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Parikh vectors
4EGG_1 8SLX_1 6JYD_1 Letter Amino acid
7 57 17 S Serine
17 35 18 T Threonine
17 32 10 N Asparagine
12 29 12 D Aspartic acid
3 34 15 Q Glutamine
18 43 22 G Glycine
12 36 9 K Lycine
5 17 17 M Methionine
11 42 32 A Alanine
5 40 6 R Arginine
4 13 2 C Cysteine
13 111 32 L Leucine
10 37 16 F Phenylalanine
10 42 6 P Proline
17 50 23 V Valine
5 25 14 Y Tyrosine
10 53 5 E Glutamic acid
13 17 8 H Histidine
15 31 24 I Isoleucine
3 17 8 W Tryptophan 

4EGG_1|Chains A, B, C, D, E, F|Putative acetyltransferase SACOL2570|Staphylococcus aureus subsp. aureus (93062)
>8SLX_1|Chains A, B, C, D|Transient receptor potential cation channel subfamily V member 2|Rattus norvegicus (10116)
>6JYD_1|Chain A|Chloride pumping rhodopsin|Nonlabens marinus S1-08 (1454201)
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 \(w=f(c)\)
4EGG , Knot 96 207 0.82 40 145 199 
MTEKEKMLAEKWYDANFDQTLINERLRAKVICFALNHTNPVATMMRKVLIDALFQTTTDNVSISIPFDTDYGWNVKLGKNVYVNTNCYFMDGGQITIGDNVFIGPNCGFYTATHPLNFHHRNEGFEKAGPIHIGSNTWFGGHVAVLPGVTIGEGSVIGAGSVVTKDIPPHSLAVGNPCKVVRKIDNDLPSETLNDETIKLEHHHHHH
8SLX , Knot 290 761 0.84 40 302 677 
MTSASSPPAFRLETSDGDEEGNAEVNKGKQEPPPMESPFQREDRNSSPQIKVNLNFIKRPPKNTSAPSQQEPDRFDRDRLFSVVSRGVPEELTGLLEYLRWNSKYLTDSAYTEGSTGKTCLMKAVLNLQDGVNACIMPLLQIDKDSGNPKLLVNAQCTDEFYQGHSALHIAIEKRSLQCVKLLVENGADVHLRACGRFFQKHQGTCFYFGELPLSLAACTKQWDVVTYLLENPHQPASLEATDSLGNTVLHALVMIADNSPENSALVIHMYDGLLQMGARLCPTVQLEEISNHQGLTPLKLAAKEGKIEIFRHILQREFSGPYQPLSRKFTEWCYGPVRVSLYDLSSVDSWEKNSVLEIIAFHCKSPNRHRMVVLEPLNKLLQEKWDRLVSRFFFNFACYLVYMFIFTVVAYHQPSLDQPAIPSSKATFGESMLLLGHILILLGGIYLLLGQLWYFWRRRLFIWISFMDSYFEILFLLQALLTVLSQVLRFMETEWYLPLLVLSLVLGWLNLLYYTRGFQHTGIYSVMIQKVILRDLLRFLLVYLVFLFGFAVALVSLSREARSPKAPEDNNSTVTEQPTVGQEEEPAPYRSILDASLELFKFTIGMGELAFQEQLRFRGVVLLLLLAYVLLTYVLLLNMLIALMSETVNHVADNSWSIWKLQKAISVLEMENGYWWCRRKKHREGRLLKVGTRGDGTPDERWCFRVEEVNWAAWEKTLPTLSEDPSGPGITGNKKNPTSKPGKNSASEEDHLPLQVLQSP
6JYD , Knot 129 296 0.82 40 175 283 
MASMTGGQQMGRDPNSMKNIESLFDYSAGQFEFIDHLLTMGVGVHFAALIFFLVVSQFVAPKYRIATALSCIVMVSAGLILNSQAVMWTDAYAYVDGSYQLQDLTFSNGYRYVNWMATIPCLLLQLLIVLNLKGKELFSTATWLILAAWGMIITGYVGQLYEVDDIAQLMIWGAVSTAFFVVMNWIVGTKIFKNRATMLGGTDSTITKVFWLMMFAWTLYPIAYLVPAFMNNADGVVLRQLLFTIADISSKVIYGLMITYIAIQQSAAAGYVPAQQALGRIGMDSKAALEHHHHHH

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 \(c\).

\(|P_{f(4EGG_1)}(2) \setminus P_{f(8SLX_1)}(2)|=27\), \(|P_{f(8SLX_1)}(2) \setminus P_{f(4EGG_1)}(2)|=184\). 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 1:100000111001001010001100010101101110000111011001110111000000101011100001101011001010000011011010110011111001100100110100000110011110110001111011111110110101111101100011100111101001100100011000100001010000000
Pair \(Z_2\) Length of longest common subsequence
4EGG_1,8SLX_1 211 3
4EGG_1,6JYD_1 188 8
8SLX_1,6JYD_1 189 4

Newick tree

 
[
	8SLX_1:10.11,
	[
		4EGG_1:94,6JYD_1:94
	]:8.11
]

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 \((0.85)(0.8)(\frac{968 }{\log_{20} 968}-\frac{207}{\log_{20}207})=207.\)
Status Protein1 Protein2 d d1/2
Query variables 4EGG_1 8SLX_1 263 166
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]