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Parikh vectors
7FMX_1 1GGX_1 8IYO_1 Letter Amino acid
19 23 14 K Lycine
8 7 3 M Methionine
10 13 11 F Phenylalanine
29 14 18 L Leucine
14 12 3 D Aspartic acid
9 7 8 Q Glutamine
5 8 4 H Histidine
11 9 4 R Arginine
14 6 6 N Asparagine
1 1 4 C Cysteine
10 12 6 P Proline
18 12 11 S Serine
14 19 8 V Valine
18 5 6 A Alanine
7 22 12 G Glycine
23 10 14 I Isoleucine
21 10 6 T Threonine
5 3 0 W Tryptophan
7 13 7 Y Tyrosine
15 19 18 E Glutamic acid 

7FMX_1|Chain A|Pre-mRNA-splicing factor 8|Saccharomyces cerevisiae S288C (559292)
>1GGX_1|Chains A, B, C, D|PROTEIN (FLUORESCENT PROTEIN FP583)|Discosoma sp. (86600)
>8IYO_1|Chains A, B, C, D|N-acetyltransferase domain-containing protein|Helicobacter pylori 26695 (85962)
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 \(w=f(c)\)
7FMX , Knot 117 258 0.84 40 175 251 
GAMNSSNYAELFNNDIKLFVDDTNVYRVTVHKTFEGNVATKAINGCIFTLNPKTGHLFLKIIHTSVWAGQKRLSQLAKWKTAEEVSALVRSLPKEEQPKQIIVTRKAMLDPLEVHMLDFPNIAIRPTELRLPFSAAMSIDKLSDVVMKATEPQMVLFNIYDDWLDRISSYTAFSRLTLLLRALKTNEESAKMILLSDPTITIKSYHLWPSFTDEQWITIESQMRDLILTEYGRKYNVNISALTQTEIKDIILGQNIKA
1GGX , Knot 104 225 0.83 40 150 218 
MRSSKNVIKEFMRFKVRMEGTVNGHEFEIEGEGEGRPYEGHNTVKLKVTKGGPLPFAWDILSPQFQYGSKVYVKHPADIPDYKKLSFPEGFKWERVMNFEDGGVVTVTQDSSLQDGCFIYKVKFIGVNFPSDGPVMQKKTMGWEASTERLYPRDGVLKGEIHKALKLKDGGHYLVEFKSIYMAKKPVQLPGYYYVDSKLDITSHNEDYTIVEQYERTEGRHHLFL
8IYO , Knot 79 163 0.82 38 122 157 
SMVTIKVFSPKYPTELEEFYAERIADNPLGFIQRLDLLPSISGFVQKLREHGGEFFEMREGNKLIGICGLNPINQTEAELCKFHINSAYQSQGLGQKLYESVEKYAFIKGYTKISLHVSKSQIKACNLYQKLGFVHIKEEDCVVELGEETLIFPTLFMEKILS

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 \(c\).

\(|P_{f(7FMX_1)}(2) \setminus P_{f(1GGX_1)}(2)|=104\), \(|P_{f(1GGX_1)}(2) \setminus P_{f(7FMX_1)}(2)|=79\). 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 1:111000001011000101110000100101000101011001101011010100101110110001111000100110100100101110011000010011100011101101011011011101001011101110100100111010010111101000110010000110010111011000000101111001010100001110100001101000100111000100001010110000100111100101
Pair \(Z_2\) Length of longest common subsequence
7FMX_1,1GGX_1 183 4
7FMX_1,8IYO_1 171 4
1GGX_1,8IYO_1 174 3

Newick tree

 
[
	1GGX_1:90.50,
	[
		7FMX_1:85.5,8IYO_1:85.5
	]:5.00
]

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 \((0.85)(0.8)(\frac{483 }{\log_{20} 483}-\frac{225}{\log_{20}225})=74.5\)
Status Protein1 Protein2 d d1/2
Query variables 7FMX_1 1GGX_1 100 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]