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Parikh vectors
6ETU_1 1XFI_1 7EXX_1 Letter Amino acid
13 24 24 V Valine
25 28 31 E Glutamic acid
25 22 19 G Glycine
12 2 5 H Histidine
10 5 7 M Methionine
25 28 24 A Alanine
18 19 26 R Arginine
7 6 5 C Cysteine
22 40 63 L Leucine
14 12 28 Q Glutamine
23 22 32 K Lycine
18 20 13 P Proline
19 8 18 T Threonine
15 28 28 S Serine
8 4 5 W Tryptophan
18 6 18 Y Tyrosine
19 19 28 N Asparagine
13 28 25 D Aspartic acid
20 27 19 I Isoleucine
18 19 26 F Phenylalanine 

6ETU_1|Chain A|Lysine-specific demethylase 4D|Homo sapiens (9606)
>1XFI_1|Chain A|unknown protein|Arabidopsis thaliana (3702)
>7EXX_1|Chain A|DNA phosphorothioation-dependent restriction protein DptG|Escherichia coli (562)
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 \(w=f(c)\)
6ETU , Knot 153 342 0.87 40 223 330 
METMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR
1XFI , Knot 156 367 0.83 40 195 345 
SESDSEMVPFPQLPMPIENNYRACTIPYRFPSDDPKKATPNEISWINVFANSIPSFKKRAESDITVPDAPARAEKFAERYAGILEDLKKDPESHGGPPDGILLCRLREQVLRELGFRDIFKKVKDEENAKAISLFPQVVSLSDAIEDDGKRLENLVRGIFAGNIFDLGSAQLAEVFSRDGMSFLASCQNLVPRPWVIDDLENFQAKWINKSWKKAVIFVDNSGADIILGILPFARELLRRGAQVVLAANELPSINDITCTELTEILSQLKDENGQLLGVDTSKLLIANSGNDLPVIDLSRVSQELAYLSSDADLVIVEGMGRGIETNLYAQFKCDSLKIGMVKHLEVAEFLGGRLYDCVFKFNEVQS
7EXX , Knot 182 444 0.83 40 219 413 
GSHMYPIATNLKVSNNQLDSYLPIRNKNNNIDWQIVTGLVLSYAVKYKIDTYSLEQFREDCKTHLQILIDEPAFLSVLERMYFSSQDIFRVSPLFLLFHAQFDGEKISAGSTADKRLGTLFANLMRDFSLNNPIQDKLNFIEKEMLNKLNKKLIRLGEGPFAKEQPYLPYLVTCFQSDLAFLAEHPQYLLQELTNTLRLYAFSWCAQLALNLDNWQDGEPQSKSLFFILDTEKASSERDKIKLFGYKWFARQSEKLFPVLSALEVLQVKGEEKRPLWQVYQDCLGYSDTSNRVLNELNNYIQKFISKEERDLPERDRATNLEDAFKQLLSVAVEQFQGKKTERAAVNRKYINELESQICTDFIQVRGRAGKVLVLNQDRLLLLTNLTVGKNKKLRLHELLRGFEQRGFYLDNQSTQMLVAFYERMGNVERMSDSGDAVYVRKTV

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 \(c\).

\(|P_{f(6ETU_1)}(2) \setminus P_{f(1XFI_1)}(2)|=103\), \(|P_{f(1XFI_1)}(2) \setminus P_{f(6ETU_1)}(2)|=75\). 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 1:100100010010010001111010000100100011010001100111101111001010000001001111011001101011110000000011011000011000000011000100100001000100011011010101100000010110110100110000111101100101011110001110000101001001011010010111100100100110011110001011110001111010110001111001000110111011010011100110010110110101100101100000101010101011101101000011001000
Pair \(Z_2\) Length of longest common subsequence
6ETU_1,1XFI_1 178 4
6ETU_1,7EXX_1 188 4
1XFI_1,7EXX_1 158 4

Newick tree

 
[
	6ETU_1:95.34,
	[
		1XFI_1:79,7EXX_1:79
	]:16.34
]

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 \((0.85)(0.8)(\frac{709 }{\log_{20} 709}-\frac{342}{\log_{20}342})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 6ETU_1 1XFI_1 126 124.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]