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Parikh vectors
5MDA_1 1KMM_1 6RRE_1 Letter Amino acid
6 34 29 E Glutamic acid
4 41 31 G Glycine
3 15 42 I Isoleucine
4 14 13 P Proline
2 16 33 S Serine
0 8 11 H Histidine
7 49 51 L Leucine
3 9 8 M Methionine
2 12 23 F Phenylalanine
7 20 27 T Threonine
6 41 22 A Alanine
5 19 20 Q Glutamine
5 16 35 K Lycine
1 6 1 W Tryptophan
2 15 17 Y Tyrosine
4 32 14 R Arginine
3 17 27 N Asparagine
4 22 34 D Aspartic acid
2 5 5 C Cysteine
4 33 28 V Valine 

5MDA_1|Chains A, B, C, D, H[auth E], I[auth F], J[auth G], K[auth W]|Gag protein|Human immunodeficiency virus 1 (11676)
>1KMM_1|Chains A, B, C, D|HISTIDYL-TRNA SYNTHETASE|Escherichia coli (562)
>6RRE_1|Chains A, B, C, D, E, F|Adenosine monophosphate-protein hydrolase SidD|Legionella pneumophila (446)
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 \(w=f(c)\)
5MDA , Knot 43 74 0.83 38 66 72 
TSILDIRQGPKEPFRDYVDRFYKTLRAEQASQEVKNWMTETLLVQNANPDCKTILKALGPGATLEEMMTACQGV
1KMM , Knot 178 424 0.84 40 221 398 
MAKNIQAIRGMNDYLPGETAIWQRIEGTLKNVLGSYGYSEIRLPIVEQTPLFKRAIGEVTDVVEKEMYTFEDRNGDSLTLRPEGTAGCVRAGIEHGLLYNQEQRLWYIGPMFRHERPQKGRYRQFHQLGCEVFGLQGPDIDAELIMLTARWWRALGISEHVTLELNSIGSLEARANYRDALVAFLEQHKEKLDEDCKRRMYTNPLRVLDSKNPEVQALLNDAPALGDYLDEESREHFAGLCKLLESAGIAYTVNQRLVRGLDYYNRTVFEWVTNSLGSQGTVCAGGRYDGLVEQLGGRATPAVGFAMGLERLVLLVQAVNPEFKADPVVDIYLVASGADTQSAAMALAERLRDELPGVKLMTNHGGGNFKKQFARADKWGARVAVVLGESEVANGTAVVKDLRSGEQTAVAQDSVAAHLRTLLG
6RRE , Knot 195 471 0.85 40 237 445 
MRSIITQICNGVLHGQSYQSGSNDLDKGNSEIFASSLFVHLNEQGKEIIKHKDSDDKIVIGYTKDGMAFQIVVDGFYGCERQAVFSFIDNYVLPLIDNFSLDLTRYPDSKKVTESLIHTIYSLRSKHAPLAEFTMSLCVTYQKDEQLFCAGFGIGDTGIAIKRNEGTIEQLVCHTEVDGFKDAFDNYSSANIDLVIERNSVFNTKVMPGDELVGYTYVPPMLEMTEKEFEVETVDGKKINKRIVRHLNLDPGNFDDKDPLFSQLLQVVKSKQKQLVEQAKETGQIQRFGDDFTVGRLVIPDQLLINQLRIHALSIGVSDGLLSYIKNENENKGFLGIYGFFTGADKNIEKATLYKNLIAKYQNNHFISLIILSALVSDSKTPLMTQYLVGYLDFPSKALLANKITELLLKELENPDMREILGSRLATDVIEELETKIIRYIHNPAGSDIHSTLNLWTADKIKAATNSSLTI

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 \(c\).

\(|P_{f(5MDA_1)}(2) \setminus P_{f(1KMM_1)}(2)|=19\), \(|P_{f(1KMM_1)}(2) \setminus P_{f(5MDA_1)}(2)|=174\). 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 1:00110100110011000100100010100100010011000111001010000110111111010011010011
Pair \(Z_2\) Length of longest common subsequence
5MDA_1,1KMM_1 193 4
5MDA_1,6RRE_1 211 3
1KMM_1,6RRE_1 142 4

Newick tree

 
[
	5MDA_1:10.30,
	[
		1KMM_1:71,6RRE_1:71
	]:38.30
]

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 \((0.85)(0.8)(\frac{498 }{\log_{20} 498}-\frac{74}{\log_{20}74})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 5MDA_1 1KMM_1 158 92
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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