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Parikh vectors
2NBG_1 3CGH_1 2FRF_1 Letter Amino acid
11 56 15 A Alanine
0 4 0 C Cysteine
10 28 13 E Glutamic acid
2 19 7 F Phenylalanine
1 28 2 Y Tyrosine
14 33 7 V Valine
11 17 2 R Arginine
11 27 17 L Leucine
6 25 4 P Proline
6 31 7 T Threonine
7 32 8 D Aspartic acid
13 38 15 G Glycine
5 29 9 I Isoleucine
4 29 5 S Serine
0 12 2 W Tryptophan
3 46 2 N Asparagine
5 20 6 Q Glutamine
3 5 11 H Histidine
7 28 19 K Lycine
6 9 2 M Methionine 

2NBG_1|Chain A|Translation initiation factor IF-2|Geobacillus stearothermophilus (1422)
>3CGH_1|Chain A|SusD homolog|Bacteroides thetaiotaomicron VPI-5482 (226186)
>2FRF_1|Chain A|Myoglobin|Equus caballus (9796)
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 \(w=f(c)\)
2NBG , Knot 62 125 0.79 36 95 120 
GSHMLDHLLEMILLVSEMEELKANPNRRAVGTVIEAKLDKGRGPVATLLVQAGTLKVGDPIVVGTTYGRVRAMVNDSGRRVKEAGPSMPVEITGLHDVPQAGDRFMVFEDEKKARQIGEARAQRQ
3CGH , Knot 206 516 0.83 40 251 491 
GNYENINSNPYEAPDLSADGYALGSAMNNLAGCVVSPDVNTAQFTDCLLGGPLGGYFADSNAGFTETISNFNPKDDWSRVFLKSDKIIPTLYSNLTQVKLVSQNTNDPVPYAIAQVIKVAAMHRVTDAFGPIPYSQIGANGEIATPYDSQEVTYNTFFDELNAAIATLNENSNEQLVPTADYIYKGDVKKWIRFANSLKLRLAIRIAYANPVKAQQMAEEAVNPANGGVIESNADNATWNYFETSQNPIYVATRYNQVQTSDHGGVPCLTGGDTHAAADIICYMNGYKDNRREKFFTKSEWAGQDYVGMRRGIVIPELKTTGHKYSGVNIAPTSPLYWMNAAEVAFLRAEGQAVFNFSMGGTAESFYNQGIRLSFEQWGADGVEDYLKDDVNKPTAYTDPAGTNTYQNALSNITIKWNDSADKEEKQERIIVQKWIANWQLGNEAWADFRRTGYPKLIPVKENKSGGVVDSEKGARRMPYPLDEFVSNKANVEYAIANYLHGADNMATDVWWASKK
2FRF , Knot 75 153 0.82 38 115 148 
GLSDGEWQQVLNVWGKVEADIAGHGQEVLIRLFTGHPETLEKFDKFKHLKTEAEMKASEDLKKHGTVVLTALGGILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISDAIIHVLHSKHPGDFGADAQGAMTKALELFRNDIAAKYKELGFQG

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 \(c\).

\(|P_{f(2NBG_1)}(2) \setminus P_{f(3CGH_1)}(2)|=24\), \(|P_{f(3CGH_1)}(2) \setminus P_{f(2NBG_1)}(2)|=180\). 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 1:10011001101111100100101010001110110101001011110111011010110111110001010111000100100111011101011001101100111100000100110101000
Pair \(Z_2\) Length of longest common subsequence
2NBG_1,3CGH_1 204 3
2NBG_1,2FRF_1 142 3
3CGH_1,2FRF_1 188 3

Newick tree

 
[
	3CGH_1:10.57,
	[
		2NBG_1:71,2FRF_1:71
	]:34.57
]

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 \((0.85)(0.8)(\frac{641 }{\log_{20} 641}-\frac{125}{\log_{20}125})=149.\)
Status Protein1 Protein2 d d1/2
Query variables 2NBG_1 3CGH_1 188 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]