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Parikh vectors
7FUK_1 2ICE_1 2GUI_1 Letter Amino acid
8 10 6 H Histidine
18 33 14 I Isoleucine
5 12 6 M Methionine
23 24 11 F Phenylalanine
4 4 0 W Tryptophan
11 23 6 Y Tyrosine
12 36 4 Q Glutamine
43 39 9 K Lycine
14 48 14 T Threonine
18 73 12 V Valine
14 30 19 A Alanine
18 23 8 N Asparagine
41 57 17 L Leucine
28 49 8 S Serine
20 28 11 R Arginine
18 28 14 D Aspartic acid
10 3 2 C Cysteine
32 40 14 E Glutamic acid
14 46 14 G Glycine
11 36 5 P Proline 

7FUK_1|Chain A|Cyclic GMP-AMP synthase|Homo sapiens (9606)
>2ICE_1|Chains A, E[auth D]|Complement C3 beta chain|Homo sapiens (9606)
>2GUI_1|Chain A|DNA polymerase III epsilon subunit|Escherichia coli (562)
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 \(w=f(c)\)
7FUK , Knot 149 362 0.80 40 204 345 
GASKLRAVLEKLKLSRDDISTAAGMVKGVVDHLLLRLKCDSAFRGVGLLNTGSYYEHVKISAPNEFDVMFKLEVPRIQLEEYSNTRAYYFVKFKRNPKENPLSQFLEGEILSASKMLSKFRKIIKEEINDIKDTDVIMKRKRGGSPAVTLLISEKISVDITLALESKSSWPASTQEGLRIQNWLSAKVRKQLRLKPFYLVPKHAKEGNGFQEETWRLSFSHIEKEILNNHGKSKTCCENKEEKCCRKDCLKLMKYLLEQLKERFKDKKHLDKFSSYHVKTAFFHVCTQNPQDSQWDRKDLGLCFDNCVTYFLQCLRTEKLENYFIPEFNLFSSNLIDKRSKEFLTKQIEYERNNEFPVFDEF
2ICE , Knot 245 642 0.82 40 267 583 
SPMYSIITPNILRLESEETMVLEAHDAQGDVPVTVTVHDFPGKKLVLSSEKTVLTPATNHMGNVTFTIPANREFKSEKGRNKFVTVQATFGTQVVEKVVLVSLQSGYLFIQTDKTIYTPGSTVLYRIFTVNHKLLPVGRTVMVNIENPEGIPVKQDSLSSQNQLGVLPLSWDIPELVNMGQWKIRAYYENSPQQVFSTEFEVKEYVLPSFEVIVEPTEKFYYIYNEKGLEVTITARFLYGKKVEGTAFVIFGIQDGEQRISLPESLKRIPIEDGSGEVVLSRKVLLDGVQNLRAEDLVGKSLYVSATVILHSGSDMVQAERSGIPIVTSPYQIHFTKTPKYFKPGMPFDLMVFVTNPDGSPAYRVPVAVQGEDTVQSLTQGDGVAKLSINTHPSQKPLSITVRTKKQELSEAEQATRTMQALPYSTVGNSNNYLHLSVLRTELRPGETLNVNFLLRMDRAHEAKIRYYTYLIMNKGRLLKAGRQVREPGQDLVVLPLSITTDFIPSFRLVAYYTLIGASGQREVVADSVWVDVKDSCVGSLVVKSGQSEDRQPVPGQQMTLKIEGDHGARVVLVAVDKGVFVLNKKNKLTQSKIWDVVEKADIGCTPGSGKDYAGVFSDAGLTFTSSSGQQTAQRAELQCPQ
2GUI , Knot 90 194 0.81 38 139 182 
SIEFSTAITRQIVLDTETTGMNQIGAHYEGHKIIEIGAVEVVNRRLTGNNFHVYLKPDRLVDPEAFGVHGIADEFLLDKPTFAEVADEFMDYIRGAELVIHNAAFDIGFMDYEFSLLKRDIPKTNTFCKVTDSLAVARKMFPGKRNSLDALCARYEIDNSKRTLHGALLDAQILAEVYLAMTGGQTSMATRESG

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 \(c\).

\(|P_{f(7FUK_1)}(2) \setminus P_{f(2ICE_1)}(2)|=51\), \(|P_{f(2ICE_1)}(2) \setminus P_{f(7FUK_1)}(2)|=114\). 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 1:11001011100101000010011111011100111010000110111110010000010101100101110101101010000000100110100010001100110101101001100100110001001000011100001101110111000101010111000001110000110100110101000101011011100100101100001010100100011000100000000000000000010110011001000100000100100001001110100001000010000111010001001100100001000111010110001100000011000100000001111001
Pair \(Z_2\) Length of longest common subsequence
7FUK_1,2ICE_1 165 4
7FUK_1,2GUI_1 189 3
2ICE_1,2GUI_1 194 4

Newick tree

 
[
	2GUI_1:99.78,
	[
		7FUK_1:82.5,2ICE_1:82.5
	]:17.28
]

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 \((0.85)(0.8)(\frac{1004 }{\log_{20} 1004}-\frac{362}{\log_{20}362})=170.\)
Status Protein1 Protein2 d d1/2
Query variables 7FUK_1 2ICE_1 219 170.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]