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Parikh vectors
4GMP_1 5HVC_1 8OFY_1 Letter Amino acid
16 3 17 N Asparagine
12 2 14 E Glutamic acid
26 6 12 S Serine
16 0 14 Y Tyrosine
20 2 33 V Valine
17 4 11 D Aspartic acid
8 0 3 C Cysteine
18 7 7 Q Glutamine
22 7 17 P Proline
28 9 34 A Alanine
7 3 7 R Arginine
23 5 36 G Glycine
8 9 5 H Histidine
21 16 33 L Leucine
13 1 9 K Lycine
3 1 7 M Methionine
28 15 8 T Threonine
5 0 5 W Tryptophan
20 3 17 I Isoleucine
12 2 23 F Phenylalanine 

4GMP_1|Chain A[auth 0]|capsid protein VP0|Human enterovirus 71 (39054)
>5HVC_1|Chain A|Type I modular polyketide synthase|Mycobacterium ulcerans (1809)
>8OFY_1|Chain A|Aquaglyceroporin 2|Trypanosoma brucei brucei (5702)
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 \(w=f(c)\)
4GMP , Knot 142 323 0.84 40 205 308 
MGSQVSTQRSGSHENSNSATEGSTINYTTINYYKDSYAATAGKQSLKQDPDKFANPVKDIFTEMAAPLKSPSAEACGYSDRVAQLTIGNSTITTQEAANIIVGYGEWPSYCSDSDATAVDKPTRPDVSVNRFYTLDTKLWEKSSKGWYWKFPDVLTETGVFGQNAQFHYLYRSGFCIHVQCNASKFHQGALLVAILPEYVIGTVAGGTGTEDSHPPYKQTQPGADGFELQHPYVLDAGIPISQLTVCPHQWINLRTNNCATIIVPYINALPFDSALNHCNFGLLVVPISPLDYDQGATPVIPITITLAPMCSEFAGLRQAVTQ
5HVC , Knot 47 95 0.75 34 67 89 
GSHMRLNGLSPQQQQQTLATLVAAATATVLGHHTPESISPATAFKDLGIDSLTALELRNTLTHNTGLDLPPTLIFDHPTPHALTQHLHTRLTQSH
8OFY , Knot 134 312 0.82 40 179 293 
MQSQPDNVAYPMELQAVNKDGTVEVRVQGNVDNSSNERWDADVQKHEVAEAQEKPVGGINFWAPRELRLNYRDYVAEFLGNFVLIYIAKGAVITSLLVPDFGLLGLTIGIGVAVTMALYVSLGISGGHLNSAVTVGNAVFGDFPWRKVPGYIAAQMLGTFLGAACAYGVFADLLKAHGGGELIAFGEKGIAWVFAMYPAEGNGIFYPIFAELISTAVLLLCVCGIFDPNNSPAKGYETVAIGALVFVMVNNFGLASPLAMNPSLDFGPRVFGAILLGGEVFSHANYYFWVPLVVPFFGAILGLFLYKYFLPH

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 \(c\).

\(|P_{f(4GMP_1)}(2) \setminus P_{f(5HVC_1)}(2)|=166\), \(|P_{f(5HVC_1)}(2) \setminus P_{f(4GMP_1)}(2)|=28\). 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 1:11001000001000000010010010000100000001101100010001001101100110011111001010101000011010110001000011011110101100000001011001001010100100100011000001101011011000111100101001000110101000100100111111111001110111101000001100000111011010010110111110010101001101000001011110101111001100001111111101100001101111101011110001111001100
Pair \(Z_2\) Length of longest common subsequence
4GMP_1,5HVC_1 194 4
4GMP_1,8OFY_1 160 4
5HVC_1,8OFY_1 176 3

Newick tree

 
[
	5HVC_1:96.44,
	[
		4GMP_1:80,8OFY_1:80
	]:16.44
]

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 \((0.85)(0.8)(\frac{418 }{\log_{20} 418}-\frac{95}{\log_{20}95})=98.5\)
Status Protein1 Protein2 d d1/2
Query variables 4GMP_1 5HVC_1 128 80.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]