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Parikh vectors
6BFH_1 6OOB_1 8FWP_1 Letter Amino acid
13 51 27 L Leucine
1 30 16 S Serine
7 26 22 T Threonine
13 16 9 Y Tyrosine
13 22 19 R Arginine
5 12 6 H Histidine
5 34 7 P Proline
14 25 17 D Aspartic acid
7 15 10 Q Glutamine
14 29 27 I Isoleucine
13 38 13 K Lycine
7 18 3 M Methionine
5 32 15 F Phenylalanine
4 3 2 W Tryptophan
10 37 16 V Valine
3 7 2 C Cysteine
22 30 27 E Glutamic acid
8 26 13 G Glycine
9 18 13 A Alanine
5 18 15 N Asparagine 

6BFH_1|Chain A|Aminoglycoside acetyltransferase|Escherichia coli (562)
>6OOB_1|Chain A|Cytochrome P450 3A4|Homo sapiens (9606)
>8FWP_1|Chain A|Cell division control protein 10|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
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 \(w=f(c)\)
6BFH , Knot 84 178 0.81 40 131 174 
MLEKKRVSFRPMNEDDLVLMLKWLTDDRVLEFYDGRDKKHTQKTIREHYTEQWADEIYRVIIEYDTIPIGYAQIYRIQGELFDEYNYHETEEKIYAMDQFIGEPEYWNMGIGAEYCRVVCQYLRTEMDADAVILDPRKNNLRAVRAYQKAGFKIIKELPEHELHEGKKEDCVLMEWRV
6OOB , Knot 203 487 0.86 40 248 467 
MAYLYGTHSHGLFKKLGIPGPTPLPFLGNILSYHKGFCMFDMECHKKYGKVWGFYDGQQPVLAITDPDMIKTVLVKECYSVFTNRRPFGPVGFMKSAISIAEDEEWKRLRSLLSPTFTSGKLKEMVPIIAQYGDVLVRNLRREAETGKPVTLKDVFGAYSMDVITSTSFGVNIDSLNNPQDPFVENTKKLLRFDFLDPFFLSITVFPFLIPILEVLNICVFPREVTNFLRKSVKRMKESRLEDTQKHRVDFLQLMIDSQNSKETESHKALSDLELVAQSIIFIFAGYETTSSVLSFIMYELATHPDVQQKLQEEIDAVLPNKAPPTYDTVLQMEYLDMVVNETLRLFPIAMRLERVCKKDVEINGMFIPKGVVVMIPSYALHRDPKYWTEPEKFLPERFSKKNKDNIDPYIYTPFGSGPRNCIGMRFALMNMKLALIRVLQNFSFKPCKETQIPLKLSLGGLLQPEKPVVLKVESRDGTVSGAHHHH
8FWP , Knot 123 279 0.82 40 179 268 
MADLNCGFQFNIMVVGQSGLGKSTLINTLFASHLIDSATGDDISALPVTKTTEMKISTHTLVEDRVRLNINVIDTPGFGDFIDNSKAWEPIVKYIKEQHSQYLRKELTAQRERFITDTRVHAILYFLQPNGKELSRLDVEALKRLTEIANVIPVIGKSDTLTLDERTEFRELIQNEFEKYNFKIYPYDSEELTDEELELNRSVRSIIPFAVVGSENEIEINGETFRGRKTRWSAINVEDINQCDFVYLREFLIRTHLQDLIETTSYIHYEGFRARQLIA

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 \(c\).

\(|P_{f(6BFH_1)}(2) \setminus P_{f(6OOB_1)}(2)|=42\), \(|P_{f(6OOB_1)}(2) \setminus P_{f(6BFH_1)}(2)|=159\). 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 1:1100001010110000111110110000110100100000000010000000110010011100001111010100101011000000000001011001110100101111100001100010001010111101000010110100011101100110001001000001110101
Pair \(Z_2\) Length of longest common subsequence
6BFH_1,6OOB_1 201 4
6BFH_1,8FWP_1 176 4
6OOB_1,8FWP_1 169 4

Newick tree

 
[
	6BFH_1:97.55,
	[
		8FWP_1:84.5,6OOB_1:84.5
	]:13.05
]

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 \((0.85)(0.8)(\frac{665 }{\log_{20} 665}-\frac{178}{\log_{20}178})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 6BFH_1 6OOB_1 179 121.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]