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Parikh vectors
5OLV_1 8EKN_1 2OPS_1 Letter Amino acid
33 41 40 I Isoleucine
22 35 13 F Phenylalanine
60 42 25 A Alanine
19 30 18 N Asparagine
16 30 36 Q Glutamine
17 45 36 P Proline
31 49 37 V Valine
18 24 8 H Histidine
48 82 47 L Leucine
17 54 18 S Serine
16 38 40 T Threonine
7 22 19 W Tryptophan
13 39 21 Y Tyrosine
17 40 17 R Arginine
21 50 24 D Aspartic acid
14 6 3 C Cysteine
16 66 46 E Glutamic acid
21 70 32 G Glycine
20 45 56 K Lycine
8 11 6 M Methionine 

5OLV_1|Chain A|Adenosine receptor A2a,Soluble cytochrome b562,Adenosine receptor A2a|Homo sapiens (9606)
>8EKN_1|Chains A, B|Chaetomium alpha glucosidase|Thermochaetoides thermophila (759272)
>2OPS_1|Chain A|Reverse transcriptase/ribonuclease H|HIV-1 M:B_HXB2R (11706)
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 \(w=f(c)\)
5OLV , Knot 173 434 0.80 40 226 394 
ADYKDDDDGAPPIMGSSVYITVELAIAVLAILGNVLVCWAVWLNSNLQNVTNYFVVSLAAADILVGVLAIPFAITISTGFCAACHGCLFIACFVLVLAQSSIFSLLAIAIDRYIAIAIPLRYNGLVTGTRAAGIIAICWVLSFAIGLTPMLGWNNCGQPKEGKAHSQGCGEGQVACLFEDVVPMNYMVYFNFFACVLVPLLLMLGVYLRIFAAARRQLADLEDNWETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMKDFRHGFDILVGQIDDALKLANEGKVKEAQAAAEQLKTTRNAYIQKYLERARSTLQKEVHAAKSAAIIAGLFALCWLPLHIINCFTFFCPDCSHAPLWLMYLAIVLAHTNSVVNPFIYAYRIREFRQTFRKIIRSHVLRQQEPFKAAAHHHHHHHHHH
8EKN , Knot 311 819 0.85 40 312 757 
MGILPSPGMPALLSLVSLLSVLLMGCVAETGVEGESILHSEIGRLNNQSLLWGPYRPNIYFGTRPRIGKSLMTGLMWGKIESYTDFQHTVRYTCEQNEGMKGYGWDEYDPRRGGIQSIHDIQNGLDITTSFVKIPGGAHGGSWAARIKGTLNDDAPKDQKTIVVFYVSQEGENSELEAVPSENEFGYEGDVILKGRSEALGNYKLVVTKGKGVIPQSDHDLSRLRGPGQTVVQSLTYPDEVLWQAKPILFQQLKAGIDWLVENKYDVADPPPPWQVYLLANKPGSGNVHIVQKVFEGDFEFDILFSSESAGKEVTSKDLEREVKQATEVFGERFARVFDLKAPFQGDNYKKFGKSMFSNLIGGIGYFYGHSLVDRSYAPEYDEENEGFWEDAAEARARHQEALEGPYELFTSIPSRPFFPRGFLWDEGFHLLPIADWDIDLALEIIKSWYNLMDEDGWIAREQILGAEARSKVPKEFQTQYPHYANPPTLFLVLDNFVERLRKNNASQPVVKDNLSLDETLSTASVDNPEVGLEYLRRLYPLLRRQFDWFRKTQAGDIKSYDREAYSTKEAYRWRGRTVSHCLTSGLDDYPRPQPPHPGELHVDLMSWVGVMVKSLISIGSLLGATEDVEFYTKVLDAIEHNLDDLHWSEKEGCYCDATIDEFEEHKLVCHKGYISLFPFLTGLLKPDSPKLGKLLALIGDESELWSPYGLRSLSKKDEFYGTAENYWRSPVWININYLAIVQLYNIATQDGPYKETARDLYTRLRKNIVETVYRNWEETGFAWEQYNPETGKGQRTQHFTGWTSLVVKIMSGHHHHHH
2OPS , Knot 216 542 0.83 40 250 501 
ISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKKKKSVTVLDVGDAYFSVPLDEDFRKYTAFTIPSINNETPGIRYQYNVLPQGWKGSPAIFQSSMTKILEPFRKQNPDIVIYQYMDDLCVGSDLEIGQHRTKIEELRQHLLRWGLTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDIQKLVGKLNWASQIYPGIKVRQLCKLLRGTKALTEVIPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMRGAHTNDVKQLTEAVQKITTESIVIWGKTPKFKLPIQKETWETWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIVGAETFYVDGAANRETKLGKAGYVTNRGRQKVVTLTDTTNQKTELQAIYLALQDSGLEVNIVTDSQYALGIIQAQPDQSESELVNQIIEQLIKKEKVYLAWVPAHKGIG

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 \(c\).

\(|P_{f(5OLV_1)}(2) \setminus P_{f(8EKN_1)}(2)|=35\), \(|P_{f(8EKN_1)}(2) \setminus P_{f(5OLV_1)}(2)|=121\). 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 1:10000000111111100101010111111111101110111110001001000111011110111111111111010011011001011110111111000110111111000111111100011101001111111011101111101111100010100101000101010110110011110011010111011111111111010111110001101000100100010110010011010011001011110100101101000010010100100110111101001101100101001011100100000101000100100010001011001111111111011110110010110100001111110111111000011011101001001000100110001100001101110000000000
Pair \(Z_2\) Length of longest common subsequence
5OLV_1,8EKN_1 156 6
5OLV_1,2OPS_1 172 5
8EKN_1,2OPS_1 138 4

Newick tree

 
[
	5OLV_1:86.02,
	[
		8EKN_1:69,2OPS_1:69
	]:17.02
]

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 \((0.85)(0.8)(\frac{1253 }{\log_{20} 1253}-\frac{434}{\log_{20}434})=212.\)
Status Protein1 Protein2 d d1/2
Query variables 5OLV_1 8EKN_1 271 203.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]