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Parikh vectors
8JER_1 3AOE_1 5RYJ_1 Letter Amino acid
8 45 22 V Valine
21 19 31 I Isoleucine
5 7 7 M Methionine
15 12 24 F Phenylalanine
4 27 20 P Proline
1 8 8 W Tryptophan
7 0 6 C Cysteine
19 34 43 L Leucine
19 15 32 K Lycine
15 11 38 S Serine
15 21 39 T Threonine
7 15 16 Y Tyrosine
8 34 16 R Arginine
14 20 21 D Aspartic acid
15 30 37 E Glutamic acid
18 46 29 G Glycine
12 7 17 H Histidine
16 50 17 A Alanine
15 13 17 N Asparagine
6 10 23 Q Glutamine 

8JER_1|Chain A|Guanine nucleotide-binding protein G(o) subunit alpha|Homo sapiens (9606)
>3AOE_1|Chains A, B|Glutamate dehydrogenase|Thermus thermophilus (262724)
>5RYJ_1|Chain A|Phosphatidylinositol 3,4,5-trisphosphate 5-phosphatase 1|Homo sapiens (9606)
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 \(w=f(c)\)
8JER , Knot 106 240 0.80 40 162 229 
MGHHHHHHENLYFQGTLSAEERAALERSKAIEKNLKEDGISAAKDVKLLLLGADNSGKSTIVKQMKIIHGGSGGSGGTTGIVETHFTFKNLHFRLFDVGGQRSERKKWIHCFEDVTAIIFCVDLSDYNRMHESLMLFDSICNNKFFIDTSIILFLNKKDLFGEKIKKSPLTICFPEYTGPNTYEDAAAYIQAQFESKNRSPNKEIYCHMTCATDTNNAQVIFDAVTDIIIANNLRGCGLY
3AOE , Knot 172 424 0.81 38 205 388 
MKSEPLSYLGKDGGPWEIFTEQVDRVVPYLGRLAPLAESLKRPKRVLIVDVPVRLDDGSVAYFEGYRVHHNTARGPAKGGVRYHPEVTLSEVMALAGWMTIKNAAVGLPYGGGKGGIRVDPRKLSPGELERLTRRYTSEIGILLGPDRDIPAPDVNTGEREMAWMMDTYSMNVGRTVPGVVTGKPIALGGSLGRRDATGRGVFITAAAAAEKIGLQVEGARVAIQGFGNVGNAAARAFHDHGARVVAVQDHTGTVYNEAGIDPYDLLRHVQEFGGVRGYPKAEPLPAADFWGLPVEFLVPAALEKQITEQNAWRIRARIVAEGANGPTTPAADDILLEKGVLVVPDVIANAGGVTVSYFEWVQDFNSYFWTEEEINARLERVLRNAFEAVWQVAQEKKIPLRTAAYVVAATRVLEARALRGLYP
5RYJ , Knot 195 463 0.86 40 252 447 
SMEQPEPDMITIFIGTWNMGNAPPPKKITSWFLSKGQGKTRDDSADYIPHDIYVIGTQEDPLSEKEWLEILKHSLQEITSVTFKTVAIHTLWNIRIVVLAKPEHENRISHICTDNVKTGIANTLGNKGAVGVSFMFNGTSLGFVNSHLTSGSEKKLRRNQNYMNILRFLALGDKKLSPFNITHRFTHLFWFGDLNYRVDLPTWEAETIIQKIKQQQYADLLSHDQLLTERREQKVFLHFEEEEITFAPTYRFERLTRDKYAYTKQKATGMKYNLPSWCDRVLWKSYPLVHVVCQSYGSTSDIMTSDHSPVFATFEAGVTSQFVSKNGPGTVDSQGQIEFLRCYATLKTKSQTKFYLEFHSSCLESFVKSQEGENEEGSEGELVVKFGETLPKLKPIISDPEYLLDQHILISIKSSDSDESYGEGCIALRLEATETQLPIYTPLTHHGELTGHFQGEIKLQTSQ

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 \(c\).

\(|P_{f(8JER_1)}(2) \setminus P_{f(3AOE_1)}(2)|=70\), \(|P_{f(3AOE_1)}(2) \setminus P_{f(8JER_1)}(2)|=113\). 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 1:110000000010101010100011100001100010001101100101111110001000110010110110110110011100010100101011011100000001100100101111010100000100011110010000111000111110000111001000110101100011000001110101010000001000100010010000010111011001111001010110
Pair \(Z_2\) Length of longest common subsequence
8JER_1,3AOE_1 183 4
8JER_1,5RYJ_1 186 3
3AOE_1,5RYJ_1 165 5

Newick tree

 
[
	8JER_1:95.28,
	[
		3AOE_1:82.5,5RYJ_1:82.5
	]:12.78
]

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 \((0.85)(0.8)(\frac{664 }{\log_{20} 664}-\frac{240}{\log_{20}240})=118.\)
Status Protein1 Protein2 d d1/2
Query variables 8JER_1 3AOE_1 151 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]