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Parikh vectors
7FHV_1 2WXF_1 8EEF_1 Letter Amino acid
25 50 41 G Glycine
10 31 11 M Methionine
6 42 15 F Phenylalanine
10 16 17 W Tryptophan
11 55 32 V Valine
2 23 3 C Cysteine
10 34 26 I Isoleucine
10 61 22 S Serine
11 34 18 N Asparagine
4 75 30 E Glutamic acid
6 65 21 K Lycine
19 42 13 P Proline
13 27 13 Y Tyrosine
47 63 41 A Alanine
30 39 41 D Aspartic acid
20 41 16 Q Glutamine
16 30 9 H Histidine
30 122 29 L Leucine
9 36 31 T Threonine
27 54 20 R Arginine 

7FHV_1|Chains A, B|Polysaccharide lyase|Stenotrophomonas maltophilia K279a (522373)
>2WXF_1|Chain A|PHOSPHATIDYLINOSITOL-4,5-BISPHOSPHATE 3-KINASE CATALYTIC SUBUNIT DELTA ISOFORM|MUS MUSCULUS (10090)
>8EEF_1|Chains A, B|Amine oxidase|Corynebacterium ammoniagenes (1697)
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 \(w=f(c)\)
7FHV , Knot 131 316 0.79 40 179 297 
ACPAPPPGQPDIRAIGYYTDKAGSVIDPALQQQNKDATAPLDRYAADVARMSDDYLRNGDPAAAQCTLSWLGAWADDGAMLGQMIRVNNDQSFYMRQWMLDAVAMAYLKVHDQANPQQRARIDPWLQKLARANLAYWDNPKRRRNNHYYWGGLGVLATGLATDDDALWQAGHAAFQKGIDDIQDDGSLPLEMARGQRALHYHDYALAPLVMMAELARLRGQDWYASRNHAIDRLARRVIEGSRDPAWFNQHTGAAQLPLQASGWVEFYRLRSPDGGVFDAAHARGPFHSPRLGGDLTLMATHGIVRTPLRHHHHHH
2WXF , Knot 354 940 0.86 40 322 842 
GGDRVKKLINSQISLLIGKGLHEFDSLRDPEVNDFRTKMRQFCEEAAAHRQQLGWVEWLQYSFPLQLEPSARGWRAGLLRVSNRALLVNVKFEGSEESFTFQVSTKDMPLALMACALRKKATVFRQPLVEQPEEYALQVNGRHEYLYGNYPLCHFQYICSCLHSGLTPHLTMVHSSSILAMRDEQSNPAPQVQKPRAKPPPIPAKKPSSVSLWSLEQPFSIELIEGRKVNADERMKLVVQAGLFHGNEMLCKTVSSSEVNVCSEPVWKQRLEFDISVCDLPRMARLCFALYAVVEKAKKARSTKKKSKKADCPIAWANLMLFDYKDQLKTGERCLYMWPSVPDEKGELLNPAGTVRGNPNTESAAALVIYLPEVAPHPVYFPALEKILELGRHGERGRITEEEQLQLREILERRGSGELYEHEKDLVWKMRHEVQEHFPEALARLLLVTKWNKHEDVAQMLYLLCSWPELPVLSALELLDFSFPDCYVGSFAIKSLRKLTDDELFQYLLQLVQVLKYESYLDCELTKFLLGRALANRKIGHFLFWHLRSEMHVPSVALRFGLIMEAYCRGSTHHMKVLMKQGEALSKLKALNDFVKVSSQKTTKPQTKEMMHMCMRQETYMEALSHLQSPLDPSTLLEEVCVEQCTFMDSKMKPLWIMYSSEEAGSAGNVGIIFKNGDDLRQDMLTLQMIQLMDVLWKQEGLDLRMTPYGCLPTGDRTGLIEVVLHSDTIANIQLNKSNMAATAAFNKDALLNWLKSKNPGEALDRAIEEFTLSCAGYCVATYVLGIGDRHSDNIMIRESGQLFHIDFGHFLGNFKTKFGINRERVPFILTYDFVHVIQQGKTNNSEKFERFRGYCERAYTILRRHGLLFLHLFALMRAAGLPELSCSKDIQYLKDSLALGKTEEEALKHFRVKFNEALRESWKTKVNWLAHNVSKDNRQ
8EEF , Knot 183 449 0.83 40 239 423 
MSKNKVVIIGAGFAGLVAARELQTAGIEYEILEAKDRIGGRAWTEERMGRPLELGATWVHWFQAHTWTEIMRYGQRTEITASPSGNDAHWVTDGKVVKGTEDDLDEKLTAAMGVTYEGSEEYFPNPHDPLWVLSDDFDGPAEVRERFLSDDQTNAIDLVKEAGFDQETIDLVDAFWCAGYIGDPYTGSALMAKQWGALSDNRYRVMEDITLKWKLNNGMRSLYDGIAGDLNTDIRLNTPVAKVEHHDNGATVTTESGEVIEASAVICTVPVGALSNIEFSPALPDAVQSVIDDKWNSQGAKIWIKIKGHHRFLGYAPKPAKMSVVRSEYFMDDDTTILVGFGYDNTNIDLNSIEDAQAVINQWRDDLEVVDTTGHNWVADKWAGQAWGTLRKGQFTQGWSLFDDIDSQLFFAGSDYAYGWRGVCVDGALEKGMTTARQVINSMRETKEQ

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 \(c\).

\(|P_{f(7FHV_1)}(2) \setminus P_{f(2WXF_1)}(2)|=37\), \(|P_{f(2WXF_1)}(2) \setminus P_{f(7FHV_1)}(2)|=180\). 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 1:1011111101010111000001101101110000001011100011011010000100101111000101111110011111011010000010100111011111010100010100010101110011010110100100000000011111111011100001110110111001100100010111011010011000001111111110110101001010000110011001101000111100001110111010111010010010111101101011100101110101110011100110000000
Pair \(Z_2\) Length of longest common subsequence
7FHV_1,2WXF_1 217 4
7FHV_1,8EEF_1 174 4
2WXF_1,8EEF_1 145 5

Newick tree

 
[
	7FHV_1:10.55,
	[
		8EEF_1:72.5,2WXF_1:72.5
	]:33.05
]

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 \((0.85)(0.8)(\frac{1256 }{\log_{20} 1256}-\frac{316}{\log_{20}316})=246.\)
Status Protein1 Protein2 d d1/2
Query variables 7FHV_1 2WXF_1 318 210.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]