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Parikh vectors
3RRY_1 2LPR_1 8HWU_1 Letter Amino acid
4 13 4 N Asparagine
11 9 0 Q Glutamine
11 32 5 G Glycine
8 2 9 K Lycine
8 20 3 S Serine
11 12 2 R Arginine
14 2 4 D Aspartic acid
3 4 5 P Proline
9 4 1 Y Tyrosine
15 19 3 V Valine
11 25 1 A Alanine
13 4 1 E Glutamic acid
11 10 1 L Leucine
5 6 2 F Phenylalanine
11 18 2 T Threonine
3 6 10 C Cysteine
3 1 1 H Histidine
11 8 3 I Isoleucine
4 1 1 M Methionine
0 2 0 W Tryptophan 

3RRY_1|Chain A|GTPase HRas|Homo sapiens (9606)
>2LPR_1|Chain A|ALPHA-LYTIC PROTEASE|Lysobacter enzymogenes (69)
>8HWU_1|Chain A|LL-TIL|Lepidobatrachus laevis (8376)
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 \(w=f(c)\)
3RRY , Knot 79 166 0.81 38 130 163 
MTEYKLVVVGAGGVGKSALTIQLIQNHFVDEYDPTIEDSYRKQVVIDGETCLLDILDTAGQEEYSAMRDQYMRTGEGFLCVFAINNTKSFEDIHQYREQIKRVKDSDDVPMVLVGNKCDLAARTVESRQAQDLARSYGIPYIETSAKTRQGVEDAFYTLVREIRQH
2LPR , Knot 92 198 0.82 40 127 188 
ANIVGGIEYSINNASLCSVGFSVTRGATKGFVTAGHCGTVNATARIGGAVVGTFAARVFPGNDRAWVSLTSAQTLLPRVANGSSFVTVRGSTEAAVGAAVCRSGRTTGYQCGTITAKNVTANYAEGAVRGLTQGNACAGRGDSGGSWITSAGQAQGVMSGGNVQSNGNNCGIPASQRSSLFERLQPILSQYGLSLVTG
8HWU , Knot 33 58 0.77 36 50 56 
GSMIRCPKDKIYKFCGSPCPPSCKDLTPNCIAVCKKGCFCRDGTVDNNHGKCVKKENC

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 \(c\).

\(|P_{f(3RRY_1)}(2) \setminus P_{f(2LPR_1)}(2)|=80\), \(|P_{f(2LPR_1)}(2) \setminus P_{f(3RRY_1)}(2)|=77\). 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 1:1000011111111110011010110001100001010000000111010001101100110000011000010010111011110000010010000001001000001111111000011100100001001100011101000100001100110011001000
Pair \(Z_2\) Length of longest common subsequence
3RRY_1,2LPR_1 157 3
3RRY_1,8HWU_1 150 2
2LPR_1,8HWU_1 155 3

Newick tree

 
[
	2LPR_1:78.97,
	[
		3RRY_1:75,8HWU_1:75
	]:3.97
]

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 \((0.85)(0.8)(\frac{364 }{\log_{20} 364}-\frac{166}{\log_{20}166})=59.5\)
Status Protein1 Protein2 d d1/2
Query variables 3RRY_1 2LPR_1 74 69.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]