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Parikh vectors
5VJS_1 5JUP_1 2IHD_1 Letter Amino acid
2 1 2 W Tryptophan
2 17 8 V Valine
7 10 10 R Arginine
0 0 2 C Cysteine
27 13 15 E Glutamic acid
3 9 3 P Proline
3 5 8 D Aspartic acid
0 2 4 M Methionine
5 7 9 T Threonine
9 6 2 I Isoleucine
38 19 17 L Leucine
17 27 11 K Lycine
1 7 12 F Phenylalanine
7 16 11 A Alanine
1 8 4 N Asparagine
20 7 4 G Glycine
5 2 9 H Histidine
5 9 14 S Serine
1 5 4 Y Tyrosine
43 6 6 Q Glutamine 

5VJS_1|Chain A|Reaction Center Maquette|synthetic construct (32630)
>5JUP_10|Chain J|eL6 (yeast L6)|Saccharomyces cerevisiae (4932)
>2IHD_1|Chain A|Regulator of G-protein signaling 8|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 \(w=f(c)\)
5VJS , Knot 60 196 0.53 36 70 115 
GSPELRQEHQQLAQEFQQLLQEIQQLGRELLKGELQGIKQLREASEKARNPEKKSVLQKILEDEEKHIELLETLQQTGQEAQQLLQELQQTGQELWQLGGSGGPELRQKHQQLAQKIQQLLQKHQQLGAKILEDEEKHIELLETILGGSGGDELRELLKGELQGIKQYRELQQLGQKAQQLVQKLQQTGQKLWQLG
5JUP , Knot 82 176 0.80 38 120 169 
MSAQKAPKWYPSEDVAALKKTRKAARPQKLRASLVPGTVLILLAGRFRGKRVVYLKHLEDNTLLISGPFKVNGVPLRRVNARYVIATSTKVSVEGVNVEKFNVEYFAKEKLTKKEKKEANLFPEQQNKEIKAERVEDQKVVDKALIAEIKKTPLLKQYLSASFSLKNGDKPHMLKF
2IHD , Knot 74 155 0.80 40 112 145 
MHHHHHHSSGVDLGTENLYFQSMLKRLSTEEATRWADSFDVLLSHKYGVAAFRAFLKTEFSEENLEFWLACEEFKKTRSTAKLVSKAHRIFEEFVDVQAPREVNIDFQTREATRKNLQEPSLTCFDQAQGKVHSLMEKDSYPRFLRSKMYLDLLS

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 \(c\).

\(|P_{f(5VJS_1)}(2) \setminus P_{f(5JUP_1)}(2)|=37\), \(|P_{f(5JUP_1)}(2) \setminus P_{f(5VJS_1)}(2)|=87\). 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 1:1010100000011001001100100110011010101100100100010010000110011000000101100100010010011001000100110111011101000000110010011000001110110000001011001111011001001101010110000010011001001100100010011011
Pair \(Z_2\) Length of longest common subsequence
5VJS_1,5JUP_1 124 3
5VJS_1,2IHD_1 130 3
5JUP_1,2IHD_1 132 4

Newick tree

 
[
	2IHD_1:66.62,
	[
		5VJS_1:62,5JUP_1:62
	]:4.62
]

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 \((0.85)(0.8)(\frac{372 }{\log_{20} 372}-\frac{176}{\log_{20}176})=58.6\)
Status Protein1 Protein2 d d1/2
Query variables 5VJS_1 5JUP_1 68 59.5
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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