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Parikh vectors
8PWD_1 3UIL_1 9DFM_1 Letter Amino acid
20 8 27 E Glutamic acid
13 16 24 G Glycine
39 12 40 L Leucine
11 2 7 M Methionine
16 16 18 A Alanine
15 16 18 R Arginine
21 6 15 D Aspartic acid
17 7 13 Q Glutamine
14 12 21 P Proline
7 6 16 T Threonine
11 2 20 F Phenylalanine
8 8 12 Y Tyrosine
1 5 2 W Tryptophan
4 6 1 C Cysteine
15 8 29 I Isoleucine
14 2 16 K Lycine
39 11 35 S Serine
7 9 19 N Asparagine
13 6 13 H Histidine
17 13 24 V Valine 

8PWD_1|Chain A|Vitamin D3 receptor A|Danio rerio (7955)
>3UIL_1|Chains A, B, C, D|Peptidoglycan recognition protein 1|Camelus dromedarius (9838)
>9DFM_1|Chains A, B|PrnB|Flavobacteriales bacterium (2021391)
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 \(w=f(c)\)
8PWD , Knot 133 302 0.83 40 181 284 
GSHMLSDEQMQIINSLVEAHHKTYDDSYSDFVRFRPPVREGPVTRSASRAASLHSLSDASSDSFNHSPESVDTKLNFSNLLMMYQDSGSPDSSEEDQQSRLSMLPHLADLVSYSIQKVIGFAKMIPGFRDLTAEDQIALLKSSAIEIIMLRSNQSFSLEDMSWSCGGPDFKYCINDVTKAGHTLELLEPLVKFQVGLKKLKLHEEEHVLLMAICLLSPDRPGVQDHVRIEALQDRLCDVLQAYIRIQHPGGRLLYAKMIQKLADLRSLNEEHSKQYRSLSFQPEHSMQLTPLVLEVFGSEVS
3UIL , Knot 84 171 0.84 40 125 165 
EDPPACGSIVPRREWRALASECRERLTRPVRYVVVSHTAGSHCDTPASCAQQAQNVQSYHVRNLGWCDVGYNFLIGEDGLVYEGRGWNIKGAHAGPTWNPISIGISFMGNYMNRVPPPRALRAAQNLLACGVALGALRSNYEVKGHRDVQPTLSPGDRLYEIIQTWSHYRA
9DFM , Knot 151 370 0.80 40 204 347 
MGSSHHHHHHSSGENLYFQGMISNEISKLDPLNLDAFFNQLPSLNQNLEVSLLIDKLREITKSYLPTTFSINDALAATRDLGMIMSSVRKLGIQPVSAVSDLEVFLETLSEITNMVPRETSYHYGPWNPIGERERRFTHFPDERGLIEGVRIAIPGIELAIREINQLSNLSLNDPAFESLAKSAALHVYQAVDGIGETIKKTDPYVFSHELRPFFDPIRIGGKSYIGAGGGQIPLFVVDVKLWLGNHSPNSEYVSFIKDSVFYLPPELRPICVDSLLEPSVINQKFAEFGSVEITDQVIKGMESLLSVIQVLLKFRKPHFQLAQRTLSKENRGNYTTGSAGYTNSFNHMVLEFTIEVEKQIRAVLAPYRS

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 \(c\).

\(|P_{f(8PWD_1)}(2) \setminus P_{f(3UIL_1)}(2)|=115\), \(|P_{f(3UIL_1)}(2) \setminus P_{f(8PWD_1)}(2)|=59\). 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 1:10011000010110011010000000000011010111001110001001101001001000010001001000101001111000010100000000001011101101100010011111011111001010001111000110111100000101001010011101000100100110010110111010111001010000011111101101001110001010110001001101010100111011010110011010010000000000101010001010111101110010
Pair \(Z_2\) Length of longest common subsequence
8PWD_1,3UIL_1 174 4
8PWD_1,9DFM_1 153 4
3UIL_1,9DFM_1 177 4

Newick tree

 
[
	3UIL_1:91.19,
	[
		8PWD_1:76.5,9DFM_1:76.5
	]:14.69
]

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 \((0.85)(0.8)(\frac{473 }{\log_{20} 473}-\frac{171}{\log_{20}171})=88.6\)
Status Protein1 Protein2 d d1/2
Query variables 8PWD_1 3UIL_1 113 87
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]