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Parikh vectors
2PZV_1 5RFN_1 6LKO_1 Letter Amino acid
2 3 7 W Tryptophan
4 11 22 Y Tyrosine
13 27 43 V Valine
5 21 26 N Asparagine
9 9 34 E Glutamic acid
4 17 15 F Phenylalanine
3 16 34 S Serine
7 29 40 L Leucine
4 24 20 T Threonine
7 17 26 D Aspartic acid
13 26 43 G Glycine
3 7 24 H Histidine
7 11 21 I Isoleucine
9 11 20 R Arginine
3 12 9 C Cysteine
7 10 19 M Methionine
8 13 22 P Proline
12 17 38 A Alanine
10 14 16 Q Glutamine
1 11 23 K Lycine 

2PZV_1|Chains A, B, C, D|Steroid Delta-isomerase|Pseudomonas putida (303)
>5RFN_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>6LKO_1|Chain A|Asparaginyl endopeptidase|Clitoria ternatea (43366)
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 \(w=f(c)\)
2PZV , Knot 65 131 0.80 40 110 125 
MNLPTAQEVQGLMARYIELVDVGDIEAIVQMYADDATVENPFGQPPIHGREQIAAFYRQGLGGGKVRACLTGPVRASHNGCGAMPFRVEMVWNGQPCALDVIDVMRFDEHGRIQTMQAYWSEVNLSVREPQ
5RFN , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
6LKO , Knot 200 502 0.82 40 259 467 
MVSAIVLYVLLAAAAHSAFAAAMGHHHHHHSSGVDLGTENLYFQSMARLNPQKEWDSVIRLPTEPVDADTDEVGTRWAVLVAGSNGYENYRHQADVCHAYQLLIKGGLKEENIVVFMYDDIAWHELNPRPGVIINNPRGEDVYAGVPKDYTGEDVTAENLFAVILGDRSKVKGGSGKVINSKPEDRIFIFYSNHGGAGVLGMPNEQILYAMDFIDVLKKKHASGGYREMVIYVEACESGSLFEGIMPKDLNVFVTTASNAQENSWVTYCPGTEPSPPPEYTTCLGDLYSVAWMEDSESHNLRRETVNQQYRSVKERTSNFKDYAMGSHVMQYGDTNITAEKLYLFQGFDPATVNLPPHNGRIEAKMEVVHQRDAELLFMWQMYQRSNHLLGKKTHILKQIAETVKHRNHLDGSVELIGVLLYGPGKGSPVLQSVRDPGLPLVDNWACLKSMVRVFESHCGSLTQYGMKHMRAFANICNSGVSESSMEEACMVACGGHDAGHL

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 \(c\).

\(|P_{f(2PZV_1)}(2) \setminus P_{f(5RFN_1)}(2)|=44\), \(|P_{f(5RFN_1)}(2) \setminus P_{f(2PZV_1)}(2)|=130\). 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 1:10110100101111001011011010111010100101001110111010001111000111110101010111010001011111010111010101101101101000101001010100101010010
Pair \(Z_2\) Length of longest common subsequence
2PZV_1,5RFN_1 174 3
2PZV_1,6LKO_1 203 3
5RFN_1,6LKO_1 173 4

Newick tree

 
[
	2PZV_1:97.05,
	[
		5RFN_1:86.5,6LKO_1:86.5
	]:10.55
]

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 \((0.85)(0.8)(\frac{437 }{\log_{20} 437}-\frac{131}{\log_{20}131})=91.6\)
Status Protein1 Protein2 d d1/2
Query variables 2PZV_1 5RFN_1 117 83.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]