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Parikh vectors
5RUL_1 8FFR_1 2FGY_1 Letter Amino acid
6 32 30 E Glutamic acid
5 27 18 F Phenylalanine
0 3 3 W Tryptophan
7 20 23 Y Tyrosine
3 7 9 C Cysteine
9 23 38 D Aspartic acid
19 33 48 L Leucine
5 17 27 P Proline
11 35 33 S Serine
15 33 49 A Alanine
3 10 19 Q Glutamine
13 26 18 K Lycine
2 12 7 M Methionine
6 29 28 T Threonine
22 30 31 V Valine
16 19 27 N Asparagine
13 31 35 G Glycine
6 14 23 H Histidine
6 26 37 I Isoleucine
2 23 39 R Arginine 

5RUL_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8FFR_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V|Nucleoprotein|Rabies virus CVS-11 (11294)
>2FGY_1|Chains A, B|carboxysome shell polypeptide|Halothiobacillus neapolitanus (927)
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 \(w=f(c)\)
5RUL , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL
8FFR , Knot 188 450 0.85 40 248 431 
MDADKIVFKVNNQVVSLKPEIIVDQHEYKYPAIKDLKKPCITLGKAPDLNKAYKSVLSGMSAAKLDPDDVCSYLAAAMQFFEGTCPEDWTSYGIVIARKGDKITPGSLVEIKRTDVEGNWALTGGMELTRDPTVPEHASLVGLLLSLYRLSKISGQNTGNYKTNIADRIEQIFETAPFVKIVEHHTLMTTHKMCANWSTIPNFRFLAGTYDMFFSRIEHLYSAIRVGTVVTAYEDCSGLVSFTGFIKQINLTAREAILYFFHKNFEEEIRRMFEPGQETAVPHSYFIHFRSLGLSGKSPYSSNAVGHVFNLIHFVGCYMGQVRSLNATVIAACAPHEMSVLGGYLGEEFFGKGTFERRFFRDEKELQEYEAAELTKTDVALADDGTVNSDDEDYFSGETRSPEAVYTRIMMNGGRLKRSHIRRYVSVSSNHQARPNSFAEFLNKTYSSDS
2FGY , Knot 216 542 0.83 40 252 512 
MSYYHHHHHHDYDIPTTENLYFQGAMGSMNTRNTRSKQRAPCGVSSSVKPRLDLIEQAPNPVYDRHPACITLPERTCRHPLTDLEANEQLGRCEDSVKNRFDRVIPFLQVVAGIPLGLDHVTRVQELAQSSLGHTLPEELLKDNWISGHNLKGIFGYATAKALTAATEQFSRKIMSEKDDSASAIGFFLDCGFHAVDISPCADGRLKGLLPYILRLPLTAFTYRKAYAGSMFDIEDDLAQWEKNELRRYREGVPNTADQPTRYLKIAVYHFSTSDPTHSGCAAHGSNDRAALEAALTQLMKFREAVENAHCCGASIDILLIGVDTDTDAIRVHIPDSKGFLNPYRYVDNTVTYAQTLHLAPDEARVIIHEAILNANRSDGWAKGNGVASEGMRRFIGQLLINNLSQIDYVVNRHGGRYPPNDIGHAERYISVGDGFDEVQIRNLAYYAHLDTVEENAIDVDVGITIFTKLNLSRGLPIPIAIHYRYDPNVPGSRERTVVKARRIYNAIKERFSSLDEQNLLQFRLSVQAQDIGSPIEEVASA

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 \(c\).

\(|P_{f(5RUL_1)}(2) \setminus P_{f(8FFR_1)}(2)|=27\), \(|P_{f(8FFR_1)}(2) \setminus P_{f(5RUL_1)}(2)|=156\). 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 1:0110010101010001010010110010010101110110101001111111100100011010000011001110111001101001100010111101001001011001000100001111111011111101100101010010001011110001000110011
Pair \(Z_2\) Length of longest common subsequence
5RUL_1,8FFR_1 183 4
5RUL_1,2FGY_1 191 5
8FFR_1,2FGY_1 140 4

Newick tree

 
[
	5RUL_1:10.14,
	[
		8FFR_1:70,2FGY_1:70
	]:30.14
]

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 \((0.85)(0.8)(\frac{619 }{\log_{20} 619}-\frac{169}{\log_{20}169})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 5RUL_1 8FFR_1 163 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]