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Parikh vectors
4PKU_1 7QMR_1 5XDL_1 Letter Amino acid
19 31 23 V Valine
43 8 26 E Glutamic acid
21 46 21 G Glycine
12 2 8 H Histidine
14 22 18 P Proline
5 6 6 W Tryptophan
21 17 13 Y Tyrosine
36 22 21 D Aspartic acid
0 2 6 C Cysteine
51 20 25 I Isoleucine
50 19 23 K Lycine
19 37 20 A Alanine
3 2 12 M Methionine
40 58 18 S Serine
19 22 8 F Phenylalanine
20 51 14 T Threonine
23 3 17 R Arginine
40 12 8 N Asparagine
28 12 12 Q Glutamine
55 27 32 L Leucine 

4PKU_1|Chain A|Lethal factor|Bacillus anthracis (1392)
>7QMR_1|Chain A|Endothiapepsin|Cryphonectria parasitica (5116)
>5XDL_1|Chain A|Epidermal growth factor receptor|Homo sapiens (9606)
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 \(w=f(c)\)
4PKU , Knot 199 519 0.80 38 228 458 
SNALSRYEKWEKIKQHYQHWSDSLSEEGRGLLKKLQIPIEPKKDDIIHSLSQEEKELLKRIQIDSSDFLSTEEKEFLKKLQIDIRDSLSEEEKELLNRIQVDSSNPLSEKEKEFLKKLKLDIQPYDINQRLQDTGGLIDSPSINLDVRKQYKRDIQNIDALLHQSIGSTLYNKIYLYENMNINNLTATLGADLVDSTDNTKINRGIFNEFKKNFKYSISSNYMIVDINERPALDNERLKWRIQLSPDTRAGYLENGKLILQRNIGLEIKDVQIIKQSEKEYIRIDAKVVPKSKIDTKIQEAQLNINQEWNKALGLPKYTKLITFNVHNRYASNIVESAYLILNEWKNNIQSDLIKKVTNYLVDGNGRFVFTDITLPNIAEQYTHQDEIYEQVHSKGLYVPESRSILLHGPSKGVELRNDSEGFIHEFGHAVDDYAGYLLDKNQSDLVTNSKKFIDIFKEEGSNLTSYGRTNEAEFFAEAFRLMHSTDHAERLKVQKNAPKTFQFINDQIKFIINSLVPR
7QMR , Knot 170 419 0.81 40 190 380 
MSSPLKNALVTAMLAGGALSSPTKQHVGIPVNASPEVGPGKYSFKQVRNPNYKFNGPLSVKKTYLKYGVPIPAWLEDAVQNSTSGLAERSTGSATTTPIDSLDDAYITPVQIGTPAQTLNLDFDTGSSDLWVFSSETTASEVDGQTIYTPSKSTTAKLLSGATWSISYGDGSSSSGDVYTDTVSVGGLTVTGQAVESAKKVSSSFTEDSTIDGLLGLAFSTLNTVSPTQQKTFFDNAKASLDSPVFTADLGYHAPGTYNFGFIDTTAYTGSITYTAVSTKQGFWEWTSTGYAVGSGTFKSTSIDGIADTGTTLLYLPATVVSAYWAQVSGAKSSSSVGGYVFPCSATLPSFTFGVGSARIVIPGDYIDFGPISTGSSSCFGGIQSSAGIGINIFGDVALKAAFVVFNGATTPTLGFASK
5XDL , Knot 147 331 0.86 40 211 325 
GSTSGEAPNQALLRILKETEFKKIKVLGSGAFGTVYKGLWIPEGEKVKIPVAIKELREATSPKANKEILDEAYVMASVDNPHVCRLLGICLTSTVQLITQLMPFGCLLDYVREHKDNIGSQYLLNWCVQIAKGMNYLEDRRLVHRDLAARNVLVKTPQHVKITDFGRAKLLGAEEKEYHAEGGKVPIKWMALESILHRIYTHQSDVWSYGVTVWELMTFGSKPYDGIPASEISSILEKGERLPQPPICTIDVYMIMVKCWMIDADSRPKFRELIIEFSKMARDPQRYLVIQGDERMHLPSPTDSNFYRALMDEEDMDDVVDADEYLIPQQG

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 \(c\).

\(|P_{f(4PKU_1)}(2) \setminus P_{f(7QMR_1)}(2)|=91\), \(|P_{f(7QMR_1)}(2) \setminus P_{f(4PKU_1)}(2)|=53\). 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 1:001100000100100000010001000101110010111010000110010000001100101000011000000110010101000100000011001010000110000001100101010100100010001111001010101000000010010111000110010001010001010010101110110000000100111001000100010000111010001110000101010101000110100101110001110100101100000001010101110001000100101010001001111100001101010000100110010111001000100011001000110101011100101101100000000100010001101100001110110011010000011100110110001101100000011000001101100010010001000010111011011000001001010001100101100010111001110
Pair \(Z_2\) Length of longest common subsequence
4PKU_1,7QMR_1 144 4
4PKU_1,5XDL_1 157 4
7QMR_1,5XDL_1 165 4

Newick tree

 
[
	5XDL_1:83.17,
	[
		4PKU_1:72,7QMR_1:72
	]:11.17
]

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 \((0.85)(0.8)(\frac{938 }{\log_{20} 938}-\frac{419}{\log_{20}419})=137.\)
Status Protein1 Protein2 d d1/2
Query variables 4PKU_1 7QMR_1 167 152.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]