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Parikh vectors
9JCO_1 1PLW_1 5ICE_1 Letter Amino acid
22 0 28 K Lycine
30 0 25 A Alanine
28 0 26 D Aspartic acid
19 0 7 Q Glutamine
25 0 31 I Isoleucine
27 0 34 L Leucine
18 0 19 T Threonine
16 0 15 N Asparagine
16 2 24 G Glycine
7 1 16 M Methionine
18 1 10 F Phenylalanine
20 0 20 S Serine
31 0 9 R Arginine
7 0 6 C Cysteine
22 0 18 E Glutamic acid
8 0 14 P Proline
6 0 12 H Histidine
4 0 6 W Tryptophan
16 1 14 Y Tyrosine
21 0 18 V Valine 

9JCO_1|Chain A|Guanine nucleotide-binding protein G(s) subunit alpha|Homo sapiens (9606)
>1PLW_1|Chain A|Met-enkephalin 1|null
>5ICE_1|Chain A|(S)-norcoclaurine 6-O-methyltransferase|Thalictrum flavum subsp. glaucum (150095)
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 \(w=f(c)\)
9JCO , Knot 157 361 0.85 40 219 345 
MGCTLSAEDKAAVERSKMIEKQLQKDKQVYRATHRLLLLGADNSGKSTIVKQMRIYHVNGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDSARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTSGIFETKFQVDKVNFHMFDVGAQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNDFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRDEFLRISTASGDGRHYCYPHFTCSVDTENARRIFNDCRDIIQRMHLRQYELL
1PLW , Knot 4 5 0.42 8 4 3 
YGGFM
5ICE , Knot 152 352 0.84 40 213 340 
GAMVMINKENLSSQAKLWNFIYGFADSLVLKSAVQLDLANIIHNHGSPMTLSELSLHLPSQPVNQDALYRVLRYLVHMKLFTKSSIDGELRYGLAPPAKFLVKGWDKCMLGAILTITDKDFMAPWHYLKEGILNDGSTSTAFEKALGTNIWDYMAEHPEKNQLFNEGMANDTRLIMSALVKECSSMFDGITTIVDVGGGTGTAVRNIAKAFPHIKCTVYDLPHVIADSPGYTEINSIQGDMFKYIPNADAIMMKCILHDWDDKECIEILKRCKDAVPRDGGKVIIIDIILDVKSEHPYTKMRLTLDLDMMLNTGGKERTEEEWKKLIHDAGYKGYKITHISAVQSVIEAYPY

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 \(c\).

\(|P_{f(9JCO_1)}(2) \setminus P_{f(1PLW_1)}(2)|=217\), \(|P_{f(1PLW_1)}(2) \setminus P_{f(9JCO_1)}(2)|=2\). 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 1:1100101000111000011000100000100100011111100010001100101001010000000000111000010011111011101010110010100100111111110011101011111001100011010100000001000110010010011010011000011000100011100010100101011011100000001100100101111110000000100110010011000110010111110000111001111000100011011000010010101100101001001100011010010101000001010001000010011000001100101000011
Pair \(Z_2\) Length of longest common subsequence
9JCO_1,1PLW_1 219 3
9JCO_1,5ICE_1 156 3
1PLW_1,5ICE_1 209 2

Newick tree

 
[
	1PLW_1:11.08,
	[
		9JCO_1:78,5ICE_1:78
	]:37.08
]

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 \((0.85)(0.8)(\frac{366 }{\log_{20} 366}-\frac{5}{\log_{20}5})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 9JCO_1 1PLW_1 155 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]