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Parikh vectors
2DCP_1 5KYB_1 5RGJ_1 Letter Amino acid
4 17 11 Y Tyrosine
9 12 17 A Alanine
10 14 11 R Arginine
12 25 9 E Glutamic acid
8 12 11 I Isoleucine
5 14 21 N Asparagine
7 30 17 D Aspartic acid
3 16 24 T Threonine
1 7 12 C Cysteine
10 34 29 L Leucine
2 10 10 M Methionine
5 14 17 F Phenylalanine
5 13 13 P Proline
20 17 16 S Serine
2 21 14 Q Glutamine
10 22 26 G Glycine
4 14 7 H Histidine
12 31 11 K Lycine
1 2 3 W Tryptophan
10 26 27 V Valine 

2DCP_1|Chain A|hypothetical protein (RAFL09-17-B18)|Arabidopsis thaliana (3702)
>5KYB_1|Chains A, B|Ubiquitin carboxyl-terminal hydrolase 7|Homo sapiens (9606)
>5RGJ_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
2DCP , Knot 68 140 0.80 40 102 130 
GSSGSSGESYWRSRMIDAVTSDEDKVAPVYKLEEICDLLRSSHVSIVKEFSEFILKRLDNKSPIVKQKALRLIKYAVGKSGSEFRREMQRNSVAVRNLFHYKGHPDPLKGDALNKAVRETAHETISAIFSEENGSGPSSG
5KYB , Knot 152 351 0.84 40 211 343 
GSHMKKHTGYVGLKNQGATCYMNSLLQTLFFTNQLRKAVYMMPTEGDDSSKSVPLALQRVFYELQHSDKPVGTKKLTKSFGWETLDSFMQHDVQELCRKLLDNVENKMKGTCVEGTIPKLFRGKMVSYIQCKEVDYRSDRREDYYDIQLSIKGKKNIFESFVDYVAVEQLDGDNKYDAGEHGLQEAEKGVKFLTLPPVLHLQLMRFMYDPQTDQNIKINDRFEFPEQLPLDEFLQKTDPKDPANYILHAVLVHSGDNHGGHYVVYLNPKGDGKWCKFDDDVVSRCTKEEAIEHNYGGHDDDLSVRHCTNAYMLVYIRESKLSEVLQAVTDHDIPQQLVERLQEEKRIEAQK
5RGJ , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ

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 \(c\).

\(|P_{f(2DCP_1)}(2) \setminus P_{f(5KYB_1)}(2)|=36\), \(|P_{f(5KYB_1)}(2) \setminus P_{f(2DCP_1)}(2)|=145\). 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 1:10010010001000110110000001111001001001100001011001001110010000111000110110011100100100010000111001100010101101011001100010001011100001011001
Pair \(Z_2\) Length of longest common subsequence
2DCP_1,5KYB_1 181 3
2DCP_1,5RGJ_1 194 3
5KYB_1,5RGJ_1 181 4

Newick tree

 
[
	5RGJ_1:94.88,
	[
		2DCP_1:90.5,5KYB_1:90.5
	]:4.38
]

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 \((0.85)(0.8)(\frac{491 }{\log_{20} 491}-\frac{140}{\log_{20}140})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 2DCP_1 5KYB_1 136 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]