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Parikh vectors
1EEK_1 4ORK_1 1TPL_1 Letter Amino acid
0 24 32 L Leucine
0 14 21 F Phenylalanine
0 3 13 P Proline
0 15 18 S Serine
3 18 20 T Threonine
0 26 23 Y Tyrosine
1 24 18 N Asparagine
0 35 36 E Glutamic acid
0 6 20 M Methionine
4 5 6 C Cysteine
0 24 25 D Aspartic acid
0 5 17 Q Glutamine
0 4 14 H Histidine
0 0 1 W Tryptophan
0 9 32 V Valine
2 8 41 A Alanine
2 16 37 G Glycine
0 35 29 I Isoleucine
0 25 27 K Lycine
0 9 26 R Arginine 

1EEK_1|Chain A[auth 1]|5'-D(*CP*GP*CP*AP*TP*(DFT)P*GP*TP*TP*AP*CP*C)-3'|
>4ORK_1|Chains A, B, C, D|Bifunctional AAC/APH|Staphylococcus aureus (1280)
>1TPL_1|Chains A, B|TYROSINE PHENOL-LYASE|Citrobacter intermedius (66695)
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 \(w=f(c)\)
1EEK , Knot 8 12 0.55 10 11 10 
CGCATNGTTACC
4ORK , Knot 127 305 0.79 38 171 285 
MEYRYDDNATNVKAMKYLIEHYFDNFKVDSIEIIGSGYDSVAYLVNNEYIFKTKFSTNKKKGYAKEKAIYNFLNTNLETNVKIPNIEYSYISDELSILGYKEIKGTFLTPEIYSTMSEEEQNLLKRDIASFLRQMHGLDYTDISECTIDNKQNVLEEYILLRETIYNDLTDIEKDYIESFMERLNATTVFEGKKCLCHNDFSCNHLLLDGNNRLTGIIDFGDSGIIDEYCDFIYLLEDSEEEIGTNFGEDILRMYGNIDIEKAKEYQDIVEEYYPIETIVYGIKNIKQEFIENGRKEIYKRTYKD
1TPL , Knot 193 456 0.86 40 248 435 
MNYPAEPFRIKSVETVSMIPRDERLKKMQEAGYNTFLLNSKDIYIDLLTDSGTNAMSDKQWAGMMMGDEAYAGSENFYHLERTVQELFGFKHIVPTHQGRGAENLLSQLAIKPGQYVAGNMYFTTTRYHQEKNGAVFVDIVRDEAHDAGLNIAFKGDIDLKKLQKLIDEKGAENIAYICLAVTVNLAGGQPVSMANMRAVRELTAAHGIKVFYDATRCVENAYFIKEQEQGFENKSIAEIVHEMFSYADGCTMSGKKDCLVNIGGFLCMNDDEMFSSAKELVVVYEGMPSYGGLAGRDMEAMAIGLREAMQYEYIEHRVKQVRYLGDKLKAAGVPIVEPVGGHAVFLDARRFCEHLTQDEFPAQSLAASIYVETGVRSMERGIISAGRNNVTGEHHRPKLETVRLTIPRRVYTYAHMDVVADGIIKLYQHKEDIRGLKFIYEPKQLRFFTARFDYI

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 \(c\).

\(|P_{f(1EEK_1)}(2) \setminus P_{f(4ORK_1)}(2)|=6\), \(|P_{f(4ORK_1)}(2) \setminus P_{f(1EEK_1)}(2)|=166\). 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 1:010100100100
Pair \(Z_2\) Length of longest common subsequence
1EEK_1,4ORK_1 172 3
1EEK_1,1TPL_1 245 2
4ORK_1,1TPL_1 159 3

Newick tree

 
[
	1EEK_1:11.26,
	[
		4ORK_1:79.5,1TPL_1:79.5
	]:33.76
]

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 \((0.85)(0.8)(\frac{317 }{\log_{20} 317}-\frac{12}{\log_{20}12})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 1EEK_1 4ORK_1 125 65
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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