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Parikh vectors
3PLA_1 4MMU_1 6IUB_1 Letter Amino acid
14 1 13 F Phenylalanine
21 7 23 S Serine
2 1 0 W Tryptophan
28 2 8 R Arginine
11 5 12 Q Glutamine
6 1 8 M Methionine
34 5 24 I Isoleucine
41 8 34 L Leucine
26 8 21 K Lycine
14 4 9 Y Tyrosine
21 4 13 V Valine
38 6 19 A Alanine
14 0 10 H Histidine
0 2 1 C Cysteine
32 6 12 E Glutamic acid
20 3 19 G Glycine
14 1 13 P Proline
11 9 13 T Threonine
19 7 11 N Asparagine
22 2 13 D Aspartic acid 

3PLA_1|Chains A, B, K|Pre mRNA splicing protein|Sulfolobus solfataricus (2287)
>4MMU_1|Chain A|Fusion glycoprotein F2|Human respiratory syncytial virus A2 (11259)
>6IUB_1|Chains A, B|SpoOJ regulator (Soj)|Helicobacter pylori (strain ATCC 700392 / 26695) (85962)
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 \(w=f(c)\)
3PLA , Knot 160 388 0.82 38 212 364 
MVKIYLIEHVIGAVAYDENGNIVDYITNPRDLGKITEELLNNEKGIPFSATVELLKKVNPQEVVVENEAEVPKLQALGYRVSYEPYSKVSRIFRESLPKVAIDIKFASNEEDYYNFLHELSLEYTRRKLRSAAQKRDLLAIQAVRAMDDIDKTINLFSERLREWYSIHFPELDKLIEDHEEYATIVSRFGDRGFLTIDSLKELGFNEQRINRILDAAKKSIGADISEDDLSAMRMIANTILDLYNIRRNLNNYLEGVMKEVAPNVTALVGPALGARLLSIAGSLDELAKMPASTIQVLGAEKALFRALRSGGRPPKHGIIFQYPAIHTSPRWQRGKIARALAAKLAIAARVDAFSGRFIGDQLNEQLKKRIDEIKEKFAQHHHHHHHH
4MMU , Knot 43 82 0.77 38 70 79 
QNITEEFYQSTCSAVSKGYLSALRTGWYTSVITIELSNIKENKCNGTDAKVKLIKQELDKYKNAVTELQLLMQSTPATNNRA
6IUB , Knot 123 276 0.83 38 168 264 
MRGSHHHHHHGSMMSEIIAVANQKGGVGKTTTAVNLAASLAVHEKKILLIDFDPQANATSSLGFRRDKIDYDIYHVLIGRKQISQVILKTQMPFLDLVPSNLGLAGFEKTFYDSQDENKRGELMLKNALESVVGLYDYIIIDSPPALGPLTINSLSAAHSVIIPIQCEFFALEGTKLLLNTIRMLQKSTNPKLKIRGFLPTMHVPQLNLTKGVLAELFKYFDSEFFRDSATGEYIMIPKSVKLAESPSFGKPILLYDIKSNGSIAYQKLAQSILQG

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 \(c\).

\(|P_{f(3PLA_1)}(2) \setminus P_{f(4MMU_1)}(2)|=166\), \(|P_{f(4MMU_1)}(2) \setminus P_{f(3PLA_1)}(2)|=24\). 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 1:1101011001111110000101100100100110100011000011110101011001010011100010110101110010001000100110001101110101100000000110010100000010011000011110110110010001011000100100101101001100000010110011001110100100111000010011011000111010000101101110011010010001000101110011101011111111101101110100110111001011110011101100110110011110011100010100101101111011111010110101110010001000100100011000000000
Pair \(Z_2\) Length of longest common subsequence
3PLA_1,4MMU_1 190 4
3PLA_1,6IUB_1 152 6
4MMU_1,6IUB_1 162 3

Newick tree

 
[
	4MMU_1:92.00,
	[
		3PLA_1:76,6IUB_1:76
	]:16.00
]

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 \((0.85)(0.8)(\frac{470 }{\log_{20} 470}-\frac{82}{\log_{20}82})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 3PLA_1 4MMU_1 144 85.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]