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Parikh vectors
6XAD_1 1QCY_1 4JLJ_1 Letter Amino acid
0 1 7 W Tryptophan
15 4 12 Y Tyrosine
11 10 13 N Asparagine
1 1 2 C Cysteine
24 14 28 L Leucine
17 3 11 P Proline
21 10 13 R Arginine
9 8 13 Q Glutamine
19 13 13 G Glycine
11 4 12 H Histidine
19 15 13 I Isoleucine
8 14 17 K Lycine
24 12 28 S Serine
32 18 13 V Valine
19 13 11 A Alanine
22 16 29 E Glutamic acid
6 3 7 M Methionine
15 8 12 F Phenylalanine
16 15 15 T Threonine
16 11 11 D Aspartic acid 

6XAD_1|Chain A|ArrX|Chrysiogenes arsenatis (309797)
>1QCY_1|Chain A|I-DOMAIN OF INTEGRIN ALPHA1BETA1|Homo sapiens (9606)
>4JLJ_1|Chains A, B|Deoxycytidine kinase|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 \(w=f(c)\)
6XAD , Knot 134 305 0.83 38 185 288 
HHHHHHDYDIPTTENLYFQGAMGSSVKPIPVDLSERSTRLATVDSPGVFRVAVSSMISPLETMKGYGPVLSYIEQQTGRKVELVQRRTYREVNELIRENKIDLAFICTYSFVEAELFGARPVAVPQVEGNPYYQAVVITRRDSGINSLEELRNKRFAFTDPMSFSGHIALRGELVKVDRTPETFFASTFYTYSHDNSLRAVYDGIVDGATIDSLVFRSSNILYPEIGAALQVVHVSPLVGAPPVVVSPGLSEEDYQLIRRAFLNMHNEPLGKQALDTLFIDRFVMVNSGHYDYIREIAGKIEVVE
1QCY , Knot 90 193 0.81 40 135 185 
TQLDIVIVLDGSNSIYPWDSVTAFLNDLLKRMDIGPKQTQVGIVQYGENVTHEFNLNKYSSTEEVLVAAKKIVQRGGAQTMTALGTDTARKEAFTEARGARRGVKKVMVIVTDGESHDNHRLKKVIQDCEDENIQRFSIAILGSYNRGNLSTEKFVEEIKSIASEPTEKHFFNVSDELALVTIVKTLGERIFA
4JLJ , Knot 126 280 0.84 40 186 268 
MGSSHHHHHHSSGLVPRGSHMATPPKRSSPSFSASSEGTRIKKISIEGNIAAGKSTFVNILKQLSEDWEVVPEPVARWSNVQSTQDEFEELTMEQKNGGNVLQMMYEKPERWSFTFQTYACLSRIRAQLASLNGKLKDAEKPVLFFERSVYSDRYIFASNLYESESMNETEWTIYQDWHDWMNNQFGQSLELDGIIYLQATPETCLHRIYLRGRNEEQGIPLEYLEKLHYKHESWLLHRTLKTNFDYLQEVPILTLDVNEDFKDKYESLVEKVKEFLSTL

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 \(c\).

\(|P_{f(6XAD_1)}(2) \setminus P_{f(1QCY_1)}(2)|=98\), \(|P_{f(1QCY_1)}(2) \setminus P_{f(6XAD_1)}(2)|=48\). 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 1:00000000011000010101111001011110100000011010011110111001101100101011110010000100101100000001001100001011110000110101111011111010101000111100000110010010000111001101010111010110100010011100100000000101100111011010011100001101011111011010111111111101110000001100111010001110011001110011110010000100111010110
Pair \(Z_2\) Length of longest common subsequence
6XAD_1,1QCY_1 146 4
6XAD_1,4JLJ_1 161 6
1QCY_1,4JLJ_1 163 3

Newick tree

 
[
	4JLJ_1:83.49,
	[
		6XAD_1:73,1QCY_1:73
	]:10.49
]

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 \((0.85)(0.8)(\frac{498 }{\log_{20} 498}-\frac{193}{\log_{20}193})=88.6\)
Status Protein1 Protein2 d d1/2
Query variables 6XAD_1 1QCY_1 110 89
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]