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Parikh vectors
7JVF_1 1AEN_1 3FEV_1 Letter Amino acid
2 1 8 C Cysteine
0 9 2 Q Glutamine
5 26 4 G Glycine
0 14 3 Y Tyrosine
1 10 4 R Arginine
0 19 4 N Asparagine
3 25 2 L Leucine
0 18 3 F Phenylalanine
1 15 3 P Proline
2 14 7 T Threonine
2 17 3 A Alanine
0 25 4 D Aspartic acid
0 20 2 E Glutamic acid
0 6 0 H Histidine
1 14 2 V Valine
1 6 1 W Tryptophan
1 10 4 I Isoleucine
0 24 5 K Lycine
1 6 1 M Methionine
0 15 3 S Serine 

7JVF_1|Chain A|Prochlorosin 2.10|Prochlorococcus marinus str. MIT 9313 (74547)
>1AEN_1|Chain A|CYTOCHROME C PEROXIDASE|Saccharomyces cerevisiae (4932)
>3FEV_1|Chains A, B, C|Fusion of Muscarinic toxin 1, Muscarinic m1-toxin1|Dendroaspis angusticeps (8618)
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 \(w=f(c)\)
7JVF , Knot 14 21 0.67 24 20 19 
AGGTIPALMXGCGWLTGLCVR
1AEN , Knot 133 294 0.85 40 194 287 
MKTLVHVASVEKGRSYEDFQKVYNAIALKLREDDEYDNYIGYGPVLVRLAWHISGTWDKHDNTGGSYGGTYRFKKEFNDPSNAGLQNGFKFLEPIHKEFPWISSGDLFSLGGVTAVQEMQGPKIPWRCGRVDTPEDTTPDNGRLPDADKDAGYVRTFFQRLNMNDREVVALMGAHALGKTHLKNSGYEGPGGAANNVFTNEFYLNLLNEDWKLEKNDANNEQWDSKSGYMMLPTDYSLIQDPKYLSIVKEYANDQDKFFKDFSKAFEKLLEDGITFPKDAPSPFIFKTLEEQGL
3FEV , Knot 38 65 0.81 38 61 63 
LTCVTSKSIFGITTENCPDGQNLCFKRWQYISPRMYDFTRGCAATCPKAEYRDVINCCGTDKCNK

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 \(c\).

\(|P_{f(7JVF_1)}(2) \setminus P_{f(1AEN_1)}(2)|=10\), \(|P_{f(1AEN_1)}(2) \setminus P_{f(7JVF_1)}(2)|=184\). 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 1:111011111010111011010
Pair \(Z_2\) Length of longest common subsequence
7JVF_1,1AEN_1 194 3
7JVF_1,3FEV_1 69 2
1AEN_1,3FEV_1 199 3

Newick tree

 
[
	1AEN_1:11.69,
	[
		7JVF_1:34.5,3FEV_1:34.5
	]:77.19
]

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 \((0.85)(0.8)(\frac{315 }{\log_{20} 315}-\frac{21}{\log_{20}21})=97.4\)
Status Protein1 Protein2 d d1/2
Query variables 7JVF_1 1AEN_1 127 68
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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