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Parikh vectors
8CRS_1 6GNI_1 2FRA_1 Letter Amino acid
36 44 7 I Isoleucine
28 79 13 L Leucine
37 31 17 K Lycine
25 30 7 R Arginine
45 38 24 G Glycine
11 18 10 H Histidine
9 11 9 C Cysteine
30 62 17 S Serine
9 6 4 W Tryptophan
21 28 16 Y Tyrosine
38 35 12 E Glutamic acid
17 15 3 M Methionine
19 39 7 F Phenylalanine
11 37 7 Q Glutamine
20 48 8 P Proline
20 52 8 T Threonine
33 40 17 V Valine
26 62 16 A Alanine
15 46 12 N Asparagine
30 46 11 D Aspartic acid 

8CRS_1|Chains A, C|Nitrogenase molybdenum-iron protein alpha chain|Azotobacter vinelandii (354)
>6GNI_1|Chain A|Protein transport protein SEC23|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>2FRA_1|Chains A, B|cathepsin S|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 \(w=f(c)\)
8CRS , Knot 197 480 0.84 40 252 456 
MTGMSREEVESLIQEVLEVYPEKARKDRNKHLAVNDPAVTQSKKCIISNKKSQPGLMTIRGCAYAGSKGVVWGPIKDMIHISHGPVGCGQYSRAGRRNYYIGTTGVNAFVTMNFTSDFQEKDIVFGGDKKLAKLIDEVETLFPLNKGISVQSECPIGLIGDDIESVSKVKGAELSKTIVPVRCEGFRGVSQSLGHHIANDAVRDWVLGKRDEDTTFASTPYDVAIIGDYNIGGDAWSSRILLEEMGLRCVAQWSGDGSISEIELTPKVKLNLVHCYRSMNYISRHMEEKYGIPWMEYNFFGPTKTIESLRAIAAKFDESIQKKCEEVIAKYKPEWEAVVAKYRPRLEGKRVMLYIGGLRPRHVIGAYEDLGMEVVGTGYEFAHNDDYDRTMKEMGDSTLLYDDVTGYEFEEFVKRIKPDLIGSGIKEKFIFQKMGIPFREMHSWDYSGPYHGFDGFAIFARDMDMTLNNPCWKKLQAPWE
6GNI , Knot 294 767 0.84 40 301 696 
DFETNEDINGVRFTWNVFPSTRSDANSNVVPVGCLYTPLKEYDELNVAPYNPVVCSGPHCKSILNPYCVIDPRNSSWSCPICNSRNHLPPQYTNLSQENMPLELQSTTIEYITNKPVTVPPIFFFVVDLTSETENLDSLKESIITSLSLLPPNALIGLITYGNVVQLHDLSSETIDRCNVFRGDREYQLEALTEMLTGQKPTGPGGAASHLPNAMNKVTPFSLNRFFLPLEQVEFKLNQLLENLSPDQWSVPAGHRPLRATGSALNIASLLLQGCYKNIPARIILFASGPGTVAPGLIVNSELKDPLRSHHDIDSDHAQHYKKACKFYNQIAQRVAANGHTVDIFAGCYDQIGMSEMKQLTDSTGGVLLLTDAFSTAIFKQSYLRLFAKDEEGYLKMAFNGNMAVKTSKDLKVQGLIGHASAVKKTDANNISESEIGIGATSTWKMASLSPYHSYAIFFEIANTAANSNPMMSAPGSADRPHLAYTQFITTYQHSSGTNRIRVTTVANQLLPFGTPAIAASFDQEAAAVLMARIAVHKAETDDGADVIRWLDRTLIKLCQKYADYNKDDPQSFRLAPNFSLYPQFTYYLRRSQFLSVFNNSPDETAFYRHIFTREDTTNSLIMIQPTLTSFSMEDDPQPVLLDSISVKPNTILLLDTFFFILIYHGEQIAQWRKAGYQDDPQYADFKALLEEPKLEAAELLVDRFPLPRFIDTEAGGSQARFLLSKLNPSDNYQDMARGGSTIVLTDDVSLQNFMTHLQQVAVSGQA
2FRA , Knot 104 225 0.83 40 152 214 
ILPDSVDWREKGCVTEVKYQGSCGACWAFSAVGALEAQLKLKTGKLVSLSAQNLVDCSTEKYGNKGCNGGFMTTAFQYIIDNKGIDSDASYPYKAMDQKCQYDSKYRAATCSKYTELPYGREDVLKEAVANKGPVSVGVDARHPSFFLYRSGVYYEPSCTQNVNHGVLVVGYGDLNGKEYWLVKNSWGHNFGEEGYIRMARNKGNHCGIASFPSYPEIGHHHHHH

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 \(c\).

\(|P_{f(8CRS_1)}(2) \setminus P_{f(6GNI_1)}(2)|=46\), \(|P_{f(6GNI_1)}(2) \setminus P_{f(8CRS_1)}(2)|=95\). 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 1:101100001001100110101001000000011100111000000110000001111010101011001111111001101001111010000110000011001101110101000100001111100011011001001111001101000011111100100100101101000111100011011000110011001100111100000001100100111110001110110001110011100110101010100101010101011000001001000100001111100011110001001011110100010000001110001010111100010101001110111101001111000111011101001100000000100110001100010100100110010101110110001110011111001001000110011011111100101010010100101110
Pair \(Z_2\) Length of longest common subsequence
8CRS_1,6GNI_1 141 4
8CRS_1,2FRA_1 190 4
6GNI_1,2FRA_1 197 4

Newick tree

 
[
	2FRA_1:10.05,
	[
		8CRS_1:70.5,6GNI_1:70.5
	]:33.55
]

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 \((0.85)(0.8)(\frac{1247 }{\log_{20} 1247}-\frac{480}{\log_{20}480})=197.\)
Status Protein1 Protein2 d d1/2
Query variables 8CRS_1 6GNI_1 253 205
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]