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Parikh vectors
3RKC_1 8ZXP_1 5CVO_1 Letter Amino acid
12 44 36 A Alanine
7 17 36 N Asparagine
1 2 14 C Cysteine
4 24 37 K Lycine
17 41 34 S Serine
18 29 41 T Threonine
3 20 14 W Tryptophan
6 9 29 R Arginine
5 11 19 Q Glutamine
5 55 29 I Isoleucine
13 59 51 L Leucine
1 15 11 M Methionine
8 12 14 Y Tyrosine
3 19 24 E Glutamic acid
11 45 35 G Glycine
5 47 14 F Phenylalanine
13 41 44 V Valine
6 15 43 D Aspartic acid
2 16 16 H Histidine
8 18 21 P Proline 

3RKC_1|Chains A, B|Capsid protein|Hepatitis E virus (1678143)
>8ZXP_1|Chains A, B, C|Trimethylamine transporter|Myroides profundi (480520)
>5CVO_1|Chains A, D|WD repeat-containing protein 48|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 \(w=f(c)\)
3RKC , Knot 72 148 0.81 40 113 146 
SRPFSVLRANDVLWLSLTAAEYDQTTYGSSTNPMYVSDTVTFVNVATGAQGVSRSLDWSKVTLDGRPLTTIQQYSKTFFVLPLRGKLSFWEAGTTKAGYPYNYNTTASDQILIENAPGHRVCISTYTTNLGSGPVSISAVGVLAPHSA
8ZXP , Knot 209 539 0.81 40 235 492 
MFKKLLDNKNLVINPPVFITSILLIVALILTCVLFPEKVGVWFPAAQLAVTSNFGWFFVVTVNVILIFAIYLAFSKFGRIRLGGDDAEPEFTKASWFAMLFSTGMGIGIMFFSIAEPVSHFFNTPRPVDTDIEAAVQAMQFTSLHWGLHAWGIYAMVGLALAFFGFNRKLPMTFRSLFYPFWGERIHGWWGHIIDILSALATVFGLSTSLGLGVIQITAGLEYLYGWEISPMMQAGIILFVIGIATISVFSGLDKGVKILSNANMYIAASFMLLIFILGPTLFIMKGYVENTGAYLANFIDISTWNDTYLGSGWQNVWTIFYWAWWIAWSPFVGSFIARISKGRTVKEFVLGVLIVPGLITLLWMNVFGGSALHTILSGDVTMIAAVKADVSTALFVFLENFPFTKFLSIVAIILIFSFFITSSDSGSLVVDNITSGSNGESPVWQRVFWSFAQGIIAIVLLWGGGLDALQTAVIITGLPFAVILLVMCYSLQKGLKEELAKSSKKAKSKEEKSYKEIIAELLDEPQSKHHHHHHHHHH
5CVO , Knot 225 677 0.72 42 270 530 
MAAHHRQNTAGRRKVQVSYVIRDEVEKYNRNGVNALQLDPALNRLFTAGRDSIIRIWSVNQHKQDPYIASMEHHTDWVNDIVLCCNGKTLISASSDTTVKVWNAHKGFCMSTLRTHKDYVKALAYAKDKELVASAGLDRQIFLWDVNTLTALTASNNTVTTSSLSGNKDSIYSLAMNQLGTIIVSGSTEKVLRVWDPRTCAKLMKLKGHTDNVKALLLNRDGTQCLSGSSDGTIRLWSLGQQRCIATYRVHDEGVWALQVNDAFTHVYSGGRDRKIYCTDLRNPDIRVLICEEKAPVLKMELDRSADPPPAIWVATTKSTVNKWTLKGIHNFRASGDYDNDCTNPITPLCTQPDQVIKGGASIIQCHILNDKRHILTKDTNNNVAYWDVLKACKVEDLGKVDFEDEIKKRFKMVYVPNWFSVDLKTGMLTITLDESDCFAAWVSAKDAGFSSPDGSDPKLNLGGLLLQALLEYWPRTHVNPMDEEENEVNHVNGEQENRVQKGNGYFQVPPHTPVIFGEAGGRTLFRLLCRDSGGETESMLLNETVPQWVIDITVDKNMPKFXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

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 \(c\).

\(|P_{f(3RKC_1)}(2) \setminus P_{f(8ZXP_1)}(2)|=27\), \(|P_{f(8ZXP_1)}(2) \setminus P_{f(3RKC_1)}(2)|=149\). 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 1:0011011010011110101100000001000011010001011011011011000101001010101100100000011111101010110110001101000000100011100111001010000001101110101111111001
Pair \(Z_2\) Length of longest common subsequence
3RKC_1,8ZXP_1 176 4
3RKC_1,5CVO_1 219 3
8ZXP_1,5CVO_1 179 3

Newick tree

 
[
	5CVO_1:10.69,
	[
		3RKC_1:88,8ZXP_1:88
	]:15.69
]

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 \((0.85)(0.8)(\frac{687 }{\log_{20} 687}-\frac{148}{\log_{20}148})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 3RKC_1 8ZXP_1 194 122
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]