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Parikh vectors
3UQE_1 2OAU_1 3MYD_1 Letter Amino acid
16 9 15 Q Glutamine
23 10 36 E Glutamic acid
17 32 38 I Isoleucine
37 33 24 A Alanine
13 19 17 R Arginine
4 0 0 C Cysteine
11 17 15 N Asparagine
5 11 11 F Phenylalanine
1 3 1 W Tryptophan
5 8 9 M Methionine
4 6 6 Y Tyrosine
12 16 19 D Aspartic acid
28 25 21 G Glycine
5 7 5 H Histidine
23 24 18 S Serine
16 9 21 T Threonine
29 33 21 V Valine
31 29 44 L Leucine
13 9 29 K Lycine
16 6 15 P Proline 

3UQE_1|Chains A, B|6-phosphofructokinase isozyme 2|Escherichia coli (83333)
>2OAU_1|Chains A, B, C, D, E, F, G|Small-conductance mechanosensitive channel|Escherichia coli (562)
>3MYD_1|Chain A|Flagellar biosynthesis protein flhA|Helicobacter pylori (210)
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 \(w=f(c)\)
3UQE , Knot 135 309 0.83 40 173 291 
MVRIYTLTLAPSLDSATITPQIDPEGKLRCTAPVFEPGGGGINVARAIAHLGGSATAIFPAGGATGEHLVSLLADENVPVATVEAKDWTRQNLHVHVEASGEQYRFVMPGAALNEDEFRQLEEQVLEIESGAILVISGSLPPGVKLEKLTQLISAAQKQGIRCIVDSSGEALSAALAIGNIELVKPNQKELSALVNRELTQPDDVRKAAQEIVNSGKAKRVVVSLGPQGALGVDSENCIQVVPPPVKSQSTVGAGDSMVGAMTLKLAENASLEEMVRFGVAAGSAATLNQGTRLCSHDDTQKIYAYLSR
2OAU , Knot 127 306 0.79 38 171 284 
MGSSHHHHHHSSGLVPRGSHMEDLNVVDSINGAGSWLVANQALLLSYAVNIVAALAIIIVGLIIARMISNAVNRLMISRKIDATVADFLSALVRYGIIAFTLIAALGRVGVQTASVIAVLGAAGLAVGLALQGSLSNLAAGVLLVMFRPFRAGEYVDLGGVAGTVLSVQIFSTTMRTADGKIIVIPNGKIIAGNIINFSREPVRRNEFIIGVAYDSDIDQVKQILTNIIQSEDRILKDREMTVRLNELGASSINFVVRVWSNSGDLQNVYWDVLERIKREFDAAGISFPYPQMDVNFKRVKEDKAA
3MYD , Knot 151 365 0.81 38 198 342 
GPLGSTRAKTQEEIKREEEQAIDEVLKIEFLELALGYQLISLADMKQGGDLLERIRGIRKKIASDYGFLMPQIRIRDNLQLPPTHYEIKLKGIVIGEGMVMPDKFLAMNTGFVNREIEGIPTKEPAFGMDALWIDAKNKEEAIIQGYTIIDPSTVIATHTSELVKKYAEDFITKDEVKSLLERLAKDYPTIVEESKKIPTGAIRSVLQALLHEKIPIKDMLTILETITDIAPLVQNDVNILTEQVRARLSRVITNAFKSEDGRLKFLTFSTDSEQFLLNKLRENGTSKSLLLNVGELQKLIEGVSEEAMKVLQKGIAPVILIVEPNLRKALSNQMEQARIDVVVLSHAELDPNSNFEALGTIHIN

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 \(c\).

\(|P_{f(3UQE_1)}(2) \setminus P_{f(2OAU_1)}(2)|=73\), \(|P_{f(2OAU_1)}(2) \setminus P_{f(3UQE_1)}(2)|=71\). 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 1:110100101110100101010101010100011110111111011011101110101111111101001101110001111010100100001010101010000111111110000100100011010011111101011111010010011011000110011000101101111110101101000010111000100100100110011001010011101110111110000010111111000001111001111101011001010011011111101101001001000000001010100
Pair \(Z_2\) Length of longest common subsequence
3UQE_1,2OAU_1 144 5
3UQE_1,3MYD_1 127 4
2OAU_1,3MYD_1 145 4

Newick tree

 
[
	2OAU_1:74.94,
	[
		3UQE_1:63.5,3MYD_1:63.5
	]:11.44
]

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 \((0.85)(0.8)(\frac{615 }{\log_{20} 615}-\frac{306}{\log_{20}306})=86.1\)
Status Protein1 Protein2 d d1/2
Query variables 3UQE_1 2OAU_1 107 105
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]