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Parikh vectors
6EAY_1 1MWP_1 8WGO_1 Letter Amino acid
9 6 66 E Glutamic acid
12 6 58 G Glycine
3 3 20 H Histidine
7 6 71 I Isoleucine
8 3 49 Y Tyrosine
17 3 36 A Alanine
17 6 91 L Leucine
1 3 8 M Methionine
18 6 33 T Threonine
2 2 16 W Tryptophan
14 8 49 V Valine
9 3 51 D Aspartic acid
10 6 68 K Lycine
33 2 56 S Serine
9 6 51 N Asparagine
5 6 8 C Cysteine
15 8 29 Q Glutamine
8 3 48 F Phenylalanine
10 6 28 P Proline
8 4 33 R Arginine 

6EAY_1|Chain A[auth L]|CA45 light chain|Macaca fascicularis (9541)
>1MWP_1|Chain A|AMYLOID A4 PROTEIN|Homo sapiens (9606)
>8WGO_1|Chains A, C[auth B]|Protein kinase domain-containing protein|Streptococcus pneumoniae (1313)
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 \(w=f(c)\)
6EAY , Knot 99 215 0.82 40 148 204 
DIQMTQSPSSLSASVGDTVTITCRASQSISNNLAWYQQRPRRAPQLLIYAASNLASGVPSRFSGSGSGTDFTLTISSLQAEDFAAYYCQQHNTLPLTFGGGTKVEIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGECS
1MWP , Knot 54 96 0.85 40 86 94 
LLAEPQIAMFCGRLNMHMNVQNGKWDSDPSGTKTCIDTKEGILQYCQEVYPELQITNVVEANQPVTIQNWCKRGRKQCKTHPHFVIPYRCLVGEFV
8WGO , Knot 329 869 0.85 40 299 799 
MDYNFNLEHPFFFTNNDYSTDTSIKYQASLPFNWHEVMNNDEWVYQYPIGKFVERQGWKIHISSEYNSSHELLQDVAKICHEMRIPFKHLSTEDKFIMRNGKLVSRGFSGKFITCYPNQNELESVLQRLESALKQYNGPYILSDKRWDEAPIYLRYGVFRPSRDDEKKVAIDELIVGDEVVKDERLPVFKIPKGIVPPDFLNKWLDKKDKKQGDFPFIIDNAIRFSNSGGIYNARLKEDGKKIILKEARPYTGLGFDGTYSSEKLASECKALKILNEWSEAPKIYWHGKIWEHTFLGIEHMKGVPLNRWVTNNFPLYEVVDKTKDYLLRVSKIVEKLIDLTNKFHSENVYHQDLHLGNILVKDEDEISIIDWEQAVFSNDEKVVHKVAAPGFRAWRETLPSEIDWYGIRQIAHYLYMPLVTTSDLTYNYVSQTRIEGKKLFESLGYTREHIDYVESLLSYLDSKCPQIENISRKKVLKPMHEIRTIESEQDIQDFIIKLLRGFTLTYGQWRKEFQSRFFPVHYYGLNFNQGIAFSDLAILWSYQQLAKKVKNFKFDDYYEIRTQVINEAVNNFKKSSLSGLFDGKIGTIWLIYEFGEIDRAVELFTTHFIEIFENSQNKNLYSGQAGILLVGLYFLSKGEIDNKLGEEILIRLREYTLNYIENPETFCKVGASDVQSNDPYENFGGLLYGHAGVAWLFGEAYKLTGESIYKNGLELAVDKELVAYKVDSNNSLQYSQGHRLLPYLATGSAGLLLLINRNKEILSSKYLKYLTSLERATDVVFCVLPGLFNGFCGLEVANNIYSDIDDNFSGQKKLIEQLYRYLCVIEEGFVIAGDNGLKITTDIASGFAGVAIGLVSIMDNKLTILPQI

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 \(c\).

\(|P_{f(6EAY_1)}(2) \setminus P_{f(1MWP_1)}(2)|=113\), \(|P_{f(1MWP_1)}(2) \setminus P_{f(6EAY_1)}(2)|=51\). 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 1:01010001001010110010100010001000111000010011011101100110111001010101001010100101001110000000011101111001010001111011111100001001010110110010100101010100110010000010000000000010001010010000001010010001100110001001000
Pair \(Z_2\) Length of longest common subsequence
6EAY_1,1MWP_1 164 3
6EAY_1,8WGO_1 199 4
1MWP_1,8WGO_1 245 3

Newick tree

 
[
	8WGO_1:11.84,
	[
		6EAY_1:82,1MWP_1:82
	]:37.84
]

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 \((0.85)(0.8)(\frac{311 }{\log_{20} 311}-\frac{96}{\log_{20}96})=67.5\)
Status Protein1 Protein2 d d1/2
Query variables 6EAY_1 1MWP_1 84 62
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]