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Parikh vectors
5HJO_1 7RGJ_1 5CYO_1 Letter Amino acid
24 10 15 N Asparagine
34 6 10 Q Glutamine
60 11 14 G Glycine
23 11 23 K Lycine
43 8 13 F Phenylalanine
23 3 2 W Tryptophan
34 16 27 I Isoleucine
58 8 10 P Proline
34 6 7 Y Tyrosine
59 11 19 V Valine
49 7 7 A Alanine
21 3 6 M Methionine
39 9 18 T Threonine
58 4 18 R Arginine
58 9 15 D Aspartic acid
7 0 4 C Cysteine
46 7 24 E Glutamic acid
32 2 9 H Histidine
90 12 19 L Leucine
65 14 14 S Serine 

5HJO_1|Chains A, C|Neutral alpha-glucosidase AB|Mus musculus (10090)
>7RGJ_1|Chains A, B|Dihydrofolate reductase type 1|Escherichia coli (562)
>5CYO_1|Chains A, B|Septin-9|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 \(w=f(c)\)
5HJO , Knot 322 857 0.84 40 312 793 
VDRSNFKTCDESSFCKRQRSIRPGLSPYRALLDTLQLGPDALTVHLIHEVTKVLLVLELQGLQKNMTRIRIDELEPRRPRYRVPDVLVADPPTARLSVSGRDDNSVELTVAEGPYKIILTAQPFRLDLLEDRSLLLSVNARGLMAFEHQRAPREPGAWEETFKTHSDSKPYGPTSVGLDFSLPGMEHVYGIPEHADSLRLKVTEGGEPYRLYNLDVFQYELNNPMALYGSVPVLLAHSFHRDLGIFWLNAAETWVDISSNTPQTDIRWMSESGIIDVFLMLGPSVFDVFRQYASLTGTQALPPLFSLGYHQSRWNYRDEADVLEVDQGFDDHNMPCDVIWLDIEHADGKRYFTWDPTRFPQPLNMLEHLASKRRKLVAIVDPHIKVDSGYRVHEELRNHGLYVKTRDGSDYEGWCWPGSASYPDFTNPRMRAWWSNMFSFDNYEGSAPNLYVWNDMNEPSVFNGPEVTMLKDAVHYGGWEHRDIHNIYGLYVHMATADGLIQRSGGIERPFVLSRAFFSGSQRFGAVWTGDNTAEWDHLKISIPMCLSLALVGLSFCGADVGGFFKNPEPELLVRWYQMGAYQPFFRAHAHLDTGRREPWLLASQYQDAIRDALFQRYSLLPFWYTLFYQAHKEGFPVMRPLWVQYPEDMSTFSIEDQFMLGDALLIHPVSDAGAHGVQVYLPGQEEVWYDIQSYQKHHGPQTLYLPVTLSSIPVFQRGGTIVPRWMRVRRSSDCMKDDPITLFVALSPQGTAQGELFLDDGHTFNYQTRHEFLLRRFSFSGSTLVSSSADPKGHLETPIWIERVVIMGAGKPAAVVLQTKGSPESRLSFQHDPETSVLILRKPGVSVASDWSIHLR
7RGJ , Knot 82 157 0.88 38 134 155 
MKLSLMVAISKNGVIGNGPDIPWSAKGEQLLFKAITYNQWLLVGRKTFESMGALPNRKYAVVTRSSFTSDNENVLIFPSIKDALTNLKKITDHVIVSGGGEIYKSLIDQVDTLHISTIDIEPEGDVYFPEIPSNFRPVFTQDFASNINYSYQIWQKG
5CYO , Knot 118 274 0.80 40 179 262 
QGFEFNIMVVGQSGLGKSTLINTLFKSKISRKSVQPTSEERIPKTIEIKSITHDIEEKGVRMKLTVIDTPGFGDHINNENCWQPIMKFINDQYEKYLQEEVNINRKKRIPDTRVHCCLYFIPATGHSLRPLDIEFMKRLSKVVNIVPVIAKADTLTLEERVHFKQRITADLLSNGIDVYPQKEFDEDSEDRLVNEKFREMIPFAVVGSDHEYQVNGKRILGRKTKWGTIEVENTTHCEFAYLRDLLIRTHMQNIKDITSSIHFEAYRVKRLNEG

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 \(c\).

\(|P_{f(5HJO_1)}(2) \setminus P_{f(7RGJ_1)}(2)|=200\), \(|P_{f(7RGJ_1)}(2) \setminus P_{f(5HJO_1)}(2)|=22\). 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 1:10000100000001000000101110100111001011101101011001001111101011000100101001010010001101111011010101010000010101101100111010110101100001110101011111000011001111000100000001011001110101111001011100100101010011010010010110001001111010111111001000111111011001101000010001011000111011111110110110001010100111111011000001000001011010011000011001111010010100010101001101101100110000011111010101001001000100011010000100001101110100101001010111001101000010110101100100101101101011001100111000010010110101101011100011100111100111010001111101000101001010111010111111010110111110010101110100111001110101010010001111100000110011100001111100110010001111101111001001001010001111011110110011101101011100011001000000011001011101001111001101110110100000010001101111101010101011100100100000001110010101001100010101010011110011111110111111000101000101000100011110011101100101010
Pair \(Z_2\) Length of longest common subsequence
5HJO_1,7RGJ_1 222 4
5HJO_1,5CYO_1 195 4
7RGJ_1,5CYO_1 169 4

Newick tree

 
[
	5HJO_1:11.32,
	[
		5CYO_1:84.5,7RGJ_1:84.5
	]:25.82
]

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 \((0.85)(0.8)(\frac{1014 }{\log_{20} 1014}-\frac{157}{\log_{20}157})=235.\)
Status Protein1 Protein2 d d1/2
Query variables 5HJO_1 7RGJ_1 299 175.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]