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Parikh vectors
7MEK_1 7TIB_1 7OYU_1 Letter Amino acid
3 42 21 N Asparagine
10 57 28 D Aspartic acid
8 25 8 Q Glutamine
7 57 18 G Glycine
3 15 11 H Histidine
2 70 31 L Leucine
2 16 5 M Methionine
4 27 14 F Phenylalanine
11 52 19 E Glutamic acid
10 75 18 S Serine
5 7 8 W Tryptophan
5 17 14 Y Tyrosine
11 62 35 A Alanine
0 40 10 R Arginine
0 6 2 C Cysteine
5 53 13 I Isoleucine
12 90 30 K Lycine
8 41 21 P Proline
12 56 16 T Threonine
11 52 30 V Valine 

7MEK_1|Chains A, B|Uncharacterized protein YcnI|Bacillus subtilis (1423)
>7TIB_1|Chain A|Replication factor C subunit 1|Saccharomyces cerevisiae (4932)
>7OYU_1|Chain A|Putrescine-binding periplasmic protein PotF|Escherichia coli (strain K12) (83333)
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 \(w=f(c)\)
7MEK , Knot 66 129 0.82 36 104 125 
HVSVKPAESAAGSWETYTMKVPSEKNLPTTKVVLKMPKDVEFQQYEPIPGWKVSTQKHDDKSVSVTWEATDGGIQEGQFQQFTFVAKNPDKAEEAAWDAYQYYKDGSIVEWTGDEDADTPHSITNITSA
7TIB , Knot 317 860 0.83 40 280 771 
MVNISDFFGKNKKSVRSSTSRPTRQVGSSKPEVIDLDTESDQESTNKTPKKMPVSNVIDVSETPEGEKKLPLPAKRKASSPTVKPASSKKTKPSSKSSDSASNITAQDVLDKIPSLDLSNVHVKENAKFDFKSANSNADPDEIVSEIGSFPEGKPNCLLGLTIVFTGVLPTLERGASEALAKRYGARVTKSISSKTSVVVLGDEAGPKKLEKIKQLKIKAIDEEGFKQLIAGMPAEGGDGEAAEKARRKLEEQHNIATKEAELLVKKEEERSKKLAATRVSGGHLERDNVREEDKLWTVKYAPTNLQQVCGNKGSVMKLKNWLANWENSKKNSFKHAGKDGSGVFRAAMLYGPPGIGKTTAAHLVAQELGYDILEQNASDVRSKTLLNAGVKNALDNMSVVGYFKHNEEAQNLNGKHFVIIMDEVDGMSGGDRGGVGQLAQFCRKTSTPLILICNERNLPKMRPFDRVCLDIQFRRPDANSIKSRLMTIAIREKFKLDPNVIDRLIQTTRGDIRQVINLLSTISTTTKTINHENINEISKAWEKNIALKPFDIAHKMLDGQIYSDIGSRNFTLNDKIALYFDDFDFTPLMIQENYLSTRPSVLKPGQSHLEAVAEAANCISLGDIVEKKIRSSEQLWSLLPLHAVLSSVYPASKVAGHMAGRINFTAWLGQNSKSAKYYRLLQEIHYHTRLGTSTDKIGLRLDYLPTFRKRLLDPFLKQGADAISSVIEVMDDYYLTKEDWDSIMEFFVGPDVTTAIIKKIPATVKSGFTRKYNSMTHPVAIYRTGSTIGGGGVGTSTSTPDFEDVVDADDNPVPADDEETQDSSTDLKKDKLIKQKAKPTKRKTATSKPGGSKKRKTKA
7OYU , Knot 150 352 0.83 40 201 329 
AEQKTLHIYNWTDYIAPDTVANFEKETGIKVVYDVFDSNEVLEGKLMAGSTGFDLVVPSASYLERQLTAGVFQPLDKSKLPEWKNLDPELLKLVAKHDPDNKFAMPYMWATTGIGYNVDKVKAVLGENAPVDSWDLILKPENLEKLKSCGVSFLDDPEEVFATVLNYLGKDPNSTKADDYTGPATDLLLKLRPNIRYFHSSQYINDLANGDICVAIGWAGSVWQASNRAKEAKNGVNVSFSIPKEGAMAWFDVFAMPADAKNKDEAYQFLNYLLRPDVVAHISDHVFYANANKAATPLVSAEVRENPGIYPPADVRAKLFTQKVQDPKIDRVRTRAWTKVKSGKLEHHHHHH

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 \(c\).

\(|P_{f(7MEK_1)}(2) \setminus P_{f(7TIB_1)}(2)|=19\), \(|P_{f(7TIB_1)}(2) \setminus P_{f(7MEK_1)}(2)|=195\). 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 1:010101100111010000101100001100011101100101000011111010000000001010101001110010100101110010010011101000000101101010001001001001001
Pair \(Z_2\) Length of longest common subsequence
7MEK_1,7TIB_1 214 4
7MEK_1,7OYU_1 159 3
7TIB_1,7OYU_1 151 4

Newick tree

 
[
	7MEK_1:99.73,
	[
		7OYU_1:75.5,7TIB_1:75.5
	]:24.23
]

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 \((0.85)(0.8)(\frac{989 }{\log_{20} 989}-\frac{129}{\log_{20}129})=238.\)
Status Protein1 Protein2 d d1/2
Query variables 7MEK_1 7TIB_1 295 169
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]