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Parikh vectors
6IMQ_1 1BLR_1 7DKE_1 Letter Amino acid
2 0 8 H Histidine
4 10 70 L Leucine
0 3 16 M Methionine
2 7 36 S Serine
2 14 49 V Valine
7 6 103 A Alanine
2 5 46 D Aspartic acid
6 3 4 C Cysteine
3 13 49 K Lycine
0 2 26 Y Tyrosine
4 3 34 Q Glutamine
6 17 32 E Glutamic acid
4 5 29 F Phenylalanine
2 3 10 W Tryptophan
1 4 33 P Proline
1 13 31 T Threonine
3 7 32 R Arginine
0 7 33 N Asparagine
1 8 60 G Glycine
1 7 19 I Isoleucine 

6IMQ_1|Chains A, B, C, D|Protein PML|Homo sapiens (9606)
>1BLR_1|Chain A|CELLULAR RETINOIC ACID BINDING PROTEIN-TYPE II|Homo sapiens (9606)
>7DKE_1|Chains A, B|Dipeptidyl-peptidase|Stenotrophomonas maltophilia (strain R551-3) (391008)
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 \(w=f(c)\)
6IMQ , Knot 32 51 0.82 34 47 48 
GSRQIVDAQAVCTRCKESADFWCFECEQLLCAKCFEAHQWFLKHEARPLAE
1BLR , Knot 66 137 0.79 38 105 131 
PNFSGNWKIIRSENFEELLKVLGVNVMLRKIAVAAASKPAVEIKQEGDTFYIKTSTTVRTTEINFKVGEEFEEQTVDGRPCKSLVKWESENKMVCEQKLLKGEGPKTSWTRELTNDGELILTMTADDVVCTRVYVRE
7DKE , Knot 273 720 0.83 40 263 646 
MRSNLLAFSIVASLGLAQVAHAAEGMWVPQQLPEIAGPLQKAGLKLSPEQLANLTGDPMGAVVALGGCTASFVSPQGLVVTNHHCAYGAIQLNSTAQKNLIKDGFNAPTLKDELSAGPNARVFVLDQITDVTAQAKAAIAGAGNDPLARSRALDAFDKAQVAACEADAGFRCRLYSFSGGNTYRLFRNMEIKDVRLVYAPPGSVGKFGGDVDNWMWPRHTGDFSFYRAYVGKDGKPAAFAADNVPYQPKHFLKFADQPLGADDFVMVAGYPGRTNRYALAGEFNETASFTYPTIAKHYNAVLKMIADAGKADADVKVKYAATAASMNNVAKNYLGQLEGFKRIDAAGQKQAEEAAVLAWLKKQGAAGKPALAAHAQLLKHLDTSKSTRERDLFVGQFNNTSAVGAAITLYRLSIERSKPDAEREAGYQERDLTTIEGGLKQMDRRYVAKMDQQLQTYWLDQYVALPAAQRDNEVLNKWLAGSDAAAVKSLVNKLGGTELGSLDTRLKWFKADRAAFEASNDPAIQYAVAVMPALLKQEEQKKIREGESLTARPLYLQAVADYKKSQGEFVYPDANLSLRITFGNVMGYGKDGVKYTPFTTLEGVAAKETGEDPFDSPKALLDAVKAKRYGGLEDKRLGSVPVNFLSNLDITGGNSGSPVLDANGKLVGLAFDGNWESVSSNWVFDPVMTRMIAVDSRYMQWIMQEVAPAPQLLKELNLAK

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 \(c\).

\(|P_{f(6IMQ_1)}(2) \setminus P_{f(1BLR_1)}(2)|=33\), \(|P_{f(1BLR_1)}(2) \setminus P_{f(6IMQ_1)}(2)|=91\). 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 1:100011010110000000101101000011010010100111000101110
Pair \(Z_2\) Length of longest common subsequence
6IMQ_1,1BLR_1 124 4
6IMQ_1,7DKE_1 238 4
1BLR_1,7DKE_1 196 4

Newick tree

 
[
	7DKE_1:12.67,
	[
		6IMQ_1:62,1BLR_1:62
	]:58.67
]

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 \((0.85)(0.8)(\frac{188 }{\log_{20} 188}-\frac{51}{\log_{20}51})=46.7\)
Status Protein1 Protein2 d d1/2
Query variables 6IMQ_1 1BLR_1 59 40
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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