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Parikh vectors
1WWR_1 9JOF_1 7MXI_1 Letter Amino acid
6 35 9 N Asparagine
4 1 5 C Cysteine
0 16 11 Q Glutamine
10 28 17 G Glycine
11 25 6 I Isoleucine
15 26 10 K Lycine
4 12 3 M Methionine
6 17 9 F Phenylalanine
6 11 19 P Proline
13 33 21 S Serine
4 25 22 T Threonine
8 10 19 R Arginine
11 10 14 H Histidine
12 27 18 A Alanine
3 20 8 D Aspartic acid
20 38 15 E Glutamic acid
18 39 15 L Leucine
1 6 5 W Tryptophan
6 24 4 Y Tyrosine
13 20 17 V Valine 

1WWR_1|Chains A, B, C, D|tRNA adenosine deaminase TadA|Aquifex aeolicus (63363)
>9JOF_1|Chains A, B|endo-1.3-fucanase|Wenyingzhuangia fucanilytica (1790137)
>7MXI_1|Chains A, B|IgE Fc|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 \(w=f(c)\)
1WWR , Knot 80 171 0.80 38 127 166 
MGSSHHHHHHSSGLVPRGSHMGKEYFLKVALREAKRAFEKGEVPVGAIIVKEGEIISKAHNSVEELKDPTAHAEMLAIKEACRRLNTKYLEGCELYVTLEPCIMCSYALVLSRIEKVIFSALDKKHGGVVSVFNILDEPTLNHRVKWEYYPLEEASELLSEFFKKLRNNII
9JOF , Knot 172 423 0.82 40 215 398 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSSTKNGIIELEQETEEELEQNQEIDYSNYPEFSWDTMPLYMHVRKNTAYTDEEINYLASFPLITLEKSQAQNTYGSTEEGTLATASAIKLKNNKAKVLYYRNVVINWGNYKNDDEFISKNPSALLKNQNNELVYMPNGSTPFFDITKSFVQEYWLKSVEDMVATPNIDGTFIDANIKVLVPSFFSSKVGVNKQAEIENSYFSMMSRLKESLSNNLILANIIRVRPEFEENGLEYLGYFNGSYLEGFDSEAFGMSNAEYLVEGIEATQKAAQSGKIITMTLGLGEAIDNNTGIDDQREDVDLNDEELNKRVDYLLAIFLICAEKYSYVYLHDGYLATNSAVWLHQFDQYKKALGAPLGKAIKNGYIYTRKFENLDVWLNLETQTATLTWKE
7MXI , Knot 107 247 0.79 40 163 236 
APMAEGGGQNHHHHHHHHGGENLYFQGGSCADSNPRGVSAYLSRPSPFDLFIRKSPTITCLVVDLAPSKGTVNLTWSRASGKPVNHSTRKEEKQRNGTLTVTSTLPVGTRDWIEGETYQCRVTHPHLPRALMRSTTKTSGPRAAPEVYAFATPEWPGSRDKRTLACLIQNFMPEDISVQWLHNEVQLPDARHSTTQPRKTKGSGFFVFSRLEVTRAEWEQKDEFICRAVHEAASPSQTVQRAVSVNP

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 \(c\).

\(|P_{f(1WWR_1)}(2) \setminus P_{f(9JOF_1)}(2)|=39\), \(|P_{f(9JOF_1)}(2) \setminus P_{f(1WWR_1)}(2)|=127\). 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 1:110000000000111101001100011011100100110010111111110010110010001001001010101111001000100001010010101010110001111001001110110000111101101100101000101000110010011001100100011
Pair \(Z_2\) Length of longest common subsequence
1WWR_1,9JOF_1 166 21
1WWR_1,7MXI_1 168 6
9JOF_1,7MXI_1 178 6

Newick tree

 
[
	7MXI_1:87.68,
	[
		1WWR_1:83,9JOF_1:83
	]:4.68
]

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 \((0.85)(0.8)(\frac{594 }{\log_{20} 594}-\frac{171}{\log_{20}171})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 1WWR_1 9JOF_1 147 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]