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Parikh vectors
3LPG_1 8OUP_1 8OFG_1 Letter Amino acid
40 13 4 D Aspartic acid
46 6 13 G Glycine
20 5 5 H Histidine
29 8 18 I Isoleucine
13 4 4 M Methionine
24 5 7 P Proline
35 9 12 T Threonine
29 4 10 R Arginine
25 12 7 F Phenylalanine
26 9 14 S Serine
19 3 1 W Tryptophan
32 8 3 Q Glutamine
40 6 19 E Glutamic acid
24 4 4 Y Tyrosine
53 12 10 V Valine
9 2 0 C Cysteine
28 6 6 N Asparagine
40 12 18 L Leucine
27 16 12 K Lycine
46 18 14 A Alanine 

3LPG_1|Chains A, B|Beta-glucuronidase|Escherichia coli (83333)
>8OUP_1|Chains A, B, C, D|Nerve hemoglobin|Spisula solidissima (6584)
>8OFG_1|Chains A, B|Diadenylate cyclase|Streptococcus pneumoniae R6 (171101)
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 \(w=f(c)\)
3LPG , Knot 251 605 0.88 40 281 574 
SHMLRPVETPTREIKKLDGLWAFSLDRENCGIDQRWWESALQESRAIAVPGSFNDQFADADIRNYAGNVWYQREVFIPKGWAGQRIVLRFDAVTHYGKVWVNNQEVMEHQGGYTPFEADVTPYVIAGKSVRITVCVNNELNWQTIPPGMVITDENGKKKQSYFHDFFNYAGIHRSVMLYTTPNTWVDDITVVTHVAQDCNHASVDWQVVANGDVSVELRDADQQVVATGQGTSGTLQVVNPHLWQPGEGYLYELCVTAKSQTECDIYPLRVGIRSVAVKGEQFLINHKPFYFTGFGRHEDADLRGKGFDNVLMVHDHALMDWIGANSYRTSHYPYAEEMLDWADEHGIVVIDETAAVGFNLSLGIGFEAGNKPKELYSEEAVNGETQQAHLQAIKELIARDKNHPSVVMWSIANEPDTRPQGAREYFAPLAEATRKLDPTRPITCVNVMFCDAHTDTISDLFDVLCLNRYYGWYVQSGDLETAEKVLEKELLAWQEKLHQPIIITEYGVDTLAGLHSMYTDMWSEEYQCAWLDMYHRVFDRVSAVVGEQVWNFADFATSQGILRVGGNKKGIFTRDRKPKSAAFLLQKRWTGMNFGEKPQQGGKQ
8OUP , Knot 80 162 0.83 40 129 159 
MADPCPVTKLTKAEKDAVANSWAALKQDWKTIGADFFVKLFETYPNIKAYFKSFDNMDMSEIKQSPKLRAHSINFCHGLNSFIQSLDEPDVLVILVQKLTVNHFRRKIAVDRFQEAFALYVSYAQDHAKFDDFTAAAWTKTLKVVADVIGGHMQTLQKSGGE
8OFG , Knot 85 181 0.81 38 130 173 
GSHMASMNAPISAEEQMIRAFVKSVEYMSPRKIGALVAIQRVRTLQEYISTGIPLDAKISAELLINIFIPNTPLHDGAVIIKEERIAVTSAYLPLTKNTGISKEFGTRHRAAIGLSEVSDALTFVVSEETGGISITYNGRFKHNLTLDEFETELREILLPKEEVGLSFKERLLGGWKHEKK

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 \(c\).

\(|P_{f(3LPG_1)}(2) \setminus P_{f(8OUP_1)}(2)|=182\), \(|P_{f(8OUP_1)}(2) \setminus P_{f(3LPG_1)}(2)|=30\). 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 1:00110110010001001011111010000011000110011000011111101000110101000110110000111101111001110101100010111000011000110011010101011110010101010001010011111110000100000010011001110001110001001100101100110000010101011101010101001000111010100101011010110110101001010100000001011011100111010011100011010111000010101011001111000111011110000000010100110110001111100011111010111110110010010000110100001010110011100000101111011001000101100011111010001010011001011100100001001101101000011010010100100110001111000100111100011001111001000110000001110100011001011110011011011000111011100011100000100111110001011011001001100
Pair \(Z_2\) Length of longest common subsequence
3LPG_1,8OUP_1 212 3
3LPG_1,8OFG_1 211 3
8OUP_1,8OFG_1 147 3

Newick tree

 
[
	3LPG_1:11.49,
	[
		8OFG_1:73.5,8OUP_1:73.5
	]:40.99
]

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 \((0.85)(0.8)(\frac{767 }{\log_{20} 767}-\frac{162}{\log_{20}162})=170.\)
Status Protein1 Protein2 d d1/2
Query variables 3LPG_1 8OUP_1 220 138
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]