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Parikh vectors
8RGA_1 8SXF_1 5JIG_1 Letter Amino acid
46 38 6 A Alanine
19 28 13 D Aspartic acid
1 0 0 C Cysteine
40 23 9 E Glutamic acid
47 42 6 G Glycine
18 22 3 P Proline
16 15 4 N Asparagine
26 19 2 Q Glutamine
76 42 16 L Leucine
3 0 2 W Tryptophan
15 8 5 Y Tyrosine
13 3 10 H Histidine
47 20 5 I Isoleucine
32 25 7 K Lycine
23 11 4 F Phenylalanine
26 22 11 R Arginine
19 5 2 M Methionine
47 28 7 S Serine
41 20 10 T Threonine
44 32 5 V Valine 

8RGA_1|Chains A, B|Multidrug resistance ABC transporter ATP-binding/permease protein BmrA|Bacillus subtilis (1423)
>8SXF_1|Chains A, B, E|Probable carboxyl-terminal protease|Pseudomonas aeruginosa (287)
>5JIG_1|Chain A|Ubiquitin and WLM domain-containing metalloprotease SPCC1442.07c|Schizosaccharomyces pombe (strain 972 / ATCC 24843) (284812)
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 \(w=f(c)\)
8RGA , Knot 226 599 0.80 40 236 551 
MSSSHHHHHHMPTKKQKSKSKLKPFFALVRRTNPSYGKLAFALALSVVTTLVSLLIPLLTKQLVDGFSMSNLSGTQIGLIALVFFVQAGLSAYATYALNYNGQKIISGLRELLWKKLIKLPVSYFDTNASGETVSRVTNDTMVVKELITTHISGFITGIISVIGSLTILFIMNWKLTLLVLVVVPLAALILVPIGRKMFSISRETQDETARFTGLLNQILPEIRLVKASNAEDVEYGRGKMGISSLFKLGVREAKVQSLVGPLISLVLMAALVAVIGYGGMQVSSGELTAGALVAFILYLFQIIMPMGQITTFFTQLQKSIGATERMIEILAEEEEDTVTGKQIENAHLPIQLDRVSFGYKPDQLILKEVSAVIEAGKVTAIVGPSGGGKTTLFKLLERFYSPTAGTIRLGDEPVDTYSLESWREHIGYVSQESPLMSGTIRENICYGLERDVTDAEIEKAAEMAYALNFIKELPNQFDTEVGERGIMLSGGQRQRIAIARALLRNPSILMLDAATSSLDSQSEKSVQQALEVLMEGRTTIVIAHRLSTVVDADQLLFVEKGEITGRGTHHELMASHGLYRDFAEQQLKMNADLENKAG
8SXF , Knot 168 403 0.83 36 209 377 
GSHMSAPLPLDELRTFAEVLDRVKAAYVEPVDDKTLLENAIKGMLSNLDPHSAYLGPEDFAELQESTSGEFGGLGIEVGSEDGFIKVVSPIDDTPAARAGIQPGDLIVQIDGKPTKGQSMTEAVDSMRGKAGSPITLTIVRDGGRPFDVELKRAIIKVKSVKSQVLEPGYAYLRITQFQVNTGEEVVKALNQLRKDNKGRLKGLVLDLRNNPGGVLQSAVEVADAFLTKGLIVYTKGRIANSELRFSADPADPSDKVPLVVLINGGSAAAAEIVAGALQDQKRAILMGTDSFGKGSVQTVLPLNNDRALKLTTALYYTPNGRSIQAQGIVPDIEVGRAKVTQERSSFEGFKEADLQGHLANGNGGADRPTGKRAAPSERPQDSDYQLSQALSLLKGLSVTRGN
5JIG , Knot 65 127 0.82 38 103 123 
GSIYTFNELVVLDYPHKDRALRYLERLRDDTGIKKIMDSHRWTVPLLSEMDPAEHTRHDSKTLGLNHNQGAHIELRLRTDRYDGFRDYKTVKSTLIHELTHNVHGEHDSSFWELFRQLTKEADAADL

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 \(c\).

\(|P_{f(8RGA_1)}(2) \setminus P_{f(8SXF_1)}(2)|=68\), \(|P_{f(8SXF_1)}(2) \setminus P_{f(8RGA_1)}(2)|=41\). 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 1:10000000001100000000010111111000010010111111101100110111111000110110100101001111111111011101010011000100110110011100110111001000101001001000011100110001011101110111010111110101011111111111111111100110100000000101011100111010110100100100101011100110111001010011111101111111111110111010010101111111110110111111010011001000111000110111000000101001001011101001011001001110010111011010111110111000110110010010110101100110000100100011010000111010100010011000100101001101101101100110010001100111101100001111011100101111011000100000001001101110100011110010011010011110010101010000111001100011000101010100011
Pair \(Z_2\) Length of longest common subsequence
8RGA_1,8SXF_1 109 4
8RGA_1,5JIG_1 193 3
8SXF_1,5JIG_1 178 4

Newick tree

 
[
	5JIG_1:10.46,
	[
		8RGA_1:54.5,8SXF_1:54.5
	]:47.96
]

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 \((0.85)(0.8)(\frac{1002 }{\log_{20} 1002}-\frac{403}{\log_{20}403})=158.\)
Status Protein1 Protein2 d d1/2
Query variables 8RGA_1 8SXF_1 199 164
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]