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Parikh vectors
8VZE_1 5RCK_1 6CPG_1 Letter Amino acid
18 2 3 M Methionine
41 22 14 P Proline
24 58 21 S Serine
29 51 13 T Threonine
23 12 9 N Asparagine
23 22 11 D Aspartic acid
55 46 15 G Glycine
30 27 34 L Leucine
20 19 21 K Lycine
17 22 12 F Phenylalanine
8 6 4 W Tryptophan
20 17 12 Y Tyrosine
37 31 15 V Valine
72 37 17 A Alanine
3 2 4 C Cysteine
32 8 25 E Glutamic acid
8 2 10 H Histidine
27 3 21 R Arginine
22 12 11 Q Glutamine
41 20 13 I Isoleucine 

8VZE_1|Chains A, B|Oxalyl-CoA decarboxylase|Thermoflexaceae bacterium (2739507)
>5RCK_1|Chain A|Endothiapepsin|Cryphonectria parasitica (5116)
>6CPG_1|Chains A, C[auth D]|Aurora kinase A|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 \(w=f(c)\)
8VZE , Knot 220 550 0.84 40 247 514 
MATITGAQIVARALKQQGVEHMFGIVGIPVIPIAAAAQREGIKFYGFHNEQSASYAAGAVGYLTGRPGVCLAVSGPGMIHGIAGLANAWSNNWPMILIGGANDSYQNGQGAFQEAPQIESARPYVKYAARPDSIARLPFYVEQAVRTTIYGRPGAAYLDLPGDIISGSVEEAEVTYPGRCPEPPRTLAPEENIKAALAALKTAERPLVIVGKGAAYSRAEDEVRKFVETTGLPYLASPMGKGVIPDDHPQSIAAARSFALQNTDLVVLMGARMNWMMHFGQPPRWNEKVRVIQMDISAEEIGTNVPTEVALVGDAKAITAQLNAELEKNPWSYPSETTWWTGLQSKIAANAATVEPMFNDDSVPMGYYRVLRDIRDLIPKDAIISNEGASTMDIGRTVLPNFFARTRLDAATFGTMGVGAGQAIAAASVHPDRKIICVQGDSAFGFGGMEVETAARYGLNITFVIINNNGIGGGPDELDPAHIPPGAYTPQAHYEKMADVYGGKGYFVTTPDQLKPVLEEAMAQPKPAIVNIMISNKSGRKPQEFGWLTR
5RCK , Knot 170 419 0.81 40 190 380 
MSSPLKNALVTAMLAGGALSSPTKQHVGIPVNASPEVGPGKYSFKQVRNPNYKFNGPLSVKKTYLKYGVPIPAWLEDAVQNSTSGLAERSTGSATTTPIDSLDDAYITPVQIGTPAQTLNLDFDTGSSDLWVFSSETTASEVDGQTIYTPSKSTTAKLLSGATWSISYGDGSSSSGDVYTDTVSVGGLTVTGQAVESAKKVSSSFTEDSTIDGLLGLAFSTLNTVSPTQQKTFFDNAKASLDSPVFTADLGYHAPGTYNFGFIDTTAYTGSITYTAVSTKQGFWEWTSTGYAVGSGTFKSTSIDGIADTGTTLLYLPATVVSAYWAQVSGAKSSSSVGGYVFPCSATLPSFTFGVGSARIVIPGDYIDFGPISTGSSSCFGGIQSSAGIGINIFGDVALKAAFVVFNGATTPTLGFASK
6CPG , Knot 127 285 0.84 40 177 277 
SVDESKKRQWALEDFEIGRPLGKGKFGNVYLAREKQSKFILALKVLFKAQLEKAGVEHQLRREVEIQSHLRHPNILRLYGYFHDATRVYLILEYAPLGTVYRELQKLSKFDEQRTATYITELANALSYCHSKRVIHRDIKPENLLLGSAGELKIADFGWSVHAPSSRRTTLCGTLDYLPPEMIEGRMHDEKVDLWSLGVLCYEFLVGKPPFEANTYQETYKRISRVEFTFPDFVTEGARDLISRLLKHNPSQRPMLREVLEHPWITANSSKPSNCQNKESASKQS

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 \(c\).

\(|P_{f(8VZE_1)}(2) \setminus P_{f(5RCK_1)}(2)|=112\), \(|P_{f(5RCK_1)}(2) \setminus P_{f(8VZE_1)}(2)|=55\). 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 1:1101011011101100011001111111111111111000110101100000100111111010101110111011111011111101100011111111100000010111001101001010100110100110111010011000101011110101110110101001010011001011001110001011111100100111111011100010001001100011101101110111100010011110011100001111111010111011011010001011010101001100110011111010110101010100011001000011011000111011010111000011110001100100111001110001100101100111011100010110110111111011111010100011010100111111101001100110101111000111111001011011111001010000110101101011001001011100111010111101110000100100111100
Pair \(Z_2\) Length of longest common subsequence
8VZE_1,5RCK_1 167 5
8VZE_1,6CPG_1 180 4
5RCK_1,6CPG_1 181 3

Newick tree

 
[
	6CPG_1:92.39,
	[
		8VZE_1:83.5,5RCK_1:83.5
	]:8.89
]

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 \((0.85)(0.8)(\frac{969 }{\log_{20} 969}-\frac{419}{\log_{20}419})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 8VZE_1 5RCK_1 188 164
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]