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Parikh vectors
2ABY_1 2CBX_1 6SPL_1 Letter Amino acid
6 19 19 R Arginine
8 27 16 G Glycine
8 13 15 Y Tyrosine
4 35 26 A Alanine
9 5 15 Q Glutamine
4 6 11 M Methionine
7 11 13 F Phenylalanine
4 22 18 T Threonine
2 3 5 W Tryptophan
6 23 22 V Valine
13 22 23 E Glutamic acid
10 7 13 H Histidine
8 17 22 I Isoleucine
14 21 42 L Leucine
14 6 19 K Lycine
4 9 14 N Asparagine
10 13 27 D Aspartic acid
0 3 3 C Cysteine
4 20 17 P Proline
11 17 21 S Serine 

2ABY_1|Chain A|hypothetical protein TA0743|Thermoplasma acidophilum (2303)
>2CBX_1|Chains A, B, C|5'-FLUORO-5'-DEOXYADENOSINE SYNTHASE|STREPTOMYCES CATTLEYA (29303)
>6SPL_1|Chain A|Mitogen-activated protein kinase 14|Mus musculus (10090)
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 \(w=f(c)\)
2ABY , Knot 70 146 0.79 38 115 139 
MGTSHHHHHHSSGRENLYFQSHMQKGLEIAFQTINGLDESLVQALAGVTASDFPDLDIKYNIFLVDLYGQKYFRILFQSKKLSELHPEERKKVREKFDENSRMQYSELMTKYHDLKKQGKIKDRPVKEVHEEYDLWEDPIWQYIGS
2CBX , Knot 130 299 0.82 40 184 289 
MAANSTRRPIIAFMSDLGTTDDSVAQCKGLMYSICPDVTVVDVCHSMTPWDVEEGARYIVDLPRFFPEGTVFATTTYPATGTTTRSVAVRIKQAAKGGARGQWAGSGAGFERAEGSYIYIAPNNGLLTTVLEEHGYLEAYEVTSPKVIPEQPEPTFYSREMVAIPSAHLAAGFPLSEVGRPLEDHEIVRFNRPAVEQDGEALVGVVSAIDHPFGNVWTNIHRTDLEKAGIGYGARLRLTLDGVLPFEAPLTPTFADAGEIGNIAIYLNSRGYLSIARNAASLAYPYHLKEGMSARVEAR
6SPL , Knot 157 361 0.85 40 223 348 
GMSQERPTFYRQELNKTIWEVPERYQNLSPVGSGAYGSVCAAFDTKTGHRVAVKKLSRPFQSIIHAKRTYRELRLLKHMKHENVIGLLDVFTPARSLEEFNDVYLVTHLMGADLNNIVKCQKLTDDHVQFLIYQILRGLKYIHSADIIHRDLKPSNLAVNEDSELKILDFGLARHTDDEMTGYVATRWYRAPEIMLNWMHYNQTVDIWSVGCIMAELLTGRTLFPGTDHIDQLKLILRLVGTPGAELLKKISSESARNYIQSLAQMPKMNFANVFIGANPLAVDLLEKMLVLDSDKRITAAQALAHAYFAQYHDPDDEPVADPYDQSFESRDLLIDEWKSLTYDEVISFVPPPLDQEEMES

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 \(c\).

\(|P_{f(2ABY_1)}(2) \setminus P_{f(2CBX_1)}(2)|=56\), \(|P_{f(2CBX_1)}(2) \setminus P_{f(2ABY_1)}(2)|=125\). 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 1:11000000000010001010001001101110010110001101111101001101010001111010100010111000010010100000100010000010000110000010001010001100100000110011100110
Pair \(Z_2\) Length of longest common subsequence
2ABY_1,2CBX_1 181 3
2ABY_1,6SPL_1 182 4
2CBX_1,6SPL_1 173 4

Newick tree

 
[
	2ABY_1:92.12,
	[
		2CBX_1:86.5,6SPL_1:86.5
	]:5.62
]

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 \((0.85)(0.8)(\frac{445 }{\log_{20} 445}-\frac{146}{\log_{20}146})=88.9\)
Status Protein1 Protein2 d d1/2
Query variables 2ABY_1 2CBX_1 113 83
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]