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Parikh vectors
5UPF_1 6IYV_1 3MVX_1 Letter Amino acid
15 7 27 Q Glutamine
18 8 21 F Phenylalanine
7 9 9 W Tryptophan
38 30 25 V Valine
49 9 28 K Lycine
3 6 10 M Methionine
29 17 26 S Serine
30 22 29 T Threonine
25 20 52 A Alanine
26 18 23 N Asparagine
5 1 0 C Cysteine
36 29 25 G Glycine
14 10 21 R Arginine
13 8 10 H Histidine
29 15 32 I Isoleucine
28 9 19 Y Tyrosine
30 18 28 D Aspartic acid
38 12 32 E Glutamic acid
45 8 47 L Leucine
21 16 38 P Proline 

5UPF_1|Chains A, B|Nicotinamide phosphoribosyltransferase|Homo sapiens (9606)
>6IYV_1|Chains A, B|L,D-transpeptidase 2|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>3MVX_1|Chains A, B|Nickel-binding periplasmic protein|Escherichia coli (83333)
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 \(w=f(c)\)
5UPF , Knot 203 499 0.84 40 241 472 
MNPAAEAEFNILLATDSYKVTHYKQYPPNTSKVYSYFECREKKTENSKLRKVKYEETVFYGLQYILNKYLKGKVVTKEKIQEAKDVYKEHFQDDVFNEKGWNYILEKYDGHLPIEIKAVPEGFVIPRGNVLFTVENTDPECYWLTNWIETILVQSWYPITVATNSREQKKILAKYLLETSGNLDGLEYKLHDFGYRGVSSQETAGIGASAHLVNFKGTDTVAGLALIKKYYGTKDPVPGYSVPAAEHSTITAWGKDHEKDAFEHIVTQFSSVPVSVVSDSYDIYNACEKIWGEDLRHLIVSRSTQAPLIIRPDSGNPLDTVLKVLEILGKKFPVTENSKGYKLLPPYLRVIQGDGVDINTLQEIVEGMKQKMWSIENIAFGSGGGLLQKLTRDLLNCSFKCSYVVTNGLGINVFKDPVADPNKRSKKGRLSLHRTPAGNFVTLEEGKGDLEEYGQDLLHTVFKNGKVTKSYSFDEIRKNAQLNIELEAAHHDYKDDDDK
6IYV , Knot 121 272 0.83 40 180 262 
SNAQLTFQTSSPAHLTMPYVMPGDGEVVGVGEPVAIRFDENIADRGAAEKAIKITTNPPVEGAFYWLNNREVRWRPEHFWKPGTAVDVAVNTYGVDLGEGMFGEDNVQTHFTIGDEVIATADDNTKILTVRVNGEVVKSMPTSMGKDSTPTANGIYIVGSRYKHIIMDSSTYGVPVNSPNGYRTDVDWATQISYSGVFVHSAPWSVGAQGHTNTSHGCLNVSPSNAQWFYDHVKRGDIVEVVNTVGGTLPGIDGLGDWNIPWDQWRAGNAKA
3MVX , Knot 202 502 0.83 38 246 472 
AAPDEITTAWPVNVGPLNPHLYTPNQMFAQSMVYEPLVKYQADGSVIPWLAKSWTHSEDGKTWTFTLRDDVKFSNGEPFDAEAAAENFRAVLDNRQRHAWLELANQIVDVKALSKTELQITLKSAYYPFLQELALPRPFRFIAPSQFKNHETMNGIKAPIGTGPWILQESKLNQYDVFVRNENYWGEKPAIKKITFNVIPDPTTRAVAFETGDIDLLYGNEGLLPLDTFARFSQNPAYHTQLSQPIETVMLALNTAKAPTNELAVREALNYAVNKKSLIDNALYGTQQVADTLFAPSVPYANLGLKPSQYDPQKAKALLEKAGWTLPAGKDIREKNGQPLRIELSFIGTDALSKSMAEIIQADMRQIGADVSLIGEEESSIYARQRDGRFGMIFHRTWGAPYDPHAFLSSMRVPSHADFQAQQGLADKPLIDKEIGEVLATHDETQRQALYRDILTRLHDEAVYLPISYISMMVVSKPELGNIPYAPIATEIPFEQIKPVKP

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 \(c\).

\(|P_{f(5UPF_1)}(2) \setminus P_{f(6IYV_1)}(2)|=112\), \(|P_{f(6IYV_1)}(2) \setminus P_{f(5UPF_1)}(2)|=51\). 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 1:1011101010111100000100000011000010001000000000001001000001101100110001010110000100100100001000110001100110000101110101110111110101110100001000110011001110010110110000000011100110001010110001001100110000011111010110101000111111100001000111100111100001011100000011001100100111011000001001000111001001110000011111010010110011011011100111000001001111010110101101001001101100011010011110111110010001100010000110011110110011101000000101010001110110100101010001001100110010100000100100010101010110000000000
Pair \(Z_2\) Length of longest common subsequence
5UPF_1,6IYV_1 163 4
5UPF_1,3MVX_1 129 4
6IYV_1,3MVX_1 176 3

Newick tree

 
[
	6IYV_1:90.57,
	[
		5UPF_1:64.5,3MVX_1:64.5
	]:26.07
]

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 \((0.85)(0.8)(\frac{771 }{\log_{20} 771}-\frac{272}{\log_{20}272})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 5UPF_1 6IYV_1 172 132
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]