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Parikh vectors
5WWL_1 4DKL_1 6MGJ_1 Letter Amino acid
15 26 21 R Arginine
23 14 19 E Glutamic acid
4 6 10 H Histidine
11 25 20 F Phenylalanine
13 40 75 T Threonine
10 33 54 A Alanine
13 16 44 D Aspartic acid
10 10 14 Q Glutamine
5 16 18 M Methionine
8 17 38 P Proline
9 20 36 Y Tyrosine
8 21 94 G Glycine
23 43 38 L Leucine
16 23 29 K Lycine
18 37 49 V Valine
8 26 51 N Asparagine
0 14 0 C Cysteine
10 43 38 I Isoleucine
10 25 70 S Serine
1 9 29 W Tryptophan 

5WWL_1|Chain A[auth M]|Centromere protein mis12|Schizosaccharomyces pombe (strain 972 / ATCC 24843) (284812)
>4DKL_1|Chain A|Mu-type opioid receptor, lysozyme chimera|Mus musculus (10090)
>6MGJ_1|Chains A, B, C, D, E, F, G, H|Xyloglucanase|Paenibacillus odorifer (189426)
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 \(w=f(c)\)
5WWL , Knot 101 215 0.84 38 156 208 
MLVELLEFTPLSFIDDVINITNQLLYKGVNGVDKAFSQTRFAKKAPQEIEEGLHKFEVLFESVVDRYYDGFEVYTLRNIFSYPPELKGYMRTFGKDVDYSITTEQDAAMDQAIQEAAEKLVVKMQLRRDLRMRLSRKREKKTEIEKHLERISFLNKVPENWQVTLPETTDFLLDQLGNLQHAVKRVVEASPTVHSREVDERITYLEKGYERLSNP
4DKL , Knot 187 464 0.82 40 233 432 
GSHSLCPQTGSPSMVTAITIMALYSIVCVVGLFGNFLVMYVIVRYTKMKTATNIYIFNLALADALATSTLPFQSVNYLMGTWPFGNILCKIVISIDYYNMFTSIFTLCTMSVDRYIAVCHPVKALDFRTPRNAKIVNVCNWILSSAIGLPVMFMATTKYRQGSIDCTLTFSHPTWYWENLLKICVFIFAFIMPVLIITVCYGLMILRLKSVRNIFEMLRIDEGLRLKIYKNTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNTNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRRAALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYEKDRNLRRITRMVLVVVAVFIVCWTPIHIYVIIKALITIPETTFQTVSWHFCIALGYTNSCLNPVLYAFLDENFKRCFREFCIPTSSTIEQP
6MGJ , Knot 282 747 0.83 38 268 666 
GMSEPYTWKSVVTGAGGGFVPGIIFNQTEPNLIYARTDIGGAYRWNQATSSWVSISDSVGWVDWNKNGVDALATDPIDPNKVYMATGTYTNHWDNNGQIMRSNDRGNTWQTTPLPFKVGGNMPGRSAGERLVIDPNKNNILYFGARSGNGLWKSIDSGVTWSKVTSFPNVGTYIQNPTLDYGNDLVGLSWITFDKSTGTLGNATQTIYVGVADTASSVYRSTDGGVTWTALAGQPTGFLPHHGELSSTGDLYITYSNGVGPYDGSKGEVWKYNKTSGAWTNISPTTGTDNWYGFGGLALDAQHPNTLMVSSLNAWWPDEVIFRSTNGGATWSRIWDWGNYPERTYKFSMDITAAPWLDHGTTSTSLDPSPKLGWMMGDLEIDPFNSNRMMYGTGATIYGSNNLTSWDTGGKVNISVMAKGVEETAVLGLISPPTGTSHLITALGDVSGFRHEDLSVAPTKFQTSPSWATTMSIDYAELSPSYMVRVGSADKEKTPSMKSIGISNDGGVNWYMPNSEPSNGTKTTVGHGQVAVSASGNSILWSTSDIGVYYSKTSGNSWTASAGLPAGAKIASDRVNPNKYYGFYAGTFYVSVDGGATFTATGASGFPTNNVAGLQPNEAQISMKAVPGIEGDIWFAGGNTVENKYGLWHSTNSGASFTKLTNVEEADLIGYGKAAPGQTYMSLYTVAKIDGVRGVFRSDDVGATWVRINDDAHQYAKINMAITGDPRIYGRVYLGTNGRGTLYADPV

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 \(c\).

\(|P_{f(5WWL_1)}(2) \setminus P_{f(4DKL_1)}(2)|=49\), \(|P_{f(4DKL_1)}(2) \setminus P_{f(5WWL_1)}(2)|=126\). 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 1:11101101011011001101000110011011001100001100110010011001011100110000011010010011001101010100110010001000001110011001100111010100010101000000000100010010110011001010110000111001101001100110101010000100010010010001001
Pair \(Z_2\) Length of longest common subsequence
5WWL_1,4DKL_1 175 4
5WWL_1,6MGJ_1 190 4
4DKL_1,6MGJ_1 159 5

Newick tree

 
[
	5WWL_1:94.94,
	[
		4DKL_1:79.5,6MGJ_1:79.5
	]:15.44
]

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 \((0.85)(0.8)(\frac{679 }{\log_{20} 679}-\frac{215}{\log_{20}215})=130.\)
Status Protein1 Protein2 d d1/2
Query variables 5WWL_1 4DKL_1 164 119
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]