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Parikh vectors
1NDF_1 7UOQ_1 7QZA_1 Letter Amino acid
25 5 4 N Asparagine
8 2 1 C Cysteine
35 10 16 Q Glutamine
38 11 7 K Lycine
34 6 20 P Proline
43 15 12 V Valine
30 19 24 G Glycine
19 14 6 H Histidine
57 9 42 L Leucine
20 4 4 M Methionine
22 4 6 Y Tyrosine
34 7 21 D Aspartic acid
41 13 17 S Serine
7 4 5 W Tryptophan
42 14 37 A Alanine
27 5 18 R Arginine
31 10 17 E Glutamic acid
31 15 6 I Isoleucine
27 3 14 F Phenylalanine
25 10 10 T Threonine 

1NDF_1|Chains A, B|Carnitine Acetyltransferase|Mus musculus (10090)
>7UOQ_1|Chain A|Integrase|Human immunodeficiency virus 1 (11676)
>7QZA_1|Chains A, B, C, D, E, F, G, H|Dyp-type peroxidase family protein|Pseudomonas putida (303)
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 \(w=f(c)\)
1NDF , Knot 241 596 0.86 40 284 573 
AHQDALPRLPVPPLQQSLDYYLKALQPIVSEEEWAHTKQLVDEFQTSGGVGERLQKGLERRAKKMENWLSEWWLKTAYLQFRQPVVIYSSPGVILPKQDFVDLQGQLRFAAKLIEGVLDFKSMIDNETLPVEFLGGQPLCMNQYYQILSSCRVPGPKQDSVVNFLKSKRPPTHITVVHNYQFFELDVYHSDGTPLTSDQIFVQLEKIWNSSLQSNKEPVGILTSNHRNTWAKAYNNLIKDKVNRESVNSIQKSIFTVCLDKQVPRVSDDVYRNHVAGQMLHGGGSKFNSGNRWFDKTLQFIVAEDGSCGMVYEHAAAEGPPIVALVDHVMEYTKKPELVRSPMVPLPMPKKLRFNITPEIKNDIEKAKQNLSIMIQDLDIMMLTFHHFGKDFPKSEKLSPDAFIQVALQLAYYRIYGQACATYESASLRMFHLGRTDTIRSASIDSLAFVKGMGDSTVPEQQKVELLRKAVQAHRAYTDRAIRGEAFDRHLLGLKLQAIEDLVSMPDIFMDTSYAIAMHFNLSTSQVPAKTDCVMFFGPVVPDGYGICYNPMEAHINFSVSAYNSCAETNAARMAHYLEKALLDMRTLLQNHPRAK
7UOQ , Knot 81 180 0.78 40 134 174 
MGSSHHHHHHSSGLVPRGSHMHGQVDSSPGIWQLDCTHLEGKVILVAVHVASGYIEAEVIPAETGQETAYFLLKLAGRWPVKTVHTDNGSNFTSTTVKAACWWAGIKQEDGIPYNPQSQGVIESMNKELKKIIGQVRDQAEHLKTAVQMAVFIHNHKRKGGIGGYSAGERIVDIIATDIQ
7QZA , Knot 119 287 0.78 40 165 266 
MPFQQGLLATPVPAHARHLFFTLQSPEALPAALDALLPQVDGKQLLLGVGAPLAKALGREIPGLRPFPLLDAAVENPSTQHALWLWLRGDERGDLLLRTQALEQALAPALSLADSVDGFLHRGGYDLTGYEDGTENPTDEEAVQAAIAADGSSFAAFQLWKHDLQYFKSLPQADQDNIIGRRLSDNEKLDDAPASAHVKRTAQESFEPEAFMVRRSVSWADQRGAGLAFVALGKSFEAFEVQLRRMSGLEDGIIDGLYRFSRPLTGGYYWCPPMSETGVDLSPLLRA

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 \(c\).

\(|P_{f(1NDF_1)}(2) \setminus P_{f(7UOQ_1)}(2)|=172\), \(|P_{f(7UOQ_1)}(2) \setminus P_{f(1NDF_1)}(2)|=22\). 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 1:10001110111111000100010110111000011000011001000111100100110001001001100111001010100111100011111100011010101011101101110100110000111011110110100000110000111100001101100001100101100001101010000101100001110100110001000001111100000001101000110001000010010001101010001101000100001110110111001001001100010111100100111000111011111111001100000101100111111110010101010100010010001011100101111010011001100001010111011101100010101010000101011011000010010100111101110001100001011001101001000011010110001111010110011011011100001111010100001110000111111111010110001101010101010000100011011001001110100110001010
Pair \(Z_2\) Length of longest common subsequence
1NDF_1,7UOQ_1 194 4
1NDF_1,7QZA_1 185 4
7UOQ_1,7QZA_1 165 4

Newick tree

 
[
	1NDF_1:98.52,
	[
		7QZA_1:82.5,7UOQ_1:82.5
	]:16.02
]

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 \((0.85)(0.8)(\frac{776 }{\log_{20} 776}-\frac{180}{\log_{20}180})=166.\)
Status Protein1 Protein2 d d1/2
Query variables 1NDF_1 7UOQ_1 218 138
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]