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Parikh vectors
2FXA_1 1QRG_1 2EGH_1 Letter Amino acid
7 7 24 Q Glutamine
13 5 11 F Phenylalanine
8 11 22 T Threonine
5 1 3 W Tryptophan
11 28 31 V Valine
13 19 52 A Alanine
7 5 23 R Arginine
6 5 14 H Histidine
18 5 16 K Lycine
12 13 16 N Asparagine
1 1 9 C Cysteine
10 16 16 I Isoleucine
7 5 19 M Methionine
11 14 23 D Aspartic acid
21 19 21 E Glutamic acid
10 16 25 G Glycine
18 6 45 L Leucine
8 15 20 P Proline
14 14 28 S Serine
7 8 6 Y Tyrosine 

2FXA_1|Chains A, B, C, D|Protease production regulatory protein hpr|Bacillus subtilis (1423)
>1QRG_1|Chain A|CARBONIC ANHYDRASE|Methanosarcina thermophila (2210)
>2EGH_1|Chains A, B|1-deoxy-D-xylulose 5-phosphate reductoisomerase|Escherichia coli str. K12 substr. (316407)
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 \(w=f(c)\)
2FXA , Knot 98 207 0.84 40 158 202 
GHMNRVEPPYDVKEALVFTQKMAQLSKALWKSIEKDWQQWLKPYDLNINEHHILWIAYQLNGASISEIAKFGVMHVSTAFNFSKKLEERGYLRFSKRLNDKRNTYVQLTEEGTEVFWSLLEEFDPTRNAVFKGSQPLYHLFGKFPEVAEMMCMIRHIYGDDFMEIFETSLTNIDNDFESVNGKLKKKAKDSAADEPAEELEPVNSGS
1QRG , Knot 97 213 0.81 40 137 206 
QEITVDEFSNIRENPVTPWNPEPSAPVIDPTAYIDPQASVIGEVTIGANVMVSPMASIRSDEGMPIFVGDRSNVQDGVVLHALETINEEGEPIEDNIVEVDGKEYAVYIGNNVSLAHQSQVHGPAAVGDDTFIGMQAFVFKSKVGNNCVLEPRSAAIGVTIPDGRYIPAGMVVTSQAEADKLPEVTDDYAYSHTNEAVVYVNVHLAEGYKETS
2EGH , Knot 176 424 0.83 40 224 396 
MRGSHHHHHHGKQLTILGSTGSIGCSTLDVVRHNPEHFRVVALVAGKNVTRMVEQCLEFSPRYAVMDDEASAKLLKTMLQQQGSRTEVLSGQQAACDMAALEDVDQVMAAIVGAAGLLPTLAAIRAGKTILLANKESLVTCGRLFMDAVKQSKAQLLPVDSEHNAIFQSLPQPIQHNLGYADLEQNGVVSILLTGSGGPFRETPLRDLATMTPDQACRHPNWSMGRKISVDSATMMNKGLEYIEARWLFNASASQMEVLIHPQSVIHSMVRYQDGSVLAQLGEPDMRTPIAHTMAWPNRVNSGVKPLDFCKLSALTFAAPDYDRYPCLKLAMEAFEQGQAATTALNAANEITVAAFLAQQIRFTDIAALNLSVLEKMDMREPQCVDDVLSVDANAREVARKEVMRLASSACDLGTPGRPAAKLN

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 \(c\).

\(|P_{f(2FXA_1)}(2) \setminus P_{f(1QRG_1)}(2)|=97\), \(|P_{f(1QRG_1)}(2) \setminus P_{f(2FXA_1)}(2)|=76\). 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 1:101001011001001111000110100111001000100110100101000011111001011010011011110100110100010001010100010000000101000100111011001010001110100110011101101101101100101001101100010010001001010100010001100110010110010
Pair \(Z_2\) Length of longest common subsequence
2FXA_1,1QRG_1 173 3
2FXA_1,2EGH_1 182 4
1QRG_1,2EGH_1 179 4

Newick tree

 
[
	2EGH_1:91.46,
	[
		2FXA_1:86.5,1QRG_1:86.5
	]:4.96
]

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 \((0.85)(0.8)(\frac{420 }{\log_{20} 420}-\frac{207}{\log_{20}207})=62.5\)
Status Protein1 Protein2 d d1/2
Query variables 2FXA_1 1QRG_1 79 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]