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Parikh vectors
5CJM_1 6ZTQ_1 3MWQ_1 Letter Amino acid
46 1 10 R Arginine
3 1 16 C Cysteine
36 6 8 P Proline
48 10 20 T Threonine
88 8 24 A Alanine
26 3 14 Q Glutamine
36 6 10 E Glutamic acid
60 3 9 G Glycine
14 0 8 H Histidine
41 11 6 I Isoleucine
23 9 6 F Phenylalanine
62 2 10 D Aspartic acid
16 4 12 Y Tyrosine
59 4 18 V Valine
26 4 20 K Lycine
65 22 4 L Leucine
19 6 8 M Methionine
34 7 33 S Serine
10 4 0 W Tryptophan
29 4 20 N Asparagine 

5CJM_1|Chain A|Malate synthase G|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>6ZTQ_1|Chain A|NADH-ubiquinone oxidoreductase chain 3|Mus musculus (10090)
>3MWQ_1|Chain A|Ribonuclease pancreatic, LINKER, Ribonuclease pancreatic|Bos taurus (9913)
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 \(w=f(c)\)
5CJM , Knot 277 741 0.82 40 268 652 
MTDRVSVGNLRIARVLYDFVNNEALPGTDIDPDSFWAGVDKVVADLTPQNQALLNARDELQAQIDKWHRRRVIEPIDMDAYRQFLTEIGYLLPEPDDFTITTSGVDAEITTTAGPQLVVPVLNARFALNAANARWGSLYDALYGTDVIPETDGAEKGPTYNKVRGDKVIAYARKFLDDSVPLSSGSFGDATGFTVQDGQLVVALPDKSTGLANPGQFAGYTGAAESPTSVLLINHGLHIEILIDPESQVGTTDRAGVKDVILESAITTIMDFEDSVAAVDAADKVLGYRNWLGLNKGDLAAAVDKDGTAFLRVLNRDRNYTAPGGGQFTLPGRSLMFVRNVGHLMTNDAIVDTDGSEVFEGIMDALFTGLIAIHGLKASDVNGPLINSRTGSIYIVKPKMHGPAEVAFTCELFSRVEDVLGLPQNTMKIGIMDEERRTTVNLKACIKAAADRVVFINTGFLDRTGDEIHTSMEAGPMVRKGTMKSQPWILAYEDHNVDAGLAAGFSGRAQVGKGMWTMTELMADMVETKIAQPRAGASTAWVPSPTAATLHALHYHQVDVAAVQQGLAGKRRATIEQLLTIPLAKELAWAPDEIREEVDNNCQSILGYVVRWVDQGVGASKVPDIHDVALMEDRATLRISSQLLANWLRHGVITSADVRASLERMAPLVDRQNAGDVAYRPMAPNFDDSIAFLAAQELILSGAQQPNGYTEPILHRRRREFKARAAEKPAPSDRAGDDAAR
6ZTQ , Knot 57 115 0.78 38 85 107 
MNLYTVIFINILLSLTLILVAFWLPQMNLYSEKANPYECGFDPTSSARLPFSMKFFLVAITFLLFDLEIALLLPLPWAIQTIKTSTMMIMAFILVTILSLGLAYEWTQKGLEWTE
3MWQ , Knot 68 256 0.49 38 112 128 
KETAAAKFERQHMDSSTSAASSSNYCNQMMKSRNLTKDRCKPVNTFVHESLADVQAVCSQKNVACKNGQTNCYQSYSTMSITDCRETGSSKYPNCAYKTTQANKHIIVACEGNPYVPVHFDASVSGRSGRSGKETAAAKFERQHMDSSTSAASSSNYCNQMMKSRNLTKDRCKPVNTFVHESLADVQAVCSQKNVACKNGQTNCYQSYSTMSITDCRETGSSKYPNCAYKTTQANKHIIVACEGNPYVPVHFDASV

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 \(c\).

\(|P_{f(5CJM_1)}(2) \setminus P_{f(6ZTQ_1)}(2)|=204\), \(|P_{f(6ZTQ_1)}(2) \setminus P_{f(5CJM_1)}(2)|=21\). 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 1:100010110101101100110001111001010011111001110101000111010001010100100001101101010001100110111010010100011010100011101111110101110110101101001101001110001100110000101001110100110001110010110101101001011111100001110110111001110010011110011010111010001100001110011100110011010001111011001110001111001011111000101110110000000111110101110011110011011000111000100110111011101111101101001011110000101011010101110111000110010011111000101111000000010101010111001111001110001001000101111100101000111110000010111111101010110111010011101100011010111001111010110101100001011110011110001010011011110011111001000100000011101101100111100110100111100010101000111011001110010101010011111000011011001111010001111110011101100101000111000000101011001110001100110
Pair \(Z_2\) Length of longest common subsequence
5CJM_1,6ZTQ_1 225 4
5CJM_1,3MWQ_1 224 3
6ZTQ_1,3MWQ_1 145 3

Newick tree

 
[
	5CJM_1:12.67,
	[
		3MWQ_1:72.5,6ZTQ_1:72.5
	]:50.17
]

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 \((0.85)(0.8)(\frac{856 }{\log_{20} 856}-\frac{115}{\log_{20}115})=208.\)
Status Protein1 Protein2 d d1/2
Query variables 5CJM_1 6ZTQ_1 260 149.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]