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Parikh vectors
1QQB_1 7FUO_1 7BRS_1 Letter Amino acid
0 43 20 K Lycine
4 14 77 A Alanine
0 20 66 R Arginine
0 18 64 D Aspartic acid
4 14 71 G Glycine
0 18 39 I Isoleucine
0 41 96 L Leucine
0 12 59 Q Glutamine
0 32 61 E Glutamic acid
0 28 61 S Serine
0 11 31 Y Tyrosine
0 18 47 N Asparagine
4 10 16 C Cysteine
0 8 34 H Histidine
0 5 24 M Methionine
0 4 39 W Tryptophan
0 18 64 V Valine
0 23 38 F Phenylalanine
0 11 62 P Proline
5 14 56 T Threonine 

1QQB_1|Chain A[auth M]|5'-D(*TP*AP*CP*GP*CP*AP*AP*TP*CP*GP*AP*TP*TP*GP*CP*GP*T)-3'|
>7FUO_1|Chain A|Cyclic GMP-AMP synthase|Homo sapiens (9606)
>7BRS_1|Chains A, B, C, D|Beta-galactosidase|Escherichia coli K-12 (83333)
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 \(w=f(c)\)
1QQB , Knot 10 17 0.55 8 12 15 
TACGCAATCGATTGCGT
7FUO , Knot 149 362 0.80 40 204 345 
GASKLRAVLEKLKLSRDDISTAAGMVKGVVDHLLLRLKCDSAFRGVGLLNTGSYYEHVKISAPNEFDVMFKLEVPRIQLEEYSNTRAYYFVKFKRNPKENPLSQFLEGEILSASKMLSKFRKIIKEEINDIKDTDVIMKRKRGGSPAVTLLISEKISVDITLALESKSSWPASTQEGLRIQNWLSAKVRKQLRLKPFYLVPKHAKEGNGFQEETWRLSFSHIEKEILNNHGKSKTCCENKEEKCCRKDCLKLMKYLLEQLKERFKDKKHLDKFSSYHVKTAFFHVCTQNPQDSQWDRKDLGLCFDNCVTYFLQCLRTEKLENYFIPEFNLFSSNLIDKRSKEFLTKQIEYERNNEFPVFDEF
7BRS , Knot 382 1025 0.86 40 327 923 
SMTMITDSLAVVLQRRDWENPGVTQLNRLAAHPPFASWRNSEEARTDRPSQQLRSLNGEWRFAWFPAPEAVPESWLECDLPEADTVVVPSNWQMHGYDAPIYTNVTYPITVNPPFVPTENPTGCYSLTFNVDESWLQEGQTRIIFDGVNSAFHLWCNGRWVGYGQDSRLPSEFDLSAFLRAGENRLAVMVLRWSDGSYLEDQDMWRMSGIFRDVSLLHKPTTQISDFHVATRFNDDFSRAVLEAEVQMCGELRDYLRVTVSLWQGETQVASGTAPFGGEIIDERGGYADRVTLRLNVENPKLWSAEIPNLYRAVVELHTADGTLIEAEACDVGFREVRIENGLLLLNGKPLLIRGVNRHEHHPLHGQVMDEQTMVQDILLMKQNNFNAVRCSHYPNHPLWYTLCDRYGLYVVDEANIETHGMVPMNRLTDDPRWLPAMSERVTRMVQRDRNHPSVIIWSLGNESGHGANHDALYRWIKSVDPSRPVQYEGGGADTTATDIICPMYARVDEDQPFPAVPKWSIKKWLSLPGETRPLILCQYAHAMGNSLGGFAKYWQAFRQYPRLQGGFVWDWVDQSLIKYDENGNPWSAYGGDFGDTPNDRQFCMNGLVFADRTPHPALTEAKHQQQFFQFRLSGQTIEVTSEYLFRHSDNELLHWMVALDGKPLASGEVPLDVAPQGKQLIELPELPQPESAGQLWLTVRVVQPNATAWSEAGHISAWQQWRLAENLSVTLPAASHAIPHLTTSEMDFCIELGNKRWQFNRQSGFLSQMWIGDKKQLLTPLRDQFTRAPLDNDIGVSEATRIDPNAWVERWKAAGHYQAEAALLQCTADTLADAVLITTAHAWQHQGKTLFISRKTYRIDGSGQMAITVDVEVASDTPHPARIGLNCQLAQVAERVNWLGLGPQENYPDRLTAACFDRWDLPLSDMYTPYVFPSENGLRCGTRELNYGPHQWRGDFQFNISRYSQQQLMETSHRHLLHAEEGTWLNIDGFHMGIGGDDSWSPSVSAEFQLSAGRYHYQLVWCQK

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 \(c\).

\(|P_{f(1QQB_1)}(2) \setminus P_{f(7FUO_1)}(2)|=7\), \(|P_{f(7FUO_1)}(2) \setminus P_{f(1QQB_1)}(2)|=199\). 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 1:01010110011001010
Pair \(Z_2\) Length of longest common subsequence
1QQB_1,7FUO_1 206 2
1QQB_1,7BRS_1 319 3
7FUO_1,7BRS_1 195 5

Newick tree

 
[
	1QQB_1:14.44,
	[
		7FUO_1:97.5,7BRS_1:97.5
	]:46.94
]

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 \((0.85)(0.8)(\frac{379 }{\log_{20} 379}-\frac{17}{\log_{20}17})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 1QQB_1 7FUO_1 147 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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