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Parikh vectors
1MYT_1 3ZXL_1 2EDX_1 Letter Amino acid
14 56 14 G Glycine
5 31 13 P Proline
8 37 18 V Valine
2 43 10 R Arginine
1 2 1 C Cysteine
3 14 3 Q Glutamine
6 15 3 H Histidine
10 22 3 I Isoleucine
15 11 3 K Lycine
8 17 9 E Glutamic acid
6 23 0 F Phenylalanine
7 47 9 T Threonine
1 17 4 W Tryptophan
21 48 7 A Alanine
6 26 1 N Asparagine
6 34 8 D Aspartic acid
17 33 3 L Leucine
3 9 2 M Methionine
5 29 20 S Serine
2 28 3 Y Tyrosine 

1MYT_1|Chain A|MYOGLOBIN|Thunnus albacares (8236)
>3ZXL_1|Chains A, B|HIAXHD3|HUMICOLA INSOLENS (85995)
>2EDX_1|Chain A|Protein tyrosine phosphatase, receptor type, F|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 \(w=f(c)\)
1MYT , Knot 72 146 0.82 40 107 141 
ADFDAVLKCWGPVEADYTTMGGLVLTRLFKEHPETQKLFPKFAGIAQADIAGNAAISAHGATVLKKLGELLKAKGSHAAILKPLANSHATKHKIPINNFKLISEVLVKVMHEKAGLDAGGQTALRNVMGIIIADLEANYKELGFSG
3ZXL , Knot 221 542 0.85 40 253 502 
MPRQASTFTNPVLWEDHPDLEVFRVGSVFYYSSSTFAYSPGAPVLKSYDLVHWTPVTHSVPRLNFGSNYDLPSGTPGAYVKGIWASTLRYRRSNDRFYWYGCVEGRTYLWTSPGGNALANNGEVPPSAWNWQHTATIDNCYADAGLLIDDDDTMYIAYGNPTINVAQLSPDGTRQVRVQQRVYAHPQGQTVAGARMYKIRGNYYILVTRPADAEYVLRSTTGSPFGPYEARTLVSRIQGPLANAGFAHQGGIVDAPDGTWHYVAFMDAYPGGRIPVVAPLRWTADGWPEVVTDSQGRWGTSYPIPVRGAKNATEGLASTDLDEFRGTRFSEHWEWNHNPDTSKFTLLGGNEGGLILRTATVTGDLFAARNTLTRRIAGPKASGIFRLDVRGMRDGDRAGAVLFRDRAAYIGVWKQGNEARIVMVDDLRLNEDGWRTASTGRVAANGPVIDTNAQQDIWLRIDADITPAFGTNTERTTTFYYSIDGGRTYTRLGPAFAMTNSWRYFTGYRFGVFNFSTKSLGGEVKVKGFKMNMILEHHHHHH
2EDX , Knot 65 134 0.79 38 95 127 
GSSGSSGTIEARTAQSTPSAPPQKVMCVSMGSTTVRVSWVPPPADSRNGVITQYSVAYEAVDGEDRGRHVVDGISREHSSWDLVGLEKWTEYRVWVRAHTDVGPGPESSPVLVRTDEDVPSGPPRKVESGPSSG

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 \(c\).

\(|P_{f(1MYT_1)}(2) \setminus P_{f(3ZXL_1)}(2)|=26\), \(|P_{f(3ZXL_1)}(2) \setminus P_{f(1MYT_1)}(2)|=172\). 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 1:10101110011110100001111110011000100001110111110101110111010110110011011010100111101110001000011100101100111011000111011100110011111110101000011101
Pair \(Z_2\) Length of longest common subsequence
1MYT_1,3ZXL_1 198 4
1MYT_1,2EDX_1 140 3
3ZXL_1,2EDX_1 194 4

Newick tree

 
[
	3ZXL_1:10.70,
	[
		1MYT_1:70,2EDX_1:70
	]:35.70
]

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 \((0.85)(0.8)(\frac{688 }{\log_{20} 688}-\frac{146}{\log_{20}146})=154.\)
Status Protein1 Protein2 d d1/2
Query variables 1MYT_1 3ZXL_1 194 121.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]