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Parikh vectors
5GLN_1 3EIM_1 1WTS_1 Letter Amino acid
33 25 0 D Aspartic acid
22 8 0 H Histidine
34 36 6 G Glycine
9 2 0 M Methionine
26 26 0 S Serine
23 28 0 Y Tyrosine
17 22 0 V Valine
10 10 0 R Arginine
1 0 4 C Cysteine
16 8 0 E Glutamic acid
13 10 0 F Phenylalanine
22 8 0 P Proline
18 25 0 T Threonine
9 19 0 N Asparagine
17 16 0 L Leucine
16 11 0 K Lycine
7 3 0 W Tryptophan
22 28 3 A Alanine
5 13 0 Q Glutamine
24 18 0 I Isoleucine 

5GLN_1|Chains A, B|Glycoside hydrolase family 43|uncultured bacterium (77133)
>3EIM_1|Chain A|Thermolysin|Bacillus thermoproteolyticus (1427)
>1WTS_1|Chain A|RNA (5'-R(*GP*GP*AP*CP*CP*2MGP*GP*MA6P*MA6P*GP*GP*UP*CP*C)-3')|null
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 \(w=f(c)\)
5GLN , Knot 147 344 0.83 40 203 326 
MGSSHHHHHHSSGLVPRGSHMEPLVTHIYTADPSAHVFDGKVYIYPSHDIDAGTPENDMGDHFDMRDYHVLSMNSIPGEVTDHGVALDIKDIPWAGRQLWAPDAASKDGKYYLYFPAKDKEDIFRIGVAVSDSPAGPFKPESEPIKGSYSIDPAVFKDDDGKYYMYFGGIWGGQLQRWTTGEYAGHDASKTDLEQDDAPAIGPRIALMSDDMLSFAEPVKEISIVDEQGNPILGGDHDRRFFEAAWMHKYNGTYYLSYSTGDTHYIVYATGDNPYGPFTYRGVILNPVIGWTNHHSIVEFNGKWYLFYHDSSLSGGKTHLRCIKVTELTHNADGTIETISPYIE
3EIM , Knot 131 316 0.79 38 180 301 
ITGTSTVGVGRGVLGDQKNINTTYSTYYYLQDNTRGDGIFTYDAKYRTTLPGSLWADADNQFFASYDAPAVDAHYYAGVTYDYYKNVHNRLSYDGNNAAIRSSVHYSQGYNNAFWNGSEMVYGDGDGQTFIPLSGGIDVVAHELTHAVTDYTAGLIYQNESGAINEAISDIFGTLVEFYANKNPDWEIGEDVYTPGISGDSLRSMSDPAKYGDPDHYSKRYTGTQDNGGVHINSGIINKAAYLISQGGTHYGVSVVGIGRDKLGKIFYRALTQYLTPTSNFSQLRAAAVQSATDLYGSTSQEVASVKQAFDAVGVK
1WTS , Knot 7 14 0.44 8 9 11 
GGACCGGAAGGUCC

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 \(c\).

\(|P_{f(5GLN_1)}(2) \setminus P_{f(3EIM_1)}(2)|=91\), \(|P_{f(3EIM_1)}(2) \setminus P_{f(5GLN_1)}(2)|=68\). 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 1:11000000000011110100101110010010101011010101010001011010001100101000011010011101000111101001111100111101100010001011100000110111110001111101000110100010111100001000101111111010010010011001000010000111111011110001101101100101100010111110000011011110000100010000100001101010010111000111101111100000110101010110000010110001001010010001010100101010
Pair \(Z_2\) Length of longest common subsequence
5GLN_1,3EIM_1 159 4
5GLN_1,1WTS_1 206 2
3EIM_1,1WTS_1 181 2

Newick tree

 
[
	1WTS_1:10.10,
	[
		5GLN_1:79.5,3EIM_1:79.5
	]:22.60
]

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 \((0.85)(0.8)(\frac{660 }{\log_{20} 660}-\frac{316}{\log_{20}316})=95.2\)
Status Protein1 Protein2 d d1/2
Query variables 5GLN_1 3EIM_1 122 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]