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Parikh vectors
8EGM_1 7RSZ_1 7QIC_1 Letter Amino acid
19 12 11 R Arginine
10 6 19 M Methionine
31 20 58 V Valine
37 18 62 A Alanine
13 35 9 N Asparagine
2 14 4 C Cysteine
14 16 2 H Histidine
12 16 14 P Proline
16 22 21 S Serine
17 29 26 T Threonine
1 7 9 W Tryptophan
16 14 9 D Aspartic acid
30 28 31 G Glycine
19 25 35 I Isoleucine
28 23 47 L Leucine
6 22 19 K Lycine
8 10 9 Q Glutamine
19 25 19 E Glutamic acid
12 12 28 F Phenylalanine
6 8 6 Y Tyrosine 

8EGM_1|Chains A, B, C, D, E, F, G, H|UDP-N-acetylmuramate--L-alanine ligase|Pseudomonas aeruginosa PAO1 (208964)
>7RSZ_1|Chains A, B, C, D|HIV-1 gp120 Clade C1086|Human immunodeficiency virus 1 (11676)
>7QIC_1|Chain A|Divalent metal cation transporter|Eggerthella lenta (84112)
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 \(w=f(c)\)
8EGM , Knot 129 316 0.78 40 178 294 
MAHHHHHHMRRIHFVGIGGAGMCGIAEVLLNLGYEVSGSDLKASAVTERLEKFGAQIFIGHQAENADGADVLVVSSAINRANPEVASALERRIPVVPRAEMLAELMRYRHGIAVAGTHGKTTTTSLIASVFAAGGLDPTFVIGGRLNAAGTNAQLGASRYLVAEADESDASFLHLQPMVAVVTNIDADHMATYGGDFNKLKKTFVEFLHNLPFYGLAVMCVDDPVVREILPQIARPTVTYGLSEDADVRAINIRQEGMRTWFTVLRPEREPLDVSVNMPGLHNVLNSLATIVIATDEGISDEAIVQGLSGFQGVGR
7RSZ , Knot 156 362 0.84 40 209 337 
VWKEAKTTLFCASDAKAYEKEVHNVWATHACVPTDPNPQEMVLANVTENFNMWKNDMVEQMHEDIISLWDESLKPCVKLTGGSAITQACPKVSFDPIPLHYCAPAGFAILKCNNKTFNGTGPCRNVSTVQCTHGIKPVVSTQLLLNGSLAEEEIIIRSENLTNNAKTIIVHLNESVNIVCTRPNNGGSGSGGNIRQAHCNINESKWNNTLQKVGEELAKHFPSKTIKFEPSSGGDLEITTHSFNCRGEFFYCNTSDLFNGTYRNGTYNHTGRSSNGTITLQCKIKQIINMWQEVGRAIYAPPIEGEITCNSNITGLLLLRDGGNDDNDTETFRPGGGDMRDNWRSELYKYKVVEIKHHHHHH
7QIC , Knot 176 438 0.81 40 201 390 
MMMKKNEKLELRDVAVDAAESTELVEVPEKKQPKKIWLLLAALGPGIVTAMAGNDAGGISTYSTVGAKFGFATLWVIPIMCVLLIVVQMTAARMGAVTGKGFAALIRERFGIRLTALAMLALLIGNVATTFSEFAGIASGMEMFGVSKYLSVPVAAVAVWLLVVGGSYKRVEKVFLILSLVFVTYIVAAFMAQPNWEEALTSTVVPHIVNDQSFVSLVIAMIGTTIAPWMMFFNQSNVVEKGVTVKDLFSQKVDVVAGTIAACLVAWFIIVTTGAVLFPQGIEIESAADAARALAPFAGHYAEALFAIGLIAASFLAACVLPLTTAFVICEAFGWEAGVSFKWKEAPLFKSIFTFVIAFSAVVVLIPNIDLMGVMLTAQFVNGLILPVLLVFMAIIAADKRVMGAYRSRIVSRVLIWLTVGIVTVLTAALLVMQVLGI

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 \(c\).

\(|P_{f(8EGM_1)}(2) \setminus P_{f(7RSZ_1)}(2)|=67\), \(|P_{f(7RSZ_1)}(2) \setminus P_{f(8EGM_1)}(2)|=98\). 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 1:1100000010010111111111011101110110010100101011000100111011110010010110111100110010101101100011111010111011000011111100100000011101111111010111110101110010111000111010000101101011111100101001100110100100011011001110111110100111001110110101001100010101101000110011011010001101010111100110011011110001100011101101101110
Pair \(Z_2\) Length of longest common subsequence
8EGM_1,7RSZ_1 165 6
8EGM_1,7QIC_1 155 4
7RSZ_1,7QIC_1 194 4

Newick tree

 
[
	7RSZ_1:93.85,
	[
		8EGM_1:77.5,7QIC_1:77.5
	]:16.35
]

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 \((0.85)(0.8)(\frac{678 }{\log_{20} 678}-\frac{316}{\log_{20}316})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 8EGM_1 7RSZ_1 127 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]