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Parikh vectors
4FNN_1 6CPI_1 3MJC_1 Letter Amino acid
16 9 54 G Glycine
11 6 25 S Serine
5 2 10 W Tryptophan
13 6 35 V Valine
7 3 12 Q Glutamine
12 3 67 L Leucine
2 5 3 K Lycine
2 2 8 M Methionine
2 3 7 F Phenylalanine
6 1 2 C Cysteine
6 0 23 H Histidine
8 3 8 I Isoleucine
6 0 26 T Threonine
16 4 91 A Alanine
16 2 33 R Arginine
9 1 4 N Asparagine
6 1 33 D Aspartic acid
8 7 27 E Glutamic acid
12 2 25 P Proline
8 1 3 Y Tyrosine 

4FNN_1|Chains A, B, C, D|Peptidoglycan recognition protein 1|Camelus dromedarius (9838)
>6CPI_1|Chain A|SH3 and multiple ankyrin repeat domains protein 1|Homo sapiens (9606)
>3MJC_1|Chains A, B|AmphB|Streptomyces nodosus (40318)
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 \(w=f(c)\)
4FNN , Knot 84 171 0.84 40 125 165 
EDPPACGSIVPRREWRALASECRERLTRPVRYVVVSHTAGSHCDTPASCAQQAQNVQSYHVRNLGWCDVGYNFLIGEDGLVYEGRGWNIKGAHAGPTWNPISIGISFMGNYMNRVPPPRALRAAQNLLACGVALGALRSNYEVKGHRDVQPTLSPGDRLYEIIQTWSHYRA
6CPI , Knot 34 61 0.76 36 51 59 
MVPGRSFMAVKSYQAQAEGEISLSKGEKIKVLSIGEGGFWEGQVKGRVGWFPSDCLEEVAN
3MJC , Knot 187 496 0.78 40 185 423 
MGSSHHHHHHSSGLVPRGSHMDALRYHIEWNRVAEPGTARPAGRLLAVISPDHAGAPWVTAVLDALGPDTVRFEAKGTDRAAWAAQLAQLREDEGEFHAVVSLLAAAEALHTDHGSVPLGLAQTLLLAQALGDAGLTAPLWCLTRGGVAAGRGDVLSSPVQGALWGLGRVIGLEHPDRWGGLIDLPETVDTRAAARLTGLLADAGGEDQLAIRGSGVLARRLAHAAPAVPGSGKRPPVHGSVLVTGGTGGIGGRVARRLAEQGAAHLVLTSRRGADAPGAAELRAELEQLGVRVTIAACDAADREALAALLAELPEDAPLTAVFHSAGVAHDDAPVADLTLGQLDALMRAKLTAARHLHELTADLDLDAFVLFSSGAAVWGSGGQPGYAAANAYLDALAEHRRSLGLTASSVAWGTWGEVGMATDPEVHDRLVRQGVLAMEPEHALGALDQMLENDDTAAAITLMDWEMFAPAFTANRPSALLSTVPEAVSALSDE

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 \(c\).

\(|P_{f(4FNN_1)}(2) \setminus P_{f(6CPI_1)}(2)|=101\), \(|P_{f(6CPI_1)}(2) \setminus P_{f(4FNN_1)}(2)|=27\). 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 1:001110101110001011100000010011001110001100000110010010010000100111001100111100111001011010110111010110111011100100111101101100111011111110000010100010101011001001100100001
Pair \(Z_2\) Length of longest common subsequence
4FNN_1,6CPI_1 128 3
4FNN_1,3MJC_1 160 5
6CPI_1,3MJC_1 178 3

Newick tree

 
[
	3MJC_1:90.45,
	[
		4FNN_1:64,6CPI_1:64
	]:26.45
]

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 \((0.85)(0.8)(\frac{232 }{\log_{20} 232}-\frac{61}{\log_{20}61})=56.5\)
Status Protein1 Protein2 d d1/2
Query variables 4FNN_1 6CPI_1 71 47
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]