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Parikh vectors
7UPA_1 9EXA_1 8QWL_1 Letter Amino acid
8 9 8 Y Tyrosine
6 3 7 Q Glutamine
11 30 17 L Leucine
6 11 5 F Phenylalanine
5 7 1 W Tryptophan
14 10 20 S Serine
10 10 16 T Threonine
3 14 19 R Arginine
4 8 9 N Asparagine
1 4 9 H Histidine
10 4 8 K Lycine
10 17 9 A Alanine
7 5 13 E Glutamic acid
10 10 14 V Valine
3 3 5 M Methionine
3 5 18 P Proline
5 3 9 D Aspartic acid
3 4 11 C Cysteine
13 13 15 G Glycine
6 18 6 I Isoleucine 

7UPA_1|Chains A[auth B], F[auth C], H|Fab 1H8 heavy chain|Mus musculus (10090)
>9EXA_1|Chains A, B|Membrane protein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8QWL_1|Chain A|Cellular tumor antigen p53|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 \(w=f(c)\)
7UPA , Knot 74 138 0.88 40 115 136 
MEFGLSWIFLAAILKGVQCEVQLQQSGPELVKPGASVKISCKASGYSFTGYTMNWVKQSHGKNLEWIGLINPFIGGTRYNQKFKGKATLTVDKSSRTAYMELLSLTSEDSAVYYCAREADYDWYFDVWGAGTTVTVSS
9EXA , Knot 89 188 0.82 40 131 182 
LEQWNLVIGFLFLTWICLLQFAYANRNRFLYIIKLIFLWLLWPVTLACFVLAAVYRINWITGGIAIAMACLVGLMWLSYFIASFRLFARTRSMWSFNPETNILLNVPLHGTILTRPLLESELVIGAVILRGHLRIAGHHLGRCDIKDLPKEITVATSRTLSYYKLGASQRVAGDSGFAAYSRYRIGNY
8QWL , Knot 102 219 0.83 40 150 211 
SSSVPSQKTYQGSYGFRLGFLHSGTAKSVTCTYSPALNKLFCQLAKTCPVQLWVDSTPPPGTRVRAMAICKQSQHMTEVVRRCPHHERCSDSDGLAPPQHLIRVEGNLRAEYLDDRNTFRHSVVVPYEPPEVGSDCTTIHYNYMCYSSCMGGMNRRPILTIITLEDSSGNLLGRDSFEVRVCACPGRDRRTEEENLRKKGEPHHELPPGSTKRALPNNT

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 \(c\).

\(|P_{f(7UPA_1)}(2) \setminus P_{f(9EXA_1)}(2)|=65\), \(|P_{f(9EXA_1)}(2) \setminus P_{f(7UPA_1)}(2)|=81\). 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 1:101110111111110110001010001101101110101000101001010010110000100101111101111100000010101010100000010101101000001100010010001010111110010100
Pair \(Z_2\) Length of longest common subsequence
7UPA_1,9EXA_1 146 3
7UPA_1,8QWL_1 177 3
9EXA_1,8QWL_1 171 3

Newick tree

 
[
	8QWL_1:91.20,
	[
		7UPA_1:73,9EXA_1:73
	]:18.20
]

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 \((0.85)(0.8)(\frac{326 }{\log_{20} 326}-\frac{138}{\log_{20}138})=57.7\)
Status Protein1 Protein2 d d1/2
Query variables 7UPA_1 9EXA_1 71 62.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]