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Parikh vectors
3ARR_1 1LIN_1 4ORH_1 Letter Amino acid
30 4 11 S Serine
34 2 5 Y Tyrosine
14 1 0 H Histidine
29 2 13 P Proline
68 11 14 G Glycine
12 0 3 W Tryptophan
33 6 11 N Asparagine
6 0 1 C Cysteine
34 9 12 L Leucine
30 8 9 K Lycine
19 9 8 M Methionine
40 12 8 T Threonine
13 6 9 R Arginine
27 21 13 E Glutamic acid
16 6 5 Q Glutamine
29 8 7 I Isoleucine
21 8 3 F Phenylalanine
36 7 12 V Valine
47 11 4 A Alanine
46 17 5 D Aspartic acid 

3ARR_1|Chain A|Chitinase A|Vibrio harveyi (669)
>1LIN_1|Chain A|CALMODULIN|Bos taurus (9913)
>4ORH_1|Chains A, D[auth E], H[auth I]|Ubiquitin-conjugating enzyme E2 variant 2|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 \(w=f(c)\)
3ARR , Knot 230 584 0.83 40 258 537 
APTAPSIDMYGSNNLQFSKIELAMETTSGYNDMVKYHELAKIKVKFNQWSGTSGDTYNVYFDGVKVATGAITGSQTTASFEYGQGGLYQMEIEACDATGCSKSAPVEITIADTDGSHLKPLTMNVDPNNKSYNTDPSIVMGTYFVEWGIYGRDYTVDNMPVDNLTHILYGFIPICGPNESVKSVGGNSFNALQTACRGVNDYEVVIHDPWAAYQKSFPQAGHEYSTPIKGNYAMLMALKQRNPDLKIIPSIGGWTLSDPFYDFVDKKNRDTFVASVKKFLKTWKFYDGVDIDWEFPGGGGAAADKGDPVNDGPAYIALMRELRVMLDELEAETGRTYELTSAIGVGYDKIEDVDYADAVQYMDYIFAMTYDFYGGWNNVPGHQTALYCGSFMRPGQCDGGGVDENGEPYKGPAYTADNGIQLLLAQGVPANKLVLGTAMYGRGWEGVTPDTLTDPNDPMTGTATGKLKGSTAQGVWEDGVIDYKGIKSFMLGANNTGINGFEYGYDAQAEAPWVWNRSTGELITFDDHRSVLAKGNYAKSLGLAGLFSWEIDADNGDILNAMHEGMAGGVVTPPNRRSHHHHHH
1LIN , Knot 68 148 0.76 36 97 133 
ADQLTEEQIAEFKEAFSLFDKDGDGTITTKELGTVMRSLGQNPTEAELQDMINEVDADGNGTIDFPEFLTMMARKMKDTDSEEEIREAFRVFDKDGNGYISAAELRHVMTNLGEKLTDEEVDEMIREADIDGDGQVNYEEFVQMMTAK
4ORH , Knot 78 153 0.85 38 123 149 
GPLGSPEFMAVSTGVKVPRNFRLLEELEEGQKGVGDGTVSWGLEDDEDMTLTRWTGMIIGPPRTNYENRIYSLKVECGPKYPEAPPSVRFVTKINMNGINNSSGMVDARSIPVLAKWQNSYSIKVVLQELRRLMMSKENMKLPQPPEGQTYNN

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 \(c\).

\(|P_{f(3ARR_1)}(2) \setminus P_{f(1LIN_1)}(2)|=192\), \(|P_{f(1LIN_1)}(2) \setminus P_{f(3ARR_1)}(2)|=31\). 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 1:11011010101000101001011100001000110000110101010010100100001010110110111010000101001011100101010010100001110101100010010110101010000000010111100110111010000100111001001101111101100010011100101100100110000111001111000011011000001101001111110000101011101111010011001100000001110100110010100110101011111111100101100111011110010111001010010000100111110001001001011001001111000101110011100011001011011000111100010100111001001101111011110011110110101101101001001001101010101010010111001110001100111110001101100100101011111000010110100000111010010011111110101010010110110011111110110000000000
Pair \(Z_2\) Length of longest common subsequence
3ARR_1,1LIN_1 223 3
3ARR_1,4ORH_1 197 4
1LIN_1,4ORH_1 144 4

Newick tree

 
[
	3ARR_1:11.14,
	[
		4ORH_1:72,1LIN_1:72
	]:42.14
]

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 \((0.85)(0.8)(\frac{732 }{\log_{20} 732}-\frac{148}{\log_{20}148})=165.\)
Status Protein1 Protein2 d d1/2
Query variables 3ARR_1 1LIN_1 211 131.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]