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Parikh vectors
7XEL_1 3NPM_1 2DAF_1 Letter Amino acid
3 1 0 C Cysteine
32 14 6 E Glutamic acid
23 15 7 K Lycine
16 20 14 S Serine
14 15 3 R Arginine
11 8 0 M Methionine
14 12 3 F Phenylalanine
14 12 7 P Proline
30 18 6 T Threonine
30 15 10 G Glycine
26 15 12 I Isoleucine
34 38 11 L Leucine
4 2 0 W Tryptophan
15 8 3 Y Tyrosine
35 22 15 V Valine
40 34 1 A Alanine
17 15 3 N Asparagine
25 21 6 D Aspartic acid
13 14 7 Q Glutamine
23 11 4 H Histidine 

7XEL_1|Chain A|Cysteine desulfurase SufS|Bacillus subtilis subsp. subtilis str. 168 (224308)
>3NPM_1|Chain A|Aspartate carbamoyltransferase catalytic chain|Escherichia coli (83333)
>2DAF_1|Chain A|FLJ35834 protein|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 \(w=f(c)\)
7XEL , Knot 181 419 0.87 40 225 404 
MGHMNITDIREQFPILHQQVNGHDLVYLDSAATSQKPRAVIETLDKYYNQYNSNVHRGVHTLGTRATDGYEGAREKVRKFINAKSMAEIIFTKGTTTSLNMVALSYARANLKPGDEVVITYMEHHANIIPWQQAVKATGATLKYIPLQEDGTISLEDVRETVTSNTKIVAVSHVSNVLGTVNPIKEMAKIAHDNGAVIVVDGAQSTPHMKIDVQDLDCDFFALSSHKMCGPTGVGVLYGKKALLENMEPAEFGGEMIDFVGLYESTWKELPWKFEAGTPIIAGAIGLGAAIDFLEEIGLDEISRHEHKLAAYALERFRQLDGVTVYGPEERAGLVTFNLDDVHPHDVATVLDAEGIAVRAGHHCAQPLMKWLDVTATARASFYLYNTEEEIDKLVEALQKTKEYFTNVFVDLEHHHHHH
3NPM , Knot 138 310 0.85 40 190 299 
ANPLYQKHIISINDLSRDDLNLVLATAAKLKANPQPELLKHKVIASAFFEASTRTRLSFETSMHRLGASVVGFSDSANTSLGKKGETLADTISVISTYVDAIVMRHPQEGAARLATEFSGNVPVLNAGDGSNQHPTQTLLDLFTIQETQGRLDNLHVAMVGDLKYGRTVHSLTQALAKFDGNRFYFIAPDALAMPQYILDMLDEKGIAWSLHSSIEEVMAEVDILYMTRVQKERLDPSEYCNVKAQFVLRASDLHNAKANMKVLHPLPRVDEIATDVDKTPHAWYFQQAGNGIFARQALLALVLNRDLVL
2DAF , Knot 55 118 0.74 34 88 109 
GSSGSSGQESVEDSLATVKVVLIPVGQEIVIPFKVDTILKYLKDHFSHLLGIPHSVLQIRYSGKILKNNETLVQHGVKPQEIVQVEIFSTNPDLYPVRRIDGLTDVSQIITVSGPSSG

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 \(c\).

\(|P_{f(7XEL_1)}(2) \setminus P_{f(3NPM_1)}(2)|=90\), \(|P_{f(3NPM_1)}(2) \setminus P_{f(7XEL_1)}(2)|=55\). 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 1:11010100100011110001010011010011000010111001000000000010011001100100100110001001101001101110010000101111001010101100111001000101111001101011010011100010101001000100000111100100111010110011011000111111011000101010100100011110000101101111101001110010110111011011110000100111010110111111111111101100111001000000111011001001011010110001111010100101001101101011110110001011101101010101010100000010011011000000100111010000000
Pair \(Z_2\) Length of longest common subsequence
7XEL_1,3NPM_1 145 3
7XEL_1,2DAF_1 197 3
3NPM_1,2DAF_1 178 3

Newick tree

 
[
	2DAF_1:99.98,
	[
		7XEL_1:72.5,3NPM_1:72.5
	]:27.48
]

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 \((0.85)(0.8)(\frac{729 }{\log_{20} 729}-\frac{310}{\log_{20}310})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 7XEL_1 3NPM_1 145 125.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]