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Parikh vectors
6MQM_1 7YPG_1 9ESE_1 Letter Amino acid
6 2 16 R Arginine
6 3 19 Q Glutamine
7 5 25 E Glutamic acid
2 6 18 H Histidine
17 4 21 P Proline
13 3 30 A Alanine
8 5 22 D Aspartic acid
5 1 8 C Cysteine
19 13 27 G Glycine
6 6 18 I Isoleucine
21 5 50 L Leucine
6 2 20 F Phenylalanine
7 1 12 Y Tyrosine
32 7 30 S Serine
7 0 6 W Tryptophan
8 4 14 N Asparagine
12 14 30 K Lycine
0 1 11 M Methionine
18 5 14 T Threonine
22 9 27 V Valine 

6MQM_1|Chains A, D, G, J|antibody Fab heavy chain|Macaca mulatta (9544)
>7YPG_1|Chains A, B, C, D, E, F|Isoform Tau-E of Microtubule-associated protein tau|Homo sapiens (9606)
>9ESE_1|Chains A, B|Indoleamine 2,3-dioxygenase 1|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 \(w=f(c)\)
6MQM , Knot 100 222 0.81 38 144 209 
QVQLVESGPGLVRPSETLSLTCAVSGASIRTKAWWSWIRQPPGKGLEWIGYVSGGWDLPNYNPSLKNRVIILKDTSLNQFSLRLTSVTAADTALYYCAREGPEDFDVWGPGFLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCD
7YPG , Knot 46 96 0.73 38 71 87 
MIKHVPGGGSVQIVYKPVDLSKVTSKCGSLGNIHHKPGGGQVEVKSEKLDFKDRVQSKIGSLDNITHVPGGGNKKIETHKLTFRENAKAKTDHGAE
9ESE , Knot 181 418 0.87 40 237 392 
MGSSHHHHHHSAALEVLFQGPHMWTISKEYHIDEEVGFALPNPQENLPDFYNDWMFIAKHLPDLIESGQLRERVEKLNMLSIDHLTDHKSQRLARLVLGCITMAYVWGKGHGDVRKVLPRNIAVPYCQLSKKLELPPILVYADCVLANWKKKDPNKPLTYENMDVLFSFRDGDCSKGFFLVSLLVEIAAASAIKVIPTVFKAMQMQERDTLLKALLEIASCLEKALQVFHQIHDHVNPKAFFSVLRIYLSGWKGNPQLSDGLVYEGFWEDPKEFAGGSAGQSSVFQCFDVLLGIQQTAGGGHAAQFLQDMRRYMPPAHRNFLCSLESNPSVREFVLSKGDAGLREAYDACVKALVSLRSYHLQIVTKYILIPASQQPKENKTSEDPSKLEAKGTGGTDLMNFLKTVRSTTEKSLLKEG

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 \(c\).

\(|P_{f(6MQM_1)}(2) \setminus P_{f(7YPG_1)}(2)|=110\), \(|P_{f(7YPG_1)}(2) \setminus P_{f(6MQM_1)}(2)|=37\). 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 1:010110011111010001010011011010001110110011101101110101110110001010001111000010010101001011001100010011001011111111010010001101111110000001101111011000110110101001110011001111100011001001101100011000010010001000010001010000
Pair \(Z_2\) Length of longest common subsequence
6MQM_1,7YPG_1 147 3
6MQM_1,9ESE_1 193 3
7YPG_1,9ESE_1 204 3

Newick tree

 
[
	9ESE_1:10.50,
	[
		6MQM_1:73.5,7YPG_1:73.5
	]:33.00
]

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 \((0.85)(0.8)(\frac{318 }{\log_{20} 318}-\frac{96}{\log_{20}96})=69.5\)
Status Protein1 Protein2 d d1/2
Query variables 6MQM_1 7YPG_1 87 60.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]