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Parikh vectors
8BCY_1 5JEU_1 9NEZ_1 Letter Amino acid
51 0 12 P Proline
41 1 5 T Threonine
60 0 22 V Valine
44 0 13 D Aspartic acid
30 0 2 H Histidine
142 0 18 L Leucine
29 0 9 Y Tyrosine
51 0 7 Q Glutamine
55 0 9 R Arginine
36 0 6 N Asparagine
29 4 13 C Cysteine
48 4 24 G Glycine
66 0 10 S Serine
67 1 17 A Alanine
37 0 19 I Isoleucine
65 0 24 K Lycine
26 0 8 M Methionine
46 0 11 F Phenylalanine
17 0 5 W Tryptophan
78 0 27 E Glutamic acid 

8BCY_1|Chain A|Phosphatidylinositol 4,5-bisphosphate 3-kinase catalytic subunit delta isoform|Homo sapiens (9606)
>5JEU_1|Chain A|DNA (5'-D(*TP*CP*GP*GP*CP*GP*CP*CP*GP*A)-3')|synthetic construct (32630)
>9NEZ_1|Chains A, E|Sulfhydrogenase 1 subunit delta|Pyrococcus furiosus (2261)
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 \(w=f(c)\)
8BCY , Knot 380 1018 0.86 40 323 906 
NQSVVVDFLLPTGVYLNFPVSRNANLSTIKQLLWHRAQYEPLFHMLSGPEAYVFTCINQTAEQQELEDEQRRLCDVQPFLPVLRLVAREGDRVKKLINSQISLLIGKGLHEFDSLCDPEVNDFRAKMCQFCEEAAARRQQLGWEAWLQYSFPLQLEPSAQTWGPGTLRLPNRALLVNVKFEGSEESFTFQVSTKDVPLALMACALRKKATVFRQPLVEQPEDYTLQVNGRHEYLYGSYPLCQFQYICSCLHSGLTPHLTMVHSSSILAMRDEQSNPAPQVQKPRAKPPPIPAKKPSSVSLWSLEQPFRIELIQGSKVNADERMKLVVQAGLFHGNEMLCKTVSSSEVSVCSEPVWKQRLEFDINICDLPRMARLCFALYAVIEKAKKARSTKKKSKKADCPIAWANLMLFDYKDQLKTGERCLYMWPSVPDEKGELLNPTGTVRSNPNTDSAAALLICLPEVAPHPVYYPALEKILELGRHSECVHVTEEEQLQLREILERRGSGELYEHEKDLVWKLRHEVQEHFPEALARLLLVTKWNKHEDVAQMLYLLCSWPELPVLSALELLDFSFPDCHVGSFAIKSLRKLTDDELFQYLLQLVQVLKYESYLDCELTKFLLDRALANRKIGHFLFWHLRSEMHVPSVALRFGLILEAYCRGSTHHMKVLMKQGEALSKLKALNDFVKLSSQKTPKPQTKELMHLCMRQEAYLEALSHLQSPLDPSTLLAEVCVEQCTFMDSKMKPLWIMYSNEEAGSGGSVGIIFKNGDDLRQDMLTLQMIQLMDVLWKQEGLDLRMTPYGCLPTGDRTGLIEVVLRSDTIANIQLNKSNMAATAAFNKDALLNWLKSKNPGEALDRAIEEFTLSCAGYCVATYVLGIGDRHSDNIMIRESGQLFHIDFGHFLGNFKTKFGINRERVPFILTYDFVHVIQQGKTNNSEKFERFRGYCERAYTILRRHGLLFLHLFALMRAAGLPELSCSKDIQYLKDSLALGKTEEEALKHFRVKFNEALRESWKTKVNWL
5JEU , Knot 6 10 0.46 8 6 8 
TCGGCGCCGA
9NEZ , Knot 119 261 0.84 40 169 251 
MGKVRIGFYALTSCYGCQLQLAMMDELLQLIPNAEIVCWFMIDRDSIEDEKVDIAFIEGSVSTEEEVELVKKIRENAKIVVAVGACAVQGGVQSWSEKPLEELWKKVYGDAKVKFQPKKAEPVSKYIKVDYNIYGCPPEKKDFLYALGTFLIGSWPEDIDYPVCLECRLNGHPCILLEKGEPCLGPVTRAGCNARCPGFGVACIGCRGAIGYDVAWFDSLAKVFKEKGMTKEEIIERMKMFNGHDERVEKMVEKIFSGGEQ

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 \(c\).

\(|P_{f(8BCY_1)}(2) \setminus P_{f(5JEU_1)}(2)|=320\), \(|P_{f(5JEU_1)}(2) \setminus P_{f(8BCY_1)}(2)|=3\). 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 1:0001110111101101011100010100100111001000111011011010110010001000010000001001011111101110010010011000101111011001001001010010101001000111000011101110001110101010011110101100111101010100001010100001111111011000101100111001000010101000010100110010010001001101010110000111100000011101001010111111001001011010011010110100101000101110111101001100010000101000111000101010100110110101110111001001000000000100111110111100000100100010111011000101101010100010000111111011011101100111001101100000101000001010011000101010000001110100010001101110111100100000110110110011011110110110101100011011100100100001100110110110000010001001110011100011011110100010110111011111010001000010111001011001011001101000001010000110101000101011001001101001110101000011000101111100000110110111110010010001101011011011100011010101010110100011101110000110101000011101110001110110000110110011001010011001100111110000001110001011010110111010001110000111110001101100100000001001010000100110001111101111101111101000001001000111100000110010101001100010001011
Pair \(Z_2\) Length of longest common subsequence
8BCY_1,5JEU_1 323 2
8BCY_1,9NEZ_1 192 4
5JEU_1,9NEZ_1 169 2

Newick tree

 
[
	8BCY_1:14.43,
	[
		9NEZ_1:84.5,5JEU_1:84.5
	]:60.93
]

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 \((0.85)(0.8)(\frac{1028 }{\log_{20} 1028}-\frac{10}{\log_{20}10})=293.\)
Status Protein1 Protein2 d d1/2
Query variables 8BCY_1 5JEU_1 378 191
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]