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Parikh vectors
1LUX_1 1DYM_1 3SLN_1 Letter Amino acid
0 18 16 D Aspartic acid
0 14 7 I Isoleucine
0 15 19 F Phenylalanine
0 28 35 T Threonine
5 21 14 A Alanine
0 22 16 N Asparagine
4 18 5 C Cysteine
0 7 4 W Tryptophan
0 12 20 Q Glutamine
0 10 10 H Histidine
0 23 7 K Lycine
0 19 8 Y Tyrosine
0 23 25 V Valine
0 24 26 P Proline
0 17 21 S Serine
0 17 13 R Arginine
0 30 12 E Glutamic acid
5 47 25 G Glycine
0 26 23 L Leucine
0 11 5 M Methionine 

1LUX_1|Chain A|5'-R(*CP*CP*AP*GP*AP*(OMC)P*UP*(OMG)P*AP*AP*GP*AP*UP*(5MC)P*UP*GP*G)-3'|null
>1DYM_1|Chain A|ENDOGLUCANASE I|HUMICOLA INSOLENS (85995)
>3SLN_1|Chains A, B, C, D, E, F, G, H, I, J|Capsid|Norovirus Hu/GII.4/2004/NL (311380)
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 \(w=f(c)\)
1LUX , Knot 8 17 0.44 8 11 13 
CCAGACUGAAGAUCUGG
1DYM , Knot 174 402 0.86 40 239 386 
QKPGETKEVHPQLTTFRCTKRGGCKPATNFIVLDSLSHPIHRAEGLGPGGCGDWGNPPPKDVCPDVESCAKNCIMEGIPDYSQYGVTTNGTSLRLQHILPDGRVPSPRVYLLDKTKRRYEMLHLTGFEFTFDVDATKLPCGMNSALYLSEMHPTGAKSKYNPGGAYYGTGYCDAQCFVTPFINGLGNIEGKGSCCNAMDIWEANSRASHVAPHTCNKKGLYLCEGEECAFEGVCDKNGCGWNNYRVNVTDYYGRGEEFKVNTLKPFTVVTQFLANRRGKLEKIHRFYVQDGKVIESFYTNKEGVPYTNMIDDEFCEATGSRKYMELGATQGMGEALTRGMVLAMSIWWDQGGNMEWLDHGEAGPCAKGEGAPSNIVQVEPFPEVTYTNLRWGEIGSTYQELQ
3SLN , Knot 133 311 0.81 40 185 297 
ESRTKPFTVPILTVEEMTNSRFPIPLEKLFTGPSSAFVVQPQNGRCTTDGALLGTTQLSPVDICTFRGDVTHIAGTQNYTMNLASQNWNNYDPTEEIPAPLGTPDFVGKIQGVLTQTTRRDGSTRGHKATVSTGSVHFTPKLGSVQFSTDTSNDFETGQNTRFTPVGVVQDGSTTHQNEPQQWVLPDYSGRDSHNVHLAPAVAPTFPGEQLLFFRSTMPGCSGYPNMNLDCLLPQEWVQHFYQESAPAQSDVALLRFVNPDTGRVLFECKLHKSGYVTVAHTGQHDLVIPPNGYFRFDSWVNQFYTLAPMG

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 \(c\).

\(|P_{f(1LUX_1)}(2) \setminus P_{f(1DYM_1)}(2)|=6\), \(|P_{f(1DYM_1)}(2) \setminus P_{f(1LUX_1)}(2)|=234\). 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 1:00111001111100011
Pair \(Z_2\) Length of longest common subsequence
1LUX_1,1DYM_1 240 2
1LUX_1,3SLN_1 192 2
1DYM_1,3SLN_1 170 5

Newick tree

 
[
	1LUX_1:11.48,
	[
		3SLN_1:85,1DYM_1:85
	]:30.48
]

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 \((0.85)(0.8)(\frac{419 }{\log_{20} 419}-\frac{17}{\log_{20}17})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 1LUX_1 1DYM_1 171 88.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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