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Parikh vectors
4QZB_1 5LQQ_1 2QWG_1 Letter Amino acid
40 65 18 L Leucine
22 38 9 F Phenylalanine
20 50 30 T Threonine
23 50 23 V Valine
15 28 7 H Histidine
15 34 25 I Isoleucine
31 62 35 S Serine
10 43 18 Y Tyrosine
25 48 23 R Arginine
12 18 5 M Methionine
7 31 18 C Cysteine
15 58 22 P Proline
5 13 14 W Tryptophan
10 37 26 N Asparagine
24 53 22 D Aspartic acid
32 51 21 E Glutamic acid
25 50 30 G Glycine
28 51 17 K Lycine
25 28 17 A Alanine
16 19 8 Q Glutamine 

4QZB_1|Chain A|DNA nucleotidylexotransferase|Mus musculus (10090)
>5LQQ_1|Chain A|Ectonucleotide pyrophosphatase/phosphodiesterase family member 2|Rattus norvegicus (10116)
>2QWG_1|Chain A|NEURAMINIDASE|Influenza A virus (11320)
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 \(w=f(c)\)
4QZB , Knot 169 400 0.84 40 225 376 
MGSSHHHHHHSSGLVPRGSHMSPSPVPGSQNVPAPAVKKISQYACQRRTTLNNYNQLFTDALDILAENDELRENEGSCLAFMRASSVLKSLPFPITSMKDTEGIPCLGDKVKSIIEGIIEDGESSEAKAVLNDERYKSFKLFTSVFGVGLKTAEKWFRMGFRTLSKIQSDKSLRFTQMQKAGFLYYEDLVSCVNRPEAEAVSMLVKEAVVTFLPDALVTMTGGFRRGKMTGHDVDFLITSPEATEDEEQQLLHKVTDFWKQQGLLLYCDILESTFEKFKQPSRKVDALDHFQKCFLILKLDHGRVHSEKSGQQEGKGWKAIRVDLVMCPYDRRAFALLGWTGSRQFERDLRRYATHERKMMLDNHALYDRTKRVFLEAESEEEIFAHLGLDYIEPWERNA
5LQQ , Knot 323 827 0.87 40 322 765 
AEWDEGPPTVLSDSPWTNTSGSCKGRCFELQEVGPPDCRCDNLCKSYSSCCHDFDELCLKTARGWECTKDRCGEVRNEENACHCSEDCLSRGDCCTNYQVVCKGESHWVDDDCEEIKVPECPAGFVRPPLIIFSVDGFRASYMKKGSKVMPNIEKLRSCGTHAPYMRPVYPTKTFPNLYTLATGLYPESHGIVGNSMYDPVFDASFHLRGREKFNHRWWGGQPLWITATKQGVRAGTFFWSVSIPHERRILTILQWLSLPDNERPSVYAFYSEQPDFSGHKYGPFGPEMTNPLREIDKTVGQLMDGLKQLRLHRCVNVIFVGDHGMEDVTCDRTEFLSNYLTNVDDITLVPGTLGRIRAKSINNSKYDPKTIIAALTCKKPDQHFKPYMKQHLPKRLHYANNRRIEDIHLLVDRRWHVARKPLDVYKKPSGKCFFQGDHGFDNKVNSMQTVFVGYGPTFKYRTKVPPFENIELYNVMCDLLGLKPAPNNGTHGSLNHLLRTNTFRPTMPDEVSRPNYPGIMYLQSEFDLGCTCDDKVEPKNKLEEFNKRLHTKGSTKERHLLYGRPAVLYRTSYDILYHTDFESGYSEIFLMPLWTSYTISKQAEVSSIPEHLTNCVRPDVRVSPGFSQNCLAYKNDKQMSYGFLFPPYLSSSPEAKYDAFLVTNMVPMYPAFKRVWAYFQRVLVKKYASERNGVNVISGPIFDYNYDGLRDTEDEIKQYVEGSSIPVPTHYYSIITSCLDFTQPADKCDGPLSVSSFILPHRPDNDESCASSEDESKWVEELMKMHTARVRDIEHLTGLDFYRKTSRSYSEILTLKTYLHTYESEI
2QWG , Knot 165 388 0.84 40 225 372 
RDFNNLTKGLCTINSWHIYGKDNAVRIGEDSDVLVTREPYVSCDPDECRFYALSQGTTIRGKHSNGTIHDRSQYRALISWPLSSPPTVYNSRVECIGWSSTSCHDGKTRMSICISGPNNNASAVIWYNRRPVTEINTWARNILRTQESECVCHNGVCPVVFTDGSATGPAETRIYYFKEGKILKWEPLAGTAKHIEECSCYGERAEITCTCKDNWQGSNRPVIRIDPVAMTHTSQYICSPVLTDNPRPNDPTVGKCNDPYPGNNNNGVKGFSYLDGVNTWLGRTISIASRSGYEMLKVPNALTDDKSKPTQGQTIVLNTDWSGYSGSFMDYWAEGECYRACFYVELIRGRPKEDKVWWTSNSIVSMCSSTEFLGQWDWPDGAKIEYFL

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 \(c\).

\(|P_{f(4QZB_1)}(2) \setminus P_{f(5LQQ_1)}(2)|=30\), \(|P_{f(5LQQ_1)}(2) \setminus P_{f(4QZB_1)}(2)|=127\). 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 1:1100000000001111010010101111000111111001000100000010000011001101110000100001001111010011001111100100001110110010011011100100001011100000001011001111110010011011100100100000101001001111000011001001010110111001110111011101011100101010010111001010000000110010011000111100011000100100100010110010001111010010100000100010110110101110100001111111010001000100010000011100011000000111010000011101110010110001
Pair \(Z_2\) Length of longest common subsequence
4QZB_1,5LQQ_1 157 4
4QZB_1,2QWG_1 186 3
5LQQ_1,2QWG_1 163 4

Newick tree

 
[
	2QWG_1:90.22,
	[
		4QZB_1:78.5,5LQQ_1:78.5
	]:11.72
]

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 \((0.85)(0.8)(\frac{1227 }{\log_{20} 1227}-\frac{400}{\log_{20}400})=215.\)
Status Protein1 Protein2 d d1/2
Query variables 4QZB_1 5LQQ_1 274 200.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]