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Parikh vectors
6JKK_1 1IUW_1 4OZQ_1 Letter Amino acid
17 19 33 D Aspartic acid
8 5 5 C Cysteine
15 34 51 G Glycine
34 46 58 L Leucine
15 12 28 F Phenylalanine
19 4 37 N Asparagine
15 9 12 H Histidine
16 6 15 M Methionine
23 21 49 S Serine
4 6 11 W Tryptophan
20 34 57 E Glutamic acid
14 19 46 T Threonine
26 28 47 V Valine
12 35 73 A Alanine
14 38 23 R Arginine
13 15 24 Q Glutamine
23 18 37 I Isoleucine
22 12 54 K Lycine
15 17 32 P Proline
16 16 28 Y Tyrosine 

6JKK_1|Chain A|Mitotic checkpoint control protein kinase BUB1|Drosophila melanogaster (7227)
>1IUW_1|Chain A|P-HYDROXYBENZOATE HYDROXYLASE|Pseudomonas aeruginosa (208964)
>4OZQ_1|Chains A, B|Chimera of Maltose-binding periplasmic protein and Kinesin family member 14 protein|Escherichia coli UMEA 3304-1 (1281237)
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 \(w=f(c)\)
6JKK , Knot 152 341 0.86 40 211 330 
GAMDPEFNPFNVELISSLLESIDFSMYIEKLPHCQLVGHVKRLHPNTHLEVHNEKFEVSKMIGKGAYGSVYVGKHLKSGKKVALKQERPTNYWEFYICLEIHSRLTSEQMIPSYAHIDYALVGNNSSVYISEFSDYGSLIGVCNKVKSVTNRNMDEYVVMHLSCQMLDIVDHLHAMGIIHADIKPDNFLLMKPICADPNEVSLQLIDFGVSIDMKLFPDNQTFNYVHHDDLFKCIEMRTGRPWTYQLDLYGLVSVMHVLLFGRYMEVVQRSPSTIWMPKTNVPRYFQRTMWENIFRTLLNIRDCRTMPNLQQLRTQLKCALAEKEKYVAEAISKFNTILQK
1IUW , Knot 163 394 0.82 40 216 373 
MKTQVAIIGAGPSGLLLGQLLHKAGIDNVILERQTPDYVLGRIRAGVLEQGMVDLLREAGVDRRMARDGLVHEGVEIAFAGQRRRIDLKRLSGGKTVTVYGQTEVTRDLMEAREACGATTVYQAAEVRLHDLQGERPYVTFERDGERLRLDCDYIAGCDGFHGISRQSIPAERLKVFERVYPFGWLGLLADTPPVSHELIYANHPRGFALCSQRSATRSRYYVQVPLSEKVEDWSDERFWTELKARLPSEVAEKLVTGPSLEKSIAPLRSFVVEPMQHGRLFLAGDAAHIVPPTGAKGLNLAASDVSTLYRLLLKAYREGRGELLERYSAICLRRIWKAERFSWWMTSVLHRFPDTDAFSQRIQQTELEYYLGSEAGLATIAENYVGLPYEEIE
4OZQ , Knot 275 720 0.83 40 276 651 
MKIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYENGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDAALAAAQTNAAAENSQVTVAVRVRPFSKREKTEKASQVVFTNGEEITVEHPDMKQVYSFIYDVSFWSFDECHPGYASQTTVYETLAAPLLDRAFEGYNTCLFAYGQTGSGKSYTMMGLNEEPGIIPRFCEDLFAQIAKKQTSEVSYHLEMSFFEVYNEKIHDLLVCKGENGQRKQPLRAREHPVSGPYVEGLSMNVVSSYSDIQSWLELGNKQRATAATGMNDKSSRSHSVFTLVMTQTKTEVVEGEEHDHRITSRINLVDLAGSERCSTAHSSGQRLKEGVSINKSLLTLGKVISALSEQANGKRVFIPYRESTLTWLLKESLGGNSKTAMIATVSPAASNIEETLSTLRYATQARLIVN

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 \(c\).

\(|P_{f(6JKK_1)}(2) \setminus P_{f(1IUW_1)}(2)|=90\), \(|P_{f(1IUW_1)}(2) \setminus P_{f(6JKK_1)}(2)|=95\). 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 1:11101010110101100110010101010011000111010010100010100001010011101101010110010010011100001000101010101000100001110010100111100001010010001011110001001000010001110100011011001011111010101001111011010100101011011101010111000010010000110010100101100010101110110111110010110001001111000110010001100110011010000011010010001001110000011011001001100
Pair \(Z_2\) Length of longest common subsequence
6JKK_1,1IUW_1 185 3
6JKK_1,4OZQ_1 177 4
1IUW_1,4OZQ_1 150 4

Newick tree

 
[
	6JKK_1:95.13,
	[
		4OZQ_1:75,1IUW_1:75
	]:20.13
]

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 \((0.85)(0.8)(\frac{735 }{\log_{20} 735}-\frac{341}{\log_{20}341})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 6JKK_1 1IUW_1 137 129.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]