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Parikh vectors
6SYK_1 5QAX_1 3DHW_1 Letter Amino acid
0 17 26 L Leucine
0 6 11 P Proline
0 16 27 V Valine
2 16 22 A Alanine
2 0 0 C Cysteine
0 15 6 E Glutamic acid
0 16 6 N Asparagine
0 7 0 H Histidine
0 11 9 F Phenylalanine
0 7 10 M Methionine
0 12 11 R Arginine
0 14 6 Q Glutamine
0 18 3 K Lycine
0 14 7 S Serine
0 12 15 T Threonine
0 10 3 W Tryptophan
0 6 8 Y Tyrosine
0 14 1 D Aspartic acid
10 15 23 G Glycine
0 17 23 I Isoleucine 

6SYK_1|Chains A, B|GCnCG|synthetic construct (32630)
>5QAX_1|Chains A, B, C, D|Beta-lactamase|Klebsiella pneumoniae (573)
>3DHW_1|Chains A, B, E, F|D-methionine transport system permease protein metI|Escherichia coli (83333)
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 \(w=f(c)\)
6SYK , Knot 7 14 0.44 6 5 7 
GCGGAGGGGAGGCG
5QAX , Knot 112 243 0.84 38 166 239 
KEWQENKSWNAHFTEHKSQGVVVLWNENKQQGFTNNLKRANQAFLPASTFKIPNSLIALDLGVVKDEHQVFKWDGQTRDIATWNRDHNLITAMKYSVVPVYQEFARQIGEARMSKMLHAFDYGNEDISGNVDSFWLDGGIRISATEQISFLRKLYHNKLHVSERSQRIVKQAMLTEANGDYIIRAKTGYSTRIEPKIGWWVGWVELDDNVWFFAMNMDMPTSDGLGLRQAITKEVLKQEKIIP
3DHW , Knot 94 217 0.77 36 124 200 
MSEPMMWLLVRGVWETLAMTFVSGFFGFVIGLPVGVLLYVTRPGQIIANAKLYRTVSAIVNIFRSIPFIILLVWMIPFTRVIVGTSIGLQAAIVPLTVGAAPFIARMVENALLEIPTGLIEASRAMGATPMQIVRKVLLPEALPGLVNAATITLITLVGYSAMGGAVGAGGLGQIGYQYGYIGYNATVMNTVLVLLVILVYLIQFAGDRIVRAVTRK

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 \(c\).

\(|P_{f(6SYK_1)}(2) \setminus P_{f(5QAX_1)}(2)|=4\), \(|P_{f(5QAX_1)}(2) \setminus P_{f(6SYK_1)}(2)|=165\). 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 1:10111111111101
Pair \(Z_2\) Length of longest common subsequence
6SYK_1,5QAX_1 169 2
6SYK_1,3DHW_1 123 4
5QAX_1,3DHW_1 160 3

Newick tree

 
[
	5QAX_1:88.12,
	[
		6SYK_1:61.5,3DHW_1:61.5
	]:26.62
]

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 \((0.85)(0.8)(\frac{257 }{\log_{20} 257}-\frac{14}{\log_{20}14})=83.5\)
Status Protein1 Protein2 d d1/2
Query variables 6SYK_1 5QAX_1 109 57
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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