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Parikh vectors
8ICS_1 3BYM_1 2ITK_1 Letter Amino acid
0 4 5 H Histidine
0 18 8 I Isoleucine
0 14 7 P Proline
0 10 24 S Serine
0 25 15 E Glutamic acid
0 11 5 N Asparagine
0 13 6 D Aspartic acid
1 3 2 C Cysteine
0 27 9 L Leucine
0 14 10 K Lycine
0 9 4 M Methionine
0 7 3 W Tryptophan
4 19 11 A Alanine
0 12 3 Y Tyrosine
0 10 8 Q Glutamine
1 16 17 G Glycine
0 11 7 F Phenylalanine
0 15 4 V Valine
0 17 13 R Arginine
2 17 6 T Threonine 

8ICS_1|Chain A[auth T]|DNA (5'-D(*CP*AP*TP*TP*AP*GP*AP*A)-3')|
>3BYM_1|Chain A|Proto-oncogene tyrosine-protein kinase LCK|Homo sapiens (9606)
>2ITK_1|Chain A|Peptidyl-prolyl cis-trans isomerase NIMA-interacting 1|Homo sapiens (9606)
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 \(w=f(c)\)
8ICS , Knot 6 8 0.52 8 7 6 
CATTAGAA
3BYM , Knot 125 272 0.85 40 189 270 
QKPWWEDEWEVPRETLKLVERLGAGQFGEVWMGYYNGHTKVAVKSLKQGSMSPDAFLAEANLMKQLQHQRLVRLYAVVTQEPIYIITEYMENGSLVDFLKTPSGIKLTINKLLDMAAQIAEGMAFIEERNYIHRDLRAANILVSDTLSCKIADFGLARLIEDNEYTAREGAKFPIKWTAPEAINYGTFTIKSDVWSFGILLTEIVTHGRIPYPGMTNPEVIQNLERGYRMVRPDNCPEELYQLMRLCWKERPEDRPTFDYLRSVLEDFFTAT
2ITK , Knot 78 167 0.79 40 122 161 
GSHGMADEEKLPPGWEKAMSRSSGRVYYFNHITNASQWERPSGNSSSGGKNGQGEPARVRCSHLLVKHSQSRRPSSWRQEKITRTKEEALELINGYIQKIKSGEEDFESLASQFSDCSSAKARGDLGAFSRGQMQKPFEDASFALRTGEMSGPVFTDSGIHIILRTE

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 \(c\).

\(|P_{f(8ICS_1)}(2) \setminus P_{f(3BYM_1)}(2)|=2\), \(|P_{f(3BYM_1)}(2) \setminus P_{f(8ICS_1)}(2)|=184\). 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 1:01001111
Pair \(Z_2\) Length of longest common subsequence
8ICS_1,3BYM_1 186 2
8ICS_1,2ITK_1 127 2
3BYM_1,2ITK_1 177 3

Newick tree

 
[
	3BYM_1:98.20,
	[
		8ICS_1:63.5,2ITK_1:63.5
	]:34.70
]

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 \((0.85)(0.8)(\frac{280 }{\log_{20} 280}-\frac{8}{\log_{20}8})=93.3\)
Status Protein1 Protein2 d d1/2
Query variables 8ICS_1 3BYM_1 121 62
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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