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Parikh vectors
1TBP_1 9KKE_1 6MJG_1 Letter Amino acid
14 103 33 V Valine
11 69 16 R Arginine
2 20 10 H Histidine
16 56 18 K Lycine
12 65 16 F Phenylalanine
6 45 16 Y Tyrosine
15 123 33 L Leucine
8 32 30 P Proline
10 68 23 T Threonine
5 56 24 D Aspartic acid
2 9 7 C Cysteine
5 48 18 Q Glutamine
11 104 26 G Glycine
17 85 25 I Isoleucine
0 8 3 W Tryptophan
17 121 40 A Alanine
4 36 15 N Asparagine
10 64 26 E Glutamic acid
5 30 13 M Methionine
10 110 28 S Serine 

1TBP_1|Chains A, B|TATA-BINDING PROTEIN|Saccharomyces cerevisiae (4932)
>9KKE_1|Chain A|ABC transporter B family member 19|Arabidopsis thaliana (3702)
>6MJG_1|Chain A|fusion protein of dbOphMA and methylated peptide|Dendrothele bispora CBS 962.96 (1314807)
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 \(w=f(c)\)
1TBP , Knot 82 180 0.78 38 127 169 
MGIVPTLQNIVATVTLGCRLDLKTVALHARNAEYNPKRFAAVIMRIREPKTTALIFASGKMVVTGAKSEDDSKLASRKYARIIQKIGFAAKFTDFKIQNIVGSCDVKFPIRLEGLAFSHGTFSSYEPELFPGLIYRMVKPKIVLLIFVSGKIVLTGAKQREEIYQAFEAIYPVLSEFRKM
9KKE , Knot 423 1252 0.80 40 317 1026 
MSETNTTDAKTVPAEAEKKKEQSLPFFKLFSFADKFDYLLMFVGSLGAIVHGSSMPVFFLLFGQMVNGFGKNQMDLHQMVHEVSRYSLYFVYLGLVVCFSSYAEIACWMYSGERQVAALRKKYLEAVLKQDVGFFDTDARTGDIVFSVSTDTLLVQDAISEKVGNFIHYLSTFLAGLVVGFVSAWKLALLSVAVIPGIAFAGGLYAYTLTGITSKSRESYANAGVIAEQAIAQVRTVYSYVGESKALNAYSDAIQYTLKLGYKAGMAKGLGLGCTYGIACMSWALVFWYAGVFIRNGQTDGGKAFTAIFSAIVGGMSLGQSFSNLGAFSKGKAAGYKLMEIINQRPTIIQDPLDGKCLDQVHGNIEFKDVTFSYPSRPDVMIFRNFNIFFPSGKTVAVVGGSGSGKSTVVSLIERFYDPNSGQILLDGVEIKTLQLKFLREQIGLVNQEPALFATTILENILYGKPDATMVEVEAAASAANAHSFITLLPKGYDTQVGERGVQLSGGQKQRIAIARAMLKDPKILLLDEATSALDASSESIVQEALDRVMVGRTTVVVAHRLCTIRNVDSIAVIQQGQVVETGTHEELIAKSGAYASLIRFQEMVGTRDFSNPSTRRTRSTRLSHSLSTKSLSLRSGSLRNLSYSYSTGADGRIEMISNAETDRKTRAPENYFYRLLKLNSPEWPYSIMGAVGSILSGFIGPTFAIVMSNMIEVFYYTDYDSMERKTKEYVFIYIGAGLYAVGAYLIQHYFFSIMGENLTTRVRRMMLSAILRNEVGWFDEDEHNSSLIAARLATDAADVKSAIAERISVILQNMTSLLTSFIVAFIVEWRVSLLILGTFPLLVLANFAQQLSLKGFAGDTAKAHAKTSMIAGEGVSNIRTVAAFNAQSKILSLFCHELRVPQKRSLYRSQTSGFLFGLSQLALYGSEALILWYGAHLVSKGVSTFSKVIKVFVVLVITANSVAETVSLAPEIIRGGEAVGSVFSVLDRQTRIDPDDADADPVETIRGDIEFRHVDFAYPSRPDVMVFRDFNLRIRAGHSQALVGASGSGKSSVIAMIERFYDPLAGKVMIDGKDIRRLNLKSLRLKIGLVQQEPALFAATIFDNIAYGKDGATESEVIDAARAANAHGFISGLPEGYKTPVGERGVQLSGGQKQRIAIARAVLKNPTVLLLDEATSALDAESECVLQEALERLMRGRTTVVVAHRLSTIRGVDCIGVIQDGRIVEQGSHSELVSRPEGAYSRLLQLQTHRI
6MJG , Knot 179 420 0.85 40 232 396 
SNAMESSTQTKPGSLIVVGTGIESIGQMTLQALSYIEAASKVFYCVIDPATEAFILTKNKNCVDLYQYYDNGKSRMDTYTQMAELMLKEVRNGLDVVGVFYGHPGVFVNPSHRALAIARSEGYQARMLPGVSAEDCLFADLCIDPSNPGCLTYEASDFLIRERPVNVHSHLILFQVGCVGIADFNFSGFDNSKFTILVDRLEQEYGPDHTVVHYIAAMMPHQDPVTDKFTIGQLREPEIAKRVGGVSTFYIPPKARKDINTDIIRLLEFLPAGKVPDKHTQIYPPNQWEPDVPTLPPYGQNEQAAITRLEAHAPPEEYQPLATSKAMTDVMTKLALDPKALAEYKADHRAFAQSVPDLTPQERAALELGDSWAIRCAMKNMPSSLLEAASQSVEEASMNGFPWVIVTGIVGVIGSVVSSA

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 \(c\).

\(|P_{f(1TBP_1)}(2) \setminus P_{f(9KKE_1)}(2)|=9\), \(|P_{f(9KKE_1)}(2) \setminus P_{f(1TBP_1)}(2)|=199\). 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 1:111110100111010110010100111010010001001111110100100011111010111011000000011000010110011111010010100111000101110101111001010000101111110011010111111101011101100000100110110111001001
Pair \(Z_2\) Length of longest common subsequence
1TBP_1,9KKE_1 208 4
1TBP_1,6MJG_1 177 4
9KKE_1,6MJG_1 125 5

Newick tree

 
[
	1TBP_1:10.49,
	[
		6MJG_1:62.5,9KKE_1:62.5
	]:42.99
]

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 \((0.85)(0.8)(\frac{1432 }{\log_{20} 1432}-\frac{180}{\log_{20}180})=330.\)
Status Protein1 Protein2 d d1/2
Query variables 1TBP_1 9KKE_1 397 226.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]