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Parikh vectors
6VKZ_1 3HQY_1 6GEW_1 Letter Amino acid
10 15 18 Q Glutamine
38 37 33 L Leucine
19 17 28 P Proline
36 17 26 D Aspartic acid
46 23 23 E Glutamic acid
13 17 10 M Methionine
27 16 34 V Valine
9 11 7 C Cysteine
18 19 20 R Arginine
17 16 13 N Asparagine
11 22 27 G Glycine
9 26 9 H Histidine
36 27 24 S Serine
3 7 3 W Tryptophan
30 12 16 Y Tyrosine
21 21 45 A Alanine
28 18 19 K Lycine
21 15 15 F Phenylalanine
20 22 18 T Threonine
29 22 22 I Isoleucine 

6VKZ_1|Chain A|DabA|Pseudo-nitzschia multiseries (37319)
>3HQY_1|Chain A|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|Rattus norvegicus (10116)
>6GEW_1|Chain A|OphA|Omphalotus olearius (72120)
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 \(w=f(c)\)
6VKZ , Knot 181 441 0.83 40 236 419 
GSHMESPKEVLSRVQDAGLTLTNPNDLYWMVDFLKEKYYDNGDYYYPIKTVCDGESIDVKFYCPFEPSLSPHYLELYGSRDERASIYETTMKKYNRINSEKTSAICTPYSSYGDTQIVAYFYSMMYYINDQTAHLKLPESEIESELIDILNDDILIYLNEFMSIFEPEDAQDLERIWDFLDFYQPYFSKVDGKIVLDEKYLVRTPSQMPLIKTICEYVSEQFAPSKNITQVIWEVVRYIKGVKDEIHIRGDKSFTLSLQEYDDFRDKVTASPMAHAVSDLTHERFSYEAYTNPAFMELENRCSEIITYFNDVCTSDRERLDEDPFNSVFILMDLDPSLNFAKSCDVVVEHAYNKMQAFLKLKEEILESASDEEERLALARMIKTREDSLIGYVLHEVCCVEDGYARDHKPLMKAFLEEEITKSLAEKVKFNPVESESVRLN
3HQY , Knot 158 380 0.82 40 226 348 
MHHHHHHSSGLVPRGSAMMHHHHHHSSGLVPRGSAMGTSEEWQGLMHFNLPARICRDIELFHFDIGPFENMWPGIFVYMIHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNNGLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLHNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKRDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLNQWEKVIRGEETAMWISGPATSKSTSEKPTRKVDD
6GEW , Knot 171 410 0.83 40 226 385 
GTSTQTKAGSLTIVGTGIESIGQMTLQALSYIEAAAKVFYCVIDPATEAFILTKNKNCVDLFQYYDNGKSRLNTYTQMSELMVREVRKGLDVVGVFYGHPGVFVNPSHRALAIAKSEGYRARMLPGVSAEDCLFADLCIDPSNPGCLTYEASDFLIRDRPVSIHSHLVLFQVGCVGIADFNFTGFDNNKFGVLVDRLEQEYGAEHPVVHYIAAMMPHQDPVTDKYTVAQLREPEIAKRVGGVSTFYIPPKARKASNLDIIRRLELLPAGQVPDKKARIYPANQWEPDVPEVEPYRPSDQAAIAQLADHAPPEQYQPLATSKAMSDVMTKLALDPKALADYKADHRAFAQSVPDLTPQERAALELGDSWAIRCAMKNMPSSLLDAARESGEEASQNGFPWVIVVGVIGVIG

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 \(c\).

\(|P_{f(6VKZ_1)}(2) \setminus P_{f(3HQY_1)}(2)|=101\), \(|P_{f(3HQY_1)}(2) \setminus P_{f(6VKZ_1)}(2)|=91\). 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 1:100100100110010011101001001011101100000001000011001001001010100110101010010101000001010000100000100000011001000010001110100110010000101011000100011011000111010011011010010010011011010010100101011100001100100111100100010001110001001110110010110001010100010101000001000101011101100100001000100011110100000011001001000000010001100111110101010110000111001000101110100011001000000111101100000011101100100100101000011101110001000110010101100001010
Pair \(Z_2\) Length of longest common subsequence
6VKZ_1,3HQY_1 192 4
6VKZ_1,6GEW_1 156 5
3HQY_1,6GEW_1 174 4

Newick tree

 
[
	3HQY_1:95.71,
	[
		6VKZ_1:78,6GEW_1:78
	]:17.71
]

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 \((0.85)(0.8)(\frac{821 }{\log_{20} 821}-\frac{380}{\log_{20}380})=118.\)
Status Protein1 Protein2 d d1/2
Query variables 6VKZ_1 3HQY_1 153 141
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]