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Parikh vectors
4CKJ_1 7OVZ_1 8EXD_1 Letter Amino acid
3 0 8 C Cysteine
9 0 33 Q Glutamine
22 1 47 E Glutamic acid
26 5 69 G Glycine
9 3 42 T Threonine
21 1 42 V Valine
21 0 42 R Arginine
7 0 27 H Histidine
24 4 32 K Lycine
16 0 46 P Proline
17 0 57 D Aspartic acid
13 2 41 I Isoleucine
24 1 51 S Serine
6 0 22 W Tryptophan
16 4 49 A Alanine
9 1 27 N Asparagine
37 4 56 L Leucine
11 0 14 M Methionine
12 0 40 F Phenylalanine
11 1 34 Y Tyrosine 

4CKJ_1|Chain A|PROTO-ONCOGENE TYROSINE-PROTEIN KINASE RECEPTOR RET|HOMO SAPIENS (9606)
>7OVZ_1|Chain A[auth M]|Maximin-1|Bombina maxima (161274)
>8EXD_1|Chains A, B, C, D|Sterylglucosidase A (SglA)|Aspergillus fumigatus Af293 (330879)
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 \(w=f(c)\)
4CKJ , Knot 137 314 0.83 40 201 306 
GPLSLSVDAFKILEDPKWEFPRKNLVLGKTLGEGEFGKVVKATAFHLKGRAGYTTVAVKMLKENASPSELRDLLSEFNVLKQVNHPHVIKLYGACSQDGPLLLIVEYAKYGSLRGFLRESRKVGPGYLGSGGSRNSSSLDHPDERALTMGDLISFAWQISQGMQYLAEMKLVHRDLAARNILVAEGRKMKISDFGLSRDVYEEDSYVKRSQGRIPVKWMAIESLFDHIYTTQSDVWSFGVLLWEIVTLGGNPYPGIPPERLFNLLKTGHRMERPDNCSEEMYRLMLQCWKQEPDKRPVFADISKDLEKMMVKRR
7OVZ , Knot 16 27 0.65 22 24 24 
GIGTKILGGVKTALKGALKELASTYAN
8EXD , Knot 301 779 0.85 40 308 725 
MGSSHHHHHHSSGLVPRGSHMGVLRLRVDGQTFRDPDNREITLRGINVAGEAKYPKKPDIPSYVSDGFFDADNVSFVGRPFSLDDAHTHFSRLRKWGYNTIRYVFTWEAIEHEGPGKYDDEWISFTIEVLRIAKQYGFYVFLDPHQDVWSRLSGGSGAPAWTLYAAGFDPRGFKKTEAALVQNTFDDPAEFPKMIWSTNYTRLVSQTMFTLFYGGRDFAPKAIIDGINIQDYLQGHFIAACRYFAQKIHEAGDIENEVVIGWESMNEPNRGLIGVQDISVIPPEQQLQLGTSPTAFQGMLTGSGRACEESTWAFGGFGPYQTGRELVDPEGETAWLPADYDDTRYGWVRDPGWKLGECLWAQHGVWDPSSDKLLRKDYFAKNPQTGEPLNYDKFTNTYFMEHYRAYRDALRSVWPEAIIFCQPPVMEVPPDLKGTVDDDPNMVHAVHYYDGITLLTKHWNRLYNVDVIGVLRGKYLTPAFAVKIGETAIRNCLRDQLKFLRDESLRYMGTHPLIFTEIGIPFDMDDRHAYKTGDYSGQVSAMDANHFAIEGSTANGFTLWLYTTSNNHEWGDNWNGEDLSIYSVDDLELPSGKLLAFENESQRDPQSPAYSESQRNTESYRVGPRDLKQALQAPSISSEISQSSQDKLGFRAAEAWVRPSPIITNGQILQYGFDLKSCVFSMRLLGEKKGLGQEAATEIFLPDFHFPDTHTVVAVSAGEWTIDYPEIHSVKFQRLRWWHPEGDHNIKIQGVKRKPGDPTVSGEELSYLEQCQGGGCSVM

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 \(c\).

\(|P_{f(4CKJ_1)}(2) \setminus P_{f(7OVZ_1)}(2)|=183\), \(|P_{f(7OVZ_1)}(2) \setminus P_{f(4CKJ_1)}(2)|=6\). 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 1:11101010110110010101100011110011010110110101101010110001110110001010010011001011001001011010110000111111100100101011100000111101101100000010010001101101101110100110011010110001110011110100101001110001000000100001011101111001100100000011011111101101110101111100110110010010010000001001110010001000111101000100111000
Pair \(Z_2\) Length of longest common subsequence
4CKJ_1,7OVZ_1 189 3
4CKJ_1,8EXD_1 179 4
7OVZ_1,8EXD_1 288 3

Newick tree

 
[
	7OVZ_1:13.79,
	[
		4CKJ_1:89.5,8EXD_1:89.5
	]:41.29
]

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 \((0.85)(0.8)(\frac{341 }{\log_{20} 341}-\frac{27}{\log_{20}27})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 4CKJ_1 7OVZ_1 126 67.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]