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Parikh vectors
3QQC_1 8PWN_1 9NJV_1 Letter Amino acid
24 59 14 A Alanine
32 17 11 R Arginine
46 16 8 E Glutamic acid
37 21 12 G Glycine
17 18 7 H Histidine
14 19 4 N Asparagine
20 21 7 D Aspartic acid
35 33 8 I Isoleucine
32 20 14 K Lycine
15 17 0 S Serine
28 31 12 V Valine
9 16 4 Q Glutamine
37 48 9 L Leucine
18 16 12 T Threonine
11 13 6 Y Tyrosine
11 14 0 C Cysteine
10 8 4 M Methionine
9 22 3 F Phenylalanine
26 17 6 P Proline
5 7 1 W Tryptophan 

3QQC_1|Chain A|DNA-directed RNA polymerase subunit b, DNA-directed RNA polymerase subunit a', DNA-directed RNA polymerase subunit A''|Pyrococcus furiosus (2261)
>8PWN_1|Chain A|Adenosine receptor A2a,Soluble cytochrome b562|Homo sapiens (9606)
>9NJV_1|Chain A[auth 13]|Large ribosomal subunit protein uL13|Escherichia coli (562)
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 \(w=f(c)\)
3QQC , Knot 176 436 0.81 40 217 407 
MKTEVWVCENCGHIALEDKRRRRVYCPVCGEEERISKVEMSYAFKLLLDELKAMVIRPKLNLSEGGKVIGSIEFGILSPQEIRKMSAVEVTVPDTYDDDGYPIEGGVMDKRMGVIDPGLRCETCGGRAGECPGHFGHIELARPVIHVGFAKTIYRILESTCRECGRIKLTDEEIEEYMKKIELARNRRSEFNEIIKEIHKKAKERMVCPHCGAPQYPIKFEKPTIYWEIRKDEQGNEYRHRLMPTEVRDWLEKIPDKDLPLLGLDPEKSRPEWMVLTVLPVPPVTARPSITLETGIRAEDDLTHKLVDIIRINNRLKQNIEAGAPQLIIEDLWDLLQYHVTTYINNEAPGVPPAKHKSGRPLKTLAQRLKGKEGRFRGNLSGGAGSGAGSGSVLARAAFEITVQHLFEAAEKGEVDNLNGVIENVLIGQKHHHHHH
8PWN , Knot 174 433 0.81 40 226 393 
DYKDDDDGAPPIMGSSVYITVELAIAVLAILGNVLVCWAVWLNSNLQNVTNYFVVSLAAADILVGVLAIPFAITISTGFCAACHGCLFIACFVLVLAQSSIFSLLAIAIDRYIAIAIPLRYNGLVTGTRAAGIIAICWVLSFAIGLTPMLGWNNCGQPKEGKAHSQGCGEGQVACLFEDVVPMNYMVYFNFFACVLVPLLLMLGVYLRIFAAARRQLADLEDNWETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMKDFRHGFDILVGQIDDALKLANEGKVKEAQAAAEQLKTTRNAYIQKYLERARSTLQKEVHAAKSAAIIAGLFALCWLPLHIINCFTFFCPDCSHAPLWLMYLAIVLAHTNSVVNPFIYAYRIREFRQTFRKIIRSHVLRQQEPFKAAAHHHHHHHHHH
9NJV , Knot 67 142 0.78 36 114 136 
MKTFTAKPETVKRDWYVVDATGKTLGRLATELARRLRGKHKAEYTPHVDTGDYIIVLNADKVAVTGNKRTDKVYYHHTGHIGGIKQATFEEMIARRPERVIEIAVKGMLPKGPLGRAMFRKLKVYAGNEHNHAAQQPQVLDI

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 \(c\).

\(|P_{f(3QQC_1)}(2) \setminus P_{f(8PWN_1)}(2)|=80\), \(|P_{f(8PWN_1)}(2) \setminus P_{f(3QQC_1)}(2)|=89\). 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 1:1000111000010111000000010011010000100101001101110010111101010100110111010111101001001011010110000001011011110001111011100000110110011011010110111011110010011000000010101000010001001011000000100110010001000110100111001101001010101000001000000111001001100110001111110100001011110111111101010101001101000100011011010001000101111011100110110001000100011111110000101100110010100101010101111011101011101110101001101100101001011100111100000000
Pair \(Z_2\) Length of longest common subsequence
3QQC_1,8PWN_1 169 6
3QQC_1,9NJV_1 169 4
8PWN_1,9NJV_1 196 3

Newick tree

 
[
	9NJV_1:93.71,
	[
		3QQC_1:84.5,8PWN_1:84.5
	]:9.21
]

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 \((0.85)(0.8)(\frac{869 }{\log_{20} 869}-\frac{433}{\log_{20}433})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 3QQC_1 8PWN_1 148 146.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]