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Parikh vectors
4FRA_1 4DLJ_1 2VII_1 Letter Amino acid
22 19 22 R Arginine
17 17 15 P Proline
13 15 7 Y Tyrosine
13 19 8 K Lycine
10 10 7 M Methionine
31 22 15 V Valine
14 27 14 D Aspartic acid
4 4 3 C Cysteine
18 23 24 E Glutamic acid
10 12 5 H Histidine
28 43 36 L Leucine
7 14 11 N Asparagine
15 15 8 Q Glutamine
15 16 18 G Glycine
16 19 9 T Threonine
6 5 2 W Tryptophan
16 25 20 A Alanine
9 22 12 I Isoleucine
18 13 12 F Phenylalanine
12 20 11 S Serine 

4FRA_1|Chain A|Histo-blood group ABO system transferase|Homo sapiens (9606)
>4DLJ_1|Chain A|Mitogen-activated protein kinase 14|Homo sapiens (9606)
>2VII_1|Chain A|PSP OPERON TRANSCRIPTIONAL ACTIVATOR|ESCHERICHIA COLI (83333)
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 \(w=f(c)\)
4FRA , Knot 132 294 0.85 40 198 286 
FMVSLPRMVYPQPKVLTPCRKDVLVVTPWLAPIVWEGTFNIDILNEQFRLQNTTIGLTVFAIKKYVAFLKLFLETAEKHFMVGHRVHYYVFTDQPAAVPRVTLGTGRQLSVLEVRAYKRWQDVSMRRMEMISDFCERRFLSEVDYLVCVDVDMEFRDHVGVEILTPLFGTLHPSFYGSSREAFTYERRPQSQAYIPKDEGDFYYMGGFFGGSVQEVQRLTRACHQAMMVDQANGIEAVWHDESHLNKYLLRHKPTKVLSPEYLWDQQLLGWPAVLRKLRFTAVPKNHQAVRNPE
4DLJ , Knot 158 360 0.86 40 223 346 
GSQERPTFYRQELNKTIWEVPERYQNLSPVGSGAYGSVCAAFDTKTGLRVAVKKLSRPFQSIIHAKRTYRELRLLKHMKHENVIGLLDVFTPARSLEEFNDVYLVTHLMGADLNNIVKCQKLTDDHVQFLIYQILRGLKYIHSADIIHRDLKPSNLAVNEDCELKILDFGLARHTDDEMTGYVATRWYRAPEIMLNWMHYNQTVDIWSVGCIMAELLTGRTLFPGTDHIDQLKLILRLVGTPGAELLKKISSESARNYIQSLTQMPKMNFANVFIGANPLAVDLLEKMLVLDSDKRITAAQALAHAYFAQYHDPDDEPVADPYDQSFESRDLLIDEWKSLTYDEVISFVPPPLDQEEMES
2VII , Knot 118 259 0.84 40 171 249 
MAEYKDNLLGEANSFLEVLEQVSHLAPLDKPVLIIGERGTGKELIASRLHYLSSRWQGPFISLNCAALNENLLDSELFGHEAGAFTGAQKRHPGRFERADGGTLFLDELATAPMMVQEKLLRVIEYGELERVGGSQPLQVNVRLVCATNADLPAMVNEGTFRADLLDRLAFDVVQLPPLRERESDIMLMAEYFAIQMCREIKLPLFPGFTERARETLLNYRWPGNIRELKNVVERSVYRHGTSDYPLDDIIIDPFKRRP

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 \(c\).

\(|P_{f(4FRA_1)}(2) \setminus P_{f(4DLJ_1)}(2)|=78\), \(|P_{f(4DLJ_1)}(2) \setminus P_{f(4FRA_1)}(2)|=103\). 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 1:111011011010101101000011110111111110101010110001010000111011110001111011100100011110010001100011111010110100101101010001001010010110010000110010011010101010001110110111101010101000011000001000101100010100111111101001001001000111100101101110000010001100010011010011000111111110010101110000110010
Pair \(Z_2\) Length of longest common subsequence
4FRA_1,4DLJ_1 181 3
4FRA_1,2VII_1 159 4
4DLJ_1,2VII_1 172 4

Newick tree

 
[
	4DLJ_1:91.01,
	[
		4FRA_1:79.5,2VII_1:79.5
	]:11.51
]

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 \((0.85)(0.8)(\frac{654 }{\log_{20} 654}-\frac{294}{\log_{20}294})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 4FRA_1 4DLJ_1 128 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]