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Parikh vectors
7KVN_1 5DQT_1 1RKT_1 Letter Amino acid
26 13 11 T Threonine
3 4 2 W Tryptophan
7 4 0 C Cysteine
15 10 10 Q Glutamine
30 15 23 E Glutamic acid
12 4 2 H Histidine
38 14 17 K Lycine
34 22 4 P Proline
18 35 15 A Alanine
22 23 13 R Arginine
37 26 13 V Valine
29 22 8 I Isoleucine
18 4 4 M Methionine
25 18 13 D Aspartic acid
26 25 11 G Glycine
32 9 12 F Phenylalanine
30 15 13 S Serine
16 9 8 Y Tyrosine
18 4 4 N Asparagine
51 29 22 L Leucine 

7KVN_1|Chains A, B|Cytochrome P450 3A4|Homo sapiens (9606)
>5DQT_1|Chains A, B, C, D, G[auth I], H[auth J], I[auth K], J[auth L]|CRISPR-associated endonuclease Cas1|Escherichia coli K12 (83333)
>1RKT_1|Chains A, B|protein yfiR|Bacillus subtilis (1423)
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 \(w=f(c)\)
7KVN , Knot 203 487 0.86 40 248 467 
MAYLYGTHSHGLFKKLGIPGPTPLPFLGNILSYHKGFCMFDMECHKKYGKVWGFYDGQQPVLAITDPDMIKTVLVKECYSVFTNRRPFGPVGFMKSAISIAEDEEWKRLRSLLSPTFTSGKLKEMVPIIAQYGDVLVRNLRREAETGKPVTLKDVFGAYSMDVITSTSFGVNIDSLNNPQDPFVENTKKLLRFDFLDPFFLSITVFPFLIPILEVLNICVFPREVTNFLRKSVKRMKESRLEDTQKHRVDFLQLMIDSQNSKETESHKALSDLELVAQSIIFIFAGYETTSSVLSFIMYELATHPDVQQKLQEEIDAVLPNKAPPTYDTVLQMEYLDMVVNETLRLFPIAMRLERVCKKDVEINGMFIPKGVVVMIPSYALHRDPKYWTEPEKFLPERFSKKNKDNIDPYIYTPFGSGPRNCIGMRFALMNMKLALIRVLQNFSFKPCKETQIPLKLSLGGLLQPEKPVVLKVESRDGTVSGAHHHH
5DQT , Knot 131 305 0.82 40 178 285 
MTWLPLNPIPLKDRVSMIFLQYGQIDVIDGAFVLIDKTGIRTHIPVGSVACIMLEPGTRVSHAAVRLAAQVGTLLVWVGEAGVRVYASGQPGGARSDKLLYQAKLALDEDLRLKVVRKMFELRFGEPAPARRSVEQLRGIEGSRVRATYALLAKQYGVTWNGRRYDPKDWEKGDTINQCISAATSCLYGVTEAAILAAGYAPAIGFVHTGKPLSFVYDIADIIKFDTVVPKAFEIARRNPGEPDREVRLACRDIFRSSKTLAKLIPLIEDVLAAGEIQPPAPPEDAQPVAIPLPVSLGDAGHRSS
1RKT , Knot 95 205 0.82 38 137 195 
MSPKVTKEHKDKRQAEILEAAKTVFKRKGFELTTMKDVVEESGFSRGGVYLYFSSTEEMFRRIIETGLDEGLRKLDKSAEHQSVWASISSYLDELTEGLRDVADTLAPVQFEYLVTAWRNEERRQYLEKRYDLFVERFSRLLQKGIDQGEFQPVQPLATIAKFFLNMNDGIIQNALYFDEEKADVSGLAESAKLYLKTVLQADEK

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 \(c\).

\(|P_{f(7KVN_1)}(2) \setminus P_{f(5DQT_1)}(2)|=119\), \(|P_{f(5DQT_1)}(2) \setminus P_{f(7KVN_1)}(2)|=49\). 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 1:1101010000111001111110111111011000011011010000001011110010011111001011001110000011000011111111001101100001001001101010010100111111001011100100010010110100111100101100001110100100100111000001101011011110101111111110110101110010011000100100001000000010110111000000000000110010111001111111000000110111001100101000100010111100111000011010010111000101111110100100001010111110111111100110001001001001110010000000101010011101100011101111010111101100101010000011101011111010011110100001010110000
Pair \(Z_2\) Length of longest common subsequence
7KVN_1,5DQT_1 168 4
7KVN_1,1RKT_1 181 4
5DQT_1,1RKT_1 143 3

Newick tree

 
[
	7KVN_1:91.97,
	[
		5DQT_1:71.5,1RKT_1:71.5
	]:20.47
]

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 \((0.85)(0.8)(\frac{792 }{\log_{20} 792}-\frac{305}{\log_{20}305})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 7KVN_1 5DQT_1 171 138
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]