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Parikh vectors
2GAC_1 3SYM_1 5XUY_1 Letter Amino acid
12 48 13 G Glycine
4 22 2 H Histidine
8 36 14 T Threonine
11 63 10 A Alanine
9 46 4 N Asparagine
2 30 7 Q Glutamine
14 65 14 E Glutamic acid
6 36 5 P Proline
4 12 2 W Tryptophan
6 63 11 R Arginine
2 9 5 C Cysteine
12 79 21 L Leucine
13 48 14 K Lycine
4 21 4 M Methionine
3 38 6 F Phenylalanine
8 29 16 S Serine
3 36 7 Y Tyrosine
9 49 8 D Aspartic acid
7 50 12 I Isoleucine
14 62 15 V Valine 

2GAC_1|Chains A, C|GLYCOSYLASPARAGINASE|Elizabethkingia meningoseptica (238)
>3SYM_1|Chain A|Glycogen phosphorylase, muscle form|Oryctolagus cuniculus (9986)
>5XUY_1|Chains A, C|Autophagy-related protein 13|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 \(w=f(c)\)
2GAC , Knot 73 151 0.80 40 113 143 
TTNKPIVLSTWNFGLHANVEAWKVLSKGGKALDAVEKGVRLVEDDPTERSVGYGGRPDRDGRVTLDACIMDENYNIGSVACMEHIKNPISVARAVMEKTPHVMLVGDGALEFALSQGFKKENLLTAESEKEWKEWLKTSQYKPIVNIENHD
3SYM , Knot 321 842 0.85 40 316 773 
SRPLSDQEKRKQISVRGLAGVENVTELKKNFNRHLHFTLVKDRNVATPRDYYFALAHTVRDHLVGRWIRTQQHYYEKDPKRIYYLSLEFYMGRTLQNTMVNLALENACDEATYQLGLDMEELEEIEEDAGLGNGGLGRLAACFLDSMATLGLAAYGYGIRYEFGIFNQKICGGWQMEEADDWLRYGNPWEKARPEFTLPVHFYGRVEHTSQGAKWVDTQVVLAMPYDTPVPGYRNNVVNTMRLWSAKAPNDFNLKDFNVGGYIQAVLDRNLAENISRVLYPNDNFFEGKELRLKQEYFVVAATLQDIIRRFKSSKFGCRDPVRTNFDAFPDKVAIQLNDTHPSLAIPELMRVLVDLERLDWDKAWEVTVKTCAYTNHTVLPEALERWPVHLLETLLPRHLQIIYEINQRFLNRVAAAFPGDVDRLRRMSLVEEGAVKRINMAHLCIAGSHAVNGVARIHSEILKKTIFKDFYELEPHKFQNKTNGITPRRWLVLCNPGLAEIIAERIGEEYISDLDQLRKLLSYVDDEAFIRDVAKVKQENKLKFAAYLEREYKVHINPNSLFDVQVKRIHEYKRQLLNCLHVITLYNRIKKEPNKFVVPRTVMIGGKAAPGYHMAKMIIKLITAIGDVVNHDPVVGDRLRVIFLENYRVSLAEKVIPAADLSEQISTAGTEASGTGNMKFMLNGALTIGTMDGANVEMAEEAGEENFFIFGMRVEDVDRLDQRGYNAQEYYDRIPELRQIIEQLSSGFFSPKQPDLFKDIVNMLMHHDRFKVFADYEEYVKCQERVSALYKNPREWTRMVIRNIATSGKFSSDRTIAQYAREIWGVEPSRQRLPAPDEKIP
5XUY , Knot 88 190 0.81 40 130 179 
METDLNSQDRKDLDKFIKFFALKTVQVIVQARLGEKICTRSSSSPTGSDWFNLAIKDIPEVTHEAKKALAGQLPAVGRSMCVEISLKTSEGDSMELEIWCLEMNEKCDKEIKVSYTVYNRLSLLLKSLLAITRVTPAYRLSRKQGHEYVILYRIYFGEVQLSGLGEGFQTVRVGTVGTPVGTITLSCAYR

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 \(c\).

\(|P_{f(2GAC_1)}(2) \setminus P_{f(3SYM_1)}(2)|=13\), \(|P_{f(3SYM_1)}(2) \setminus P_{f(2GAC_1)}(2)|=216\). 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 1:0000111100101110101011011001101101100110110001000011011010001010101011000001101101001001101101110001011111011101110011000011010000010011000000111010000
Pair \(Z_2\) Length of longest common subsequence
2GAC_1,3SYM_1 229 4
2GAC_1,5XUY_1 157 4
3SYM_1,5XUY_1 216 3

Newick tree

 
[
	3SYM_1:12.25,
	[
		2GAC_1:78.5,5XUY_1:78.5
	]:41.75
]

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 \((0.85)(0.8)(\frac{993 }{\log_{20} 993}-\frac{151}{\log_{20}151})=231.\)
Status Protein1 Protein2 d d1/2
Query variables 2GAC_1 3SYM_1 294 171
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]