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Parikh vectors
3EYW_1 9LUG_1 7MPU_1 Letter Amino acid
26 12 13 R Arginine
18 13 12 Q Glutamine
41 29 24 L Leucine
14 10 3 M Methionine
18 13 7 P Proline
21 23 12 T Threonine
12 11 4 Y Tyrosine
16 10 12 K Lycine
20 15 14 S Serine
19 28 12 V Valine
3 12 4 C Cysteine
37 26 9 G Glycine
24 7 7 H Histidine
25 11 11 I Isoleucine
14 16 9 F Phenylalanine
8 3 2 W Tryptophan
36 17 19 A Alanine
8 21 5 N Asparagine
20 17 16 D Aspartic acid
33 8 11 E Glutamic acid 

3EYW_1|Chains A, B|C-terminal domain of Glutathione-regulated potassium-efflux system protein kefC fused to full length Glutathione-regulated potassium-efflux system ancillary protein kefF|Escherichia coli (83333)
>9LUG_1|Chains A, B|ORF1a polyprotein|Severe acute respiratory syndrome coronavirus 2 (2697049)
>7MPU_1|Chains A, B|NanoRNase C|Brucella melitensis (29459)
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 \(w=f(c)\)
3EYW , Knot 170 413 0.82 40 224 389 
GSHGMRVIIAGFGRFGQITGRLLLSSGVKMVVLDHDPDHIETLRKFGMKVFYGDATRMDLLESAGAAKAEVLINAIDDPQTNLQLTEMVKEHFPHLQIIARARDVDHYIRLRQAGVEKPERETFEGALKTGRLALESLGLGPYEARERADVFRRFNIQMVEEMAMVENDTKARAAVYKRTSAMLSEIITEDREHLSLIQRHGWQGTEEGKHTGNMADEPETKPSSTSGGLVPRGSSGMILIIYAHPYPHHSHANKRMLEQARTLEGVEIRSLYQLYPDFNIDIAAEQEALSRADLIVWQHPMQWYSIPPLLKLWIDKVFSHGWAYGHGGTALHGKHLLWAVTTGGGESHFEIGAHPGFDVLSQPLQATAIYCGLNWLPPFAMHCTFICDDETLEGQARHYKQRLLEWQEAHHG
9LUG , Knot 132 302 0.83 40 194 285 
GFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLRVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMVLPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGV
7MPU , Knot 98 206 0.84 40 144 200 
SMTIRFHRNDLPNLDNYQVDAVAIDTETLGLNPHRDRLCVVQISPGDGTADVIQIEAGQKKAPNLVKLLKDRSITKIFHFGRFDLAVLAHAFGTMPQPVFCTKIASKLTRTYTDRHGLKEICSELLDVSISKQQQSSDWAAEVLSQAQLEYAASDVLYLHRLKAVLEQRLERDGRTKQAEACFKFLPTRSELDLMGWAESDIFAHS

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 \(c\).

\(|P_{f(3EYW_1)}(2) \setminus P_{f(9LUG_1)}(2)|=102\), \(|P_{f(9LUG_1)}(2) \setminus P_{f(3EYW_1)}(2)|=72\). 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 1:10011011111110110101011100110111100010010010011101101010010110011110101110110010001010011000110101110100100010100111001000010111001011100111110010001011001010110011110000010111000001110011000000101100011010001000101100100010000111110100111111010101000010001100100101101001001010101011100011001011110011010011111011100110011101011011010011111001110001011101110110011010110011011111110001100000101010000001101001001
Pair \(Z_2\) Length of longest common subsequence
3EYW_1,9LUG_1 174 4
3EYW_1,7MPU_1 182 4
9LUG_1,7MPU_1 184 3

Newick tree

 
[
	7MPU_1:92.95,
	[
		3EYW_1:87,9LUG_1:87
	]:5.95
]

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 \((0.85)(0.8)(\frac{715 }{\log_{20} 715}-\frac{302}{\log_{20}302})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 3EYW_1 9LUG_1 145 125
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]