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Parikh vectors
5UPE_1 3LZQ_1 6EKV_1 Letter Amino acid
25 12 36 A Alanine
45 10 65 L Leucine
5 0 4 C Cysteine
38 7 32 V Valine
7 2 11 W Tryptophan
30 10 48 D Aspartic acid
38 14 41 E Glutamic acid
3 6 25 M Methionine
21 12 21 P Proline
30 8 51 T Threonine
29 10 62 I Isoleucine
49 13 70 K Lycine
18 8 38 F Phenylalanine
14 3 19 R Arginine
26 8 62 N Asparagine
15 2 26 Q Glutamine
36 20 46 G Glycine
13 4 6 H Histidine
29 2 63 S Serine
28 8 24 Y Tyrosine 

5UPE_1|Chains A, B|Nicotinamide phosphoribosyltransferase|Homo sapiens (9606)
>3LZQ_1|Chains A, B|P19 protein|Campylobacter jejuni (354242)
>6EKV_1|Chain A|Toxin complex component ORF-X2|Clostridium botulinum (strain Kyoto / Type A2) (536232)
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 \(w=f(c)\)
5UPE , Knot 203 499 0.84 40 241 472 
MNPAAEAEFNILLATDSYKVTHYKQYPPNTSKVYSYFECREKKTENSKLRKVKYEETVFYGLQYILNKYLKGKVVTKEKIQEAKDVYKEHFQDDVFNEKGWNYILEKYDGHLPIEIKAVPEGFVIPRGNVLFTVENTDPECYWLTNWIETILVQSWYPITVATNSREQKKILAKYLLETSGNLDGLEYKLHDFGYRGVSSQETAGIGASAHLVNFKGTDTVAGLALIKKYYGTKDPVPGYSVPAAEHSTITAWGKDHEKDAFEHIVTQFSSVPVSVVSDSYDIYNACEKIWGEDLRHLIVSRSTQAPLIIRPDSGNPLDTVLKVLEILGKKFPVTENSKGYKLLPPYLRVIQGDGVDINTLQEIVEGMKQKMWSIENIAFGSGGGLLQKLTRDLLNCSFKCSYVVTNGLGINVFKDPVADPNKRSKKGRLSLHRTPAGNFVTLEEGKGDLEEYGQDLLHTVFKNGKVTKSYSFDEIRKNAQLNIELEAAHHDYKDDDDK
3LZQ , Knot 72 159 0.76 38 119 151 
GGEVPIGDPKELNGMEIAAVYLQPIEMEPRGIDLAASLADIHLEADIHALKNNPNGFPEGFWMPYLTIAYELKNTDTGAIKRGTLMPMVADDGPHYGANIAMEKDKKGGFGVGNYELTFYISNPEKQGFGRHVDEETGVGKWFEPFKVDYKFKYTGTPK
6EKV , Knot 281 750 0.82 40 267 673 
MNNLKPFIYYDWKKTILKNAKESYSINEIIPKTFFMELHGTKITNSTLNGTWKAWNLTDEGEGSHPVLKCIIDDGYLDMNFGASSEKIPLKNVWIKLCMKINPNSDGTYSIPEKSSSFYIKDNSLKISKDNLILDKYLNKLMLSYFKNNIKNIEMFINKSRIQTKVVGDLSLLGWNTENSVSFRTMNEFIKKDNLYPKDFKAVYSYRKMTFTATGTFDSWEMTTGADGRNIRFKCPIKSAAYDLDGDVFNSSTENFLLIQVDLTYFDSKTTINDPTGENDGKQFNLKVKTNDDKLKNVLIVTYNLTDTDGSMSSEDKDFLSLAFRNWFNDNIQQFEQIFAYILLDETAKIPEYQWLKPTQISYGSASVETANDEPDLDASIFSAMSMVENNTNSTPSHAVDNRMLQLTKTQAAFGISFPLFIEHFLKQALLSSQFISVDDIVADINTLTITNNKQIIFGKVENSDGKNVDSSLKPGKLKLSLQNNLIVLELFDLTWEQGRGVTGHFDFRQEYELTLESKSEKQIPILKVHDEPEIEYYVEEAQWKANEDMIVSAVVGTVFSMILGAGMKLAGSALSKAGKLIRSKATTIKGRKKIYINRSNVRQLRKDSGVTEMELQRINRRNSSIASEDARFISNNGTTSIQTLGDMKKKPMSTGQRIAIGVKKITGTAVMFGAVGLGMNFGEMLINYINAMENNDYSAIPGINSFMQQCIGAMQWPDKDSELKVTFGKLQGIYLLGGTLEKNNKPNSK

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 \(c\).

\(|P_{f(5UPE_1)}(2) \setminus P_{f(3LZQ_1)}(2)|=154\), \(|P_{f(3LZQ_1)}(2) \setminus P_{f(5UPE_1)}(2)|=32\). 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
5UPE_1,3LZQ_1 186 4
5UPE_1,6EKV_1 134 5
3LZQ_1,6EKV_1 192 4

Newick tree

 
[
	3LZQ_1:10.04,
	[
		5UPE_1:67,6EKV_1:67
	]:35.04
]

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 \((0.85)(0.8)(\frac{658 }{\log_{20} 658}-\frac{159}{\log_{20}159})=142.\)
Status Protein1 Protein2 d d1/2
Query variables 5UPE_1 3LZQ_1 182 117.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]