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Parikh vectors
1CDO_1 4NUP_1 4RLB_1 Letter Amino acid
29 15 25 A Alanine
26 15 17 L Leucine
21 17 10 T Threonine
5 2 7 W Tryptophan
18 16 22 D Aspartic acid
9 9 6 Q Glutamine
20 9 7 E Glutamic acid
39 14 31 G Glycine
20 21 11 I Isoleucine
31 4 14 K Lycine
8 5 5 M Methionine
20 12 13 S Serine
10 12 5 R Arginine
13 18 22 N Asparagine
17 5 8 F Phenylalanine
3 5 17 Y Tyrosine
43 16 16 V Valine
14 0 0 C Cysteine
10 3 7 H Histidine
18 19 10 P Proline 

1CDO_1|Chains A, B|ALCOHOL DEHYDROGENASE|Gadus callarias (8053)
>4NUP_1|Chains A, B, C|N-CADHERIN EC1-2|Mus musculus (10090)
>4RLB_1|Chains A, B|Carbapenem-associated resistance protein|Acinetobacter baumannii (470)
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 \(w=f(c)\)
1CDO , Knot 157 374 0.83 40 204 353 
ATVGKVIKCKAAVAWEANKPLVIEEIEVDVPHANEIRIKIIATGVCHTDLYHLFEGKHKDGFPVVLGHEGAGIVESVGPGVTEFQPGEKVIPLFISQCGECRFCQSPKTNQCVKGWANESPDVMSPKETRFTCKGRKVLQFLGTSTFSQYTVVNQIAVAKIDPSAPLDTVCLLGCGVSTGFGAAVNTAKVEPGSTCAVFGLGAVGLAAVMGCHSAGAKRIIAVDLNPDKFEKAKVFGATDFVNPNDHSEPISQVLSKMTNGGVDFSLECVGNVGVMRNALESCLKGWGVSVLVGWTDLHDVATRPIQLIAGRTWKGSMFGGFKGKDGVPKMVKAYLDKKVKLDEFITHRMPLESVNDAIDLMKHGKCIRTVLSL
4NUP , Knot 97 217 0.80 38 147 207 
DWAAVIPPINLPENSRGPFPQELVRIRSDRDKNLSLRYSVTGPGADQPPTGIFIINPISGQLSVTKPLDRELIARFHLRAHAVDINGNQVENPIDIVINVIDMNDNRPEFLHQVWNGSVPEGSKPGTYVMTVTAIDADDPNALNGMLRYRILSQAPSTPSPNMFTINNETGDIITVAAGLDREKVQQYTLIIQATDMEGNPTYGLSNTATAVITVTD
4RLB , Knot 107 253 0.78 38 154 240 
MKHHHHHHPMSDYDIPTTENLYFQGAMDEAVVHDSYAFDKNQLIPVGARAEVGTTGYGGALLWQANPYVGLALGYNGGDISWSDDLSINGTKYDMDMDNKLAYLNAEIRPWGASTNPWAQGLYVAAGAAYVDNQYDLTKNVGTNASVEIDGNRFNGGANGVSIAGNLKYDNDIAPYIGFGFAPKFSKNWGVFGEVGAYYSGNPKVSLASNNDALIGSDGRTLGKTLDDQERKIANDDKYKWLPVGKVGVNFYW

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 \(c\).

\(|P_{f(1CDO_1)}(2) \setminus P_{f(4NUP_1)}(2)|=119\), \(|P_{f(4NUP_1)}(2) \setminus P_{f(1CDO_1)}(2)|=62\). 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 1:10110110001111101001111001010110100101011101100001001101000011111110011111001111100101100111111000100010001000001011100010110100001000100110111000100001100111101010111001011101100111111001010110001111111111111110001110011110101001001011110011010000011001100100111010100110111100110001011110111110010011001101111001010111110100111011010100010100110001110010011011001001001101
Pair \(Z_2\) Length of longest common subsequence
1CDO_1,4NUP_1 181 3
1CDO_1,4RLB_1 188 4
4NUP_1,4RLB_1 167 4

Newick tree

 
[
	1CDO_1:95.00,
	[
		4NUP_1:83.5,4RLB_1:83.5
	]:11.50
]

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 \((0.85)(0.8)(\frac{591 }{\log_{20} 591}-\frac{217}{\log_{20}217})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 1CDO_1 4NUP_1 137 106.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]