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Parikh vectors
2KKJ_1 3LZX_1 4OGL_1 Letter Amino acid
4 16 5 N Asparagine
7 28 17 L Leucine
0 2 1 W Tryptophan
1 11 4 Y Tyrosine
5 27 28 A Alanine
1 27 22 G Glycine
3 26 11 I Isoleucine
2 14 23 T Threonine
2 6 11 M Methionine
6 12 10 P Proline
7 16 15 S Serine
1 17 12 D Aspartic acid
10 12 9 Q Glutamine
0 31 16 E Glutamic acid
4 26 10 K Lycine
2 26 27 V Valine
3 11 13 R Arginine
0 2 3 C Cysteine
0 7 6 H Histidine
1 15 10 F Phenylalanine 

2KKJ_1|Chain A|CREB-binding protein|Mus musculus (10090)
>3LZX_1|Chains A, B|Ferredoxin--NADP reductase 2|Bacillus subtilis (1423)
>4OGL_1|Chains A, B, C, D, E, F|Uridine phosphorylase|Vibrio cholerae O1 biovar El Tor (243277)
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 \(w=f(c)\)
2KKJ , Knot 31 59 0.71 32 48 53 
PNRSISPSALQDLLRTLKSPSSPQQQQQVLNILKSNPQLMAAFIKQRTAKYVANQPGMQ
3LZX , Knot 144 332 0.84 40 197 321 
MREDTKVYDITIIGGGPVGLFTAFYGGMRQASVKIIESLPQLGGQLSALYPEKYIYDVAGFPKIRAQELINNLKEQMAKFDQTICLEQAVESVEKQADGVFKLVTNEETHYSKTVIITAGNGAFKPRKLELENAEQYEGKNLHYFVDDLQKFAGRRVAILGGGDSAVDWALMLEPIAKEVSIIHRRDKFRAHEHSVENLHASKVNVLTPFVPAELIGEDKIEQLVLEEVKGDRKEILEIDDLIVNYGFVSSLGPIKNWGLDIEKNSIVVKSTMETNIEGFFAAGDICTYEGKVNLIASGFGEAPTAVNNAKAYMDPKARVQPLHSTSLFENK
4OGL , Knot 113 253 0.82 40 164 238 
MTKTVFHLGVTEADLNGATLAIIPGDPARVQKIAELMDNPVFLASHREYTVYRAELDGQSVVVCSTGIGGPSTSIAVEELAQLGVRTFLRVGTTGAIQPHVNVGDMIVTTGSVRLDGASLHFAPMEFPAVPDFDVATAMKAAAQESGATVHMGVTASSDTFYPGQERYDTFTGRVVRRFQGSMKEWQDMGVLNFEMESATLLTMCASSGLKAGCVAGVIINRTQKEIPDHATLKETEARSIKVVVEAARKMLK

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 \(c\).

\(|P_{f(2KKJ_1)}(2) \setminus P_{f(3LZX_1)}(2)|=23\), \(|P_{f(3LZX_1)}(2) \setminus P_{f(2KKJ_1)}(2)|=172\). 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 1:10001010110011001001001000001101100010111111000010011001110
Pair \(Z_2\) Length of longest common subsequence
2KKJ_1,3LZX_1 195 3
2KKJ_1,4OGL_1 164 3
3LZX_1,4OGL_1 169 4

Newick tree

 
[
	3LZX_1:94.10,
	[
		2KKJ_1:82,4OGL_1:82
	]:12.10
]

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 \((0.85)(0.8)(\frac{391 }{\log_{20} 391}-\frac{59}{\log_{20}59})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 2KKJ_1 3LZX_1 134 78
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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