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Parikh vectors
5NIY_1 6LAP_1 6ORG_1 Letter Amino acid
23 15 27 V Valine
14 11 19 Q Glutamine
24 8 37 E Glutamic acid
32 9 49 L Leucine
6 18 24 P Proline
11 20 24 S Serine
2 13 20 Y Tyrosine
9 15 13 N Asparagine
19 6 32 D Aspartic acid
10 6 13 H Histidine
17 23 21 T Threonine
32 6 33 A Alanine
16 12 20 R Arginine
0 3 2 C Cysteine
23 9 24 K Lycine
7 10 4 M Methionine
14 11 20 F Phenylalanine
26 12 39 G Glycine
23 14 20 I Isoleucine
1 5 11 W Tryptophan 

5NIY_1|Chain A|Signal recognition particle-docking protein FtsY|Escherichia coli (562)
>6LAP_1|Chain A|Capsid protein VP1|Echovirus E11 (12078)
>6ORG_1|Chains A, B|Glycoside hydrolase|Streptococcus pneumoniae serotype 4 (strain ATCC BAA-334 / TIGR4) (170187)
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 \(w=f(c)\)
5NIY , Knot 129 309 0.79 38 173 291 
MFARLKRSLLKTKENLGSGFISLFRGKKIDDDLFEELEEQLLIADVGVETTRKIITNLTEGASRKQLRDAEALYGLLKEEMGEILAKVDEPLNVEGKAPFVILMVGVNGVGKTTTIGKLARQFEQQGKSVMLAAGDTFRAAAVEQLQVWGQRNNIPVIAQHTGADSASVIFDAIQAAKARNIDVLIADTAGRLQNKSHLMEELKKIVRVMKKLDVEAPHEVMLTIDASTGQNAVSQAKLFHEAVGLTGITLTKLDGTAKGGVIFSVADQFGIPIRYIGVGERIEDLRPFKADDFIEALFAREDHHHHHH
6LAP , Knot 108 226 0.86 40 165 217 
SESSIENFLCRSACVYMGEYHTTNTDTSKLFASWTINARRMVQMRRKLELFTYVRFDMEVTFVITSKQDQGTQLGQDMPPLTHQIMYIPPGGPIPKSVTDYTWQTSTNPSIFWTEGNAPPRMSIPFISIGNAYSNFYDGWSHFSQNGVYGYNTLNHMGQIYVRHVNGSSPLPMTSTVRMYFKPKHVKVWVPRPPRLCQYKNASTVNFTPTNITEKRQSINYIPETV
6ORG , Knot 186 452 0.83 40 228 430 
MKKIKPHGPLPSQTQLAYLGDELAAFIHFGPNTFYDQEWGTGQEDPERFNPSQLDAREWVRVLKETGFKKLILVVKHHDGFVLYPTAHTDYSVKVSPWRRGKGDLLLEVSQAATEFDMDMGVYLSPWDAHSPLYHVDREADYNAYYLAQLKEILSNPNYGNAGKFAEVWMDGARGEGAQKVNYEFEKWFETIRDLQGDCLIFSTEGTSIRWIGNERGYAGDPLWQKVNPDKLGTEAELNYLQHGDPSGTIFSIGEADVSIRPGWFYHEDQDPKSLEELVEIYFHSVGRGTPLLLNIPPNQAGLFDAKDIERLYEFATYRNELYKEDLALGAEVSGPALSADFACRHLTDGLETSSWASDADLPIQLELDLGSPKTFDVIELREDLKLGQRIAAFHVQVEVDGVWQEFGSGHTVGYKRLLRGAVVEAQKIRVVITESQALPLLTKISLYKTPG

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 \(c\).

\(|P_{f(5NIY_1)}(2) \setminus P_{f(6LAP_1)}(2)|=99\), \(|P_{f(6LAP_1)}(2) \setminus P_{f(5NIY_1)}(2)|=91\). 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 1:111010001100000110111011010010001100100011110111000001100100110000100101101110001101110100110101011111111110111000011011001000100111111001011110010111000011111000110010111011011010010111100110100000110010011011001010110011101010010011001011001111011010010101011111011001111100111100100101101001101111000000000
Pair \(Z_2\) Length of longest common subsequence
5NIY_1,6LAP_1 190 3
5NIY_1,6ORG_1 149 4
6LAP_1,6ORG_1 173 3

Newick tree

 
[
	6LAP_1:95.68,
	[
		5NIY_1:74.5,6ORG_1:74.5
	]:21.18
]

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 \((0.85)(0.8)(\frac{535 }{\log_{20} 535}-\frac{226}{\log_{20}226})=88.5\)
Status Protein1 Protein2 d d1/2
Query variables 5NIY_1 6LAP_1 111 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]