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Parikh vectors
4UZZ_1 3GSJ_1 1RPC_1 Letter Amino acid
8 0 1 N Asparagine
6 0 1 D Aspartic acid
2 3 4 C Cysteine
2 3 4 G Glycine
3 0 0 M Methionine
2 0 1 W Tryptophan
1 0 0 R Arginine
5 0 0 H Histidine
8 0 0 K Lycine
5 0 2 F Phenylalanine
12 0 0 E Glutamic acid
10 0 1 I Isoleucine
10 0 1 L Leucine
9 0 1 S Serine
3 2 0 T Threonine
2 0 1 Y Tyrosine
5 0 2 V Valine
5 4 2 A Alanine
10 0 0 Q Glutamine
8 0 0 P Proline 

4UZZ_1|Chain A|INTRAFLAGELLAR TRANSPORT COMPLEX B PROTEIN 46 CARBOXY-TERMINAL PROTEIN|TETRAHYMENA THERMOPHILA (5911)
>3GSJ_1|Chain A|5'-D(*CP*GP*GP*AP*AP*AP*TP*TP*AP*CP*CP*G)-3'|null
>1RPC_1|Chain A|Tricyclic peptide RP 71955|actinomycete Sp9440 (66694)
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 \(w=f(c)\)
4UZZ , Knot 58 116 0.79 40 96 114 
GAASPKQIQMWINNVAEIRKTKQPHSVSYTKPMPEIDELMQEWPQEIEEILQHLKIPSEELDFNLSDFCKLACAILDIPVHDQPNESNVIESLHVLFTLYSEFKSNQHFQQNKNDG
3GSJ , Knot 7 12 0.48 8 9 10 
CGGAAATTACCG
1RPC , Knot 14 21 0.67 24 20 19 
CLGIGSCNDFAGCGYAVVCFW

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 \(c\).

\(|P_{f(4UZZ_1)}(2) \setminus P_{f(3GSJ_1)}(2)|=93\), \(|P_{f(3GSJ_1)}(2) \setminus P_{f(4UZZ_1)}(2)|=6\). 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 1:11101001011100110100000100100001110100110011001001100101100010101001001101110111000100001100101110100010000010000001
Pair \(Z_2\) Length of longest common subsequence
4UZZ_1,3GSJ_1 99 3
4UZZ_1,1RPC_1 112 2
3GSJ_1,1RPC_1 27 2

Newick tree

 
[
	4UZZ_1:60.52,
	[
		3GSJ_1:13.5,1RPC_1:13.5
	]:47.02
]

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 \((0.85)(0.8)(\frac{128 }{\log_{20} 128}-\frac{12}{\log_{20}12})=43.9\)
Status Protein1 Protein2 d d1/2
Query variables 4UZZ_1 3GSJ_1 54 29.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]