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Parikh vectors
1OAW_1 8ZTZ_1 7YUY_1 Letter Amino acid
1 32 89 D Aspartic acid
2 36 100 E Glutamic acid
4 24 134 I Isoleucine
3 50 175 L Leucine
0 18 115 F Phenylalanine
0 40 97 V Valine
2 53 138 A Alanine
1 36 113 S Serine
0 15 80 Q Glutamine
0 26 43 H Histidine
2 39 102 T Threonine
1 17 99 Y Tyrosine
1 13 77 N Asparagine
8 2 33 C Cysteine
8 61 118 G Glycine
6 14 82 K Lycine
2 13 55 M Methionine
2 32 87 P Proline
1 8 37 W Tryptophan
4 40 102 R Arginine

1OAW_1|Chain A|OMEGA-AGATOXIN IVA|Agelenopsis aperta (6908)
>8ZTZ_1|Chains A, B, C, D, E|Putative AMP-binding enzyme|Kutzneria albida DSM 43870 (1449976)
>7YUY_1|Chain A[auth F]|1,3-beta-glucan synthase component FKS1|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
1OAW , Knot 26 48 0.69 32 39 46
KKKCIAKDYGRCKWGGTPCCRGRGCICSIMGTNCECKPRLIMEGLGLA
8ZTZ , Knot 221 569 0.82 40 245 517
MNHKVHHHHHHIEGRHMITKEERAQINADPELGAGNVLHRLRAYGRPTDRPVLWTDGTWRAPDGSHPEVITLGELYEYVETYAGFYHGKGIRPRDVVGVLTASSTEFAINFMAINSLGAIPSFANAKLRPEIAREYIRRQGASGAVTDTERHEVLAGGELGFVVTAEDIRPEHRAQLPQGWPYRHDPTDPIIISHSSGTTGMPKAVPHTHQTLLYAQLHRLKLSVGGSMGRLLVALPGNHNAAMSVMMFGLLLDSPVYLQSSQRGSDVLDAIEKFKPTTVFGFSGTYGQIATSDLSTRDMSSIEAYYNTGDAAHEAHIRVLVAQGSHEEIGPDFKPVRVPGSVFTDGLGSSETGYSIFHNGHKPGSASFGRCIGKPMSFAQAAVLSEDGRPLPAGEVGRLGVRSPTLTPGYWNDSLTWHKLRLGGYWLTGDLAMQDAEGNFYHLDRAPDAIRTEAGIVFSTRTEELLLASLPELADCTVTAIAEEGVRADWDGDGVAEAYVLLQFTDGAREPGDLTGWVNEVLAGQGFPPVTRALRMDSTDVSTGVTGKVLKRVMRERARELVARADQG
7YUY , Knot 645 1876 0.86 40 376 1597
MNTDQQPYQGQTDYTQGPGNGQSQEQDYDQYGQPLYPSQADGYYDPNVAAGTEADMYGQQPPNESYDQDYTNGEYYGQPPNMAAQDGENFSDFSSYGPPGTPGYDSYGGQYTASQMSYGEPNSSGTSTPIYGNYDPNAIAMALPNEPYPAWTADSQSPVSIEQIEDIFIDLTNRLGFQRDSMRNMFDHFMVLLDSRSSRMSPDQALLSLHADYIGGDTANYKKWYFAAQLDMDDEIGFRNMSLGKLSRKARKAKKKNKKAMEEANPEDTEETLNKIEGDNSLEAADFRWKAKMNQLSPLERVRHIALYLLCWGEANQVRFTAECLCFIYKCALDYLDSPLCQQRQEPMPEGDFLNRVITPIYHFIRNQVYEIVDGRFVKRERDHNKIVGYDDLNQLFWYPEGIAKIVLEDGTKLIELPLEERYLRLGDVVWDDVFFKTYKETRTWLHLVTNFNRIWVMHISIFWMYFAYNSPTFYTHNYQQLVDNQPLAAYKWASCALGGTVASLIQIVATLCEWSFVPRKWAGAQHLSRRFWFLCIIFGINLGPIIFVFAYDKDTVYSTAAHVVAAVMFFVAVATIIFFSIMPLGGLFTSYMKKSTRRYVASQTFTAAFAPLHGLDRWMSYLVWVTVFAAKYSESYYFLVLPLRDPIRILSTTAMRCTGEYWWGAVLCKVQPKIVLGLVIATDFILFFLDTYLWYIIVNTIFSVGKSFYLGISILTPWRNIFTRLPKRIYSKILATTDMEIKYKPKVLISQVWNAIIISMYREHLLAIDHVQKLLYHQVPSEIEGKRTLRAPTFFVSQDDNNFETEFFPRDSEAERRISFFAQSLSTPIPEPLPVDNMPTFTVLTPHYAERILLSLREIIREDDQFSRVTLLEYLKQLHPVEWECFVKDTKILAEETAAYEGNENEAEKEDALKSQIDDLPFYCIGFKSAAPEYTLRTRIWASLRSQTLYRTISGFMNYSRAIKLLYRVENPEIVQMFGGNAEGLERELEKMARRKFKFLVSMQRLAKFKPHELENAEFLLRAYPDLQIAYLDEEPPLTEGEEPRIYSALIDGHCEILDNGRRRPKFRVQLSGNPILGDGKSDNQNHALIFYRGEYIQLIDANQDNYLEECLKIRSVLAEFEELNVEQVNPYAPGLRYEEQTTNHPVAIVGAREYIFSENSGVLGDVAAGKEQTFGTLFARTLSQIGGKLHYGHPDFINATFMTTRGGVSKAQKGLHLNEDIYAGMNAMLRGGRIKHCEYYQCGKGRDLGFGTILNFTTKIGAGMGEQMLSREYYYLGTQLPVDRFLTFYYAHPGFHLNNLFIQLSLQMFMLTLVNLSSLAHESIMCIYDRNKPKTDVLVPIGCYNFQPAVDWVRRYTLSIFIVFWIAFVPIVVQELIERGLWKATQRFFCHLLSLSPMFEVFAGQIYSSALLSDLAIGGARYISTGRGFATSRIPFSILYSRFAGSAIYMGARSMLMLLFGTVAHWQAPLLWFWASLSSLIFAPFVFNPHQFAWEDFFLDYRDYIRWLSRGNNQYHRNSWIGYVRMSRARITGFKRKLVGDESEKAAGDASRAHRTNLIMAEIIPCAIYAAGCFIAFTFINAQTGVKTTDDDRVNSVLRIIICTLAPIAVNLGVLFFCMGMSCCSGPLFGMCCKKTGSVMAGIAHGVAVIVHIAFFIVMWVLESFNFVRMLIGVVTCIQCQRLIFHCMTALMLTREFKNDHANTAFWTGKWYGKGMGYMAWTQPSRELTAKVIELSEFAADFVLGHVILICQLPLIIIPKIDKFHSIMLFWLKPSRQIRPPIYSLKQTRLRKRMVKKYCSLYFLVLAIFAGCIIGPAVASAKIHKHIGDSLDGVVHNLFQPINTTNNDTGSQMSTYQSHYYTHTPSLKTWSTIK

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 \(c\).

\(|P_{f(1OAW_1)}(2) \setminus P_{f(8ZTZ_1)}(2)|=20\), \(|P_{f(8ZTZ_1)}(2) \setminus P_{f(1OAW_1)}(2)|=226\). 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 1:000011000100011101000101010011100000010111011111
Pair \(Z_2\) Length of longest common subsequence
1OAW_1,8ZTZ_1 246 3
1OAW_1,7YUY_1 341 3
8ZTZ_1,7YUY_1 151 5

Newick tree

 
[
	1OAW_1:16.03,
	[
		8ZTZ_1:75.5,7YUY_1:75.5
	]:90.53
]

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 \((0.85)(0.8)(\frac{617 }{\log_{20} 617}-\frac{48}{\log_{20}48})=170.\)
Status Protein1 Protein2 d d1/2
Query variables 1OAW_1 8ZTZ_1 213 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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