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Parikh vectors
1YNG_1 8YMO_1 7EFQ_1 Letter Amino acid
0 40 24 K Lycine
0 14 0 W Tryptophan
0 28 8 Y Tyrosine
0 24 16 Q Glutamine
0 84 39 L Leucine
0 38 13 T Threonine
0 43 19 E Glutamic acid
0 14 6 H Histidine
0 22 9 M Methionine
0 52 15 F Phenylalanine
0 55 14 V Valine
0 36 8 R Arginine
0 27 18 D Aspartic acid
3 6 1 C Cysteine
0 62 21 I Isoleucine
0 41 11 P Proline
0 40 16 S Serine
11 67 17 A Alanine
0 39 9 N Asparagine
5 40 12 G Glycine 

1YNG_1|Chain A|apolipoprotein B mRNA|null
>8YMO_1|Chains A, B[auth C]|Protein OSCA1|Arabidopsis thaliana (3702)
>7EFQ_1|Chains A, B|Peroxisome proliferator-activated receptor gamma|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 \(w=f(c)\)
1YNG , Knot 10 31 0.36 8 12 17 
GGAUAUAUGAUAUAAUUUGAUCAGUAUAUCC
8YMO , Knot 295 772 0.84 40 301 711 
MATLKDIGVSAGINILTAFIFFIIFAFLRLQPFNDRVYFSKWYLRGLRSSPASGGGFAGRFVNLELRSYLKFLHWMPEALKMPERELIDHAGLDSVVYLRIYWLGLKIFAPIAMLAWAVLVPVNWTNNELELAKHFKNVTSSDIDKLTISNIPEGSNRFWAHIIMAYAFTIWTCYMLMKEYETVANMRLQFLASEGRRPDQFTVLVRNVPPDPDETVSELVEHFFLVNHPDNYLTHQVVCNANKLADLVSKKTKLQNWLDYYQLKYTRNNSQIRPITKLGCLGLCGQKVDAIEHYIAEVDKTSKEIAEERENVVNDQKSVMPASFVSFKTRWAAAVCAQTTQTRNPTEWLTEWAAEPRDIYWPNLAIPYVSLTVRRLVMNVAFFFLTFFFIIPIAFVQSLATIEGIEKVAPFLKVIIEKDFIKSLIQGLLAGIALKLFLIFLPAILMTMSKFEGFTSVSFLERRSASRYYIFNLVNVFLGSVIAGAAFEQLNSFLNQSPNQIPKTIGMAIPMKATAFITYIMVDGWAGVAGEILMLKPLIIYHLKNAFLVKTEKDREEAMNPGSIGFNTGEPQIQLYFLLGLVYAPVTPMLLPFILVFFALAYVVYRHQIINVYNQEYESAAAFWPDVHGRVITALIISQLLLMGLLGTKHAASAAPFLIALPVITIGFHRFCKGRFEPAFVRYPLQEAMMKDTLERAREPNLNLKGYLQDAYIHPVFKGGDNDDDGDMIGKLENEVIIVPTKRQSRRNTPAPSRISGESSPSLAVINGKEV
7EFQ , Knot 121 276 0.82 38 174 264 
GALNPESADLRALAKHLYDSYIKSFPLTKAKARAILTGKTTDKSPFVIYDMNSLMMGEDKIKFKHITPLQEQSKEVAIRIFQGCQFRSVEAVQEITEYAKSIPGFVNLDLNDQVTLLKYGVHEIIYTMLASLMNKDGVLISEGQGFMTREFLKSLRKPFGDFMEPKFEFAVKFNALELDDSDLAIFIAVIILSGDRPGLLNVKPIEDIQDNLLQALELQLKLNHPESSQLFAKLLQKMTDLRQIVTEHVQLLQVIKKTETDMSLHPLLQEIYKDLY

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 \(c\).

\(|P_{f(1YNG_1)}(2) \setminus P_{f(8YMO_1)}(2)|=7\), \(|P_{f(8YMO_1)}(2) \setminus P_{f(1YNG_1)}(2)|=296\). 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 1:1110101011010110001100110101000
Pair \(Z_2\) Length of longest common subsequence
1YNG_1,8YMO_1 303 2
1YNG_1,7EFQ_1 184 2
8YMO_1,7EFQ_1 163 4

Newick tree

 
[
	1YNG_1:13.85,
	[
		7EFQ_1:81.5,8YMO_1:81.5
	]:55.35
]

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 \((0.85)(0.8)(\frac{803 }{\log_{20} 803}-\frac{31}{\log_{20}31})=226.\)
Status Protein1 Protein2 d d1/2
Query variables 1YNG_1 8YMO_1 291 150
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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