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Parikh vectors
8GSG_1 3ODT_1 1CXL_1 Letter Amino acid
2 13 62 N Asparagine
0 29 40 D Aspartic acid
2 13 17 E Glutamic acid
2 19 37 I Isoleucine
0 11 32 F Phenylalanine
2 33 48 S Serine
0 13 65 A Alanine
2 10 27 Q Glutamine
2 7 33 Y Tyrosine
1 33 49 V Valine
4 7 2 C Cysteine
2 26 38 L Leucine
0 5 28 P Proline
0 9 13 W Tryptophan
0 8 12 H Histidine
1 28 65 G Glycine
0 20 29 K Lycine
0 3 13 M Methionine
1 16 56 T Threonine
0 10 20 R Arginine 

8GSG_1|Chains A, C|Insulin A chain|Homo sapiens (9606)
>3ODT_1|Chains A, B|Protein DOA1|Saccharomyces cerevisiae (4932)
>1CXL_1|Chain A|PROTEIN (CYCLODEXTRIN-GLYCOSYLTRANSFERASE)|Bacillus circulans (1397)
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 \(w=f(c)\)
8GSG , Knot 15 21 0.72 22 20 19 
GIVEQCCTSICSLYQLENYCN
3ODT , Knot 133 313 0.81 40 176 295 
GIDPFTGYQLSATLKGHDQDVRDVVAVDDSKVASVSRDGTVRLWSKDDQWLGTVVYTGQGFLNSVCYDSEKELLLFGGKDTMINGVPLFATSGEDPLYTLIGHQGNVCSLSFQDGVVISGSWDKTAKVWKEGSLVYNLQAHNASVWDAKVVSFSENKFLTASADKTIKLWQNDKVIKTFSGIHNDVVRHLAVVDDGHFISCSNDGLIKLVDMHTGDVLRTYEGHESFVYCIKLLPNGDIVSCGEDRTVRIWSKENGSLKQVITLPAISIWSVDCMSNGDIIVGSSDNLVRIFSQEKSRWASEDEIKGELRSGC
1CXL , Knot 265 686 0.84 40 266 621 
APDTSVSNKQNFSTDVIYQIFTDRFSDGNPANNPTGAAFDGTCTNLRLYCGGDWQGIINKINDGYLTGMGVTAIWISQPVENIYSIINYSGVNNTAYHGYWARDFKKTNPAYGTIADFQNLIAAAHAKNIKVIIDFAPNHTSPASSDQPSFAENGRLYDNGTLLGGYTNDTQNLFHHNGGTDFSTTENGIYKNLYDLADLNHNNSTVDVYLKDAIKMWLDLGIDGIRMDAVKHMPFGWQKSFMAAVNNYKPVFTFGQWFLGVNEVSPENHKFANESGMSLLDFRFAQKVRQVFRDNTDNMYGLKAMLEGSAADYAQVDDQVTFIDNHDMERFHASNANRRKLEQALAFTLTSRGVPAIYYGTEQYMSGGTDPDNRARIPSFSTSTTAYQVIQKLAPLRKSNPAIAYGSTQERWINNDVLIYERKFGSNVAVVAVNRNLNAPASISGLVTSLPQGSYNDVLGGLLNGNTLSVGSGGAASNFTLAAGGTAVWQYTAATATPTIGHVGPMMAKPGVTITIDGRGFGSSKGTVYFGTTAVSGADITSWEDTQIKVKIPAVAGGNYNIKVANAAGTASNVYDNFEVLSGDQVSVRFVVNNATTALGQNVYLTGSVSELGNWDPAKAIGPMYNQVVYQYPNWYYDVSVPAGKTIEFKFLKKQGSTVTWEGGSNHTFTAPSSGTATINVNWQP

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 \(c\).

\(|P_{f(8GSG_1)}(2) \setminus P_{f(3ODT_1)}(2)|=9\), \(|P_{f(3ODT_1)}(2) \setminus P_{f(8GSG_1)}(2)|=165\). 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 1:111000000100100100000
Pair \(Z_2\) Length of longest common subsequence
8GSG_1,3ODT_1 174 3
8GSG_1,1CXL_1 260 2
3ODT_1,1CXL_1 158 4

Newick tree

 
[
	8GSG_1:11.29,
	[
		3ODT_1:79,1CXL_1:79
	]:40.29
]

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 \((0.85)(0.8)(\frac{334 }{\log_{20} 334}-\frac{21}{\log_{20}21})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 8GSG_1 3ODT_1 126 67
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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