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Parikh vectors
1OZA_1 7NHR_1 9FNG_1 Letter Amino acid
12 18 23 F Phenylalanine
19 35 27 T Threonine
8 2 14 W Tryptophan
27 48 28 V Valine
23 42 36 D Aspartic acid
41 83 23 L Leucine
32 52 34 A Alanine
13 40 13 Q Glutamine
24 65 23 I Isoleucine
11 15 9 M Methionine
16 27 23 P Proline
14 39 20 N Asparagine
4 2 4 C Cysteine
34 46 32 G Glycine
3 15 17 H Histidine
29 48 16 K Lycine
19 56 35 S Serine
6 22 21 Y Tyrosine
12 33 11 R Arginine
24 39 21 E Glutamic acid 

1OZA_1|Chain A|Aspartate-semialdehyde dehydrogenase|Haemophilus influenzae (727)
>7NHR_1|Chains A, B, C, D, E, F, G, H|Putative transmembrane protein Wzc|Escherichia coli (562)
>9FNG_1|Chains A, B|Glycoside hydrolase family 71|Aspergillus nidulans FGSC A4 (227321)
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 \(w=f(c)\)
1OZA , Knot 156 371 0.83 40 205 353 
MKNVGFIGWRGMVGSVLMDRMSQENDFENLNPVFFTTSQAGQKAPVFGGKDAGDLKSAFDIEELKKLDIIVTCQGGDYTNEVYPKLKATGWDGYWVDAASALLMKDDAIIVLDPVNQHVISEGLKKGIKTFVGGNCTVSLMLMAIGGLFEKDLVEWISVATYQAASGAGAKNMRELLSQMGLLEQAVSSELKDPASSILDIERKVTAKMRADNFPTDNFGAALGGSLIPWIDKLLPETGQTKEEWKGYAETNKILGLSDNPIPVDGLCVRIGALRCHSQAFTIKLKKDLPLEEIEQIIASHNEWVKVIPNDKEITLRELTPAKVTGTLSVPVGRLRKLAMGPEYLAAFTVGDQLLWGAAEPVRRILKQLVA
7NHR , Knot 273 727 0.82 40 265 644 
MTSVTSKQSTILGSDEIDLGRVIGELIDHRKLIISITSVFTLFAILYALLATPIYETDALIQIEQKQGNAILSSLSQVLPDGQPQSAPETALLQSRMILGKTIDDLNLQIQIEQKYFPVIGRGLARLMGEKPGNIDITRLYLPDSDDISNNTPSIILTVKDKENYSINSDGIQLNGVVGTLLNEKGISLLVNEIDAKPGDQFVITQLPRLKAISDLLKSFSVADLGKDTGMLTLTLTGDNPKRISHILDSISQNYLAQNIARQAAQDAKSLEFLNQQLPKVRAELDSAEDKLNAYRKQKDSVDLNMEAKSVLDQIVNVDNQLNELTFREAEVSQLYTKEHPTYKALMEKRQTLQEEKSKLNKRVSSMPSTQQEVLRLSRDVESGRAVYLQLLNRQQELNIAKSSAIGNVRIIDNAVTDPNPVRPKKTIIIVIGVVLGLIVSVVLVLFQVFLRRGIESPEQLEEIGINVYASIPISEWLTKNARQSGKVRKNQSDTLLAVGNPADLAVEAIRGLRTSLHFAMMEAKNNVLMISGASPSAGMTFISSNLAATIAITGKKVLFIDADLRKGYAHKMFGHKNDKGLSEFLSGQAAAEMIIDKVEGGGFDYIGRGQIPPNPAELLMHPRFEQLLNWASQNYDLIIIDTPPILAVTDAAIIGRYAGTCLLVARFEKNTVKEIDVSMKRFEQSGVVVKGCILNGVVKKASSYYRYGHNHYGYSYYDKKHHHHHH
9FNG , Knot 182 430 0.85 40 247 414 
MGSSHHHHHHSSENLYFQGHSLPGANSLTIRKDSNKYVTAHFMVGIVENYTVDDWKHDMELAKETGIDAFALNCASIDSYTDKQLAYAYEAAEEVDFKVFISFDFAYWSNGDTARITSIMQTYADHPGQFQYNGAALVSTFVGDSFDWGPVKRAVDHPIFAVPNLQDPNWAGHATTSIDGAFSWYAWPTDGGNSIIKGPMTTIWDDRFRNNLKDKVYMAPVSPWFSTHFNTKNWVFICEDLPHLRWQQMLEMQPELIEIISWNDYGESHYIGPYSEAHSDDGSAQWTKDFPHDAWRIIAKPYIAAYKAGEREPTVESDQLVYWYRPTPKAVTCSKDPLGPPNGINLLEDSVFVTTLLTEPATLTVGSGSLEFSVDVDAGIVTNSFPMGVGSQAFSVTRDGEEILGGDGGLDVQDRCDYYNFNVYVGSFSA

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 \(c\).

\(|P_{f(1OZA_1)}(2) \setminus P_{f(7NHR_1)}(2)|=43\), \(|P_{f(7NHR_1)}(2) \setminus P_{f(1OZA_1)}(2)|=103\). 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 1:10011111101111011100100000100101111000011001111110011010011010010010111000110000010101010110101101101111000111110110001100110011001111000101111111111000110110110001101111001001100111100110001001100110100010101010011000111111101111100111001000001010100001111000111101101011110000011010100011100100111000011011100001010010110101010111101001111100111101100111111011001100111
Pair \(Z_2\) Length of longest common subsequence
1OZA_1,7NHR_1 146 4
1OZA_1,9FNG_1 168 4
7NHR_1,9FNG_1 148 6

Newick tree

 
[
	9FNG_1:81.10,
	[
		1OZA_1:73,7NHR_1:73
	]:8.10
]

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 \((0.85)(0.8)(\frac{1098 }{\log_{20} 1098}-\frac{371}{\log_{20}371})=191.\)
Status Protein1 Protein2 d d1/2
Query variables 1OZA_1 7NHR_1 237 178.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]