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Parikh vectors
6IJD_1 2EJN_1 7TIW_1 Letter Amino acid
19 4 17 F Phenylalanine
40 7 16 S Serine
23 10 24 T Threonine
8 0 3 W Tryptophan
6 5 11 Y Tyrosine
12 7 21 N Asparagine
11 6 10 M Methionine
3 6 12 C Cysteine
35 11 17 A Alanine
21 4 11 R Arginine
23 5 26 G Glycine
15 6 7 H Histidine
20 11 11 K Lycine
24 11 17 D Aspartic acid
11 3 14 Q Glutamine
49 17 29 L Leucine
28 6 13 P Proline
37 14 27 V Valine
28 13 9 E Glutamic acid
22 7 11 I Isoleucine 

6IJD_1|Chains A, B|UDP-glycosyltransferase 89C1|Arabidopsis thaliana (3702)
>2EJN_1|Chains A, B|Major allergen I polypeptide chain 1, chain 2|Felis catus (9685)
>7TIW_1|Chain A|3C-like proteinase nsp5|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 \(w=f(c)\)
6IJD , Knot 176 435 0.82 40 213 402 
MTTTTTKKPHVLVIPFPQSGHMVPHLDLTHQILLRGATVTVLVTPKNSSYLDALRSLHSPEHFKTLILPFPSHPCIPSGVESLQQLPLEAIVHMFDALSRLHDPLVDFLSRQPPSDLPDAILGSSFLSPWINKVADAFSIKSISFLPINAHSISVMWAQEDRSFFNDLETATTESYGLVINSFYDLEPEFVETVKTRFLNHHRIWTVGPLLPFKAGVDRGGQSSIPPAKVSAWLDSCPEDNSVVYVGFGSQIRLTAEQTAALAAALEKSSVRFIWAVRDAAKKVNSSDNSVEEDVIPAGFEERVKEKGLVIRGWAPQTMILEHRAVGSYLTHLGWGSVLEGMVGGVMLLAWPMQADHFFNTTLIVDKLRAAVRVGENRDSVPDSDKLARILAESAREDLPERVTLMKLREKAMEAIKEGGSSYKNLDELVAEMCL
2EJN , Knot 76 153 0.83 38 115 144 
MEICPAVKRDVDLFLTGTPDEYVEQVAQYKALPVVLENARILKNCVDAKMTEEDKENALSLLDKIYTSPLCVKMAETCPIFYDVFFAVANGNELLLDLSLTKVNATEPERTAMKKIQDCYVENGLISRVLDGLVMTTISSSKDCMGEHHHHHH
7TIW , 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 \(c\).

\(|P_{f(6IJD_1)}(2) \setminus P_{f(2EJN_1)}(2)|=137\), \(|P_{f(2EJN_1)}(2) \setminus P_{f(6IJD_1)}(2)|=39\). 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 1:100000001011111110010111010100011101101011101000001011001001001001111110010110110010011101110110110010011101100011001101111001101110011011010010111101001011110000011001001000001111001001010110010001100001101111111011100110001111010111000100001101111001010100011111110000101111100110010000001000111111000100011110111100111000111001001111011011111111111110100110001110010111011000001100001101110010001100101101000110110011000001001110101
Pair \(Z_2\) Length of longest common subsequence
6IJD_1,2EJN_1 176 4
6IJD_1,7TIW_1 183 4
2EJN_1,7TIW_1 181 3

Newick tree

 
[
	7TIW_1:91.98,
	[
		6IJD_1:88,2EJN_1:88
	]:3.98
]

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 \((0.85)(0.8)(\frac{588 }{\log_{20} 588}-\frac{153}{\log_{20}153})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 6IJD_1 2EJN_1 160 106
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]