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Parikh vectors
6JGM_1 4ZFM_1 1YXP_1 Letter Amino acid
19 33 7 A Alanine
16 35 0 D Aspartic acid
16 23 0 I Isoleucine
39 34 0 L Leucine
15 19 0 S Serine
16 30 0 R Arginine
8 25 0 N Asparagine
21 20 0 K Lycine
10 29 0 Y Tyrosine
17 29 0 E Glutamic acid
16 40 6 G Glycine
5 11 0 M Methionine
4 17 0 W Tryptophan
23 34 0 V Valine
3 1 6 C Cysteine
9 18 0 Q Glutamine
10 19 0 H Histidine
16 22 0 F Phenylalanine
19 25 0 P Proline
17 21 0 T Threonine 

6JGM_1|Chain A|Cyclin-dependent kinase 2|Homo sapiens (9606)
>4ZFM_1|Chains A, B, C, D|Putative 6-phospho-beta-galactobiosidase|Geobacillus stearothermophilus (1422)
>1YXP_1|Chains A, B|5'-R(*CP*UP*(5BU)P*GP*CP*UP*GP*AP*AP*GP*UP*GP*CP*AP*CP*AP*CP*AP*GP*CP*AP*AP*G)-3'|
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 \(w=f(c)\)
6JGM , Knot 135 299 0.85 40 191 285 
SMENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL
4ZFM , Knot 196 485 0.83 40 247 453 
MIHHHHHHEHRHLKPFPPEFLWGAASAAYQVEGAWNEDGKGLSVWDVFAKQPGRTFKGTNGDVAVDHYHRYQEDVALMAEMGLKAYRFSVSWSRVFPDGNGAVNEKGLDFYDRLIEELRNHGIEPIVTLYHWDVPQALMDAYGAWESRRIIDDFDRYAVTLFQRFGDRVKYWVTLNQQNIFISFGYRLGLHPPGVKDMKRMYEANHIANLANAKVIQSFRHYVPDGKIGPSFAYSPMYPYDSRPENVLAFENAEEFQNHWWMDVYAWGMYPQAAWNYLESQGLEPTVAPGDWELLQAAKPDFMGVNYYQTTTVEHNPPDGVGEGVMNTTGKKGTSTSSGIPGLFKTVRNPHVDTTNWDWAIDPVGLRIGLRRIANRYQLPILITENGLGEFDTLEPGDIVNDDYRIDYLRRHVQEIQRAITDGVDVLGYCAWSFTDLLSWLNGYQKRYGFVYVNRDDESEKDLRRIKKKSFYWYQRVIETNGAEL
1YXP , Knot 10 23 0.45 8 10 16 
CUUGCUGAAGUGCACACAGCAAG

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 \(c\).

\(|P_{f(6JGM_1)}(2) \setminus P_{f(4ZFM_1)}(2)|=52\), \(|P_{f(4ZFM_1)}(2) \setminus P_{f(6JGM_1)}(2)|=108\). 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 1:01001001001101001110010001010111100101000001110011001011001001011011011000001011101100010011010110111111100011011011110000011000101001110001110110111101111110000001101100110111100000011011011011101100011111000100110110011010011111100110001011011000100111110001001100110001000101011110111001001110101
Pair \(Z_2\) Length of longest common subsequence
6JGM_1,4ZFM_1 160 3
6JGM_1,1YXP_1 195 2
4ZFM_1,1YXP_1 251 3

Newick tree

 
[
	1YXP_1:12.26,
	[
		6JGM_1:80,4ZFM_1:80
	]:41.26
]

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 \((0.85)(0.8)(\frac{784 }{\log_{20} 784}-\frac{299}{\log_{20}299})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 6JGM_1 4ZFM_1 168 134.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]