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Parikh vectors
5YDH_1 4RDQ_1 5CGC_1 Letter Amino acid
37 22 24 R Arginine
13 17 10 Q Glutamine
37 32 13 E Glutamic acid
18 20 41 I Isoleucine
43 20 48 A Alanine
44 19 23 G Glycine
22 21 23 Y Tyrosine
33 17 19 N Asparagine
55 55 43 L Leucine
18 14 26 K Lycine
46 20 18 P Proline
37 16 28 T Threonine
33 31 21 S Serine
13 10 5 W Tryptophan
45 29 30 V Valine
27 18 14 D Aspartic acid
8 6 14 C Cysteine
12 7 12 H Histidine
6 9 12 M Methionine
29 26 20 F Phenylalanine 

5YDH_1|Chains A, B|Acetylcholinesterase|Anopheles gambiae (7165)
>4RDQ_1|Chains A, B, C, D, E|Bestrophin-1|Gallus gallus (9031)
>5CGC_1|Chain A|Metabotropic glutamate receptor 5,Endolysin,Metabotropic glutamate receptor 5|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 \(w=f(c)\)
5YDH , Knot 231 576 0.85 40 262 545 
DNDPLVVNTDKGRIRGITVDAPSGKKVDVWLGIPYAQPPVGPLRFRHPRPAEKWTGVLNTTTPPNSCVQIVDTVFGDFPGATMWNPNTPLSEDCLYINVVAPRPRPKNAAVMLWIFGGGFYSGTATLDVYDHRALASEENVIVVSLQYRVASLGFLFLGTPEAPGNAGLFDQNLALRWVRDNIHRFGGDPSRVTLFGESAGAVSVSLHLLSALSRDLFQRAILQSGSPTAPWALVSREEATLRALRLAEAVGCPHEPSKLSDAVECLRGKDPHVLVNNEWGTLGICEFPFVPVVDGAFLDETPQRSLASGRFKKTEILTGSNTEEGYYFIIYYLTELLRKEEGVTVTREEFLQAVRELNPYVNGAARQAIVFEYTDWTEPDNPNSNRDALDKMVGDYHFTCNVNEFAQRYAEEGNNVYMYLYTHRSKGNPWPRWTGVMHGDEINYVFGEPLNPTLGYTEDEKDFSRKIMRYWSNFAKTGNPNPNTASSEFPEWPKHTAHGRHYLELGLNTSFVGRGPRLRQCAFWKKYLPQLVAATSNLPGPAPPSEPCESSAFFYRPDLIVLLVSLLTATVRFIQ
4RDQ , Knot 170 409 0.83 40 227 389 
TVTYTNRVADARLGTFSQLLLQWKGSIYKLLYSEFLIFISLYFAISLVYRLILSESQRLMFEKLALYCNSYAELIPVSFVLGFYVSLVVSRWWAQYESIPWPDRIMNLVSCNVDGEDEYGRLLRRTLMRYSNLCSVLILRSVSTAVYKRFPSMEHVVRAGLMTPEEHKKFESLNSPHNKFWIPCVWFSNLAVKARNEGRIRDSVLLQGILNELNTLRSQCGRLYGYDWISIPLVYTQVVTVAVYSFFLACLIGRQFLDPEKAYPGHELDLFVPVFTFLQFFFYAGWLKVAEQLINPFGEDDDDFETNWLIDRNLQVSLMAVDEMHQDLPILEKDLYWNEPDPQPPYTAATAEYKRPSFLGSTFDISMQKEEMEFQPLEQIKENEEANHSTPLLGHLGRLLGVQSEGEEF
5CGC , Knot 179 444 0.82 40 227 406 
AASPVQYLRWGDPAPIAAVVFACLGLLATLFVTVVFIIYRDTPVVKSSSRELCYIILAGICLGYLCTFCLIAKPKQIYCYLQRIGIGLSPAMSYSALVTKTYRAARILAMSKKNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYKICTKKPRFMSACAQLVIAFILICIQLGIIVALFIMEPPDIMHDYPSIREVYLICNTTNLGVVAPLGYNGLLILACTFYAFKTRNVPANFNEAKYIAFTMYTTCIIWLAFVPIYFGSNYKIITMCFSVSLSATVALGCMFVPKVYIILAKPERNVRSAAAAHHHHHHHHHH

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 \(c\).

\(|P_{f(5YDH_1)}(2) \setminus P_{f(4RDQ_1)}(2)|=100\), \(|P_{f(4RDQ_1)}(2) \setminus P_{f(5YDH_1)}(2)|=65\). 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 1:000111100001010110101101001011111101011111101001011001011100001100010110011101111011010011000010101111010100111111111111001010101000011100001111010001101111111010111011110001110110001001110100101110011110101011011000110011100101011111100001010110110111010010010011001010010111000110111001111111011110001000110101000011010000010011100100110000110100001101100101010111001111000010010010000011001110001000100110001001001010100000010111010111010010011101101011000000010001100100110010101001000110110001010001011100011101101000111000110111100011111110010000111001011111101101010110
Pair \(Z_2\) Length of longest common subsequence
5YDH_1,4RDQ_1 165 4
5YDH_1,5CGC_1 161 4
4RDQ_1,5CGC_1 172 3

Newick tree

 
[
	4RDQ_1:85.48,
	[
		5YDH_1:80.5,5CGC_1:80.5
	]:4.98
]

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 \((0.85)(0.8)(\frac{985 }{\log_{20} 985}-\frac{409}{\log_{20}409})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 5YDH_1 4RDQ_1 196 166
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]