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Parikh vectors
1LEX_1 5XSE_1 1ATR_1 Letter Amino acid
0 22 25 L Leucine
0 7 7 M Methionine
0 15 11 P Proline
0 8 22 S Serine
0 1 1 W Tryptophan
0 7 22 R Arginine
0 10 29 D Aspartic acid
0 4 12 Q Glutamine
0 5 18 N Asparagine
4 4 2 C Cysteine
0 1 10 Y Tyrosine
0 22 32 V Valine
0 10 24 E Glutamic acid
0 13 29 K Lycine
2 8 26 T Threonine
0 10 26 I Isoleucine
0 8 18 F Phenylalanine
2 34 37 A Alanine
4 18 29 G Glycine
0 1 6 H Histidine 

1LEX_1|Chains A, B|DNA (5'-D(*CP*GP*CP*GP*AP*AP*TP*TP*CP*GP*CP*G)-3')|
>5XSE_1|Chains A, B, C|KHG/KDPG aldolase|Zymomonas mobilis subsp. mobilis (strain ATCC 31821 / ZM4 / CP4) (264203)
>1ATR_1|Chain A|HEAT-SHOCK COGNATE 70 KD PROTEIN|Bos taurus (9913)
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 \(w=f(c)\)
1LEX , Knot 6 12 0.41 8 7 8 
CGCGAATTCGCG
5XSE , Knot 96 208 0.82 40 128 201 
MRDIDSVMRLAPVMPVLVIEDIADAKPIAEALVAGGLNVLEVTLRTPCALEAIKIMKEVPGAVVGAGTVLNAKMLDQAQEAGCEFFVSPGLTADLGKHAVAQKAALLPGVANAADVMLGLDLGLDRFKFFPAENIGGLPALKSMASVFRQVRFCPTGGITPTSAPKYLENPSILCVGGSWVVPAGKPDVAKITALAKEASAFKRAAVA
1ATR , Knot 164 386 0.84 40 213 366 
MSKGPAVGIDLGTTYSCVGVFQHGKVEIIANDQGNRTTPSYVAFTDTERLIGDAAKNQVAMNPTNTVFDAKRLIGRRFDDAVVQSDMKHWPFMVVNDAGRPKVQVEYKGETKSFYPEEVSSMVLTKMKEIAEAYLGKTVTNAVVTVPAYFNDSQRQATKDAGTIAGLNVLRIINEPTAAAIAYGLDKKVGAERNVLIFDLGGGVFDVSILTIEDGIFEVKSTAGDTHLGGEDFDNRMVNHFIAEFKRKHKKDISENKRAVRRLRTACERAKRTLSSSTQASIEIDSLYEGIDFYTSITRARFEELNADLFRGTLDPVEKALRDAKLDKSQIHDIVLVGGSTRIPKIQKLLQDFFNGKELNKSINPDEAVAYGAAVQAAILSGDKSE

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 \(c\).

\(|P_{f(1LEX_1)}(2) \setminus P_{f(5XSE_1)}(2)|=4\), \(|P_{f(5XSE_1)}(2) \setminus P_{f(1LEX_1)}(2)|=125\). 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 1:010111000101
Pair \(Z_2\) Length of longest common subsequence
1LEX_1,5XSE_1 129 2
1LEX_1,1ATR_1 212 3
5XSE_1,1ATR_1 159 5

Newick tree

 
[
	1ATR_1:10.57,
	[
		1LEX_1:64.5,5XSE_1:64.5
	]:37.07
]

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 \((0.85)(0.8)(\frac{220 }{\log_{20} 220}-\frac{12}{\log_{20}12})=73.2\)
Status Protein1 Protein2 d d1/2
Query variables 1LEX_1 5XSE_1 92 48
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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