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Parikh vectors
2GCI_1 2GUB_1 1DEW_1 Letter Amino acid
26 35 0 R Arginine
26 37 0 D Aspartic acid
11 11 0 Q Glutamine
34 39 0 L Leucine
23 19 0 P Proline
18 28 0 E Glutamic acid
7 10 0 H Histidine
8 6 0 W Tryptophan
5 11 0 K Lycine
8 23 0 F Phenylalanine
17 11 0 S Serine
23 19 0 V Valine
13 8 0 M Methionine
14 16 1 T Threonine
9 9 0 Y Tyrosine
44 47 2 A Alanine
11 10 1 N Asparagine
3 1 6 C Cysteine
41 37 5 G Glycine
19 11 0 I Isoleucine 

2GCI_1|Chains A, B, C, D|probable alpha-methylacyl-CoA racemase MCR|Mycobacterium tuberculosis (1773)
>2GUB_1|Chain A|Xylose isomerase|Streptomyces rubiginosus (1929)
>1DEW_1|Chains A[auth U], C[auth X]|5'-D(*GP*CP*GP*TP*CP*CP*(3DR)P*CP*GP*AP*CP*GP*AP*CP*G)-3'|
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 \(w=f(c)\)
2GCI , Knot 154 360 0.84 40 196 342 
MAGPLSGLRVVELAGIGPGPHAAMILGDLGADVVRIDRPSSVDGISRDAMLRNRRIVTADLKSDQGLELALKLIAKADVLIEGYRPGVTERLGLGPEECAKVNDRLIYARMTGWGQTGPRSQQAGHDINYISLNGILHAIGRGDERPVPPLNLVGDFGGGSMFLLVGILAALWERQSSGKGQVVDAAMVDGSSVLIQMMWAMRATGMWTDTRGANMLDGGAPYYDTYECADGRYVAVGAIEPQFYAAMLAGLGLDAAELPPQNDRARWPELRALLTEAFASHDRDHWGAVFANSDACVTPVLAFGEVHNEPHIIERNTFYEANGGWQPMPAPRFSRTASSQPRPPAATIDIEAVLTDWDG
2GUB , Knot 159 388 0.81 40 206 357 
MNYQPTPEDRFTFGLWTVGWQGRDPFGDATRRALDPVESVRRLAELGAHGVTFHDDDLIPFGSSDSEREEHVKRFRQALDDTGMKVPMATTNLFTHPVFKDGGFTANDRDVRRYALRKTIRNIDLAVELGAETYVAWGGREGAESGGAKDVRDALDRMKEAFDLLGEYVTSQGYDIRFAIEPKPNEPRGDILLPTVGHALAFIERLERPELYGVNPEVGHEQMAGLNFPHGIAQALWAGKLFHIDLNGQNGIKYDQDLRFGAGDLRAAFWLVDLLESAGYSGPRHFDFKPPRTEDFDGVWASAAGCMRNYLILKERAAAFRADPEVQEALRASRLDELARPTAADGLQALLDDRSAFEEFDVDAAAARGMAFERLDQLAMDHLLGARG
1DEW , Knot 7 15 0.42 10 9 10 
GCGTCCNCGACGACG

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 \(c\).

\(|P_{f(2GCI_1)}(2) \setminus P_{f(2GUB_1)}(2)|=64\), \(|P_{f(2GUB_1)}(2) \setminus P_{f(2GCI_1)}(2)|=74\). 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 1:111110110110111111110111111011101101001001011000111000011010100001101110111010111010011100011111000101000110101011100110000110010010101110111010001111101110111101111111111110000010101101111010011101111101011100001101101111000000010100111111010101111111110110111000010110101110011100000011111100010101111110100010110000100101110111110100010001011110101011100101
Pair \(Z_2\) Length of longest common subsequence
2GCI_1,2GUB_1 138 4
2GCI_1,1DEW_1 201 2
2GUB_1,1DEW_1 211 2

Newick tree

 
[
	1DEW_1:11.10,
	[
		2GCI_1:69,2GUB_1:69
	]:43.10
]

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 \((0.85)(0.8)(\frac{748 }{\log_{20} 748}-\frac{360}{\log_{20}360})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 2GCI_1 2GUB_1 129 124.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]