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Parikh vectors
2QLI_1 4ILM_1 7VHY_1 Letter Amino acid
11 31 36 L Leucine
12 12 29 P Proline
3 1 5 W Tryptophan
11 11 21 A Alanine
13 9 28 D Aspartic acid
23 21 34 E Glutamic acid
11 12 18 Y Tyrosine
15 20 25 V Valine
0 1 19 C Cysteine
21 24 33 K Lycine
11 4 9 M Methionine
9 32 20 I Isoleucine
10 18 28 R Arginine
4 12 11 N Asparagine
8 5 18 Q Glutamine
12 25 29 S Serine
13 11 22 T Threonine
24 22 26 G Glycine
6 5 16 H Histidine
9 13 27 F Phenylalanine 

2QLI_1|Chains A, B|Fluorescent protein plum|Discosoma sp. LW-2004 (301246)
>4ILM_1|Chains A, C[auth B], E[auth D], G[auth F], I[auth H], K[auth J], M[auth L], O[auth N]|CRISPR-associated endoribonuclease Cas6 2|Sulfolobus solfataricus (273057)
>7VHY_1|Chains A, B|Histone acetyltransferase p300|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 \(w=f(c)\)
2QLI , Knot 105 226 0.84 38 153 217 
MVSKGEEVIKEFMRFKQHMEGSVNGHEFEIEGEGEGRPYEGTQTARLKVTKGGPLPFAWDILSPQIMYGSKAYVKHPADIPDYLKLSFPEGFKWERVMNFEDGGVVTVTQDSSLQDGEFIYKVKVRGTNFPSDGPVMQKKTMGWEASSERMYPEDGALKGEMKMRLRLKDGGHYDAEVKTTYMAKKPVQLPGAYKTDIKLDITSHNEDYTIVEQYERAEGRHSTGA
4ILM , Knot 127 289 0.83 40 168 273 
MPLIFKIGYNVIPLQDVILPTPSSKVLKYLIQSGKLIPSLKDLITSRDKYKPIFISHLGFNQRRIFQTNGNLKTITKGSRLSSIIAFSTQANVLSEVADEGIFETVYGKFHIMIESIEIVEVEKLKEEVEKHMNDNIRVRFVSPTLLSSKVLLPPSLSERYKKIHAGYSTLPSVGLIVAYAYNVYCNLIGKKEVEVRAFKFGILSNALSRIIGYDLHPVTVAIGEDSKGNLRKARGVMGWIEFDIPDERLKRRALNYLLTSSYLGIGRSRGIGFGEIRLEFRKIEEKEG
7VHY , Knot 194 454 0.87 40 259 435 
GPSLGYCCGRKLEFSPQTLCCYGKQLCTIPRDATYYSYQNRYHFCEKCFNEIQGESVSLGDDPSQPQTTINKEQFSKRKNDTLDPELFVECTECGRKMHQICVLHHEIIWPAGFVCDGCLKKSARTRKENKFSAKRLPSTRLGTFLENRVNDFLRRQNHPESGEVTVRVVHASDKTVEVKPGMKARFVDSGEMAESFPYRTKALFAFEEIDGVDLCFFGMHVQEYGSDCPPPNQRRVYISYLDSVHFFRPKCLRTAVYHEILIGYLEYVKKLGYTTGHIWACPPSEGDDYIFHCHPPDQKIPKPKRLQEWFKKMLDKAVSERIVHDYKDIFKQATEDRLTSAKELPYFEGDFWPNVLEESIKESGGSGSQKLYATMEKHKEVFFVIRLIAGPAANSLPPIVDPDPLIPCDLMDGRDAFLTLARDKHLEFSSLRRAQWSTMCMLVELHTQSQDRF

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 \(c\).

\(|P_{f(2QLI_1)}(2) \setminus P_{f(4ILM_1)}(2)|=70\), \(|P_{f(4ILM_1)}(2) \setminus P_{f(2QLI_1)}(2)|=85\). 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 1:1100100110011010001010101001010101010100100010101001111111101101011010010100110110010101101101001101001111010000010010110010101001100111100001110100001010011101010101010011000101000011001101111000010101000000001100000101000011
Pair \(Z_2\) Length of longest common subsequence
2QLI_1,4ILM_1 155 4
2QLI_1,7VHY_1 196 3
4ILM_1,7VHY_1 191 4

Newick tree

 
[
	7VHY_1:10.37,
	[
		2QLI_1:77.5,4ILM_1:77.5
	]:24.87
]

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 \((0.85)(0.8)(\frac{515 }{\log_{20} 515}-\frac{226}{\log_{20}226})=83.0\)
Status Protein1 Protein2 d d1/2
Query variables 2QLI_1 4ILM_1 103 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]