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Parikh vectors
7NGG_1 6QWC_1 3MAG_1 Letter Amino acid
10 29 13 G Glycine
21 29 34 L Leucine
13 22 20 S Serine
19 20 15 T Threonine
4 8 19 Y Tyrosine
33 41 11 A Alanine
8 10 17 N Asparagine
13 14 17 D Aspartic acid
0 2 2 C Cysteine
6 10 10 Q Glutamine
8 8 20 I Isoleucine
9 8 16 F Phenylalanine
10 13 16 P Proline
18 17 16 R Arginine
5 5 10 M Methionine
15 12 16 E Glutamic acid
4 10 22 K Lycine
4 5 3 W Tryptophan
13 18 21 V Valine
3 9 9 H Histidine 

7NGG_1|Chain A[auth C]|HTH-type transcriptional regulator EthR|Mycobacterium tuberculosis (1773)
>6QWC_1|Chain A|Beta-lactamase|Klebsiella pneumoniae (573)
>3MAG_1|Chain A|VP39|Vaccinia virus (10245)
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 \(w=f(c)\)
7NGG , Knot 97 216 0.80 38 136 207 
MTTSAASQASLPRGRRTARPSGDDRELAILATAENLLEDRPLADISVDDLAKGAGISRPTFYFYFPSKEAVLLTLLDRVVNQADMALQTLAENPADTDRENMWRTGINVFFETFGSHKAVTRAGQAARATSVEVAELWSTFMQKWIAYTAAVIDAERDRGAAPRTLPAHELATALNLMNERTLFASFAGEQPSVPEARVLDTLVHIWVTSIYGENR
6QWC , Knot 127 290 0.82 40 175 272 
MGSSHHHHHHSSGLVPRGSHMLTNLVAEPFAKLEQDFGGSIGVYAMDTGSGATVSYRAEERFPLCSSFKGFLAAAVLARSQQQAGLLDTPIRYGKNALVRWSPISEKYLTTGMTVAELSAAAVQYSDNAAANLLLKELGGPAGLTAFMRSIGDTTFRLDRWELELNSAIPGDARDTSSPRAVTESLQKLTLGSALAAPQRQQFVDWLKGNTTGNHRIRAAVPADWAVGDKTGTCGGYGTANDYAVVWPTGRAPIVLAVYTRAPNKDDKHSEAVIAAAARLALEGLGVNGQ
3MAG , Knot 132 307 0.82 40 196 298 
MDVVSLDKPFMYFEEIDNELDYEPESANEVAKKLPYQGQLKLLLGELFFLSKLQRHGILDGATVVYIGSAPGTHIRYLRDHFYNLGVIIKWMLIDGRHHDPILNGLRDVTLVTRFVDEEYLRSIKKQLHPSKIILISDVRSKRGGNEPSTADLLSNYALQNVMISILNPVASSLKWRCPFPDQWIKDFYIPHGNKMLQPFAPSYSAEMRLLSIYTGENMRLTRVTKSDAVNYEKKMYYLNKIVRNKVVVNFDYPNQEYDYFHMYFMLRTVYCNKTFPTTKAKVLFLQQSIFRFLNIPTTSTEKVSHE

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 \(c\).

\(|P_{f(7NGG_1)}(2) \setminus P_{f(6QWC_1)}(2)|=60\), \(|P_{f(6QWC_1)}(2) \setminus P_{f(7NGG_1)}(2)|=99\). 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 1:100011001011010001010100001111101001100011101010011011110010101011000111101100110010111001100110000001100110111001100011001101101001011011001100111001111010000111100111001101101100001110111001011010110011011100101000
Pair \(Z_2\) Length of longest common subsequence
7NGG_1,6QWC_1 159 4
7NGG_1,3MAG_1 174 3
6QWC_1,3MAG_1 169 4

Newick tree

 
[
	3MAG_1:87.74,
	[
		7NGG_1:79.5,6QWC_1:79.5
	]:8.24
]

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 \((0.85)(0.8)(\frac{506 }{\log_{20} 506}-\frac{216}{\log_{20}216})=83.6\)
Status Protein1 Protein2 d d1/2
Query variables 7NGG_1 6QWC_1 106 90.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]