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Parikh vectors
5VQF_1 4HVR_1 7LNA_1 Letter Amino acid
19 4 8 K Lycine
6 2 9 M Methionine
16 6 10 Y Tyrosine
23 13 9 V Valine
18 13 14 G Glycine
43 16 2 L Leucine
11 10 5 H Histidine
14 5 13 N Asparagine
13 7 10 Q Glutamine
25 11 7 E Glutamic acid
9 9 3 F Phenylalanine
14 4 12 T Threonine
28 15 8 R Arginine
15 13 6 D Aspartic acid
13 5 4 I Isoleucine
26 15 5 P Proline
31 6 5 S Serine
6 4 2 W Tryptophan
22 11 8 A Alanine
11 5 2 C Cysteine 

5VQF_1|Chains A, B, C, D|Transforming growth factor beta-1|Sus scrofa (9823)
>4HVR_1|Chain A|3-hydroxyanthranilate 3,4-dioxygenase|Cupriavidus metallidurans (266264)
>7LNA_1|Chains A, B, C|Major prion protein|Mesocricetus auratus (10036)
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 \(w=f(c)\)
5VQF , Knot 155 363 0.84 40 213 336 
GPLSTSKTIDMELVKRKRIEAIRGQILSKLRLASPPSQGDVPPGPLPEAVLALYNSTRDRVAGESVEPEPEPEADYYAKEVTRVLMVESGNQIYDKFKGTPHSLYMLFNTSELREAVPEPVLLSRAELRLLRLKLKVEQHVELYQKYSQDSWRYLSNRLLAPSDSPEWLSFDVTGVVRQWLTRREAIEGFRLSAHCSCDSKDNTLHVEINGFNSGRRGDLATIHGMNRPFLLLMATPLERAQHLHSSRHRRALDTNYCFSSTEKNCCVRQLYIDFRKDLGWKWIHEPKGYHANFCLGPCPYIWSLDTQYSKVLALYNQHNPGASAAPCCVPQALEPLPIVYYVGRKPKVEQLSNMIVRSCKCS
4HVR , Knot 88 174 0.87 40 139 169 
MLTYGAPFNFPRWIDEHAHLLKPPVGNRQVWQDSDFIVTVVGGPNHRTDYHDDPLEEFFYQLRGNAYLNLWVDGRRERADLKEGDIFLLPPHVRHSPQRPEAGSACLVIERQRPAGMLDGFEWYCDACGHLVHRVEVQLKSIVTDLPPLFESFYASEDKRRCPHCGQVHPGRAA
7LNA , Knot 70 142 0.81 40 115 136 
GQGGGTHNQWNKPSKPKTNMKHMAGAAAAGAVVGGLGGYMLGSAMSRPMMHFGNDWEDRYYRENMNRYPNQVYYRPVDQYNNQNNFVHDCVNITIKQHTVTTTTKGENFTETDIKIMERVVEQMCTTQYQKESQAYYDGRRS

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 \(c\).

\(|P_{f(5VQF_1)}(2) \setminus P_{f(4HVR_1)}(2)|=125\), \(|P_{f(4HVR_1)}(2) \setminus P_{f(5VQF_1)}(2)|=51\). 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 1:111000001010110000101101011001011011001011111110111110000000111001010101010001001001111001001000101010010111000010011101111001010110101010001010000000010010001111000101101010111001100001101101010000000000101010110010010110101100111111101100100100000001100000100000000100101010001110110010100101011101011010000001111000001110111001101101111100110010100100111000000
Pair \(Z_2\) Length of longest common subsequence
5VQF_1,4HVR_1 176 3
5VQF_1,7LNA_1 204 3
4HVR_1,7LNA_1 180 4

Newick tree

 
[
	7LNA_1:98.76,
	[
		5VQF_1:88,4HVR_1:88
	]:10.76
]

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 \((0.85)(0.8)(\frac{537 }{\log_{20} 537}-\frac{174}{\log_{20}174})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 5VQF_1 4HVR_1 134 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]