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Parikh vectors
3OSD_1 7NCU_1 4FKR_1 Letter Amino acid
16 16 14 S Serine
9 9 10 Y Tyrosine
16 16 23 V Valine
5 15 16 R Arginine
24 11 16 D Aspartic acid
12 8 16 F Phenylalanine
12 11 16 I Isoleucine
18 26 39 L Leucine
5 7 5 M Methionine
8 7 4 W Tryptophan
17 18 19 A Alanine
28 17 16 G Glycine
6 12 10 H Histidine
18 21 17 E Glutamic acid
22 15 21 K Lycine
9 14 19 P Proline
18 8 8 N Asparagine
2 0 3 C Cysteine
13 6 9 Q Glutamine
7 16 17 T Threonine 

3OSD_1|Chain A|putative glycosyl hydrolase|Bacteroides thetaiotaomicron (226186)
>7NCU_1|Chains A, B|Glutathione S-transferase GliG|Aspergillus fumigatus A1163 (451804)
>4FKR_1|Chain A|Cyclin-dependent kinase 2|Homo sapiens (9606)
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 \(w=f(c)\)
3OSD , Knot 117 265 0.82 40 177 255 
GGSAQEEQSANEVAVSYSKSLKAAEMDSLQLPVDADGYITIFDGKTFNGWRGYGKDRVPSKWTIEDGCIKFNGSGGGEAQDGDGGDLIFAHKFKNFELEMEWKVSKGGNSGIFYLAQEVTSKDKDGNDVLEPIYISAPEYQVLDNDNHPDAKLGKDNNRQSASLYDMIPAVPQNAKPFGEWNKAKIMVYKGTVVHGQNDENVLEYHLWTKQWTDLLQASKFSQDKWPLAFELLNNCGGENHEGFIGMQDHGDDVWFRNIRVKVLD
7NCU , Knot 111 253 0.81 38 171 241 
MSGSHHHHHHSGSMSERPSDLVVNRLVLFVVKGTATSTHNTVKPLILLEELGVPHDIYVVEKVSAPWFSEINPHKMVPAILDRSPDGRDTLRAWESTSTLMYIADAYDKDGTFGGRNVQERSEINNWLTLHTAALGPTAGYWLYFYKLHPEKLPKTIEKLRSNITVQYDILERRLNEPGQQYLALKDRPTIADIATLPFAMKSTAELFGLEFEKWPKLQEWSVRMGEREAVKRAWQRVAGFGHGEKEYGMLEA
4FKR , Knot 135 299 0.85 42 192 285 
XMENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL

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 \(c\).

\(|P_{f(3OSD_1)}(2) \setminus P_{f(7NCU_1)}(2)|=89\), \(|P_{f(7NCU_1)}(2) \setminus P_{f(3OSD_1)}(2)|=83\). 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 1:1101000001001110000010110100101110101010110100101101010001100101001010101011101001011011110010010101010100110011101100100000010011011010110001100000101011000000010100111111001011101001011100101101000001100011000100110100100001111101100011000011111000100111001010110
Pair \(Z_2\) Length of longest common subsequence
3OSD_1,7NCU_1 172 4
3OSD_1,4FKR_1 165 3
7NCU_1,4FKR_1 175 3

Newick tree

 
[
	7NCU_1:88.12,
	[
		3OSD_1:82.5,4FKR_1:82.5
	]:5.62
]

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 \((0.85)(0.8)(\frac{518 }{\log_{20} 518}-\frac{253}{\log_{20}253})=75.6\)
Status Protein1 Protein2 d d1/2
Query variables 3OSD_1 7NCU_1 97 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]