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Parikh vectors
6DYH_1 8UEF_1 4OGZ_1 Letter Amino acid
5 14 11 Q Glutamine
5 7 8 H Histidine
2 16 23 S Serine
17 17 33 A Alanine
3 11 23 R Arginine
3 12 11 C Cysteine
12 11 30 K Lycine
4 13 22 P Proline
3 24 19 T Threonine
4 27 29 V Valine
6 21 21 N Asparagine
12 17 32 D Aspartic acid
2 17 11 F Phenylalanine
1 3 14 W Tryptophan
2 11 32 I Isoleucine
10 29 44 L Leucine
3 10 10 M Methionine
2 11 18 Y Tyrosine
8 9 33 E Glutamic acid
2 26 49 G Glycine 

6DYH_1|Chains A, B[auth D]|Soluble cytochrome b562|Escherichia coli (562)
>8UEF_1|Chain A|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
>4OGZ_1|Chains A, B|Putative alpha-galactosidase/melibiase|Bacteroides fragilis (272559)
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 \(w=f(c)\)
6DYH , Knot 54 106 0.79 40 80 101 
ADLEDNMETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMWDFRHGFDHLVYHIDDALKLANEGKVKEAQAAAEQLKCHCNACHQKYR
8UEF , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
4OGZ , Knot 197 473 0.85 40 259 454 
GSEVRTAIFEGKPCINPPHVVGNYPATPFLFYIPTSGERPIKWHAENLPKGLKLDKETGIIKGKVVEKGTYKVMLKAENALGTDTQELLINIGDELLLTPPMGWNSWNTFGRHLTEELLLQTADAMVENGMRDLGYAYINIDDFWQLPERGADGHIQIDKTKFPRGIKYVADYLHERGFKLGIYSDAADKTCGGVCGSYGYEEIDARDFASWGVDLLKYDYCNAPAGRVEAMERYEKMGRALRATDRSIVFSICEWGQREPWKWAKKVGGHLWRVSGDIGDLWNRSTDEKGGLRGILNILEINAPLSEYARPGGWNDPDMLVVGIGGKSKSIGYESEGCTNEQYQSHFALWCMMASPLLCGNDVRQMNDSTLQILLNKDLIAINQDPLGIQAERAIRADHYDVWVKPLSDGSKAIACLNRISGPVDVELNVKTVEGLSLDRVYDVIEGSLVAEASTGWVVKLAPGECKVFICK

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 \(c\).

\(|P_{f(6DYH_1)}(2) \setminus P_{f(8UEF_1)}(2)|=37\), \(|P_{f(8UEF_1)}(2) \setminus P_{f(6DYH_1)}(2)|=153\). 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 1:1010001001000101100100110100110010111101001011010000100101101001100110010011011001010010111001000001000000
Pair \(Z_2\) Length of longest common subsequence
6DYH_1,8UEF_1 190 4
6DYH_1,4OGZ_1 217 3
8UEF_1,4OGZ_1 181 4

Newick tree

 
[
	6DYH_1:10.52,
	[
		8UEF_1:90.5,4OGZ_1:90.5
	]:15.02
]

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 \((0.85)(0.8)(\frac{412 }{\log_{20} 412}-\frac{106}{\log_{20}106})=93.0\)
Status Protein1 Protein2 d d1/2
Query variables 6DYH_1 8UEF_1 119 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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