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Parikh vectors
3EUE_1 5OVT_1 4YHL_1 Letter Amino acid
23 15 31 S Serine
16 9 25 T Threonine
0 2 4 W Tryptophan
12 11 13 Y Tyrosine
11 5 8 Q Glutamine
32 16 13 L Leucine
18 8 15 K Lycine
10 4 4 M Methionine
13 4 7 N Asparagine
11 5 3 H Histidine
12 7 12 P Proline
24 15 21 V Valine
12 9 6 F Phenylalanine
12 14 7 D Aspartic acid
13 1 5 C Cysteine
27 17 21 G Glycine
18 10 2 I Isoleucine
27 27 13 A Alanine
17 9 4 R Arginine
20 13 8 E Glutamic acid 

3EUE_1|Chain A|Uridine phosphorylase 1|Homo sapiens (9606)
>5OVT_1|Chains A, B, C, D, E, F, G|BPH|Thiobacillus denitrificans (36861)
>4YHL_1|Chain A[auth H]|aDabi-Fab2b heavy chain|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 \(w=f(c)\)
3EUE , Knot 141 328 0.83 38 197 308 
MRGSHHHHHHGSPGLQEFMAATGANAEKAESHNDCPVRLLNPNIAKMKEDILYHFNLTTSRHNFPALFGDVKFVCVGGSPSRMKAFIRCVGAELGLDCPGRDYPNICAGTDRYAMYKVGPVLSVSHGMGIPSISIMLHELIKLLYYARCSNVTIIRIGTSGGIGLEPGTVVITEQAVDTCFKAEFEQIVLGKRVIRKTDLNKKLVQELLLCSAELSEFTTVVGNTMCTLDFYEGQGRLDGALCSYTEKDKQAYLEAAYAAGVRNIEMESSVFAAMCSACGLQAAVVCVTLLNRLEGDQISSPRNVLSEYQQRPQRLVSYFIKKKLSKA
5OVT , Knot 93 201 0.81 40 145 194 
TTVTIVRKDGRIAIAADTLTKWGGGKESADYVANHEKIIRVGDSYVAITGSATFKLILADYFASLDEPPQLDSVARIFCVWNTLHGALKEHYYLQAGEDKEDDLESSRMDVLIANPRGIFGVAAHRTVQEFSKFYAYGSGSPYALGAMYAAYRAPSLDAEAVARLGVMAAAEFHDESGLPVQSFVMELSPDVGSGENLYFQ
4YHL , Knot 94 222 0.76 40 141 205 
QVQLVQSGAEVKKPGASVKVSCKASGYTFTDYYMHWVRQAPGQGLEWMGETNPRNGGTTYNEKFKGKATMTRDTSTSTAYMELSSLRSEDTAVYYCTIGTSGYDYFDYWGQGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSC

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 \(c\).

\(|P_{f(3EUE_1)}(2) \setminus P_{f(5OVT_1)}(2)|=98\), \(|P_{f(5OVT_1)}(2) \setminus P_{f(3EUE_1)}(2)|=46\). 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 1:1010000000101110011110110100100000011011010110100011001010000001111110101101110100101110011101110011000101011000011001111101001111101011100110110010000101101100111110110111000110001010100111100110000100011001110010100100111001001010010101011100000000010101101111001010001111100101101111010110010100100100110000001001100110001001
Pair \(Z_2\) Length of longest common subsequence
3EUE_1,5OVT_1 144 3
3EUE_1,4YHL_1 180 5
5OVT_1,4YHL_1 164 4

Newick tree

 
[
	4YHL_1:90.30,
	[
		3EUE_1:72,5OVT_1:72
	]:18.30
]

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 \((0.85)(0.8)(\frac{529 }{\log_{20} 529}-\frac{201}{\log_{20}201})=94.6\)
Status Protein1 Protein2 d d1/2
Query variables 3EUE_1 5OVT_1 117 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]