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Parikh vectors
5RPP_1 6WAF_1 7BOV_1 Letter Amino acid
22 10 12 T Threonine
2 4 2 W Tryptophan
14 12 19 D Aspartic acid
5 5 15 M Methionine
17 4 11 Y Tyrosine
4 7 6 H Histidine
9 5 21 P Proline
36 11 26 S Serine
7 9 21 Q Glutamine
12 14 11 R Arginine
17 15 14 N Asparagine
5 3 4 C Cysteine
5 15 41 E Glutamic acid
33 6 17 G Glycine
11 14 23 I Isoleucine
14 21 41 L Leucine
33 15 35 A Alanine
6 10 17 F Phenylalanine
19 17 27 V Valine
8 8 29 K Lycine 

5RPP_1|Chain A|Proteinase K|Parengyodontium album (37998)
>6WAF_1|Chains A, B|LuxR family transcriptional regulator|Vibrio vulnificus (672)
>7BOV_1|Chain A|Uncharacterized UDP-glucosyltransferase YjiC|Bacillus subtilis subsp. subtilis str. 168 (224308)
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 \(w=f(c)\)
5RPP , Knot 120 279 0.80 40 170 266 
AAQTNAPWGLARISSTSPGTSTYYYDESAGQGSCVYVIDTGIEASHPEFEGRAQMVKTYYYSSRDGNGHGTHCAGTVGSRTYGVAKKTQLFGVKVLDDNGSGQYSTIIAGMDFVASDKNNRNCPKGVVASLSLGGGYSSSVNSAAARLQSSGVMVAVAAGNNNADARNYSPASEPSVCTVGASDRYDRRSSFSNYGSVLDIFGPGTDILSTWIGGSTRSISGTSMATPHVAGLAAYLMTLGKTTAASACRYIADTANKGDLSNIPFGTVNLLAYNNYQA
6WAF , Knot 96 205 0.83 40 146 197 
MDSIAKRPRTRLSPLKRKQQLMEIALEVFARRGIGRGGHADIAEIAQVSVATVFIYFPTREDLVDEVLNHVVRQFSNFLSDNIDLDLHAKENIANITNAMIELVVQDNHWLKVWFEWSASTRDEVWPLFVTTNRTNQLLVQNMFIKAIERGEVCDQHNPEDLANLFHGICYSLFVQANRTNNTAELSKLVSSYLDMLCIYKREHE
7BOV , Knot 164 392 0.83 40 213 361 
MKKYHISMINIPAYGHVNPTLALVEKLCEKGHRVTYATTEEFAPAVQQAGGEALIYHTSLNIDPKQIREMMEKNDAPLSLLKESLSILPQLEELYKDDQPDLIIYDFVALAGKLFAEKLNVPVIKLCSSYAQNESFQLGNEDMLKKIREAEAEFKAYLEQEKLPAVSFEQLAVPEALNIVFMPKSFQIQHETFDDRFCFVGPSLGERKEKESLLIDKDDRPLMLISLGTAFNAWPEFYKMCIKAFRDSSWQVIMSVGKTIDPESLEDIPANFTIRQSVPQLEVLEKADLFISHGGMNSTMEAMNAGVPLVVIPQMYEQELTANRVDELGLGVYLPKEEVTVSSLQEAVQAVSSDQELLSRVKNMQKDVKEAGGAERAAAEIEAFMKKSAVPQ

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 \(c\).

\(|P_{f(5RPP_1)}(2) \setminus P_{f(6WAF_1)}(2)|=106\), \(|P_{f(6WAF_1)}(2) \setminus P_{f(5RPP_1)}(2)|=82\). 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 1:110001111110100001100000000011010010110011010010101010110000000001010100011011000011100001111011000101000011111011100000000101111010111100001001110100011111111100010100001100101001110000000001000101101111100110011110000101001101011111101101100011010001100100101001111010111000001
Pair \(Z_2\) Length of longest common subsequence
5RPP_1,6WAF_1 188 3
5RPP_1,7BOV_1 195 4
6WAF_1,7BOV_1 175 3

Newick tree

 
[
	5RPP_1:98.36,
	[
		6WAF_1:87.5,7BOV_1:87.5
	]:10.86
]

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 \((0.85)(0.8)(\frac{484 }{\log_{20} 484}-\frac{205}{\log_{20}205})=81.0\)
Status Protein1 Protein2 d d1/2
Query variables 5RPP_1 6WAF_1 104 90.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]