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Parikh vectors
7UMD_1 3PNL_1 7DSP_1 Letter Amino acid
10 30 26 A Alanine
0 16 7 Q Glutamine
0 17 22 I Isoleucine
0 40 49 L Leucine
0 19 37 S Serine
0 9 11 Y Tyrosine
0 26 19 D Aspartic acid
10 7 5 C Cysteine
0 13 14 N Asparagine
0 19 22 E Glutamic acid
13 39 29 G Glycine
0 9 20 H Histidine
0 14 28 K Lycine
0 6 6 M Methionine
0 15 19 P Proline
0 4 4 W Tryptophan
0 13 6 R Arginine
0 9 13 F Phenylalanine
0 26 21 T Threonine
0 26 25 V Valine 

7UMD_1|Chain A|RNA (40-MER)|Dengue virus 1 (11053)
>3PNL_1|Chain A|PTS-dependent dihydroxyacetone kinase, dihydroxyacetone-binding subunit dhaK|Escherichia coli (83333)
>7DSP_1|Chains A, B|Anthranilate phosphoribosyltransferase|Saccharomyces cerevisiae S288C (559292)
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 \(w=f(c)\)
7UMD , Knot 14 40 0.43 8 13 24 
GGACGUGGACCGACUUCGGUCGGAAGGAAACUUAACGUCC
3PNL , Knot 151 357 0.82 40 203 341 
GSKKLINDVQDVLDEQLAGLAKAHPSLTLHQDPVYVTRADAPVAGKVALLSGGGSGHEPMHCGYIGQGMLSGACPGEIFTSPTPDKIFECAMQVDGGEGVLLIIKNYTGDILNFETATELLHDSGVKVTTVVIDDDVAVKDSLYTAGRRGVANTVLIEKLVGAAAERGDSLDACAELGRKLNNQGHSIGIALGACTVPAAGKPSFTLADNEMEFGVGIHGEPGIDRRPFSSLDQTVDEMFDTLLVNGSYHRTLRFWDYQQGSWQEEQQTKQPLQSGDRVIALVNNLGATPLSELYGVYNRLTTRCQQAGLTIERNLIGAYCTSLDMTGFSITLLKVDDETLALWDAPVHTPALNWGK
7DSP , Knot 159 383 0.82 40 204 366 
HHHMSEATLLSYTKKLLASPPQLSSTDLHDALLVILSLLQKCDTNSDESLSIYTKVSSFLTALRVTKLDHKAEYIAEAAKAVLRHSDLVDLPLPKKDELHPEDGPVILDIVGTGGDGQNTFNVATSAAIVASGIQGLKICKHGGKASTSNSGAGDLIGTLGCDMFKVNSSTVPKLWPDNTFMFLLAPFFHHGMGHVSKIRKFLGIPTVFNVLGPLLHPVSHVNKRILGVYSKELAPEYAKAAALVYPGSETFIVWGHVGLDEVSPIGKTTVWHIDPTSSELKLKTFQLEPSMFGLEEHELSKCASYGPKENARILKEEVLSGKYHLGDNNPIYDYILMNTAVLYCLSQGHQNWKEGIIKAEESIHSGNALRSLEHFIDSVSSL

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 \(c\).

\(|P_{f(7UMD_1)}(2) \setminus P_{f(3PNL_1)}(2)|=7\), \(|P_{f(3PNL_1)}(2) \setminus P_{f(7UMD_1)}(2)|=197\). 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 1:1110101110011000011001111111110001101000
Pair \(Z_2\) Length of longest common subsequence
7UMD_1,3PNL_1 204 4
7UMD_1,7DSP_1 209 3
3PNL_1,7DSP_1 177 3

Newick tree

 
[
	7UMD_1:10.72,
	[
		3PNL_1:88.5,7DSP_1:88.5
	]:19.22
]

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 \((0.85)(0.8)(\frac{397 }{\log_{20} 397}-\frac{40}{\log_{20}40})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 7UMD_1 3PNL_1 146 79
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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