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Parikh vectors
2QCS_1 8XET_1 9FTB_1 Letter Amino acid
35 34 11 K Lycine
19 17 12 D Aspartic acid
26 19 15 E Glutamic acid
21 18 18 I Isoleucine
25 8 13 F Phenylalanine
13 11 14 P Proline
14 17 6 Y Tyrosine
20 23 14 V Valine
23 23 25 A Alanine
2 0 5 C Cysteine
8 4 13 H Histidine
15 19 11 T Threonine
6 4 6 W Tryptophan
16 8 9 N Asparagine
15 10 12 Q Glutamine
22 16 13 G Glycine
32 25 19 L Leucine
6 3 4 M Methionine
17 18 16 S Serine
15 6 12 R Arginine 

2QCS_1|Chain A|cAMP-dependent protein kinase, alpha-catalytic subunit|Mus musculus (10090)
>8XET_1|Chain A|Iron-hydroxamate ABC transporter substrate-binding protein|Streptococcus pyogenes SSI-1 (193567)
>9FTB_1|Chain A|NeuA|Aeromonas caviae (648)
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 \(w=f(c)\)
2QCS , Knot 152 350 0.84 40 209 337 
GNAAAAKKGSEQESVKEFLAKAKEDFLKKWETPSQNTAQLDQFDRIKTLGTGSFGRVMLVKHKESGNHYAMKILDKQKVVKLKQIEHTLNEKRILQAVNFPFLVKLEFSFKDNSNLYMVMEYVAGGEMFSHLRRIGRFSEPHARFYAAQIVLTFEYLHSLDLIYRDLKPENLLIDQQGYIQVTDFGFAKRVKGRTWTLCGTPEYLAPEIILSKGYNKAVDWWALGVLIYEMAAGYPPFFADQPIQIYEKIVSGKVRFPSHFSSDLKDLLRNLLQVDLTKRFGNLKNGVNDIKNHKWFATTDWIAIYQRKVEAPFIPKFKGPGDTSNFDDYEEEEIRVSINEKCGKEFTEF
8XET , Knot 123 283 0.81 38 177 271 
SNTASKSLSPMPQIAGVTYYGDIPKQPKRVVSLASTYTGYLKKLDMNLVGVTSYDKKNPILAKTVKKAKQVAATDLEAITTLKPDLIVVGSTEENIKQLAEIAPVISIEYRKRDYLQVLSDFGRIFNKEGKAKKWLKDWKTKTAAYEKEVKAVTGDKATFTIMGLYEKDVYLFGKDWGRGGEIIHQAFHYDAPEKVKTEVFKQGYLSLSQEVLPDYIGDYVVIAAEDDKTGSALYESKLWQSIPAVKKHHVIKVNANVFYFTDPLSLEYQLETLREAILSSEN
9FTB , Knot 109 248 0.80 40 166 240 
MGSSHHHHHHSSGLVPRGSHMNIAIIPARGGSKRIPRKNIKPFHSKPMIAWSILAAKKAGCFERIIVSTDDAEIAAVALEYGAEVPFTRPAEIANDYATTGEVISHAINWLINQQGQVPENVCCLYATAPFVEPDDLCQGLELLTFNKECQFVFSATRFSFPIQRAIKLDESGWVSMFHPEYQLTRSQDLEEAYHDAGQFYWGKANAWLNKLPIFAVHTQVVLLPSHRVQDIDTQDDWLRAEKLFTLR

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 \(c\).

\(|P_{f(2QCS_1)}(2) \setminus P_{f(8XET_1)}(2)|=93\), \(|P_{f(8XET_1)}(2) \setminus P_{f(2QCS_1)}(2)|=61\). 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 1:10111100100000100111010001100100100001010010010011010110111100000100011011000011010010001000011011011111010101000001011100111101100100110100101010110111010010010110001010011100010101001111001010010101010011101110010001101111111100111101111100110100011010101100100010011001101010001101001100100001110001111000010111110101110000100000001010100001001001
Pair \(Z_2\) Length of longest common subsequence
2QCS_1,8XET_1 154 3
2QCS_1,9FTB_1 173 4
8XET_1,9FTB_1 145 4

Newick tree

 
[
	2QCS_1:84.78,
	[
		8XET_1:72.5,9FTB_1:72.5
	]:12.28
]

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 \((0.85)(0.8)(\frac{633 }{\log_{20} 633}-\frac{283}{\log_{20}283})=97.7\)
Status Protein1 Protein2 d d1/2
Query variables 2QCS_1 8XET_1 122 110.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]