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Parikh vectors
3GVV_1 3FVY_1 6FQJ_1 Letter Amino acid
26 43 18 V Valine
24 43 10 R Arginine
12 32 28 K Lycine
5 10 4 W Tryptophan
12 31 13 Y Tyrosine
30 57 25 G Glycine
11 27 16 I Isoleucine
10 8 10 M Methionine
17 37 13 D Aspartic acid
6 6 5 C Cysteine
22 34 3 Q Glutamine
26 66 20 E Glutamic acid
39 61 19 A Alanine
43 84 20 L Leucine
16 39 7 F Phenylalanine
11 50 17 S Serine
6 19 10 N Asparagine
9 11 2 H Histidine
26 33 7 P Proline
16 37 17 T Threonine 

3GVV_1|Chain A|Uroporphyrinogen decarboxylase|Homo sapiens (9606)
>3FVY_1|Chain A|Dipeptidyl-peptidase 3|Homo sapiens (9606)
>6FQJ_1|Chains A, B, C, D, E, F, G, H|Glutamate receptor 2,Glutamate receptor 2|Rattus norvegicus (10116)
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 \(w=f(c)\)
3GVV , Knot 153 367 0.82 40 199 342 
MEANGLGPQGFPELKNDTFLRAAWGEETDYTPVWCMRQAGRYLPEFRETRAAQDFFSTCRSPEACCELTLQPLRRFPLDAAIIFSDILVVPQALGMEVTMVPGKGPSFPEPLREEQDLERLRDPEVVASELGYVFQAITLTRQRLAGRVPLIGFAGAPWTLMTYMVEGGGSSTMAQAKRWLYQRPQASHQLLRILTDALVPYLVGQVVAGAQALQLYESHAGHLGPQLFNKFALPYIRDVAKQVKARLREAGLAPVPMIIFAKDGHFALEELAQAGYEVVGLDWTVAPKKARECVGKTVTLQGNLDPCALYASEEEIGQLVKQMLDDFGPHRYIANLGHGLYPDMDPEHVGAFVDAVHKHSRLLRQN
3FVY , Knot 281 728 0.84 40 278 663 
GAMADTQYILPNDIGVSSLDCREAFRLLSPTERLYAYHLSRAAWYGGLAVLLQTSPEAPYIYALLSRLFRAQDPDQLHQHALAEGLTEEEYQAFLVYAAGVYSNMGNYKSFGDTKFVPNLPKEKLERVILGSEAAQQHPEEVRGLWQTCGELMFSLEPRLRHLGLGKEGITTYFSGNCTMEDAKLAQDFLDSQNLSAYNTRLFKEVDGEGKPYYEVRLASVLGSEPSLDSEVTSKLKSYEFRGSPFQVTRGDYAPILQKVVEQLEKAKAYAANSHQGQMLAQYIESFTQGSIEAHKRGSRFWIQDKGPIVESYIGFIESYRDPFGSRGEFEGFVAVVNKAMSAKFERLVASAEQLLKELPWPPTFEKDKFLTPDFTSLDVLTFAGSGIPAGINIPNYDDLRQTEGFKNVSLGNVLAVAYATQREKLTFLEEDDKDLYILWKGPSFDVQVGLHELLGHGSGKLFVQDEKGAFNFDQETVINPETGEQIQSWYRSGETWDSKFSTIASSYEECRAESVGLYLCLHPQVLEIFGFEGADAEDVIYVNWLNMVRAGLLALEFYTPEAFNWRQAHMQARFVILRVLLEAGEGLVTITPTTGSDGRPDARVRLDRSKIRSVGKPALERFLRRLQVLKSTGDVAGGRALYEGYATVTDAPPECFLTLRDTVLLRKESRKLIVQPNTRLEGSDVQLLEYEASAAGLIRSFSERFPEDGPELEEILTQLATADARFW
6FQJ , Knot 122 264 0.86 40 173 254 
GANKTVVVTTILESPYVMMKKNHEMLEGNERYEGYCVDLAAEIAKHCGFKYKLTIVGDGKYGARDADTKIWNGMVGELVYGKADIAIAPLTITLVREEVIDFSKPFMSLGISIMIKKGTPIESAEDLSKQTEIAYGTLDSGSTKEFFRRSKIAVFDKMWTYMRSAEPSVFVRTTAEGVARVRKSKGKYAYLLESTMNEYIEQRKPCDTMKVCGNLDSKGYGIATPKGSSLGNAVNLAVLKLNEQGLLDKLKNKWWYDKGECGSG

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 \(c\).

\(|P_{f(3GVV_1)}(2) \setminus P_{f(3FVY_1)}(2)|=44\), \(|P_{f(3FVY_1)}(2) \setminus P_{f(3GVV_1)}(2)|=123\). 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 1:1010111101110100001101111000000111010011001101000011001100000101000101011001110111110011111011110101111011011011000001001001011100110110110100001110111111111110110011011100011010011000101000110110011110111011111011010000110111011001111010011001010100111111111111001011100110110011110101110010001100101010101011010000110110011001110001101101101010100111110110000011000
Pair \(Z_2\) Length of longest common subsequence
3GVV_1,3FVY_1 167 5
3GVV_1,6FQJ_1 178 4
3FVY_1,6FQJ_1 191 4

Newick tree

 
[
	6FQJ_1:95.06,
	[
		3GVV_1:83.5,3FVY_1:83.5
	]:11.56
]

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 \((0.85)(0.8)(\frac{1095 }{\log_{20} 1095}-\frac{367}{\log_{20}367})=192.\)
Status Protein1 Protein2 d d1/2
Query variables 3GVV_1 3FVY_1 240 178.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]