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Parikh vectors
2KBU_1 7MPM_1 3RIE_1 Letter Amino acid
1 3 5 M Methionine
2 9 14 F Phenylalanine
3 5 10 P Proline
1 10 5 Q Glutamine
2 10 17 G Glycine
0 4 9 C Cysteine
1 14 26 I Isoleucine
3 14 18 S Serine
1 11 15 T Threonine
1 17 12 A Alanine
3 15 2 R Arginine
2 10 32 K Lycine
2 4 17 Y Tyrosine
1 13 21 V Valine
2 6 17 N Asparagine
0 17 12 D Aspartic acid
1 24 20 L Leucine
1 2 3 W Tryptophan
2 12 24 E Glutamic acid
1 7 4 H Histidine 

2KBU_1|Chain A|Peptidyl-prolyl cis-trans isomerase NIMA-interacting 1|null
>7MPM_1|Chains A, C, E, G, I, K, M, O|NanoRNase C|Bartonella henselae (38323)
>3RIE_1|Chains A, B, C|Spermidine synthase|Plasmodium falciparum (36329)
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 \(w=f(c)\)
2KBU , Knot 22 31 0.81 38 30 29 
KLPPGWEKRMSXRVYYFNHITNASQFERPSG
7MPM , Knot 97 207 0.83 40 149 198 
SMTEIRVHQGDLPNLDNYRIDAVAVDTETLGLQPHRDRLCVVQLSSGDGTADVIQIAKGQKSAPNLVRLLSDRDITKIFHFGRFDLAILAHTFGVMPDVVFCTKIASKLTRTYTDRHGLKEICGELLNVNISKQQQSSDWAAETLSRAQIEYAASDVLYLHRLKDIFEERLKREERESVAKACFQFLPMRANLDLLGWSEIDIFAHS
3RIE , Knot 123 283 0.81 40 179 270 
GSKKWFSEFSIMWPGQAFSLKIKKILYETKSKYQNVLVFESTTYGKVLVLDGVIQLTEKDEFAYHEMMTHVPMTVSKEPKNVLVVGGGDGGIIRELCKYKSVENIDICEIDETVIEVSKIYFKNISCGYEDKRVNVFIEDASKFLENVTNTYDVIIVDSSDPIGPAETLFNQNFYEKIYNALKPNGYCVAQCESLWIHVGTIKNMIGYAKKLFKKVEYANISIPTYPCGCIGILCCSKTDTGLTKPNKKLESKEFADLKYYNYENHSAAFKLPAFLLKEIENI

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 \(c\).

\(|P_{f(2KBU_1)}(2) \setminus P_{f(7MPM_1)}(2)|=19\), \(|P_{f(7MPM_1)}(2) \setminus P_{f(2KBU_1)}(2)|=138\). 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 1:0111110001000100100100100100101
Pair \(Z_2\) Length of longest common subsequence
2KBU_1,7MPM_1 157 2
2KBU_1,3RIE_1 181 3
7MPM_1,3RIE_1 172 4

Newick tree

 
[
	3RIE_1:91.30,
	[
		2KBU_1:78.5,7MPM_1:78.5
	]:12.80
]

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 \((0.85)(0.8)(\frac{238 }{\log_{20} 238}-\frac{31}{\log_{20}31})=70.2\)
Status Protein1 Protein2 d d1/2
Query variables 2KBU_1 7MPM_1 89 51
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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