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Parikh vectors
4LBD_1 2QXV_1 7UKP_1 Letter Amino acid
1 11 9 W Tryptophan
6 20 39 V Valine
16 19 54 G Glycine
19 15 18 T Threonine
20 15 21 E Glutamic acid
10 15 8 H Histidine
38 31 46 L Leucine
13 22 7 K Lycine
17 29 40 S Serine
15 18 39 A Alanine
15 30 27 D Aspartic acid
6 12 6 C Cysteine
14 9 2 M Methionine
10 14 22 F Phenylalanine
17 12 27 P Proline
16 20 27 R Arginine
4 20 12 N Asparagine
5 12 22 Y Tyrosine
10 11 21 Q Glutamine
15 26 10 I Isoleucine 

4LBD_1|Chain A|RETINOIC ACID RECEPTOR GAMMA|Homo sapiens (9606)
>2QXV_1|Chain A|Embryonic ectoderm development|Mus musculus (10090)
>7UKP_1|Chains A, C|Integrin alpha-IIb heavy chain|Homo sapiens (9606)
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 \(w=f(c)\)
4LBD , Knot 118 267 0.82 40 173 257 
MGSSHHHHHHSSGLVPRGSHMDSYELSPQLEELITKVSKAHQETFPSLCQLGKYTTNSSADHRVQLDLGLWDKFSELATKCIIKIVEFAKRLPGFTGLSIADQITLLKAACLDILMLRICTRYTPEQDTMTFSDGLTLNRTQMHNAGFGPLTDLVFAFAGQLLPLEMDDTETGLLSAICLICGDRMDLEEPEKVDKLQEPLLEALRLYARRRRPSQPYMFPRMLMKITDLRGISTKGAERAITLKMEIPGPMPPLIREMLENPEMFE
2QXV , Knot 158 361 0.86 40 224 354 
SFKCVNSLKEDHNQPLFGVQFNWHSKEGDPLVFATVGSNRVTLYECHSQGEIRLLQSYVDADADENFYTCAWTYDSNTSHPLLAVAGSRGIIRIINPITMQCIKHYVGHGNAINELKFHPRDPNLLLSVSKDHALRLWNIQTDTLVAIFGGVEGHRDEVLSADYDLLGEKIMSCGMDHSLKLWRINSKRMMNAIKESYDYNPNKTNRPFISQKIHFPDFSTRDIHRNYVDCVRWLGDLILSKSCENAIVCWKPGKMEDDIDKIKPSESNVTILGRFDYSQCDIWYMRFSMDFWQKMLALGNQVGKLYVWDLEVEDPHKAKCTTLTHHKCGAAIRQTSFSRDSSILIAVCDDASIWRWDRLR
7UKP , Knot 181 457 0.80 40 219 411 
LNLDPVQLTFYAGPNGSQFGFSLDFHKDSHGRVAIVVGAPRTLGPSQEETGGVFLCPWRAEGGQCPSLLFDLRDETRNVGSQTLQTFKARQGLGASVVSWSDVIVACAPWQHWNVLEKTEEAEKTPVGSCFLAQPESGRRAEYSPCRGNTLSRIYVENDFSWDKRYCEAGFSSVVTQAGELVLGAPGGYYFLGLLAQAPVADIFSSYRPGILLWHVSSQSLSFDSSNPEYFDGYWGYSVAVGEFDGDLNTTEYVVGAPTWSWTLGAVEILDSYYQRLHRLRGEQMASYFGHSVAVTDVNGDGRHDLLVGAPLYMESRADRKLAEVGRVYLFLQPRGPHALGAPSLLLTGTQLYGRFGSAIAPLGDLDRDGYNDIAVAAPYGGPSGRGQVLVFLGQSEGLRSRPSQVLDSPFPTGSAFGFSLRGAVDIDDNGYPDLIVGAYGANQVAVYRAQPVVKAS

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 \(c\).

\(|P_{f(4LBD_1)}(2) \setminus P_{f(2QXV_1)}(2)|=67\), \(|P_{f(2QXV_1)}(2) \setminus P_{f(4LBD_1)}(2)|=118\). 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 1:110000000000111101001000010101001100100100001101001100000001000101011110010011000110110110011110110110010110110101111010000010000101001101000010011111100111111101111010000011101101101001010010010010011101101010000100101110111010010110001100110101011111111100110010110
Pair \(Z_2\) Length of longest common subsequence
4LBD_1,2QXV_1 185 4
4LBD_1,7UKP_1 178 4
2QXV_1,7UKP_1 171 4

Newick tree

 
[
	4LBD_1:92.45,
	[
		7UKP_1:85.5,2QXV_1:85.5
	]:6.95
]

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 \((0.85)(0.8)(\frac{628 }{\log_{20} 628}-\frac{267}{\log_{20}267})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 4LBD_1 2QXV_1 132 112.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]