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Parikh vectors
8COL_1 9ETO_1 1JOR_1 Letter Amino acid
18 21 5 S Serine
2 5 1 W Tryptophan
23 26 9 V Valine
9 16 6 N Asparagine
12 2 0 C Cysteine
13 27 8 D Aspartic acid
6 13 6 Q Glutamine
33 30 10 G Glycine
11 10 3 H Histidine
13 11 4 M Methionine
14 11 6 P Proline
54 24 14 A Alanine
21 15 5 R Arginine
18 24 10 T Threonine
15 23 5 I Isoleucine
11 21 23 K Lycine
14 12 3 F Phenylalanine
5 15 7 Y Tyrosine
23 21 12 E Glutamic acid
26 37 12 L Leucine 

8COL_1|Chains A[auth AAA], B[auth BBB], C[auth CCC], D[auth DDD]|Putative L-asparaginase II protein|Rhizobium etli (29449)
>9ETO_1|Chains A, B|4-hydroxytryptamine kinase|Psilocybe cubensis (181762)
>1JOR_1|Chain A|staphylococcal nuclease|Staphylococcus aureus (1280)
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 \(w=f(c)\)
8COL , Knot 144 341 0.82 40 192 327 
GIDPFTMTNPVTVEVTRGLLVESRHRGAVAVVDGDGKLFFSLGDIDTAVFPRSACKAMQALPLVESGAADAYGFGDKELALACASHNGEEEHVALAASMLSRAGRNVEALECGAHWSMNQKVLIQQARSLDAPTALHNNCSGKHAGFICACCHRDIDPKGYVGYEHPLQVEIRAVMERLTGAVLGAESCGTDGCSIPTYAMPLRNLAHGFARMATGTGLEPLRAKASRRLIEACMAEPFYVAGSGRACTKLMQIAPGRIFVKTGAEGVFCAAIPEKGIGISLKSEDGATRAAEAMVAATLARFFETEETVHAALMAFAAMPMRNWNGIHVGDIRATSVFSA
9ETO , Knot 156 364 0.84 40 218 347 
GHMAFDLKTEDGLITYLTKHLSLDVDTSGVKRLSGGFVNVTWRIKLNAPYQGHTSIILKHAQPHMSTDEDFKIGVERSVYEYQAIKLMMANREVLGGVDGIVSVPEGLNYDLENNALIMQDVGKMKTLLDYVTAKPPLATDIARLVGTEIGGFVARLHNIGRERRDDPEFKFFSGNIVGRTTSDQLYQTIIPNAAKYGVDDPLLPTVVKDLVDDVMHSEETLVMADLWSGNILLQLEEGNPSKLQKIYILDWELCKYGPASLDLGYFLGDCYLISRFQDEQVGTTMRQAYLQSYARTSKHSINYAKVTAGIAAHIVMWTDFMQWGSEEERINFVKKGVAAFHDARGNNDNGEITSTLLKESSTA
1JOR , Knot 70 149 0.78 38 109 141 
ATSTKKLHKEPATLIKAIDGDTVKLMYKGQPMTFRLLLVDTPETKHPKKGVEKYGPEASAFTKKMVENAKKIEVEFDKGQRTDKYGRGLAYIYADGKMVNEALVRQGLAKVAYVYKPNNTHEQLLRKSEAQAKKEKLNIWSEDNADSGQ

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 \(c\).

\(|P_{f(8COL_1)}(2) \setminus P_{f(9ETO_1)}(2)|=75\), \(|P_{f(9ETO_1)}(2) \setminus P_{f(8COL_1)}(2)|=101\). 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 1:11011010011010100111100000111111010101110110100111100100110111110011101011100011110100010000111110110011001011001101010001110010010110110000010011110100000101010110001101010111001011111100010010011001111001101110110101101101010001101011011011101010001101111011100110111011110011110100001100110111110110110000010111111111110010110110101001101
Pair \(Z_2\) Length of longest common subsequence
8COL_1,9ETO_1 176 3
8COL_1,1JOR_1 171 3
9ETO_1,1JOR_1 173 4

Newick tree

 
[
	9ETO_1:87.82,
	[
		8COL_1:85.5,1JOR_1:85.5
	]:2.32
]

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 \((0.85)(0.8)(\frac{705 }{\log_{20} 705}-\frac{341}{\log_{20}341})=99.8\)
Status Protein1 Protein2 d d1/2
Query variables 8COL_1 9ETO_1 128 123
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]