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Parikh vectors
3IDH_1 2GYO_1 3OYW_1 Letter Amino acid
30 14 5 R Arginine
16 8 3 Q Glutamine
17 21 4 I Isoleucine
31 19 5 S Serine
3 3 1 W Tryptophan
42 27 11 G Glycine
42 26 12 L Leucine
19 9 10 F Phenylalanine
9 11 7 P Proline
13 11 13 N Asparagine
16 8 2 H Histidine
12 6 2 Y Tyrosine
37 27 10 V Valine
23 8 1 M Methionine
23 25 4 T Threonine
25 43 14 A Alanine
27 19 9 D Aspartic acid
12 5 6 C Cysteine
48 17 7 E Glutamic acid
25 10 8 K Lycine 

3IDH_1|Chain A|Glucokinase|Homo sapiens (9606)
>2GYO_1|Chains A, B|3-oxoacyl-[acyl-carrier-protein] synthase 3|Escherichia coli (562)
>3OYW_1|Chain A|Galectin-1|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 \(w=f(c)\)
3IDH , Knot 192 470 0.83 40 241 440 
MGHHHHHHENLYFQGMKKEKVEQILAEFQLQEEDLKKVMRRMQKEMDRGLRLETHEEASVKMLPTYVRSTPEGSEVGDFLSLDLGGTNFRVMLVKVGEGEEGQWSVKTKHQMYSIPEDAMTGTAEMLFDYISECISDFLDKHQMKHKKLPLGFTFSFPVRHEDIDKGILLNWTKGFKASGAEGNNVVGLLRDAIKRRGDFEMDVVAMVNDTVATMISCYYEDHQCEVGMIVGTGCNACYMEEMQNVELVEGDEGRMCVNTEWGAFGDSGELDEFLLEYDRLVDESSANPGQQLYEKLIGGKYMGELVRLVLLRLVDENLLFHGEASEQLRTRGAFETRFVSQVESDTGDRKQIYNILSTLGLRPSTTDCDIVRRACESVSTRAAHMCSAGLAGVINRMRESRSEDVMRITVGVDGSVYKLHPSFKERFHASVRRLTPSCEITFIESEEGSGRGAALVSAVACKKACMLGQ
2GYO , Knot 139 317 0.84 40 185 300 
MYTKIIGTGSYLPEQVRTNADLEKMVDTSDEWIVTRTGIRERHIAAPNETVSTMGFEAATRAIEMAGIEKDQIGLIVVATTSATHAFPSAACQIQSMLGIKGCPAFDVAAACAGFTYALSVADQYVKSGAVKYALVVGSDVLARTCDPTDRGTIIIFGDGAGAAVLAASEEPGIISTHLHADGSYGELLTLPNADRVNPENSIHLTMAGNEVFKVAVTELAHIVDETLAANNLDRSQLDWLVPHQANLRIISATAKKLGMSMDNVVVTLDRHGNTSAASVPCALDEAVRDGRIKPGQLVLLEAFGGGFTWGSALVRF
3OYW , Knot 67 134 0.81 40 105 130 
ACGLVASNLNLKPGECLRVRGEVAPDAKSFVLNLGKDSNNLCLHFNPRFNAHGDANTIVCNSKDGGAWGTEQREAVFPFQPGSVAEVCITFDQANLTVKLPDGYEFKFPNRLNLEAINYMAADGDFKIKCVAFD

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 \(c\).

\(|P_{f(3IDH_1)}(2) \setminus P_{f(2GYO_1)}(2)|=106\), \(|P_{f(2GYO_1)}(2) \setminus P_{f(3IDH_1)}(2)|=50\). 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 1:11000000001010110000100111010100001001100100010011010000010101110010001010011011010111001011110110100101010000010011001101010111001000100110000100001111101011100001001111010011010110100111110011000101010111110001101100000000001111110100100100100101101001010100011111001010011100001100001011001000111100110110111101100011101010001000111000110010000100001001100111010000001100100010001101001111111001000000011010111010100101010001010100101000101100001010111110111000101110
Pair \(Z_2\) Length of longest common subsequence
3IDH_1,2GYO_1 156 4
3IDH_1,3OYW_1 200 3
2GYO_1,3OYW_1 160 4

Newick tree

 
[
	3OYW_1:94.36,
	[
		3IDH_1:78,2GYO_1:78
	]:16.36
]

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 \((0.85)(0.8)(\frac{787 }{\log_{20} 787}-\frac{317}{\log_{20}317})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 3IDH_1 2GYO_1 160 133
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]