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Parikh vectors
7UGG_1 2ERC_1 1ENN_1 Letter Amino acid
41 9 0 R Arginine
38 26 0 N Asparagine
56 12 0 E Glutamic acid
45 26 0 I Isoleucine
49 30 0 K Lycine
25 9 0 Y Tyrosine
54 6 2 A Alanine
51 17 0 S Serine
13 2 2 C Cysteine
26 8 0 Q Glutamine
105 21 0 L Leucine
47 19 0 F Phenylalanine
35 5 0 P Proline
12 1 0 W Tryptophan
43 13 0 D Aspartic acid
15 9 0 H Histidine
25 5 0 M Methionine
51 9 2 T Threonine
43 11 0 V Valine
34 6 3 G Glycine 

7UGG_1|Chains A, B, C, D|Transient receptor potential cation channel subfamily V member 3|Mus musculus (10090)
>2ERC_1|Chains A, B|RRNA METHYL TRANSFERASE|Bacillus subtilis (1423)
>1ENN_1|Chains A, B|DNA (5'-D(*GP*CP*GP*AP*AP*TP*TP*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 \(w=f(c)\)
7UGG , Knot 307 808 0.84 40 310 734 
MGAHSKEMAPLMGKRTTAPGGNPVVLTEKRPADLTPTKKSAHFFLEIEGFEPNPTVTKTSPPIFSKPMDSNIRQCLSGNCDDMDSPQSPQDDVTETPSNPNSPSANLAKEEQRQKKKRLKKRIFAAVSEGCVEELRELLQDLQDLCRRRRGLDVPDFLMHKLTASDTGKTCLMKALLNINPNTKEIVRILLAFAEENDILDRFINAEYTEEAYEGQTALNIAIERRQGDITAVLIAAGADVNAHAKGVFFNPKYQHEGFYFGETPLALAACTNQPEIVQLLMENEQTDITSQDSRGNNILHALVTVAEDFKTQNDFVKRMYDMILLRSGNWELETMRNNDGLTPLQLAAKMGKAEILKYILSREIKEKPLRSLSRKFTDWAYGPVSSSLYDLTNVDTTTDNSVLEIIVYNTNIDNRHEMLTLEPLHTLLHTKWKKFAKYMFFLSFCFYFFYNITLTLVSYYRPREDEDLPHPLALTHKMSWLQLLGRMFVLIWATCISVKEGIAIFLLRPSDLQSILSDAWFHFVFFVQAVLVILSVFLYLFAYKEYLACLVLAMALGWANMLYYTRGFQSMGMYSVMIQKVILHDVLKFLFVYILFLLGFGVALASLIEKCSKDKKDCSSYGSFSDAVLELFKLTIGLGDLNIQQNSTYPILFLFLLITYVILTFVLLLNMLIALMGETVENVSKESERIWRLQRARTILEFEKMLPEWLRSRFRMGELCKVADEDFRLCLRINEVKWTEWKTHVSFLNEDPGPIRRTADLNKIQDSSRSNSKTTLYAFDELDEFPETSVLVPRGSAAAWSHPQFEK
2ERC , Knot 110 244 0.82 40 159 234 
MNEKNIKHSQNFITSKHNIDKIMTNIRLNEHDNIFEIGSGKGHFTLELVQRCNFVTAIEIDHKLCKTTENKLVDHDNFQVLNKDILQFKFPKNQSYKIFGNIPYNISTDIIRKIVFDSIADEIYLIVEYGFAKRLLNTKRSLALFLMAEVDISILSMVPREYFHPKPKVNSSLIRLNRKKSRISHKDKQKYNYFVMKWVNKEYKKIFTKNQFNNSLKHAGIDDLNNISFEQFLSLFNSYKLFNK
1ENN , Knot 6 9 0.48 8 7 7 
GCGAATTCG

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 \(c\).

\(|P_{f(7UGG_1)}(2) \setminus P_{f(2ERC_1)}(2)|=171\), \(|P_{f(2ERC_1)}(2) \setminus P_{f(7UGG_1)}(2)|=20\). 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 1:1110000111111000011110111100001101010000101110101101010100001111001100010001010000100100100010001001001010110000000000100011111001010010011001001000001101101110010100010001101110101000011011111100001100110100000100100110111000010101111111101010101111010000011011001111110000101101110000001000000100110111011001000001100100111100101010010000110110111011010110011000100011001000100110111000100100100000001101110000100000110101100110001001100111101010110010101100001000001101111000101101110111111100101001111111010010011001110111110111111011101110000110111111111101100001100111001110011100110111101111111111110110000000000000101001110110101111010100000011111111100111011111011111110010010000001101001001101001110110001011010011000101010100101001000101100011110001010010000000000010110010011000111101011110010100
Pair \(Z_2\) Length of longest common subsequence
7UGG_1,2ERC_1 191 4
7UGG_1,1ENN_1 305 2
2ERC_1,1ENN_1 164 2

Newick tree

 
[
	7UGG_1:13.07,
	[
		2ERC_1:82,1ENN_1:82
	]:57.07
]

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 \((0.85)(0.8)(\frac{1052 }{\log_{20} 1052}-\frac{244}{\log_{20}244})=217.\)
Status Protein1 Protein2 d d1/2
Query variables 7UGG_1 2ERC_1 278 176.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]