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Parikh vectors
6EKO_1 8IGN_1 6EQX_1 Letter Amino acid
24 17 31 D Aspartic acid
19 26 47 G Glycine
17 17 9 F Phenylalanine
3 3 11 W Tryptophan
11 21 29 N Asparagine
20 11 19 I Isoleucine
21 16 35 S Serine
24 17 45 A Alanine
15 11 24 R Arginine
7 14 18 Q Glutamine
20 9 22 E Glutamic acid
33 29 36 L Leucine
24 11 14 K Lycine
7 12 8 C Cysteine
9 7 21 H Histidine
8 10 6 M Methionine
9 13 21 P Proline
11 24 38 T Threonine
8 11 17 Y Tyrosine
22 27 31 V Valine 

6EKO_1|Chains A, B|Restriction endonuclease PfoI|Pseudomonas fluorescens (294)
>8IGN_1|Chains A, B|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
>6EQX_1|Chain A|Furin|Homo sapiens (9606)
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 \(w=f(c)\)
6EKO , Knot 141 312 0.86 40 199 300 
MQKYRLYEKDGSPVQDFNRFVKGWLDIEFGLKEHQPPKVFDTIRDKYNEAIEAVVLSGVAPRTAHKAALSTLTELLFGHDLAKELSARLDIQPIGVGGFRSAHSQAFAKNVGENFVNLMVYALACILKDNDDVLVDKGLPPHLKKALTLSRECRIKDTLREIKIPIEGDLCVFSRSNHCNAIVISAATRLKEVFHIGTMWALFSDVAKDEYCLNKWGLKVESSESLKDTMYVFATADMINKDGARSQGCDVERETPRNLIAMDASFFDYVFVSKMGIGHVSSDLSLKYGRESLFHELGCIIDMIEQKFDILL
8IGN , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
6EQX , Knot 198 482 0.84 40 245 450 
DVYQEPTDPKFPQQWYLSGVTQRDLNVKAAWAQGYTGHGIVVSILDDGIEKNHPDLAGNYDPGASFDVNDQDPDPQPRYTQMNDNRHGTRCAGEVAAVANNGVCGVGVAYNARIGGVRMLDGEVTDAVEARSLGLNPNHIHIYSASWGPEDDGKTVDGPARLAEEAFFRGVSQGRGGLGSIFVWASGNGGREHDSCNCDGYTNSIYTLSISSATQFGNVPWYSEACSSTLATTYSSGNQNEKQIVTTDLRQKCTESHTGTSASAPLAAGIIALTLEANKNLTWRDMQHLVVQTSKPAHLNANDWATNGVGRKVSHSYGYGLLDAGAMVALAQNWTTVAPQRKCIIDILTEPKDIGKRLEVRKTVTACLGEPNHITRLEHAQARLTLSYNRRGDLAIHLVSPMGTRSTLLAARPHDYSADGFNDWAFMTTHSWDEDPSGEWVLEIENTSEANNYGTLTKFTLVLYGTASGSLVPRGSHHHHHH

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 \(c\).

\(|P_{f(6EKO_1)}(2) \setminus P_{f(8IGN_1)}(2)|=91\), \(|P_{f(8IGN_1)}(2) \setminus P_{f(6EKO_1)}(2)|=88\). 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 1:100001000010110010011011101011100001101100100000011011110111100100111001001111001100101010101111111001000111001100110111011101100000111001111010011010000010001001011101010110000000111101100100110110111110011000001001110100000100010111010110001100010010000100111101011001110011110100010100100011001101101100010111
Pair \(Z_2\) Length of longest common subsequence
6EKO_1,8IGN_1 179 3
6EKO_1,6EQX_1 180 4
8IGN_1,6EQX_1 191 4

Newick tree

 
[
	6EQX_1:93.86,
	[
		6EKO_1:89.5,8IGN_1:89.5
	]:4.36
]

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 \((0.85)(0.8)(\frac{618 }{\log_{20} 618}-\frac{306}{\log_{20}306})=86.9\)
Status Protein1 Protein2 d d1/2
Query variables 6EKO_1 8IGN_1 110 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]