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Parikh vectors
2RNA_1 3IYM_1 3KFV_1 Letter Amino acid
3 49 29 A Alanine
3 19 17 Q Glutamine
0 6 4 M Methionine
6 6 8 Y Tyrosine
3 10 4 N Asparagine
5 22 13 D Aspartic acid
1 3 0 C Cysteine
8 17 19 E Glutamic acid
3 32 19 G Glycine
2 10 14 H Histidine
2 14 12 K Lycine
2 25 18 T Threonine
2 25 33 R Arginine
2 8 16 I Isoleucine
7 40 28 L Leucine
4 24 20 P Proline
6 52 22 S Serine
2 7 5 W Tryptophan
3 35 17 V Valine
0 30 10 F Phenylalanine 

2RNA_1|Chain A|Tyrosine-protein kinase ITK/TSK|Mus musculus (10090)
>3IYM_1|Chains A, B|Capsid protein|Penicillium stoloniferum virus S (216371)
>3KFV_1|Chain A|Tight junction protein ZO-3|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 \(w=f(c)\)
2RNA , Knot 37 64 0.80 36 57 62 
GSPEETLVIALYDYQTNDPQELALRCDEEYYLLDSSEIHWWRVQDKNGHEGYAPSSYLVEKSPN
3IYM , Knot 175 434 0.81 40 210 403 
MSSIAPTDSVSSSGKRSKPGKRERQQARSAVGSAGGKPASASKAAAFAQGGSSDPVPMPGKYPVVFSTGAGEPTRDQEFALPVHKAFPLFGSVSDKYRRNPRYAEFRAHSEFTDGVFGTHLAVSSLLRLAQQLVHAHVNMGLPLGDFAPLASSDVRIPSALASVVNQFGEFSSPSIGTRFLLRDFEHAVSRVVFLADQLWTNGNSHHIFARSWLPMSNNDGNFKTIVASRLLEFISAGDLSILPTVLEDAVLSGEVPEAWEQVKDLLGDAPGVGQVDRRDRFDFLFKSYADVGQFTTAFTTQAASDVLTELGLPWNSPSAGHLNWQYSTKQRFTFLADTWAKLSAAYSQFFELSSGLATRQSATGSHAQMVDLTSVEGVTVLKAALALSAPEFSLAACFPPSCIFVGGLTRRVVVTTSLSVSQRATEFCQMDWR
3KFV , Knot 131 308 0.81 38 181 293 
MGDSFYIRTHFELEPSPPSGLGFTRGDVFHVLDTLHPGPGQSHARGGHWLAVRMGRDLREQERGIIPNQSRAEQLASLEAAQRAVGVGPGSSAGSNARAEFWRLRGLRRGAKKTTQRSREDLSALTRQGRYPPYERVVLREASFKRPVVILGPVADIAMQKLTAEMPDQFEIAETVSRTDSPSKIIKLDTVRVIAEKDKHALLDVTPSAIERLNYVQYYPIVVFFIPESRPALKALRQWLAPASRRSTRRLYAQAQKLRKHSSHLFTATIPLNGTSDTWYQELKAIIREQQTRPIWTAEDQLHHHHHH

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 \(c\).

\(|P_{f(2RNA_1)}(2) \setminus P_{f(3IYM_1)}(2)|=28\), \(|P_{f(3IYM_1)}(2) \setminus P_{f(2RNA_1)}(2)|=181\). 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 1:1010001111100000001001110000000110000101101000010010110001100010
Pair \(Z_2\) Length of longest common subsequence
2RNA_1,3IYM_1 209 3
2RNA_1,3KFV_1 172 3
3IYM_1,3KFV_1 139 3

Newick tree

 
[
	2RNA_1:10.95,
	[
		3KFV_1:69.5,3IYM_1:69.5
	]:33.45
]

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 \((0.85)(0.8)(\frac{498 }{\log_{20} 498}-\frac{64}{\log_{20}64})=131.\)
Status Protein1 Protein2 d d1/2
Query variables 2RNA_1 3IYM_1 164 94.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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