISKG
7HGO , Knot 51 94 0.82 38 77 90 
GMPHAFKPGDLVFAKMKGYPHWPARIDDIADGAVKPPPNKYPIFFFGTHETAFLGPKDLFPYDKSKDKYGKPNKRKGFNEGLWEIQNNPHASYS

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 \(c\).

\(|P_{f(8SDE_1)}(2) \setminus P_{f(3RJV_1)}(2)|=30\), \(|P_{f(3RJV_1)}(2) \setminus P_{f(8SDE_1)}(2)|=94\). 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 1:11010010100000101000010101101000100100000010010000011100010011001000000101110000001100101110
Pair \(Z_2\) Length of longest common subsequence
8SDE_1,3RJV_1 124 4
8SDE_1,7HGO_1 116 3
3RJV_1,7HGO_1 148 3

Newick tree

 
[
	3RJV_1:71.35,
	[
		8SDE_1:58,7HGO_1:58
	]:13.35
]

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 \((0.85)(0.8)(\frac{304 }{\log_{20} 304}-\frac{92}{\log_{20}92})=66.8\)
Status Protein1 Protein2 d d1/2
Query variables 8SDE_1 3RJV_1 80 55.5
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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