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Parikh vectors
5CPL_1 4QGA_1 1WFL_1 Letter Amino acid
2 1 6 C Cysteine
24 23 4 E Glutamic acid
30 14 9 G Glycine
16 6 3 F Phenylalanine
23 6 6 P Proline
20 12 3 D Aspartic acid
18 4 2 H Histidine
6 4 1 M Methionine
15 9 1 T Threonine
11 0 0 W Tryptophan
22 18 1 V Valine
9 10 3 N Asparagine
30 16 4 L Leucine
17 9 12 S Serine
9 10 3 Y Tyrosine
17 19 0 I Isoleucine
27 14 4 R Arginine
11 7 1 Q Glutamine
9 12 8 K Lycine
55 11 3 A Alanine 

5CPL_1|Chains A, B|Xenobiotic reductase|Pseudomonas putida (303)
>4QGA_1|Chains A, B|Thymidylate kinase|Staphylococcus aureus subsp. aureus (282458)
>1WFL_1|Chain A|Zinc finger protein 216|Mus musculus (10090)
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 \(w=f(c)\)
5CPL , Knot 160 371 0.85 40 212 357 
MSALFEPYTLKDVTLRNRIAIPPMCQYMAEDGLINDWHQVHYASMARGGAGLLVVEATAVAPEGRITPGCAGIWSDAHAQAFVPVVQAIKAAGSVPGIQIAHAGRKASANRPWEGDDHIGADDARGWETIAPSAIAFGAHLPNVPRAMTLDDIARVKQDFVDAARRARDAGFEWIELHFAHGYLGQSFFSEHSNKRTDAYGGSFDNRSRFLLETLAAVREVWPENLPLTARFGVLEYDGRDEQTLEESIELARRFKAGGLDLLSVSVGFTIPETNIPWGPAFMGPIAERVRREAKLPVTSAWGFGTPQLAEAALQANQLDLVSVGRAHLADPHWAYFAAKELGVEKASWTLPAPYAHWLERYRRPHHHHHH
4QGA , Knot 98 205 0.84 38 152 201 
MSAFITFEGPEGSGKTTVINEVYHRLVKDYDVIMTREPGGVPTGEEIRKIVLEGNDMDIRTEAMLFAASRREHLVLKVIPALKEGKVVLCDRYIDSSLAYQGYARGIGVEEVRALNEFAINGLYPDLTIYLNVSAEVGRERIIKNSRDQNRLDQEDLKFHEKVIEGYQEIIHNESQRFKSVNADQPLENVVEDTYQTIIKYLEKI
1WFL , Knot 38 74 0.73 36 56 66 
GSSGSSGPSSSQSEEKAPELPKPKKNRCFMCRKKVGLTGFDCRCGNLFCGLHRYSDKHNCPYDYKAEASGPSSG

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 \(c\).

\(|P_{f(5CPL_1)}(2) \setminus P_{f(4QGA_1)}(2)|=126\), \(|P_{f(4QGA_1)}(2) \setminus P_{f(5CPL_1)}(2)|=66\). 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 1:10111010010010100011111100011001110010010010110111111110101111010101101111001010111111011011101111011011001010011010001110010110011101111110110110110100110100011011001001110110101101011001100000000010110100000111001111001110011101011110001000001000101100101111011010111011000111111111111001000101110011111010110111010010110110101101011011100111001010111101011000001000000
Pair \(Z_2\) Length of longest common subsequence
5CPL_1,4QGA_1 192 3
5CPL_1,1WFL_1 206 3
4QGA_1,1WFL_1 156 2

Newick tree

 
[
	5CPL_1:10.77,
	[
		4QGA_1:78,1WFL_1:78
	]:27.77
]

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 \((0.85)(0.8)(\frac{576 }{\log_{20} 576}-\frac{205}{\log_{20}205})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 5CPL_1 4QGA_1 135 105
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]