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Parikh vectors
1UKQ_1 1AFV_1 3GYW_1 Letter Amino acid
55 12 8 A Alanine
23 6 9 R Arginine
64 7 18 N Asparagine
67 11 7 G Glycine
43 13 17 I Isoleucine
33 2 12 F Phenylalanine
38 8 10 S Serine
44 3 15 D Aspartic acid
36 11 23 L Leucine
61 10 14 T Threonine
48 10 18 V Valine
2 0 3 C Cysteine
26 10 8 Q Glutamine
12 5 9 H Histidine
24 5 27 K Lycine
30 13 7 P Proline
13 4 2 W Tryptophan
34 2 17 Y Tyrosine
19 11 23 E Glutamic acid
14 8 2 M Methionine 

1UKQ_1|Chains A, B|Cyclomaltodextrin glucanotransferase|Bacillus sp. (1410)
>1AFV_1|Chains A, B|HUMAN IMMUNODEFICIENCY VIRUS TYPE 1 CAPSID PROTEIN|Human immunodeficiency virus 1 (11676)
>3GYW_1|Chain A|Nucleosome assembly protein 1, putative|Plasmodium falciparum (36329)
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 \(w=f(c)\)
1UKQ , Knot 267 686 0.84 40 272 622 
APDTSVSNKQNFSTDVIYQIFTDRFSDGNPANNPTGAAFDGSCTNLRLYCGGDWQGIINKINDGYLTGMGITAIWISQPVENIYSVINYSGVNNTAYHGYWARDFKKTNPAYGTMQDFKNLIDTAHAHNIKVIIDFAPNHTSPASSDDPSFAENGRLYDNGNLLGGYTNDTQNLFHHYGGTDFSTIENGIYKNLYDLADLNHNNSSVDVYLKDAIKMWLDLGVDGIRVDAVKHMPFGWQKSFMATINNYKPVFTFGEWFLGVNEISPEYHQFANESGMSLLDFRFAQKARQVFRDNTDNMYGLKAMLEGSEVDYAQVNDQVTFIDNHDMERFHTSNGDRRKLEQALAFTLTSRGVPAIYYGSEQYMSGGNDPDNRARLPSFSTTTTAYQVIQKLAPLRKSNPAIAYGSTHERWINNDVIIYERKFGNNVAVVAINRNMNTPASITGLVTSLPRGSYNDVLGGILNGNTLTVGAGGAASNFTLAPGGTAVWQYTTDATTPIIGNVGPMMAKPGVTITIDGRGFGSGKGTVYFGTTAVTGADIVAWEDTQIQVKIPAVPGGIYDIRVANAAGAASNIYDNFEVLTGDQVTVRFVINNATTALGQNVFLTGNVSELGNWDPNNAIGPMYNQVVYQYPTWYYDVSVPAGQTIEFKFLKKQGSTVTWEGGANRTFTTPTSGTATVNVNWQP
1AFV , Knot 73 151 0.80 38 121 146 
PIVQNLQGQMVHQAISPRTLNAWVKVVEEKAFSPEVIPMFSALSEGATPQDLNTMLNTVGGHQAAMQMLKETINEEAAEWDRLHPVHAGPIAPGQMREPRGSDIAGTTSTLQEQIGWMTHNPPIPVGEIYKRWIILGLNKIVRMYSPTSIL
3GYW , Knot 109 249 0.80 40 166 241 
TDLTEEQKETLKKLKLYQKEYYDYESKFEYELFLLRQKYHDLYGPIYDKRREALVGNGEAKIGTPNLPEFWLRALRNNNTVSHVIEDHDEEILVYLNDIRCDYIKKNKEKKEGFILSFYFATNPFFSNSVLTKTYHMKCVDCDNEPVLLHTEATVIDWYDNKNILKKNVVKKQHNKNSREVKTVQQTVNRDSFFHFFTSHKVPNSNVIKQLSKHEVAQLEMIIEGDYEVALTIKERIIPYAVDYYLGII

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 \(c\).

\(|P_{f(1UKQ_1)}(2) \setminus P_{f(1AFV_1)}(2)|=178\), \(|P_{f(1AFV_1)}(2) \setminus P_{f(1UKQ_1)}(2)|=27\). 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 1:11000100000100011001100010010110010111101000010100110101110010010101111011110011001001100011000100101100100001101010010011001010010111011100001100001011001010001011110000000110001100100100110001001101000000101010011011101110110101100111110001110100001110110111110010100001100011011010110010011000000101101110100100101000101100001001000010000100111101000111110010000101100100010110100000100110011110000111101000001100011100001100111111000100110101110011010000111111010010111111100101111101110000010011110111111011101010101110101010110011011011110000101011111111001011011111001000101101001010111001001110011101010011010100111110001100010100010111100101011000100101011100010010010101010101
Pair \(Z_2\) Length of longest common subsequence
1UKQ_1,1AFV_1 205 4
1UKQ_1,3GYW_1 188 3
1AFV_1,3GYW_1 185 3

Newick tree

 
[
	1UKQ_1:10.21,
	[
		3GYW_1:92.5,1AFV_1:92.5
	]:7.71
]

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 \((0.85)(0.8)(\frac{837 }{\log_{20} 837}-\frac{151}{\log_{20}151})=192.\)
Status Protein1 Protein2 d d1/2
Query variables 1UKQ_1 1AFV_1 241 145.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]