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Parikh vectors
8QKB_1 9CEA_1 1RDE_1 Letter Amino acid
7 2 0 C Cysteine
6 5 0 I Isoleucine
16 3 0 V Valine
11 1 0 N Asparagine
13 5 0 D Aspartic acid
25 4 9 G Glycine
10 6 0 L Leucine
13 1 0 Y Tyrosine
17 6 0 A Alanine
8 1 0 Q Glutamine
18 7 0 E Glutamic acid
6 2 0 M Methionine
8 3 0 F Phenylalanine
9 2 0 P Proline
19 1 0 S Serine
7 3 6 T Threonine
5 2 0 W Tryptophan
6 3 0 R Arginine
3 1 0 H Histidine
13 5 0 K Lycine 

8QKB_1|Chains A, B, C, D|Cathepsin L|Homo sapiens (9606)
>9CEA_1|Chains A, B, C, D|Chromobox protein homolog 5|Homo sapiens (9606)
>1RDE_1|Chain A|Thrombin-binding DNA aptamer|null
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 \(w=f(c)\)
8QKB , Knot 99 220 0.81 40 156 209 
APRSVDWREKGYVTPVKNQGQCGSCWAFSATGALEGQMFRKTGRLISLSEQNLVDCSGPQGNEGCNGGLMDYAFQYVQDNGGLDSEESYPYEATEESCKYNPKYSVANDAGFVDIPKQEKALMKAVATVGPISVAIDAGHESFLFYKEGIYFEPDCSSEDMDHGVLVVGYGFESTESDNNKYWLVKNSWGEEWGMGGYVKMAKDRRNHCGIASAASYPTV
9CEA , Knot 37 63 0.81 40 55 61 
IARGFERGLEPEKIIGATDSCGDLMFLMKWKDTDEADLVLAKEANVKCPQIVIAFYEERLTWH
1RDE , Knot 6 15 0.36 4 4 6 
GGTTGGTGTGGTTGG

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 \(c\).

\(|P_{f(8QKB_1)}(2) \setminus P_{f(9CEA_1)}(2)|=126\), \(|P_{f(9CEA_1)}(2) \setminus P_{f(8QKB_1)}(2)|=25\). 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 1:1100101000101011000100100111010111010110001011010000110001101001001111001100100011100000010010000000010001100111101100001110111011110111011000111000110101000000100111111011000000000011100011001111101011000000011101100101
Pair \(Z_2\) Length of longest common subsequence
8QKB_1,9CEA_1 151 3
8QKB_1,1RDE_1 156 2
9CEA_1,1RDE_1 59 1

Newick tree

 
[
	8QKB_1:86.98,
	[
		9CEA_1:29.5,1RDE_1:29.5
	]:57.48
]

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 \((0.85)(0.8)(\frac{283 }{\log_{20} 283}-\frac{63}{\log_{20}63})=71.1\)
Status Protein1 Protein2 d d1/2
Query variables 8QKB_1 9CEA_1 88 55.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]