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Parikh vectors
4FGZ_1 6GQP_1 3TEZ_1 Letter Amino acid
24 9 67 N Asparagine
5 9 2 C Cysteine
13 16 24 F Phenylalanine
14 14 56 T Threonine
5 5 7 W Tryptophan
14 12 28 Y Tyrosine
9 21 39 A Alanine
6 20 28 R Arginine
5 10 10 H Histidine
26 37 57 L Leucine
4 9 8 M Methionine
2 18 28 P Proline
5 9 31 Q Glutamine
15 20 35 G Glycine
17 21 47 D Aspartic acid
30 21 57 K Lycine
16 14 68 S Serine
10 18 40 V Valine
26 24 49 E Glutamic acid
20 15 54 I Isoleucine 

4FGZ_1|Chains A, B|Phosphoethanolamine N-methyltransferase|Plasmodium falciparum (5833)
>6GQP_1|Chain A|Vascular endothelial growth factor receptor 2|Homo sapiens (9606)
>3TEZ_1|Chain A|Protective antigen|Bacillus anthracis (1392)
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 \(w=f(c)\)
4FGZ , Knot 120 266 0.84 40 166 254 
MTLIENLNSDKTFLENNQYTDEGVKVYEFIFGENYISSGGLEATKKILSDIELNENSKVLDIGSGLGGGCMYINEKYGAHTHGIDICSNIVNMANERVSGNNKIIFEANDILTKEFPENNFDLIYSRDAILHLSLENKNKLFQKCYKWLKPTGTLLITDYCATEKENWDDEFKEYVKQRKYTLITVEEYADILTACNFKNVVSKDLSDYWNQLLEVEHKYLHENKEEFLKLFSEKKFISLDDGWSRKIKDSKRKMQRWGYFKATKN
6GQP , Knot 141 322 0.84 40 203 302 
MDPDELPLDEHCERLPYDASKWEFPRDRLKLGKPLGRGAFGQVIEADAFGIDKTATCRTVAVKMLKEGATHSEHRALMSELKILIHIGHHLNVVNLLGACTKPGGPLMVIVEFCKFGNLSTYLRSKRNEFVPYKVAPEDLYKDFLTLEHLICYSFQVAKGMEFLASRKCIHRDLAARNILLSEKNVVKICDFGLARDIYKDPDYVRKGDARLPLKWMAPETIFDRVYTIQSDVWSFGVLLWEIFSLGASPYPGVKIDEEFCRRLKEGTRMRAPDYTTPEMYQTMLDCWHGEPSQRPTFSELVEHLGNLLQANAQQDENLYFQ
3TEZ , Knot 276 735 0.82 40 269 662 
EVKQENRLLNESESSSQGLLGYYFSDLNFQAPMVVTSSTTGDLSIPSSELENIPSENQYFQSAIWSGFIKVKKSDEYTFATSADNHVTMWVDDQEVINKASNSNKIRLEKGRLYQIKIQYQRENPTEKGLDFKLYWTDSQNKKEVISSDNLQLPELKQKSSNSRKKRSTSAGPTVPDRDNDGIPDSLEVEGYTVDVKNKRTFLSPWISNIHEKKGLTKYKSSPEKWSTASDPYSDFEKVTGRIDKNVSPEARHPLVAAYPIVHVDMENIILSKNEDQSTQNTDSQTRTISKNTSTSRTHTSEVHGNAEVHASFFDIGGSVSAGFSNSNSSTVAIDHCLSLAGERTWAETMGLNTADTARLNANIRYVNTGTAPIYNVLPTTSLVLGKNQTLATIKAKENQLSQILAPNNYYPSKNLAPIALNAQDDFSSTPITMNYNQFLELEKTKQLRLDTDQVYGNIATYNFENGRVRVDTGSNWSEVLPQIQETTARIIFNGKDLNLVERRIAAVNPSDPLETTKPDMTLKEALKIAFGFNEPNGNLQYQGKDITEFDFNFDQQTSQNIKNQLAELNATNIYTVLDKIKLNAKMNILIRDKRFHYDRNNIAVGADESVVKEAHREVINSSTEGLLLNIDKDIRKILSGYIVEIEDTEGLKEVINDRYDMLCISSLRQDGKTFIDFKKYNDKLPLYISNPNYKVNVYAVTKENTIINPSENGDTSTNGIKKILIFSKKGYEIG

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 \(c\).

\(|P_{f(4FGZ_1)}(2) \setminus P_{f(6GQP_1)}(2)|=64\), \(|P_{f(6GQP_1)}(2) \setminus P_{f(4FGZ_1)}(2)|=101\). 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 1:10110010000011000000001101001111000100111010001100101000001101101111101010000110001101000110110001010001110100110001100010110000111010100000110000011010101110000100000100010001000000110100010110100100110001000100110100001000000110110000110100110001000000100110101000
Pair \(Z_2\) Length of longest common subsequence
4FGZ_1,6GQP_1 165 4
4FGZ_1,3TEZ_1 167 4
6GQP_1,3TEZ_1 160 4

Newick tree

 
[
	4FGZ_1:83.97,
	[
		6GQP_1:80,3TEZ_1:80
	]:3.97
]

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 \((0.85)(0.8)(\frac{588 }{\log_{20} 588}-\frac{266}{\log_{20}266})=90.7\)
Status Protein1 Protein2 d d1/2
Query variables 4FGZ_1 6GQP_1 116 104.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]