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Parikh vectors
3PWR_1 5UHQ_1 3ARE_1 Letter Amino acid
2 4 9 M Methionine
2 8 11 F Phenylalanine
15 8 8 I Isoleucine
13 5 20 G Glycine
1 5 8 Y Tyrosine
7 12 21 V Valine
4 4 13 R Arginine
0 0 5 C Cysteine
9 14 27 L Leucine
6 2 13 P Proline
1 4 28 S Serine
4 2 15 D Aspartic acid
3 4 9 N Asparagine
5 3 23 Q Glutamine
4 3 15 E Glutamic acid
1 0 17 H Histidine
7 2 14 K Lycine
8 8 14 T Threonine
2 1 14 W Tryptophan
5 4 18 A Alanine 

3PWR_1|Chains A, B|Protease|Human immunodeficiency virus 1 (11676)
>5UHQ_1|Chains A, B, C, D|Sugar transporter SemiSWEET|Leptospira biflexa serovar Patoc (strain Patoc 1 / ATCC 23582 / Paris) (456481)
>3ARE_1|Chain A|Antigen-presenting glycoprotein CD1d1|Mus musculus (10090)
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 \(w=f(c)\)
3PWR , Knot 53 99 0.82 38 77 95 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVIEEMSLPGRWKPKMIGGIGGFIKVRQYDQIIIEIAGHKAIGTVVVGPTPVNIIGRNLLTQIGATLNF
5UHQ , Knot 49 93 0.79 36 75 90 
MENLIGYVAAFLTTVSFLPAVLRVVMTKQTRDISRNMYIMFFLGVVLWFVYGILRSDLPIILANVVTLFFVTIILYYKLTEGNQTGSLEVLFQ
3ARE , Knot 131 302 0.82 40 189 286 
SEAQQKNYTFRCLQMSSFANRSWSRTDSVVWLGDLQTHRWSNDSATISFTKPWSQGKLSNQQWEKLQHMFQVYRVSFTRDIQELVKMMSPKEDYPIEIQLSAGCEMYPGNASESFLHVAFQGKYVVRFWGTSWQTVPGAPSWLDLPIKVLNADQGTSATVQMLLNDTCPLFVRGLLEAGKSDLEKQEKPVAWLSSVPSSAHGHRQLVCHVSGFYPKPVWVMWMRGDQEQQGTHRGDFLPNADETWYLQATLDVEAGEEAGLACRVKHSSLGGQDIILYWGSLHHILDAQKMVWNHRHHHHHH

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 \(c\).

\(|P_{f(3PWR_1)}(2) \setminus P_{f(5UHQ_1)}(2)|=62\), \(|P_{f(5UHQ_1)}(2) \setminus P_{f(3PWR_1)}(2)|=60\). 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 1:101011001110101110100111001100011001011101010111111111010000011101110011101111101101110011001110101
Pair \(Z_2\) Length of longest common subsequence
3PWR_1,5UHQ_1 122 3
3PWR_1,3ARE_1 184 3
5UHQ_1,3ARE_1 176 4

Newick tree

 
[
	3ARE_1:97.80,
	[
		3PWR_1:61,5UHQ_1:61
	]:36.80
]

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 \((0.85)(0.8)(\frac{192 }{\log_{20} 192}-\frac{93}{\log_{20}93})=32.5\)
Status Protein1 Protein2 d d1/2
Query variables 3PWR_1 5UHQ_1 41 40
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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