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Parikh vectors
4QIP_1 4TVZ_1 5RFV_1 Letter Amino acid
8 26 17 F Phenylalanine
8 20 13 P Proline
0 6 3 W Tryptophan
9 29 29 L Leucine
15 22 11 K Lycine
9 24 21 N Asparagine
9 26 17 D Aspartic acid
14 29 9 E Glutamic acid
14 23 26 G Glycine
4 8 7 H Histidine
13 33 11 I Isoleucine
9 25 24 T Threonine
7 18 11 Y Tyrosine
12 30 27 V Valine
3 15 11 R Arginine
2 10 14 Q Glutamine
1 7 10 M Methionine
11 16 17 A Alanine
0 5 12 C Cysteine
11 22 16 S Serine 

4QIP_1|Chain A|Major pollen allergen Bet v 1-A|Betula pendula (3505)
>4TVZ_1|Chains A, B|Scavenger receptor class B member 2|Homo sapiens (9606)
>5RFV_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
4QIP , Knot 77 159 0.81 36 114 149 
GVFNYETETTSVIPAARLFKAFILDGDNLFPKVAPQAISSVENIEGNGGPGTIKKISFPEGFPFKYVKDRVDEVDHTNFKYNYSVIEGGPIGDTLEKISNEIKIVATPDGGSILKISNKYHTKGDHEVKAEQVKASKEMGETLLRAVESYLLAHSDAYN
4TVZ , Knot 169 394 0.85 40 233 371 
IEKKIVLRNGTEAFDSWEKPPLPVYTQFYFFNVTNPEEILRGETPRVEEVGPYTYRELRNKANIQFGDNGTTISAVSNKAYVFERDQSVGDPKIDLIRTLNIPVLTVIEWSQVHFLREIIEAMLKAYQQKLFVTHTVDELLWGYKDEILSLIHVFRPDISPYFGLFYEKNGTNDGDYVFLTGEDSYLNFTKIVEWNGKTSLDWWITDKCNMINGTDGDSFHPLITKDEVLYVFPSDFCRSVYITFSDYESVQGLPAFRYKVPAEILANTSDNAGFCIPEGNCLGSGVLNVSICKNGAPIIMSFPHFYQADERFVSAIEGMHPNQEDHETFVDINPLTGIILKAAKRFQINIYVKKLDDFVETGDIRTMVFPVMYLNESVHIDKETASRLKSMIN
5RFV , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ

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 \(c\).

\(|P_{f(4QIP_1)}(2) \setminus P_{f(4TVZ_1)}(2)|=31\), \(|P_{f(4TVZ_1)}(2) \setminus P_{f(4QIP_1)}(2)|=150\). 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 1:111000000001111101101111010011101110110010010101111010010110111100100010010000100000110111110010010001011101011011010000000100010100101000110011011000111000100
Pair \(Z_2\) Length of longest common subsequence
4QIP_1,4TVZ_1 181 3
4QIP_1,5RFV_1 196 4
4TVZ_1,5RFV_1 193 3

Newick tree

 
[
	5RFV_1:99.40,
	[
		4QIP_1:90.5,4TVZ_1:90.5
	]:8.90
]

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 \((0.85)(0.8)(\frac{553 }{\log_{20} 553}-\frac{159}{\log_{20}159})=114.\)
Status Protein1 Protein2 d d1/2
Query variables 4QIP_1 4TVZ_1 144 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]