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Parikh vectors
4GRT_1 5CYR_1 1WYC_1 Letter Amino acid
41 49 38 G Glycine
15 20 7 M Methionine
31 38 24 S Serine
31 46 28 T Threonine
8 27 13 Q Glutamine
34 65 34 L Leucine
34 44 4 K Lycine
13 28 13 Y Tyrosine
44 54 34 V Valine
21 44 32 D Aspartic acid
15 42 29 R Arginine
17 18 11 N Asparagine
9 10 4 C Cysteine
37 58 39 A Alanine
16 18 11 H Histidine
29 20 10 I Isoleucine
14 29 9 F Phenylalanine
19 35 21 P Proline
4 12 11 W Tryptophan
29 48 20 E Glutamic acid 

4GRT_1|Chain A|GLUTATHIONE REDUCTASE|Homo sapiens (9606)
>5CYR_1|Chains A, B|RNA-dependent RNA polymerase|Thosea asigna virus (83810)
>1WYC_1|Chain A|6-aminohexanoate-dimer hydrolase|Flavobacterium sp. (239)
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 \(w=f(c)\)
4GRT , Knot 192 461 0.85 40 237 446 
VASYDYLVIGGGSGGLESAWRAAELGARAAVVESHKLGGTCVNVGCVPKKVMWNTAVHSEFMHDHADYGFPSCEGKFNWRVIKEKRDAYVSRLNAIYQNNLTKSHIEIIRGHAAFTSDPKPTIEVSGKKYTAPHILIATGGMPSTPHESQIPGASLGITSDGFFQLEELPGRSVIVGAGYIAVEMAGILSALGSKTSLMIRHDKVLRSFDSMISTNCTEELENAGVEVLKFSQVKEVKKTLSGLEVSMVTAVPGRLPVMTMIPDVDCLLWAIGRVPNTKDLSLNKLGIQTDDKGHIIVDEFQNTNVKGIYAVGDVCGKALLTPVAIAAGRKLAHRLFEYKEDSKLDYNNIPTVVFSHPPIGTVGLTEDEAIHKYGIENVKTYSTSFTPMYHAVTKRKTKCVMKMVCANKEEKVVGIHMQGLGCDEMLQGFAVAVKMGATKADFDNTVAIHPTSSEELVTLR
5CYR , Knot 269 705 0.83 40 291 647 
MSYYHHHHHHDYDIPTTENLYFQGAMGAMGIMEASNPVIAPTRLSLEAMLAERAMVARQDLAGLKRKLAGADRVLAPQSPEQCGRESAQAQARSVTSELKSAVKEAQGLEHQTLDFLEQLGEYPVCGILHGDHPVHPSGTHNNNGKVSVKRQFAAGVNTSDALTCAFRFEDSDLVRETALKTTYTDGTWAGFVQRLKMQTTRKCVQEKVSRKLLKQLFPYDPQKLVDVSGELSELVLGIKTNAIASAGPPYWRTKRDALPDMLDCVLPLLYDHIVRKDLTTLRNKHPELFLAECKNKTDRYEVESLGEKTRPYFSHPFHLSALVSVLSQSFSGALKIMTEDSTSFNAYGFSWTNGGAEDLAIWARQAGEAGKKPPRIACYGDDTDIYYRKDGKLYRICPDFKQMDGSVDATTIEAVVDYVVDAHVKQYPTARQFWEEVGKLWVEMATQSPFLIDGTKVYRKMQKDGLMTGVVGTTLFDTVKSALAYNDWADQLMFGSLNLLEEKYAIEFFKNKHGLVIKEGTWKPALVNEDPGFGELWTEQKFLGLQLKVVRRENEKVYVPNLPFEDWLTMWVTPRSKYRSKETETMRERTLFDRARGLLVTGAVFDERARGLMGAVINSTAPEVVCMRVQEGGGRGAPPAYAFLTRDGVFEFPISDGYPSYDWVVSLYSRDHPCDMPRVFPEAATLIASYRKQVMDTRVVIKEE
1WYC , Knot 163 392 0.82 40 215 376 
MNARSTGQHPARYPGAAAGEPTLDSWQEPPHNRWAFAHLGEMVPSAAVSRRPVNAPGHALARLGAIAAQLPDLEQRLEQTYTDAFLVLRGTEVVAEYYRAGFAPDDRHLLMSVSKSLCGTVVGALVDEGRIDPAQPVTEYVPELAGSVYDGPSVLQVLDMQISIDYNEDYVDPASEVQTHDRSAGWRTRRHGDPADTYEFLTTLRGDGSTGEFQYCSANTDVLAWIVERVTGLRYVEALSTYLWAKLDADRDATITVDTTGFGFANGGVSCTARDLARVGRMMLDGGVAPGGRVVSEDWVRRVLAGGSHEAMTDKGFTNTFPDGSYTRQWWCTGNERGNVSGIGIHGQNLWLDPLTDSVIVKLSSWPDPDTEHWHRLQNGILLDVSRALDAV

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 \(c\).

\(|P_{f(4GRT_1)}(2) \setminus P_{f(5CYR_1)}(2)|=49\), \(|P_{f(5CYR_1)}(2) \setminus P_{f(4GRT_1)}(2)|=103\). 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 1:11000011111101110011011011101111000011100101101100111001100011000100111000101010110000010100101100001000010110101110001010101010000110111101111001000011110111000111010011100111111011101111101110000111000011001001100000001001110110100100100010110101101111011110111010011111101100001010011100000101110010000101101110101011101111111001100110000000100001101110011110111000011000110010000001011001100000001101101000001111010111000110111111011100101000111010000011010
Pair \(Z_2\) Length of longest common subsequence
4GRT_1,5CYR_1 152 4
4GRT_1,1WYC_1 162 4
5CYR_1,1WYC_1 168 5

Newick tree

 
[
	1WYC_1:84.57,
	[
		4GRT_1:76,5CYR_1:76
	]:8.57
]

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 \((0.85)(0.8)(\frac{1166 }{\log_{20} 1166}-\frac{461}{\log_{20}461})=183.\)
Status Protein1 Protein2 d d1/2
Query variables 4GRT_1 5CYR_1 233 192
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]