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Parikh vectors
8VWY_1 2DYU_1 1FKN_1 Letter Amino acid
7 21 19 Y Tyrosine
16 17 34 V Valine
17 15 23 T Threonine
12 6 6 C Cysteine
18 7 16 Q Glutamine
12 36 34 G Glycine
9 29 23 I Isoleucine
27 23 29 L Leucine
19 10 20 F Phenylalanine
10 21 24 A Alanine
13 21 15 N Asparagine
7 8 7 H Histidine
6 8 8 M Methionine
4 8 7 W Tryptophan
9 11 19 R Arginine
12 15 21 D Aspartic acid
19 21 22 E Glutamic acid
22 21 15 K Lycine
12 21 19 P Proline
22 15 30 S Serine 

8VWY_1|Chains A, E[auth B], F[auth C]|Adhesion G protein-coupled receptor B3|Mus musculus (10090)
>2DYU_1|Chains A, B|Formamidase|Helicobacter pylori (85962)
>1FKN_1|Chains A, B|MEMAPSIN 2|Homo sapiens (9606)
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 \(w=f(c)\)
8VWY , Knot 125 273 0.85 40 187 265 
DFWCSTLVKGVIYGSYSVSEMFPKNFTNCTWTLENPDPTKYSIYLKFSKKDLSCSNFSLLAYQFDHFSHEKIKDLLRKNHSIMQLCSSKNAFVFLQYDKNFIQIRRVFPTDFPGLQKKVEEDQKSFFEFLVLNKVSPSQFGCHVLCTWLESCLKSENGRTESCGIMYTKCTCPQHLGEWGIDDQSLVLLNNVVLPLNEQTEGCLTQELQTTQVCNLTREAKRPPKEEFGMMGDHTIKSQRPRSVHEKRVPQEQADAAKFMAQTGAAALEVLFQ
2DYU , Knot 146 334 0.84 40 203 318 
MGSIGSMGKPIEGFLVAAIQFPVPIVNSRKDIDHNIESIIRTLHATKAGYPGVELIIFPEYSTQGLNTAKWLSEEFLLDVPGKETELYAKACKEAKVYGVFSIMERNPDSNKNPYNTAIIIDPQGEIILKYRKLFPWNPIEPWYPGDLGMPVCEGPGGSKLAVCICHDGMIPELAREAAYKGCNVYIRISGYSTQVNDQWILTNRSNAWHNLMYTVSVNLAGYDNVFYYFGEGQICNFDGTTLVQGHRNPWEIVTGEIYPKMADNARLSWGLENNIYNLGHRGYVAKPGGEHDAGLTYIKDLAAGKYKLPWEDHMKIKDGSIYGYPTTGGRFGK
1FKN , Knot 166 391 0.84 40 226 375 
RRGSFVEMVDNLRGKSGQGYYVEMTVGSPPQTLNILVDTGSSNFAVGAAPHPFLHRYYQRQLSSTYRDLRKGVYVPYTQGKWEGELGTDLVSIPHGPNVTVRANIAAITESDKFFINGSNWEGILGLAYAEIARPDDSLEPFFDSLVKQTHVPNLFSLQLCGAGFPLNQSEVLASVGGSMIIGGIDHSLYTGSLWYTPIRREWYYEVIIVRVEINGQDLKMDCKEYNYDKSIVDSGTTNLRLPKKVFEAAVKSIKAASSTEKFPDGFWLGEQLVCWQAGTTPWNIFPVISLYLMGEVTNQSFRITILPQQYLRPVEDVATSQDDCYKFAISQSSTGTVMGAVIMEGFYVVFDRARKRIGFAVSACHVHDEFRTAAVEGPFVTLDMEDCGYN

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 \(c\).

\(|P_{f(8VWY_1)}(2) \setminus P_{f(2DYU_1)}(2)|=85\), \(|P_{f(2DYU_1)}(2) \setminus P_{f(8VWY_1)}(2)|=101\). 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 1:011000110111010001001110010000101001010000101010000100001011100100100001001100000110100000111110000011010011100111100010000001101111001010011001100110001000010000011100000010011011100001111001111100000101000100001001000100110001111100010000100100001100010110111001111101110
Pair \(Z_2\) Length of longest common subsequence
8VWY_1,2DYU_1 186 4
8VWY_1,1FKN_1 185 3
2DYU_1,1FKN_1 169 3

Newick tree

 
[
	8VWY_1:95.34,
	[
		1FKN_1:84.5,2DYU_1:84.5
	]:10.84
]

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 \((0.85)(0.8)(\frac{607 }{\log_{20} 607}-\frac{273}{\log_{20}273})=93.8\)
Status Protein1 Protein2 d d1/2
Query variables 8VWY_1 2DYU_1 121 109
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]