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Parikh vectors
3COA_1 2WTK_1 8IGN_1 Letter Amino acid
0 14 27 V Valine
0 18 21 N Asparagine
0 43 29 L Leucine
10 17 24 T Threonine
0 1 3 W Tryptophan
0 22 11 I Isoleucine
0 9 10 M Methionine
0 11 13 P Proline
0 21 17 A Alanine
0 15 11 R Arginine
1 3 12 C Cysteine
0 31 9 E Glutamic acid
0 18 16 S Serine
0 11 11 Y Tyrosine
0 18 17 D Aspartic acid
5 8 26 G Glycine
0 7 7 H Histidine
0 32 11 K Lycine
0 20 14 Q Glutamine
0 22 17 F Phenylalanine 

3COA_1|Chains A, C[auth D]|DNA (5'-D(*DTP*DGP*DGP*DTP*DTP*DTP*DGP*DTP*DTP*DTP*DTP*DGP*DCP*DTP*DTP*DG)-3')|
>2WTK_1|Chains A, D|CALCIUM-BINDING PROTEIN 39|HOMO SAPIENS (9606)
>8IGN_1|Chains A, B|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 \(w=f(c)\)
3COA , Knot 7 16 0.40 6 6 9 
TGGTTTGTTTTGCTTG
2WTK , Knot 147 341 0.83 40 201 327 
MPFPFGKSHKSPADIVKNLKESMAVLEKQDISDKKAEKATEEVSKNLVAMKEILYGTNEKEPQTEAVAQLAQELYNSGLLSTLVADLQLIDFEGKKDVAQIFNNILRRQIGTRTPTVEYICTQQNILFMLLKGYESPEIALNCGIMLRECIRHEPLAKIILWSEQFYDFFRYVEMSTFDIASDAFATFKDLLTRHKLLSAEFLEQHYDRFFSEYEKLLHSENYVTKRQSLKLLGELLLDRHNFTIMTKYISKPENLKLMMNLLRDKSRNIQFEAFHVFKVFVANPNKTQPILDILLKNQAKLIEFLSKFQNDRTEDEQFNDEKTYLVKQIRDLKRPAQQEA
8IGN , 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 \(c\).

\(|P_{f(3COA_1)}(2) \setminus P_{f(2WTK_1)}(2)|=4\), \(|P_{f(2WTK_1)}(2) \setminus P_{f(3COA_1)}(2)|=199\). 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 1:0110001000010001
Pair \(Z_2\) Length of longest common subsequence
3COA_1,2WTK_1 203 2
3COA_1,8IGN_1 192 4
2WTK_1,8IGN_1 169 6

Newick tree

 
[
	3COA_1:10.11,
	[
		8IGN_1:84.5,2WTK_1:84.5
	]:18.61
]

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 \((0.85)(0.8)(\frac{357 }{\log_{20} 357}-\frac{16}{\log_{20}16})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 3COA_1 2WTK_1 145 75
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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