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Parikh vectors
6MEQ_1 3TRH_1 2HLJ_1 Letter Amino acid
36 8 8 E Glutamic acid
31 12 6 I Isoleucine
20 11 3 K Lycine
22 6 5 P Proline
28 12 10 T Threonine
10 0 7 Y Tyrosine
23 6 4 N Asparagine
26 14 12 G Glycine
4 5 4 M Methionine
24 9 7 S Serine
28 10 10 V Valine
15 4 9 R Arginine
33 6 10 D Aspartic acid
16 8 10 Q Glutamine
10 4 9 H Histidine
41 18 20 L Leucine
16 5 6 F Phenylalanine
0 0 2 W Tryptophan
28 29 14 A Alanine
5 2 1 C Cysteine 

6MEQ_1|Chain A|Protocadherin gamma-B3|Homo sapiens (9606)
>3TRH_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P|Phosphoribosylaminoimidazole carboxylase carboxyltransferase subunit|Coxiella burnetii (777)
>2HLJ_1|Chain A|Hypothetical protein|Pseudomonas putida (160488)
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 \(w=f(c)\)
6MEQ , Knot 174 416 0.84 38 220 384 
EPIRYAIPEELDRGSLVGNLAKDLGFGVGDLPTRNLRVIAEKKFFTVSPENGNLLVSDRIDREEICGKKSTCVLEFEMVAEKPLNFFHVTVLIQDINDNPPTFSQNITELEISELALTGATFALESAQDPDVGVNSLQQYYLSPDPHFSLIQKENLDGSRYPELVLKAPLDREEQPHHHLVLTAVDGGEPSRSCTTQIRVIVADANDNPPVFTQDMYRVNVAENLPAGSSVLKVMAIDMDEGINAEIIYAFINIGKEVRQLFKLDSKTGELTTIGELDFEERDSYTIGVEAKDGGHHTAYCKVQIDISDENDNAPEITLASESQHIQEDAELGTAVALIKTHDLDSGFNGEILCQLKGNFPFKIVQDTKNTYRLVTDGALDREQIPEYNVTITATDKGNPPLSSSKTITLHILDAA
3TRH , Knot 77 169 0.78 36 111 159 
SNAMNKIFVAILMGSDSDLSTMETAFTELKSLGIPFEAHILSAHRTPKETVEFVENADNRGCAVFIAAAGLAAHLAGTIAAHTLKPVIGVPMAGGSLGGLDALLSTVQMPGGVPVACTAIGKAGAKNAAILAAQIIALQDKSIAQKLVQQRTAKRETLKKADENLQTQL
2HLJ , Knot 78 157 0.83 40 121 152 
GMPALITYRTTVQEDWVDYNGHLRDAFYLLIFSYATDALMDRIGLDADSRGQSGNSLFTLEAHINYLHEVKLGTEVWVQTQILGFDRKRLHVYHSLHRAGFDEVLAASEQMLLHVDLAGPQSAPFGHTTVCRLNHLVEQQEGAQAPQYMGRTIKLPA

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 \(c\).

\(|P_{f(6MEQ_1)}(2) \setminus P_{f(3TRH_1)}(2)|=139\), \(|P_{f(3TRH_1)}(2) \setminus P_{f(6MEQ_1)}(2)|=30\). 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 1:01100111001001011101100111111011000101110001101010010111000100001010000011010111001101101011100100011010001001010011101101110010010111001000010101010110000101000101110111000001000111011011010000000101111010001111000100101100111100110111101001101011011101100100110100001010011010100000001110100110001000101010000001101011000001000101101111100001001101011001010111011000000001100111000011000101010001011100000101011011
Pair \(Z_2\) Length of longest common subsequence
6MEQ_1,3TRH_1 169 3
6MEQ_1,2HLJ_1 173 3
3TRH_1,2HLJ_1 134 4

Newick tree

 
[
	6MEQ_1:90.84,
	[
		3TRH_1:67,2HLJ_1:67
	]:23.84
]

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 \((0.85)(0.8)(\frac{585 }{\log_{20} 585}-\frac{169}{\log_{20}169})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 6MEQ_1 3TRH_1 149 104.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]