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Parikh vectors
4NJG_1 6XZR_1 3CYF_1 Letter Amino acid
10 0 8 T Threonine
11 0 7 R Arginine
16 0 9 D Aspartic acid
7 9 3 C Cysteine
9 0 4 Q Glutamine
11 0 15 E Glutamic acid
21 8 18 G Glycine
5 0 5 M Methionine
5 0 3 Y Tyrosine
16 0 19 V Valine
22 11 24 A Alanine
8 0 7 N Asparagine
10 0 16 K Lycine
13 0 9 P Proline
11 0 9 H Histidine
10 0 3 F Phenylalanine
11 0 9 S Serine
5 0 0 W Tryptophan
9 0 10 I Isoleucine
20 0 19 L Leucine 

4NJG_1|Chains A, B|7-carboxy-7-deazaguanine synthase|Burkholderia multivorans (395019)
>6XZR_1|Chain A[auth IN1]|RNA (5'-R(*AP*GP*UP*AP*GP*AP*AP*AP*CP*AP*AP*GP*GP*GP*CP*CP*CP*UP*G)-3')|Synthetic construct (32630)
>3CYF_1|Chain A|Protein DJ-1|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 \(w=f(c)\)
4NJG , Knot 110 230 0.86 40 167 220 
MGSSHHHHHHSSGLVPRGSHMTYAVKEIFYTLQGEGANAGRPAVFCRFAGCNLWSGREEDRAQAVCRFCDTDFVGTDGENGGKFKDADALVATIAGLWPAGEAHRFVVCTGGEPMLQLDQPLVDALHAAGFGIAIETNGSLPVLESIDWICVSPKADAPLVVTKGNELKVVIPQDNQRLADYAKLDFEYFLVQPMDGPSRDLNTKLAIDWCKRHPQWRLSMQTHKYLNIP
6XZR , Knot 16 47 0.43 8 15 31 
AGUAGAAACAAGGGUAUUUUUCUUUACUAGUCUACCCUGCUUUUGCU
3CYF , Knot 88 197 0.78 38 126 187 
MASKRALVILAKGAEEMNTVIPVDVMRRAGIKVTVAGLAGKDPVQCSRDVVICPDASLEDAKKEGPYDVVVLPGGNLGAQNLSESAAVKEILKEQENRKGLIAAICAGPTALLAHEIGFGSKVTTHPLAKDKMMNGGHYTYSENRVEKDGLILTSRGPGTSFEFALAIVEALNGKEVAAQVKAPLVLKDLEHHHHHH

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 \(c\).

\(|P_{f(4NJG_1)}(2) \setminus P_{f(6XZR_1)}(2)|=162\), \(|P_{f(6XZR_1)}(2) \setminus P_{f(4NJG_1)}(2)|=10\). 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 1:11000000000011110100100110011001010110110111100111001101000001011001000011100100110100101111011111111010011100110111010011101101111111100010111100101101010101111100100101111000001100101010011101101100010001110100001010101000001011
Pair \(Z_2\) Length of longest common subsequence
4NJG_1,6XZR_1 172 3
4NJG_1,3CYF_1 151 6
6XZR_1,3CYF_1 131 2

Newick tree

 
[
	4NJG_1:85.44,
	[
		3CYF_1:65.5,6XZR_1:65.5
	]:19.94
]

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 \((0.85)(0.8)(\frac{277 }{\log_{20} 277}-\frac{47}{\log_{20}47})=75.4\)
Status Protein1 Protein2 d d1/2
Query variables 4NJG_1 6XZR_1 104 58
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]