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Parikh vectors
7QBT_1 3UDK_1 3POS_1 Letter Amino acid
30 24 13 I Isoleucine
9 8 3 M Methionine
33 34 15 V Valine
26 24 13 T Threonine
40 29 17 L Leucine
28 15 24 K Lycine
24 20 8 P Proline
21 24 9 A Alanine
15 15 14 N Asparagine
12 6 3 C Cysteine
37 23 20 E Glutamic acid
41 34 25 G Glycine
1 7 5 W Tryptophan
28 32 18 S Serine
12 19 11 Y Tyrosine
17 18 6 R Arginine
22 22 26 D Aspartic acid
9 17 12 Q Glutamine
7 13 5 H Histidine
24 20 18 F Phenylalanine 

7QBT_1|Chains A, B, C, D|Methyl coenzyme M reductase-arginine methyltransferase Mmp10|Methanosarcina acetivorans (2214)
>3UDK_1|Chain A|Beta-secretase 1|Homo sapiens (9606)
>3POS_1|Chains A, B, C|Calreticulin|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 \(w=f(c)\)
7QBT , Knot 173 436 0.80 40 224 396 
MASWSHPQFEKSGGGGGENLYFQGHMEVVVDVGGNPGVDCKGFCKYCYFKKVKDIQPLGCKYCLPFKKGCDYCTRSVKESYSGFKSLQMVLEETANKLYFTSGEVKKFTVSGGGDLSCYPELKSLITFLSQFNTPIHLGYTSGKGFSKPDDALFYIDNGVTEVSFTVFATDPALRAEYMKDPEPEASIQVLRDFCTHCEVYGAIVLLPGINDGEVLEKTLCDLENMGAKGAILMRFANFQENGLILNNSPIIPGITPHTVSEFTEIVRSSAEKHPSIRITGTPLEDPLIGSPFAIRNVPEALLKLPRVSKKATIITGQVAASRLTEIFEALGGTVNVIPVKKDIGCLITIDDFKALDLSEVTETVFIPGRAFVHDMEIKEALRRDGVDRIVRRGPERLSVDGEMSIGMTREEVLELEVENFTELIGQINSLGLPLE
3UDK , Knot 171 404 0.84 40 230 384 
GSFVEMVDNLRGKSGQGYYVEMTVGSPPQTLNILVDTGSSNFAVGAAPHPFLHRYYQRQLSSTYRDLRKGVYVPYTQGKWEGELGTDLVSIPHGPNVTVRANIAAITESDKFFINGSNWEGILGLAYAEIARPDDSLEPFFDSLVKQTHVPNLFSLQLCGAGFPLNQSEVLASVGGSMIIGGIDHSLYTGSLWYTPIRREWYYEVIIVRVEINGQDLKMDCKEYNYDKSIVDSGTTNLRLPKKVFEAAVKSIKAASSTEKFPDGFWLGEQLVCWQAGTTPWNIFPVISLYLMGEVTNQSFRITILPQQYLRPVEDVATSQDDCYKFAISQSSTGTVMGAVIMEGFYVVFDRARKRIGFAVSACHVHDEFRTAAVEGPFVTLDMEDCGYNIPQTDESRSHHHHHH
3POS , Knot 122 265 0.85 40 177 258 
NKGSIEGREPAVYFKEQFLDGDGWTSRWIESKHKSDFGKFVLSSGKFYGDEEKDKGLQTSQDARFYALSASFEPFSNKGQTLVVQFTVKHEQNIDCGGGYVKLFPNSLDQTDMHGDSEYNIMFGPDICGPGTKKVHVIFNYKGKNVLINKDIRCKDDEFTHLYTLIVRPDNTYEVKIDNSQVESGSLEDDWDFLPGSGDPSIYAYDNFGVLGLDLWQVKSGTIFDNFLITNDEAYAEEFGNETWGVTKAAEKQMKDKQDEEQRLK

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 \(c\).

\(|P_{f(7QBT_1)}(2) \setminus P_{f(3UDK_1)}(2)|=81\), \(|P_{f(3UDK_1)}(2) \setminus P_{f(7QBT_1)}(2)|=87\). 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 1:1101001010001111100101010101110111011100011000001001001011100001110010000000100000110010111000100101001010010101110100010100110110010011011000101100100111010011001010111001110100100101010101100100000101111111110010110001001001110111110110100011110001111110100100100110001000101010101100111101111001101110110100010110101110010011011110101111000110110100101101001000111110111001010011000110011001100101010101110000110101001001110100111110
Pair \(Z_2\) Length of longest common subsequence
7QBT_1,3UDK_1 168 3
7QBT_1,3POS_1 165 3
3UDK_1,3POS_1 167 4

Newick tree

 
[
	3UDK_1:84.16,
	[
		7QBT_1:82.5,3POS_1:82.5
	]:1.66
]

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 \((0.85)(0.8)(\frac{840 }{\log_{20} 840}-\frac{404}{\log_{20}404})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 7QBT_1 3UDK_1 144 142
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]