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Parikh vectors
5FRP_1 3QBK_1 6TLU_1 Letter Amino acid
7 9 15 M Methionine
35 12 13 F Phenylalanine
24 11 24 P Proline
6 8 5 W Tryptophan
27 10 23 Y Tyrosine
7 3 2 C Cysteine
40 12 23 E Glutamic acid
30 5 26 R Arginine
23 3 12 Q Glutamine
88 44 34 L Leucine
58 24 25 S Serine
44 22 13 T Threonine
36 29 30 V Valine
49 5 17 N Asparagine
67 18 21 I Isoleucine
17 21 22 G Glycine
13 2 22 H Histidine
48 5 23 K Lycine
36 37 22 A Alanine
48 11 27 D Aspartic acid 

5FRP_1|Chains A, B|SISTER CHROMATID COHESION PROTEIN PDS5|SACCHAROMYCES CEREVISIAE (4932)
>3QBK_1|Chains A, B, C[auth D]|Halorhodopsin|Natronomonas pharaonis (2257)
>6TLU_1|Chain A[auth AAA]|Casein kinase II subunit alpha|Homo sapiens (9606)
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 \(w=f(c)\)
5FRP , Knot 268 703 0.83 40 263 642 
GAMAKGAVTKLKFNSPIISTSDQLISTNELLDRLKALHEELASLDQDNTDLTGLDKYRDALVSRKLLKHKDVGIRAFTACCLSDILRLYAPDAPYTDAQLTDIFKLVLSQFEQLGDQENGYHIQQTYLITKLLEYRSIVLLADLPSSNNLLIELFHIFYDPNKSFPARLFNVIGGILGEVISEFDSVPLEVLRLIFNKFLTYNPNEIPEGLNVTSDCGYEVSLILCDTYSNRMSRHLTKYYSEIIHEATNDDNNSRLLTVVVKLHKLVLRLWETVPELINAVIGFIYHELSSENELFRKEATKLIGQILTSYSDLNFVSTHSDTFKAWISKIADISPDVRVEWTESIPQIIATREDISKELNQALAKTFIDSDPRVRRTSVMIFNKVPVTEIWKNITNKAIYTSLLHLAREKHKEVRELCINTMAKFYSNSLNEIERTYQNKEIWEIIDTIPSTLYNLYYINDLNINEQVDSVIFEYLLPFEPDNDKRVHRLLTVLSHFDKKAFTSFFAFNARQIKISFAISKYIDFSKFLNNQESMSSSQGPIVMNKYNQTLQWLASGLSDSTKAIDALETIKQFNDERIFYLLNACVTNDIPFLTFKNCYNELVSKLQTPGLFKKYNISTGASIMPRDIAKVIQILLFRASPIIYNVSNISVLLNLSNNSDAKQLDLKRRILDDISKVNPTLFKDQIRTLKTIIKDLDDPD
3QBK , Knot 125 291 0.81 40 160 272 
MTETLPPVTESAVALQAEVTQRELFEFVLNDPLLASSLYINIALAGLSILLFVFMTRGLDDPRAKLIAVSTILVPVVSIASYTGLASGLTISVLEMPAGHFAEGSSVMLGGEEVDGVVTMWGRYLTWALSTPMILLALGLLAGSNATKLFTAITFDIAMCVTGLAAALTTSSHLMRWFWYAISCACFIVVLYILLVEWAQDAKAAGTADIFSTLKLLTVVMWLGYPIVWALGVEGVAVLPVGYTSWAYSALDIVAKYIFAFLLLNYLTSNEGVVSGSILDVPSASGAPADD
6TLU , Knot 172 399 0.86 40 218 376 
MSGPVPSRARVYTDVNTHRPREYWDYESHVVEWGNQDDYQLVRKLGRGKYSEVFEAINITNNEKVVVKILKPVKKKKIKREIKILENLRGGPNIITLADIVKDPVSRTPALVFEHVNNTDFKQLYQTLTDYDIRFYMYEILKALDYCHSMGIMHRDVKPHNVMIDHEHRKLRLIDWGLAEFYHPGQEYNVRVASRYFKGPELLVDYQMYDYSLDMWSLGCMLASMIFRKEPFFHGHDNYDQLVRIAKVLGTEDLYDYIDKYNIELDPRFNDILGRHSRKRWERFVHSENQHLVSPEALDFLDKLLRYDHQSRLTAREAMEHPYFYTVVKDQARMGSSSMPGGSTPVSSANMMSGISSVPTPSPLGPLAGSPVIAAANPLGMPVPAAAGAQQLEHHHHHH

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 \(c\).

\(|P_{f(5FRP_1)}(2) \setminus P_{f(3QBK_1)}(2)|=145\), \(|P_{f(3QBK_1)}(2) \setminus P_{f(5FRP_1)}(2)|=42\). 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 1:1111011100101001110000011000011001011000110100000010110000011100011000011101101001001101011011000101001101110010011000010010000110011000011111011000011101101100100011101101111111011001001110110111001100010011011010000100101110000000100010000001100100000000110111010011101100110110111111000100000110001001110110000010110000001011100110101010101000110111000010001001110011000101000011110011100110010001100011011000000100101001101000010010000000011011001100100100100101000100111001111010000010011011001000110011110100101011100010100110000010000111110000001011101100000110110010010000110110101000111101000000110010011110000100110111001101101111010111001001011101000001001010001100100101011000100100110010010
Pair \(Z_2\) Length of longest common subsequence
5FRP_1,3QBK_1 187 5
5FRP_1,6TLU_1 157 4
3QBK_1,6TLU_1 188 4

Newick tree

 
[
	3QBK_1:98.30,
	[
		5FRP_1:78.5,6TLU_1:78.5
	]:19.80
]

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 \((0.85)(0.8)(\frac{994 }{\log_{20} 994}-\frac{291}{\log_{20}291})=188.\)
Status Protein1 Protein2 d d1/2
Query variables 5FRP_1 3QBK_1 235 163.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]