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Parikh vectors
3EKJ_1 5XUG_1 1VHM_1 Letter Amino acid
20 6 10 F Phenylalanine
32 13 11 T Threonine
2 6 2 W Tryptophan
25 7 17 V Valine
19 14 8 N Asparagine
16 2 8 H Histidine
22 7 13 I Isoleucine
34 11 21 L Leucine
20 2 3 M Methionine
19 10 9 S Serine
20 13 10 Q Glutamine
49 16 15 G Glycine
34 4 10 E Glutamic acid
12 4 4 P Proline
19 6 9 R Arginine
38 14 14 D Aspartic acid
32 9 6 K Lycine
14 8 5 Y Tyrosine
22 10 17 A Alanine
2 0 3 C Cysteine 

3EKJ_1|Chain A|Myosin light chain kinase, Green fluorescent protein, Calmodulin chimera|artificial gene (32630)
>5XUG_1|Chains A, B|endo-1,4-beta-mannanase|Rhizopus microsporus (58291)
>1VHM_1|Chains A, B|Protein yebR|Escherichia coli (562)
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 \(w=f(c)\)
3EKJ , Knot 186 451 0.84 40 234 412 
MRGSHHHHHHGMASMTGGQQMGRDLYDDDDKDLATMVDSSRRKWNKTGHAVRAIGRLSSLENVYIMADKQKNGIKANFKIRHNIEDGGVQLAYHYQQNTPIGDGPVLLPDNHYLSTQSKLSKDPNEKRDHMVLLEFVTAAGITLGMDELYKGGTGGSMVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKQHDFFKSAMPEGYIQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNTRDQLTEEQIAEFKEAFSLFDKDGDGGITTKQLGTVMRSLGQNPTEAELQDMINEVGADGNGTIDFPQFLTMMARKMKDTDSEEEIREAFRVFGKDGNGYISAAQLRHVMTNLGEKLTDEEVDEMIREAGIDGDGQVNYEQFVQMMTAK
5XUG , Knot 81 162 0.84 38 129 158 
ADRGTETVPGLGQRKQQILNSGGGVWDLAIAMLETKNLGTDYVYGDGKTYDSANFGIFKQNWFMLRTSTSQFKGQTTNQWNNGAVLNSNLQQDIKARQESQNYYGPDKWFAGHRNGESGLSNPYTQDITNYKDAVNWIHDQLASDPKYLSDDTRFWVDVTAI
1VHM , Knot 92 195 0.83 40 147 187 
MSLLILIGWQIQVRQPSDYIMNKTEFYADLNRDFNALMAGETSFLATLANTSALLYERLTDINWAGFYLLEDDTLVLGPFQGKIACVRIPVGRGVCGTAVARNQVQRIEDVHVFDGHIACDAASNSEIVLPLVVKNQIIGVLDIDSTVFGRFTDEDEQGLRQLVAQLEKVLATTDYKKFFASVAGEGGSHHHHHH

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 \(c\).

\(|P_{f(3EKJ_1)}(2) \setminus P_{f(5XUG_1)}(2)|=144\), \(|P_{f(5XUG_1)}(2) \setminus P_{f(3EKJ_1)}(2)|=39\). 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 1:1010000000111010110011001000000011011000000100010110111010010010111000001101010100010011101100000001110111111000010000010001000000111101101111011100100110110110010011011111110101010100101010101010010101011000101111110110010011001000100100001100111010100001110001000001010101001100101011010001011100100000001000011010011011000101110000110110011001001010011001110101010110110111001000000001001101110010101011010011001100100001001100111010101000011011010
Pair \(Z_2\) Length of longest common subsequence
3EKJ_1,5XUG_1 183 3
3EKJ_1,1VHM_1 185 8
5XUG_1,1VHM_1 160 3

Newick tree

 
[
	3EKJ_1:95.66,
	[
		5XUG_1:80,1VHM_1:80
	]:15.66
]

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 \((0.85)(0.8)(\frac{613 }{\log_{20} 613}-\frac{162}{\log_{20}162})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 3EKJ_1 5XUG_1 160 107
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]