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Parikh vectors
1WQE_1 9EKQ_1 1CWZ_1 Letter Amino acid
0 10 2 R Arginine
1 37 0 N Asparagine
2 8 0 Q Glutamine
0 9 0 M Methionine
1 13 0 P Proline
2 22 0 A Alanine
4 0 1 C Cysteine
4 23 1 E Glutamic acid
0 28 0 L Leucine
0 40 0 S Serine
0 1 0 W Tryptophan
1 10 0 H Histidine
1 36 1 I Isoleucine
1 30 0 K Lycine
0 14 0 F Phenylalanine
2 25 0 V Valine
2 34 0 D Aspartic acid
1 27 3 G Glycine
1 18 0 T Threonine
0 17 0 Y Tyrosine 

1WQE_1|Chain A|OmTx3|Opisthacanthus madagascariensis (167108)
>9EKQ_1|Chain A|Lantibiotic leader peptide processing serine protease CylA|Enterococcus faecalis (1351)
>1CWZ_1|Chain A|HISTONE H3|null
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 \(w=f(c)\)
1WQE , Knot 16 23 0.72 26 19 20 
NDPCEEVCIQHTGDVKACEEACQ
9EKQ , Knot 165 402 0.82 38 204 382 
MGSSHHHHHHSQDPNSLSNNISFFIDNSQTTAIEEIESELSSEKVDYIQEIGLVSFKNLDDSDRKFIGKYFNVSEGKKLPDFKPEEVNSSILNINILNKDFKSFNWPYKKILSHIDPVKEQLGKDITIALIDSGIDRLHPNLQDNNLRLKNYVNDIELDEYGHGTQVAGVIDTIAPRVNLNSYKVMDGTDGNSINMLKAIVDATNDQVDIINVSLGSYKNMEIDDERFTVEAFRKVVNYARKNNILIVASAGNESRDISTGNEKHIPGGLESVITVGATKKSGDIADYSNYGSNVSIYGPAGGYGDNYKITGQIDAREMMMTYYPTSLVSPLGKAADFPDGYTLSFGTSLATPEVSAALAAIMSKNVDNSKDSNEVLNTLFENADSFIDKNSMLKYKEVRIK
1CWZ , Knot 7 9 0.57 12 8 7 
REGRIGGCX

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 \(c\).

\(|P_{f(1WQE_1)}(2) \setminus P_{f(9EKQ_1)}(2)|=9\), \(|P_{f(9EKQ_1)}(2) \setminus P_{f(1WQE_1)}(2)|=194\). 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 1:00100010100010101000100
Pair \(Z_2\) Length of longest common subsequence
1WQE_1,9EKQ_1 203 3
1WQE_1,1CWZ_1 27 1
9EKQ_1,1CWZ_1 202 2

Newick tree

 
[
	9EKQ_1:11.65,
	[
		1WQE_1:13.5,1CWZ_1:13.5
	]:10.15
]

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 \((0.85)(0.8)(\frac{425 }{\log_{20} 425}-\frac{23}{\log_{20}23})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 1WQE_1 9EKQ_1 159 83
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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