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Parikh vectors
9MOT_1 3GRS_1 5DDZ_1 Letter Amino acid
46 34 2 K Lycine
20 15 4 M Methionine
36 44 15 V Valine
32 17 21 R Arginine
25 11 14 Q Glutamine
23 16 10 H Histidine
47 34 22 L Leucine
49 31 17 T Threonine
35 13 14 Y Tyrosine
10 10 2 C Cysteine
49 29 23 I Isoleucine
30 14 14 F Phenylalanine
37 24 15 P Proline
11 3 1 W Tryptophan
33 17 17 N Asparagine
48 29 15 E Glutamic acid
38 43 18 G Glycine
59 31 19 S Serine
34 42 24 A Alanine
47 21 8 D Aspartic acid 

9MOT_1|Chain A|Coagulation factor Va heavy chain|Homo sapiens (9606)
>3GRS_1|Chain A|GLUTATHIONE REDUCTASE|Homo sapiens (9606)
>5DDZ_1|Chain A|Ricin|Ricinus communis (3988)
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 \(w=f(c)\)
9MOT , Knot 278 709 0.85 40 302 644 
AQLRQFYVAAQGISWSYRPEPTNSSLNLSVTSFKKIVYREYEPYFKKEKPQSTISGLLGPTLYAEVGDIIKVHFKNKADKPLSIHPQGIRYSKLSEGASYLDHTFPAEKMDDAVAPGREYTYEWSISEDSGPTHDDPPCLTHIYYSHENLIEDFNSGLIGPLLICKKGTLTEGGTQKTFDKQIVLLFAVFDESKSWSQSSSLMYTVNGYVNGTMPDITVCAHDHISWHLLGMSSGPELFSIHFNGQVLEQNHHKVSAITLVSATSTTANMTVGPEGKWIISSLTPKHLQAGMQAYIDIKNCPKKTRNLKKITREQRRHMKRWEYFIAAEEVIWDYAPVIPANMDKKYRSQHLDNFSNQIGKHYKKVMYTQYEDESFTKHTVNPNMKEDGILGPIIRAQVRDTLKIVFKNMASRPYSIYPHGVTFSPYEDEVNSSFTSGRNNTMIRAVQPGETYTYKWNILEFDEPTENDAQCLTRPYYSDVDIMRDIASGLIGLLLICKSRSLDRRGIQRAADIEQQAVFAVFDENKSWYLEDNINKFCENPDEVKRDDPKFYESNIMSTINGYVPESITTLGFCFDDTVQWHFCSVGTQNEILTIHFTGHSFIYGKRHEDTLTLFPMRGESVTVTMDNVGTWMLTSMNSSPRSKKLRLKFRDVKCIPDDDEDSYEIFEPPESTVMATRKMHDRLEPEDEESDADYDYQNRLAAALGIR
3GRS , Knot 198 478 0.85 40 246 462 
ACRQEPQPQGPPPAAGAVASYDYLVIGGGSGGLASARRAAELGARAAVVESHKLGGTCVNVGCVPKKVMWNTAVHSEFMHDHADYGFPSCEGKFNWRVIKEKRDAYVSRLNAIYQNNLTKSHIEIIRGHAAFTSDPKPTIEVSGKKYTAPHILIATGGMPSTPHESQIPGASLGITSDGFFQLEELPGRSVIVGAGYIAVEMAGILSALGSKTSLMIRHDKVLRSFDSMISTNCTEELENAGVEVLKFSQVKEVKKTLSGLEVSMVTAVPGRLPVMTMIPDVDCLLWAIGRVPNTKDLSLNKLGIQTDDKGHIIVDEFQNTNVKGIYAVGDVCGKALLTPVAIAAGRKLAHRLFEYKEDSKLDYNNIPTVVFSHPPIGTVGLTEDEAIHKYGIENVKTYSTSFTPMYHAVTKRKTKCVMKMVCANKEEKVVGIHMQGLGCDEMLQGFAVAVKMGATKADFDNTVAIHPTSSEELVTLR
5DDZ , Knot 118 275 0.80 40 175 264 
MGHHHHHHIFPKQYPIINFTTAGATVQSYTNFIRAVRGRLTTGADVRHEIPVLPNRVGLPINQRFILVELSNHAELSVTLALDVTNAYVVGYRAGNSAYFFHPDNQEDAEAITHLFTDVQNRYTFAFGGNYDRLEQLAGNLRENIELGNGPLEEAISALYYYSTGGTQLPTLARSFIICIQMISEAARFQYIEGEMRTRIRYNRRSAPDPSVITLENSWGRLSTAIQESNQGAFASPIQLQRRNGSKFSVYDVSILIPIIALMVYRCAPPPSSQF

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 \(c\).

\(|P_{f(9MOT_1)}(2) \setminus P_{f(3GRS_1)}(2)|=96\), \(|P_{f(3GRS_1)}(2) \setminus P_{f(9MOT_1)}(2)|=40\). 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 1:1010010111011010001010000101010010011000001010000100010111110101011011010100010011010101100001001100100011100100111110000001010000110000110100100000011001001111111100010100110000100011111111000001000001100101010101101010100010101111001101101010101100000010110110100001010111010111001010010111010101000100000100100000001001001111001110011111101000000001001000110000011000000001000010101000111111101010001011100110010010101101010000100010010000110110110000001011010010000100100100001011001101111111100000100011001101000111111000001010001001000100100001010000110010101100100111010001010100110000110101010011010000001011110100101010011011100100010000101010010011000000001101100011100010001010000001000000011111110
Pair \(Z_2\) Length of longest common subsequence
9MOT_1,3GRS_1 136 4
9MOT_1,5DDZ_1 185 4
3GRS_1,5DDZ_1 177 3

Newick tree

 
[
	5DDZ_1:96.87,
	[
		9MOT_1:68,3GRS_1:68
	]:28.87
]

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 \((0.85)(0.8)(\frac{1187 }{\log_{20} 1187}-\frac{478}{\log_{20}478})=183.\)
Status Protein1 Protein2 d d1/2
Query variables 9MOT_1 3GRS_1 234 196
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]