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Parikh vectors
1EEL_1 2QJD_1 7GDK_1 Letter Amino acid
0 6 11 R Arginine
4 2 12 C Cysteine
4 10 26 G Glycine
0 4 11 I Isoleucine
0 4 29 L Leucine
0 1 3 W Tryptophan
0 6 17 D Aspartic acid
0 4 14 Q Glutamine
0 1 7 H Histidine
0 12 11 K Lycine
0 1 13 P Proline
2 2 17 A Alanine
0 6 21 N Asparagine
0 9 16 S Serine
2 7 24 T Threonine
0 8 9 E Glutamic acid
0 4 10 M Methionine
0 9 17 F Phenylalanine
0 6 11 Y Tyrosine
0 8 27 V Valine 

1EEL_1|Chains A, B|5'-D(*CP*GP*CP*GP*AP*AP*TP*TP*CP*GP*CP*G)-3'|
>2QJD_1|Chains A, B|Novel immune-type receptor 10|Ictalurus punctatus (7998)
>7GDK_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 \(w=f(c)\)
1EEL , Knot 6 12 0.41 8 7 8 
CGCGAATTCGCG
2QJD , Knot 58 110 0.82 40 93 107 
MIKELHVKTVKRGENVTMECSMSKVKDKNKLAWYRQSFGKVPQYFVRYYSSNSGYKFAEGFKDSRFSMTVNDQKFDLNIIGTREDDGGEYFCGEVEGNTIKFTSGTRLQF
7GDK , 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 \(c\).

\(|P_{f(1EEL_1)}(2) \setminus P_{f(2QJD_1)}(2)|=6\), \(|P_{f(2QJD_1)}(2) \setminus P_{f(1EEL_1)}(2)|=92\). 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 1:010111000101
Pair \(Z_2\) Length of longest common subsequence
1EEL_1,2QJD_1 98 2
1EEL_1,7GDK_1 193 3
2QJD_1,7GDK_1 197 3

Newick tree

 
[
	7GDK_1:10.97,
	[
		1EEL_1:49,2QJD_1:49
	]:59.97
]

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 \((0.85)(0.8)(\frac{122 }{\log_{20} 122}-\frac{12}{\log_{20}12})=41.8\)
Status Protein1 Protein2 d d1/2
Query variables 1EEL_1 2QJD_1 55 30
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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