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Parikh vectors
3PPZ_1 6BDK_1 5GSC_1 Letter Amino acid
10 32 10 F Phenylalanine
20 30 29 S Serine
22 18 46 A Alanine
13 12 16 H Histidine
11 18 11 M Methionine
8 16 18 Y Tyrosine
22 34 22 P Proline
11 26 16 T Threonine
7 3 3 W Tryptophan
21 37 24 V Valine
18 25 12 D Aspartic acid
4 7 0 C Cysteine
8 15 16 Q Glutamine
16 26 36 G Glycine
15 29 15 I Isoleucine
31 51 32 L Leucine
18 38 18 K Lycine
20 22 13 R Arginine
16 18 12 N Asparagine
18 30 17 E Glutamic acid 

3PPZ_1|Chains A, B|Serine/threonine-protein kinase CTR1|Arabidopsis thaliana (3702)
>6BDK_1|Chain A|Cytochrome P450 3A4|Homo sapiens (9606)
>5GSC_1|Chains A, B|Beta-lactamase|Enterobacter aerogenes (548)
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 \(w=f(c)\)
3PPZ , Knot 133 309 0.82 40 187 291 
MKHHHHHHPMSDYDIPTTENLYFQGAMDGDDMDIPWCDLNIKEKIGAGSFGTVHRAEWHGSDVAVKILMEQDFHAERVNEFLREVAIMKRLRHPNIVLFMGAVTQPPNLSIVTEYLSRGSLYRLLHKSGAREQLDERRRLSMAYDVAKGMNYLHNRNPPIVHRDLKSPNLLVDKKYTVKVCDFGLSRLKASTFLSSKSAAGTPEWMAPEVLRDEPSNEKSDVYSFGVILWELATLQQPWGNLNPAQVVAAVGFKCKRLEIPRNLNPQVAAIIEGCWTNEPWKRPSFATIMDLLRPLIKSAVPPPNRSDL
6BDK , Knot 203 487 0.86 40 248 467 
MAYLYGTHSHGLFKKLGIPGPTPLPFLGNILSYHKGFCMFDMECHKKYGKVWGFYDGQQPVLAITDPDMIKTVLVKECYSVFTNRRPFGPVGFMKSAISIAEDEEWKRLRSLLSPTFTSGKLKEMVPIIAQYGDVLVRNLRREAETGKPVTLKDVFGAYSMDVITSTSFGVNIDSLNNPQDPFVENTKKLLRFDFLDPFFLSITVFPFLIPILEVLNICVFPREVTNFLRKSVKRMKESRLEDTQKHRVDFLQLMIDSQNSKETESHKALSDLELVAQSIIFIFAGYETTSSVLSFIMYELATHPDVQQKLQEEIDAVLPNKAPPTYDTVLQMEYLDMVVNETLRLFPIAMRLERVCKKDVEINGMFIPKGVVVMIPSYALHRDPKYWTEPEKFLPERFSKKNKDNIDPYIYTPFGSGPRNCIGMRFALMNMKLALIRVLQNFSFKPCKETQIPLKLSLGGLLQPEKPVVLKVESRDGTVSGAHHHH
5GSC , Knot 155 366 0.83 38 192 342 
MHHHHHHGEASPVDPLRPVVDASIQPLLKEHRIPGMAVAVLKDGKAHYFNYGVANRESGAGVSEQTLFEIGSVSKTLTATLGAYAVVKGAMQLDDKASRHAPWLKGSAFDSITMGELATYSAGGLPLQFPEEVDSSEKMRAYYRQWAPVYSPGSHRQYSNPSIGLFGHLAASSLKQPFAPLMEQTLLPGLGMHHTYVNVPKQAMASYAYGYSKEDKPIRVNPGMLADEAYGIKTSSADLLRFVKANIGGVDDKALQQAISLTHQGHYSVGGMTQGLGWESYAYPVTEQTLLAGNSAKVILEANPTAAPRESGSQVLFNKTGSTNGFGAYVAFVPARGIGIVMLANRNYPIEARIKAAHAILAQLAG

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 \(c\).

\(|P_{f(3PPZ_1)}(2) \setminus P_{f(6BDK_1)}(2)|=44\), \(|P_{f(6BDK_1)}(2) \setminus P_{f(3PPZ_1)}(2)|=105\). 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 1:100000001100001100001010111010010111001010001111011010010101001110111000101001001100111100100101111111100110101100010010100110001100010000010110011011001000011110001001011100000101001110010100110000111010111101100010000001001111110110100111010110111111100001011001010111110101000110010110110110111001111100001
Pair \(Z_2\) Length of longest common subsequence
3PPZ_1,6BDK_1 149 4
3PPZ_1,5GSC_1 159 6
6BDK_1,5GSC_1 176 4

Newick tree

 
[
	5GSC_1:86.75,
	[
		3PPZ_1:74.5,6BDK_1:74.5
	]:12.25
]

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 \((0.85)(0.8)(\frac{796 }{\log_{20} 796}-\frac{309}{\log_{20}309})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 3PPZ_1 6BDK_1 168 136
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]