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Parikh vectors
3ETQ_1 6WDH_1 7XEK_1 Letter Amino acid
8 5 11 M Methionine
7 17 14 P Proline
13 34 30 G Glycine
4 9 23 H Histidine
8 6 13 Q Glutamine
10 11 26 I Isoleucine
12 4 14 F Phenylalanine
14 10 16 S Serine
0 2 4 W Tryptophan
12 6 15 Y Tyrosine
17 29 14 R Arginine
0 2 3 C Cysteine
15 11 32 E Glutamic acid
13 26 35 V Valine
10 15 17 N Asparagine
12 10 25 D Aspartic acid
15 24 23 K Lycine
8 14 30 T Threonine
9 21 40 A Alanine
17 15 34 L Leucine 

3ETQ_1|Chains A, B|Potassium/sodium hyperpolarization-activated cyclic nucleotide-gated channel 2|Mus musculus (10090)
>6WDH_1|Chain A[auth b]|50S ribosomal protein L2|Escherichia coli (562)
>7XEK_1|Chain A|Cysteine desulfurase SufS|Bacillus subtilis subsp. subtilis str. 168 (224308)
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 \(w=f(c)\)
3ETQ , Knot 99 204 0.86 36 155 198 
LVPRGSDSSRRQYQEKYKQVEQYMSFHKLPADFRQKIHDYYEHRYQGKMFDEDSILGELNGPLREEIVNFNNRKLVASMPLFANADPNFVTAMLTKLKFEVFQPGDYIIREGTIGKKMYFIQHGVVSVLTKGNKEMKLSDGSYFGEISLLTRGRRTASVRADTYSRLYSLSVDNFNEVLEEYPMMRRAFETVAIDRLDRIGKKN
6WDH , Knot 117 271 0.80 40 171 260 
AVVKCKPTSPGRRHVVKVVNPELHKGKPFAPLLEKNSKSGGRNNNGRITTRHIGGGHKQAYRIVDFKRNKDGIPAVVERLEYDPNRSANIALVLYKDGERRYILAPKGLKAGDQIQSGVDAAIKPGNTLPMRNIPVGSTVHNVEMKPGKGGQLARSAGTYVQIVARDGAYVTLRLRSGEMRKVEADCRATLGEVGNAEHMLRVLGKAGAARWRGVRPTVRGTAMNPVDHPHGGGEGRNFGKHPVTPWGVQTKGKKTRSNKRTDKFIVRRRS
7XEK , Knot 181 419 0.87 40 225 404 
MGHMNITDIREQFPILHQQVNGHDLVYLDSAATSQKPRAVIETLDKYYNQYNSNVHRGVHTLGTRATDGYEGAREKVRKFINAKSMAEIIFTKGTTTSLNMVALSYARANLKPGDEVVITYMEHHANIIPWQQAVKATGATLKYIPLQEDGTISLEDVRETVTSNTKIVAVSHVSNVLGTVNPIKEMAKIAHDNGAVIVVDGAQSTPHMKIDVQDLDCDFFALSSHKMCGPTGVGVLYGKKALLENMEPAEFGGEMIDFVGLYESTWKELPWKFEAGTPIIAGAIGLGAAIDFLEEIGLDEISRHEHKLAAYALERFRQLDGVTVYGPEERAGLVTFNLDDVHPHDVATVLDAEGIAVRAGHHCAQPLMKWLDVTATARASFYLYNTEEEIDKLVEALQKTKEYFTNVFVDLEHHHHHH

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 \(c\).

\(|P_{f(3ETQ_1)}(2) \setminus P_{f(6WDH_1)}(2)|=74\), \(|P_{f(6WDH_1)}(2) \setminus P_{f(3ETQ_1)}(2)|=90\). 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 1:111010000000000000010001010011101000100000000010110000111010111000110100001110111110101011011100101011011001100101100101100111011001000101001001101011001000101010000010010100100110001110011001110010011000
Pair \(Z_2\) Length of longest common subsequence
3ETQ_1,6WDH_1 164 3
3ETQ_1,7XEK_1 170 3
6WDH_1,7XEK_1 168 3

Newick tree

 
[
	7XEK_1:85.31,
	[
		3ETQ_1:82,6WDH_1:82
	]:3.31
]

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 \((0.85)(0.8)(\frac{475 }{\log_{20} 475}-\frac{204}{\log_{20}204})=78.8\)
Status Protein1 Protein2 d d1/2
Query variables 3ETQ_1 6WDH_1 101 89.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]