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Parikh vectors
7QBI_1 3AIE_1 2XEU_1 Letter Amino acid
18 49 4 G Glycine
44 66 3 L Leucine
17 58 4 K Lycine
49 56 5 S Serine
23 62 4 N Asparagine
2 0 7 C Cysteine
15 30 3 P Proline
45 60 3 T Threonine
3 12 0 W Tryptophan
19 47 3 Y Tyrosine
22 63 2 D Aspartic acid
11 16 2 M Methionine
27 27 1 F Phenylalanine
14 41 2 Q Glutamine
33 41 6 I Isoleucine
12 34 2 E Glutamic acid
4 15 3 H Histidine
27 55 4 V Valine
30 80 2 A Alanine
16 32 4 R Arginine 

7QBI_1|Chains A, D[auth B]|Pheromone alpha factor receptor|Saccharomyces cerevisiae (4932)
>3AIE_1|Chains A, B, C, D, E, F, G, H|Glucosyltransferase-SI|Streptococcus mutans (1309)
>2XEU_1|Chain A|RING FINGER PROTEIN 4|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 \(w=f(c)\)
7QBI , Knot 178 431 0.83 40 221 404 
MSDAAPSLSNLFYDPTYNPGQSTINYTSIYGNGSTITFDELQGLVNSTVTQAIMFGVRCGAAALTLIVMWMTSRSRKTPIFIINQVSLFLIILHSALYFKYLLSNYSSVTYALTGFPQFISRGDVHVYGATNIIQVLLVASIETSLVFQIKVIFTGDNFKRIGLMLTSISFTLGIATVTMYFVSAVKGMIVTYNDVSATQDKYFNASTILLASSINFMSFVLVVKLILAIRSRRFLGLKQFDSFHILLIMSCQSLLVPSIIFILAYSLKPNQGTDVLTTVATLLAVLSLPLSSMWATAANNASKTNTITSDFTTSTDRFYPGTLSSFQTDSINNDAKSSLRSRLYDLYPRRKETTSDKHSERTFVSETADDIEKNQFYQLPTPTSSKNTRIGPFADASYKEGEVEPVDMYTPDTAADEEARKFWTEDNNNL
3AIE , Knot 309 844 0.82 38 280 752 
SFAQYNQVYSTDAANFEHVDHYLTAESWYRPKYILKDGKTWTQSTEKDFRPLLMTWWPDQETQRQYVNYMNAQLGIHQTYNTATSPLQLNLAAQTIQTKIEEKITAEKNTNWLRQTISAFVKTQSAWNSDSEKPFDDHLQKGALLYSNNSKLTSQANSNYRILNRTPTNQTGKKDPRYTADRTIGGYEFLLANDVDNSNPVVQAEQLNWLHFLMNFGNIYANDPDANFDSIRVDAVDNVDADLLQIAGDYLKAAKGIHKNDKAANDHLSILEAWSYNDTPYLHDDGDNMINMDNRLRLSLLYSLAKPLNQRSGMNPLITNSLVNRTDDNAETAAVPSYSFIRAHDSEVQDLIRDIIKAEINPNVVGYSFTMEEIKKAFEIYNKDLLATEKKYTHYNTALSYALLLTNKSSVPRVYYGDMFTDDGQYMAHKTINYEAIETLLKARIKYVSGGQAMRNQQVGNSEIITSVRYGKGALKATDTGDRITRTSGVVVIEGNNPSLRLKASDRVVVNMGAAHKNQAYRPLLLTTDNGIKAYHSDQEAAGLVRYTNDRGELIFTAADIKGYANPQVSGYLGVWVPVGAAADQDVRVAASTAPSTDGKSVHQNAALDSRVMFEGFSNFQAFATKKEEYTNVVIAKNVDKFAEWGVTDFEMAPQYVSSTDGSFLDSVIQNGYAFTDRYDLGISKPNKYGTADDLVKAIKALHSKGIKVMADWVPDQMYALPEKEVVTATRVDKYGTPVAGSQIKNTLYVVDGKSSGKDQQAKYGGAFLEELQAKYPELFARKQISTGVPMDPSVKIKQWSAKYFNGTNILGRGAGYVLKDQATNTYFSLVSDNTFLPKSLVNP
2XEU , Knot 38 64 0.82 38 58 62 
GAMVSCPICMDGYSEIVQNGRLIVSTECGHVFCSQCLRDSLKNANTCPTCRKKINHKRYHPIYI

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 \(c\).

\(|P_{f(7QBI_1)}(2) \setminus P_{f(3AIE_1)}(2)|=37\), \(|P_{f(3AIE_1)}(2) \setminus P_{f(7QBI_1)}(2)|=96\). 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 1:10011101001100100011000100001010100101001011100010011111100111110111111000000011111001011111100110100110000010011011101100101010110011011111010001110101110100100111110010101111010101101101111000010100000101001111001011011111011111000011110010010111110000111101111110010100100110011011111011100111011001000001000100000010110100100001000100010001001010000000000000011000100100001001101000000011111010000101011010010011000100110000001
Pair \(Z_2\) Length of longest common subsequence
7QBI_1,3AIE_1 133 4
7QBI_1,2XEU_1 223 3
3AIE_1,2XEU_1 262 3

Newick tree

 
[
	2XEU_1:13.11,
	[
		7QBI_1:66.5,3AIE_1:66.5
	]:68.61
]

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 \((0.85)(0.8)(\frac{1275 }{\log_{20} 1275}-\frac{431}{\log_{20}431})=218.\)
Status Protein1 Protein2 d d1/2
Query variables 7QBI_1 3AIE_1 274 205.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]