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Parikh vectors
5IKL_1 5APP_1 1ZLL_1 Letter Amino acid
12 2 1 Y Tyrosine
38 15 2 E Glutamic acid
8 8 0 H Histidine
27 18 2 K Lycine
17 1 2 F Phenylalanine
4 0 0 W Tryptophan
31 8 4 R Arginine
23 4 2 T Threonine
68 16 3 A Alanine
21 5 5 Q Glutamine
56 6 10 L Leucine
28 0 1 P Proline
11 2 3 M Methionine
25 8 2 S Serine
36 3 2 V Valine
18 10 2 N Asparagine
24 6 0 D Aspartic acid
6 0 3 C Cysteine
60 0 0 G Glycine
24 21 8 I Isoleucine 

5IKL_1|Chains A[auth B], B[auth D], C[auth F]|Geranyl-CoA carboxylase, beta-subunit|Pseudomonas aeruginosa (strain ATCC 15692 / PAO1 / 1C / PRS 101 / LMG 12228) (208964)
>5APP_1|Chains A, B, C|General control protein GCN4,GENERAL CONTROL PROTEIN GCN4, OUTER MEMBRANE PROTEIN 100,General control protein GCN4|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>1ZLL_1|Chains A, B, C, D, E|Cardiac phospholamban|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 \(w=f(c)\)
5IKL , Knot 216 537 0.84 40 236 493 
PAIQSELDVNGEDFARNREAMLAAVAGFRELEQKVLDKAAEARPKFEKRGQLLPRERLALLLDPGAPFLELSSLAGYKLHDDKDGTQAGGGIIAGIGYIAGVRCLVSASNSAIKGGTISPTGLKKTLRLQQIAMENKLPVVTLTESGGANLNYAAEIFVEGARGFANQARISAMGIPQVTVVHGSSTAGGAYQPGLSDYVVVVRGKAKMFLAGPPLLKAATGEIASDEELGGAELHAQVAGTAEYLAENDADGVRLAREIVGMLPWNAQLPARPARSWREPLYPVEELLGVVPADPKKPYDVREIVARIADGSEFLDFKNEFDGQTVCGHLRIEGHACGLIGNNGPITPQGAAKAAQFIQLCEQSNTPLLFLHNTTGFMVGTESERQGVIKHGSKMIQAVANARVPKLTLVVGGSYGAGNYAMCGRGLDPRFIFAWPNSRTAVMGGAQAGKVLRIVTEEKHAKEGKEADPKMLEMLETVTAQKLDSQSTALYGTASLWDDGLVDPRDSRRLLGYLLDICAEAEARPLKGNSFGVARF
5APP , Knot 48 133 0.58 32 64 96 
MKQIEDKIEEILSKIYHIENEIARIKKLIQNVDVRSTENAARSRANEQKIAENKKAIENKADKADVEKNRADIAANSRAIATFRSSSQNIAALTTKMKQIEDKIEEILSKIYHIENEIARIKKLIKLHHHHHH
1ZLL , Knot 32 52 0.81 32 45 49 
MEKVQYLTRSAIRRASTIEMPQQARQKLQNLFINFCLILICLLLICIIVMLL

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 \(c\).

\(|P_{f(5IKL_1)}(2) \setminus P_{f(5APP_1)}(2)|=186\), \(|P_{f(5APP_1)}(2) \setminus P_{f(5IKL_1)}(2)|=14\). 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 1:111000101010011000011111111100100011001101010100010111000111110111111010011100100000100111111111101111001101000110110101011000101001110001111010001110100110111011011100101011111010110100011110011100011110101011111111101101011000011110101011101001100010110110011111110101110110010011011001111111010010010011101101001101000101001010101010101111001110101110110110100000011111000011111000000111001001101110101101011111001110011010110101111110000111111011011011000001001001010110110010100100000110101011001110100000111011010101010110100111101
Pair \(Z_2\) Length of longest common subsequence
5IKL_1,5APP_1 200 3
5IKL_1,1ZLL_1 215 3
5APP_1,1ZLL_1 83 2

Newick tree

 
[
	5IKL_1:11.45,
	[
		5APP_1:41.5,1ZLL_1:41.5
	]:75.95
]

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 \((0.85)(0.8)(\frac{670 }{\log_{20} 670}-\frac{133}{\log_{20}133})=154.\)
Status Protein1 Protein2 d d1/2
Query variables 5IKL_1 5APP_1 202 118.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]