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Parikh vectors
9NDE_1 2MLB_1 3APP_1 Letter Amino acid
0 3 46 S Serine
3 13 30 T Threonine
0 0 3 W Tryptophan
0 1 14 Y Tyrosine
6 7 40 G Glycine
0 2 15 N Asparagine
0 8 4 E Glutamic acid
0 12 21 L Leucine
0 2 0 R Arginine
0 1 3 H Histidine
0 1 0 M Methionine
0 5 12 P Proline
6 0 2 C Cysteine
0 4 23 D Aspartic acid
0 0 25 Q Glutamine
0 7 13 I Isoleucine
0 7 5 K Lycine
0 1 20 F Phenylalanine
0 2 23 V Valine
6 3 24 A Alanine 

9NDE_1|Chain A|DNA (5'-D(P*GP*CP*AP*CP*CP*TP*GP*TP*AP*CP*GP*GP*AP*CP*AP*GP*TP*AP*GP*CP*A)-3')|synthetic construct (32630)
>2MLB_1|Chain A|redesigned ubiquitin|synthetic (32630)
>3APP_1|Chain A|PENICILLOPEPSIN|Penicillium janthinellum (5079)
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 \(w=f(c)\)
9NDE , Knot 11 21 0.53 8 12 17 
GCACCTGTACGGACAGTAGCA
2MLB , Knot 39 79 0.72 34 55 71 
MDTINITLPDGKTLTLTVTPEFTVKELAEEIARRLGLSPEDIKLTHNGKTLDPSLTLAEYGITPGSTITLEIKKKGGLE
3APP , Knot 130 323 0.77 36 158 296 
AASGVATNTPTANDEEYITPVTIGGTTLNLNFDTGSADLWVFSTELPASQQSGHSVYNPSATGKELSGYTWSISYGDGSSASGNVFTDSVTVGGVTAHGQAVQAAQQISAQFQQDTNNDGLLGLAFSSINTVQPQSQTTFFDTVKSSLAQPLFAVALKHQQPGVYDFGFIDSSKYTGSLTYTGVDNSQGFWSFNVDSYTAGSQSGDGFSGIADTGTTLLLLDDSVVSQYYSQVSGAQQDSNAGGYVFDCSTNLPDFSVSISGYTATVPGSLINYGPSGDGSTCLGGIQSNSGIGFSIFGDIFLKSQYVVFDSDGPQLGFAPQA

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 \(c\).

\(|P_{f(9NDE_1)}(2) \setminus P_{f(2MLB_1)}(2)|=11\), \(|P_{f(2MLB_1)}(2) \setminus P_{f(9NDE_1)}(2)|=54\). 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 1:101000101011101101101
Pair \(Z_2\) Length of longest common subsequence
9NDE_1,2MLB_1 65 2
9NDE_1,3APP_1 158 3
2MLB_1,3APP_1 145 3

Newick tree

 
[
	3APP_1:85.51,
	[
		9NDE_1:32.5,2MLB_1:32.5
	]:53.01
]

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 \((0.85)(0.8)(\frac{100 }{\log_{20} 100}-\frac{21}{\log_{20}21})=30.1\)
Status Protein1 Protein2 d d1/2
Query variables 9NDE_1 2MLB_1 36 22.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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