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Parikh vectors
4KKT_1 4ALM_1 2WXF_1 Letter Amino acid
9 11 54 R Arginine
14 24 63 A Alanine
19 14 39 D Aspartic acid
8 10 42 F Phenylalanine
29 18 61 S Serine
0 0 16 W Tryptophan
0 0 23 C Cysteine
20 24 50 G Glycine
1 11 30 H Histidine
34 19 34 I Isoleucine
10 7 42 P Proline
9 11 27 Y Tyrosine
30 22 55 V Valine
15 16 34 N Asparagine
5 10 41 Q Glutamine
19 22 75 E Glutamic acid
24 26 122 L Leucine
33 18 65 K Lycine
5 7 31 M Methionine
12 12 36 T Threonine 

4KKT_1|Chains A, B, C, D|Membrane fusion protein|Borrelia burgdorferi (224326)
>4ALM_1|Chains A, B, C, D|ENOYL-[ACYL-CARRIER-PROTEIN] REDUCTASE [NADPH]|STAPHYLOCOCCUS AUREUS (1280)
>2WXF_1|Chain A|PHOSPHATIDYLINOSITOL-4,5-BISPHOSPHATE 3-KINASE CATALYTIC SUBUNIT DELTA ISOFORM|MUS MUSCULUS (10090)
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 \(w=f(c)\)
4KKT , Knot 125 296 0.80 36 154 276 
GSHMVGDNKLDDKNIDKEKESSYRFPVIAMKVKKGILSDYLSLNGDVDTKVKADIFPDAVGKITSLRIKLGAYVQKGQIVATLDPSRPGSVYLKSPVRAPISGYILNITKKIGETVNPQSNIAVVGRIDTKQILTYVSEKYISNIKVGNDAIIEVGAYSNEKFKAKVSEISPILDSKSRTIEVYLTPIGSNLDKLIIGMFSKIKLITKRFKDVIKISREAVVEREGKKFVFKVDLESKSVQMLPITVLFEIDNIVALSGEVEENDLIVVEGMSALSNGSLINLVDTKEGLSAESNI
4ALM , Knot 120 282 0.80 36 179 271 
MKHHHHHHPMSDYDIPTTENLYFQGAMVNLENKTYVIMGIANKRSIAFGVAKVLDQLGAKLVFTYRKERSRKELEKLLEQLNQPEAHLYQIDVQSDEEVINGFEQIGKDVGNIDGVYHSIAFANMEDLRGRFSETSREGFLLAQDISSYSLTIVAHEAKKLMPEGGSIVATTYLGGEFAVQNYNVMGVAKASLEANVKYLALDLGPDNIRVNAISAGPIRTLSAKGVGGFNTILKEIEERAPLKRNVDQVEVGKTAAYLLSDLSSGVTGENIHVDSGFHAIK
2WXF , Knot 354 940 0.86 40 322 842 
GGDRVKKLINSQISLLIGKGLHEFDSLRDPEVNDFRTKMRQFCEEAAAHRQQLGWVEWLQYSFPLQLEPSARGWRAGLLRVSNRALLVNVKFEGSEESFTFQVSTKDMPLALMACALRKKATVFRQPLVEQPEEYALQVNGRHEYLYGNYPLCHFQYICSCLHSGLTPHLTMVHSSSILAMRDEQSNPAPQVQKPRAKPPPIPAKKPSSVSLWSLEQPFSIELIEGRKVNADERMKLVVQAGLFHGNEMLCKTVSSSEVNVCSEPVWKQRLEFDISVCDLPRMARLCFALYAVVEKAKKARSTKKKSKKADCPIAWANLMLFDYKDQLKTGERCLYMWPSVPDEKGELLNPAGTVRGNPNTESAAALVIYLPEVAPHPVYFPALEKILELGRHGERGRITEEEQLQLREILERRGSGELYEHEKDLVWKMRHEVQEHFPEALARLLLVTKWNKHEDVAQMLYLLCSWPELPVLSALELLDFSFPDCYVGSFAIKSLRKLTDDELFQYLLQLVQVLKYESYLDCELTKFLLGRALANRKIGHFLFWHLRSEMHVPSVALRFGLIMEAYCRGSTHHMKVLMKQGEALSKLKALNDFVKVSSQKTTKPQTKEMMHMCMRQETYMEALSHLQSPLDPSTLLEEVCVEQCTFMDSKMKPLWIMYSSEEAGSAGNVGIIFKNGDDLRQDMLTLQMIQLMDVLWKQEGLDLRMTPYGCLPTGDRTGLIEVVLHSDTIANIQLNKSNMAATAAFNKDALLNWLKSKNPGEALDRAIEEFTLSCAGYCVATYVLGIGDRHSDNIMIRESGQLFHIDFGHFLGNFKTKFGINRERVPFILTYDFVHVIQQGKTNNSEKFERFRGYCERAYTILRRHGLLFLHLFALMRAAGLPELSCSKDIQYLKDSLALGKTEEEALKHFRVKFNEALRESWKTKVNWLAHNVSKDNRQ

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 \(c\).

\(|P_{f(4KKT_1)}(2) \setminus P_{f(4ALM_1)}(2)|=53\), \(|P_{f(4ALM_1)}(2) \setminus P_{f(4KKT_1)}(2)|=78\). 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 1:10011100010000100000000011111101001110001010101000101011101110100101011101001011101010011010100110111010110100011001010001111101000011001000010010110011101110000010101001011100000010101011100100111111001011000100110100011100010011101010000101111011101001111010100001111011011001011011000011010001
Pair \(Z_2\) Length of longest common subsequence
4KKT_1,4ALM_1 131 4
4KKT_1,2WXF_1 196 4
4ALM_1,2WXF_1 193 3

Newick tree

 
[
	2WXF_1:10.73,
	[
		4KKT_1:65.5,4ALM_1:65.5
	]:40.23
]

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 \((0.85)(0.8)(\frac{578 }{\log_{20} 578}-\frac{282}{\log_{20}282})=83.3\)
Status Protein1 Protein2 d d1/2
Query variables 4KKT_1 4ALM_1 99 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]