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Parikh vectors
3BDP_1 6HPI_1 3CUI_1 Letter Amino acid
0 9 27 V Valine
0 6 14 R Arginine
2 7 46 A Alanine
0 7 14 N Asparagine
0 9 26 D Aspartic acid
0 10 14 Q Glutamine
3 11 25 G Glycine
0 19 19 L Leucine
0 9 18 K Lycine
0 5 5 M Methionine
0 10 18 S Serine
0 3 10 Y Tyrosine
2 4 4 C Cysteine
0 8 14 E Glutamic acid
0 3 5 H Histidine
0 10 8 I Isoleucine
0 7 16 F Phenylalanine
0 9 10 P Proline
3 10 15 T Threonine
0 2 7 W Tryptophan 

3BDP_1|Chain A[auth P]|DNA (5'-D(*GP*CP*AP*TP*GP*AP*TP*GP*CP*2DT)-3')|
>6HPI_1|Chain A|Interleukin-36 alpha|Homo sapiens (9606)
>3CUI_1|Chain A|Exo-beta-1,4-glucanase|Cellulomonas fimi (1708)
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 \(w=f(c)\)
3BDP , Knot 7 10 0.53 8 6 7 
GCATGATGCT
6HPI , Knot 79 158 0.84 40 130 152 
MEKALKIDTPQQGSIQDINHRVWVLQDQTLIAVPRKDRMSPVTIALISCRHVETLEKDRGNPIYLGLNGLNLCLMCAKVGDQPTLQLKEKDIMDLYNQPEPVKSFLFYHSQSGRNSTFESVAFPGWFIAVSSEGGCPLILTQELGKANTTDFGLTMLF
3CUI , Knot 136 315 0.82 40 191 295 
ATTLKEAADGAGRDFGFALDPNRLSEAQYKAIADSEFNLVVAENAMKWDATEPSQNSFSFGAGDRVASYAADTGKELYGHTLVWHSQLPDWAKNLNGSAFESAMVNHVTKVADHFEGKVASWDVVNEAFADGGGRRQDSAFQQKLGNGYIETAFRAARAADPTAKLCINDYNVEGINAKSNSLYDLVKDFKARGVPLDCVGFQSHLIVGQVPGDFRQNLQRFADLGVDVRITELDIRMRTPSDATKLATQAADYKKVVQACMQVTRCQGVTVWGITDKYSWVPDVFPGEGAALVWDASYAKKPAYAAVMEAFGAS

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 \(c\).

\(|P_{f(3BDP_1)}(2) \setminus P_{f(6HPI_1)}(2)|=4\), \(|P_{f(6HPI_1)}(2) \setminus P_{f(3BDP_1)}(2)|=128\). 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 1:1010110100
Pair \(Z_2\) Length of longest common subsequence
3BDP_1,6HPI_1 132 2
3BDP_1,3CUI_1 191 2
6HPI_1,3CUI_1 177 3

Newick tree

 
[
	3CUI_1:99.24,
	[
		3BDP_1:66,6HPI_1:66
	]:33.24
]

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 \((0.85)(0.8)(\frac{168 }{\log_{20} 168}-\frac{10}{\log_{20}10})=57.9\)
Status Protein1 Protein2 d d1/2
Query variables 3BDP_1 6HPI_1 76 40
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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