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Parikh vectors
6MGA_1 8WNT_1 6DAD_1 Letter Amino acid
33 63 11 A Alanine
25 18 5 N Asparagine
24 31 6 Q Glutamine
32 53 11 G Glycine
1 13 0 W Tryptophan
44 33 7 V Valine
35 40 21 E Glutamic acid
31 31 2 P Proline
29 63 4 S Serine
3 3 0 C Cysteine
10 11 1 H Histidine
38 90 9 L Leucine
16 29 8 K Lycine
11 23 8 F Phenylalanine
11 9 2 Y Tyrosine
22 24 6 R Arginine
38 38 17 D Aspartic acid
23 20 9 I Isoleucine
3 10 9 M Methionine
30 29 12 T Threonine 

6MGA_1|Chain A|Protocadherin-1|Homo sapiens (9606)
>8WNT_1|Chain A|Amino acid transporter heavy chain SLC3A2|Homo sapiens (9606)
>6DAD_1|Chains A, C[auth B]|Calmodulin-1|Homo sapiens (9606)
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 \(w=f(c)\)
6MGA , Knot 186 459 0.82 40 213 424 
TRVVYKVPEEQPPNTLIGSLAADYGFPDVGHLYKLEVGAPYLRVDGKTGDIFTTETSIDREGLRECQNQLPGDPCILEFEVSITDLVQNGSPRLLEGQIEVQDINDNTPNFASPVITLAIPENTNIGSLFPIPLASDRDAGPNGVASYELQAGPEAQELFGLQVAEDQEEKQPQLIVMGNLDRERWDSYDLTIKVQDGGSPPRASSALLRVTVLDTNDNAPKFERPSYEAELSENSPIGHSVIQVKANDSDQGANAEIEYTFHQAPEVVRRLLRLDRNTGLITVQGPVDREDLSTLRFSVLAKDRGTNPKSARAQVVVTVKDMNDNAPTIEIRGIGLVTHQDGMANISEDVAEETAVALVQVSDRDEGENAAVTCVVAGDVPFQLRQASETGSDSKKKYFLQTTTPLDYEKVKDYTIEIVAVDSGNPPLSSTNSLKVQVVDVNDNATDPPVGGHHHHHH
8WNT , Knot 239 631 0.81 40 249 547 
MELQPPEASIAVVSIPRQLPGSHSEAGVQGLSAGDDSETGSDCVTQAGLQLLASSDPPALASKNAEVTVETGFHHVSQADIEFLTSIDPTASASGSAGITGTMSQDTEVDMKEVELNELEPEKQPMNAASGAAMSLAGAEKNGLVKIKVAEDEAEAAAAAKFTGLSKEELLKVAGSPGWVRTRWALLLLFWLGWLGMLAGAVVIIVRAPRCRELPAQKWWHTGALYRIGDLQAFQGHGAGNLAGLKGRLDYLSSLKVKGLVLGPIHKNQKDDVAQTDLLQIDPNFGSKEDFDSLLQSAKKKSIRVILDLTPNYRGENSWFSTQVDTVATKVKDALEFWLQAGVDGFQVRDIENLKDASSFLAEWQNITKGFSEDRLLIAGTNSSDLQQILSLLESNKDLLLTSSYLSDSGSTGEHTKSLVTQYLNATGNRWCSWSLSQARLLTSFLPAQLLRLYQLMLFTLPGTPVFSYGDEIGLDAAALPGQPMEAPVMLWDESSFPDIPGAVSANMTVKGQSEDPGSLLSLFRRLSDQRSKERSLLHGDFHAFSAGPGLFSYIRHWDQNERFLVVLNFGDVGLSAGLQASDLPASASLPAKADLLLSTQPGREEGSPLELERLKLEPHEGLLLRFPYAA
6DAD , Knot 68 148 0.76 36 99 135 
ADQLTEEQIAEFKEAFSLFDKDGDGTITTKELGTVMRSLGQNPTEAELQDMINEVDADGNGTIDFPEFLTMMARKMKDTDSEEEIREAFRVFDKDGIGYISAAELRHVMTNLGEKLTDEEVDEMIREADIDGDGQVNYEEFVQMMTAK

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 \(c\).

\(|P_{f(6MGA_1)}(2) \setminus P_{f(8WNT_1)}(2)|=46\), \(|P_{f(8WNT_1)}(2) \setminus P_{f(6MGA_1)}(2)|=82\). 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 1:001100110001100111011100111011010010111101010100101100000100011000000111010110101010011001010110101010010000101101110111100001101111111000011101110001011101001111011000000010111110100001000010101001101101001110101100000110100100010100001110011010100000110101000100110110011010000111010111000010010101110001001001010111010010001101010111110000111010001100011111010000010011100111101110100100010000000110000110000100001011110010111000001010110100010011111000000
Pair \(Z_2\) Length of longest common subsequence
6MGA_1,8WNT_1 128 6
6MGA_1,6DAD_1 172 4
8WNT_1,6DAD_1 190 4

Newick tree

 
[
	6DAD_1:97.88,
	[
		6MGA_1:64,8WNT_1:64
	]:33.88
]

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 \((0.85)(0.8)(\frac{1090 }{\log_{20} 1090}-\frac{459}{\log_{20}459})=164.\)
Status Protein1 Protein2 d d1/2
Query variables 6MGA_1 8WNT_1 198 171
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]