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Parikh vectors
5MIL_1 9ASL_1 6BQH_1 Letter Amino acid
9 67 37 A Alanine
10 36 11 Q Glutamine
10 31 23 F Phenylalanine
17 71 22 S Serine
8 71 36 V Valine
14 34 25 D Aspartic acid
16 61 18 G Glycine
8 19 14 H Histidine
10 46 42 I Isoleucine
23 113 49 L Leucine
7 20 16 Y Tyrosine
6 57 17 R Arginine
9 25 26 N Asparagine
11 38 18 T Threonine
0 19 9 C Cysteine
11 70 17 E Glutamic acid
16 50 30 K Lycine
9 25 14 M Methionine
6 30 21 P Proline
2 11 7 W Tryptophan 

5MIL_1|Chains A, B|DUTPase family protein|Staphylococcus aureus (1280)
>9ASL_1|Chains A, B, C, D|DNA polymerase theta|Homo sapiens (9606)
>6BQH_1|Chain A|5-hydroxytryptamine receptor 2C,Soluble cytochrome b562|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 \(w=f(c)\)
5MIL , Knot 91 202 0.79 38 136 186 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSMTNTLTIDQLQELLQIQKEFDDRIPTLNLRDSKIAYVVEFFEWFNTLETFKNWKKKPGKPLDVQLDELADMLAFGLSIANQSGVSLKTLEKLIPSTLGKVYFNTSSIMKDFMEDFVYFGLGEEDSLSLPLNIAYNLYSIDQLIDAYKKKMKRNHERQDGTADAGKGYV
9ASL , Knot 333 894 0.84 40 304 786 
MNLLRRSGKRRRSESGSDSFSGSGGDSSASPQFLSGSVLSPPPGLGRCLKAAAAGECKPTVPDYERDKLLLANWGLPKAVLEKYHSFGVKKMFEWQAECLLLGQVLEGKNLVYSAPTSAGKTLVAELLILKRVLEMRKKALFILPFVSVAKEKKYYLQSLFQEVGIKVDGYMGSTSPSRHFSSLDIAVCTIERANGLINRLIEENKMDLLGMVVVDELHMLGDSHRGYLLELLLTKICYITRKSASCQADLASSLSNAVQIVGMSATLPNLELVASWLNAELYHTDFRPVPLLESVKVGNSIYDSSMKLVREFEPMLQVKGDEDHVVSLCYETICDNHSVLLFCPSKKWCEKLADIIAREFYNLHHQAEGLVKPSECPPVILEQKELLEVMDQLRRLPSGLDSVLQKTVPWGVAFHHAGLTFEERDIIEGAFRQGLIRVLAATSTLSSGVNLPARRVIIRTPIFGGRPLDILTYKQMVGRAGRKGVDTVGESILICKNSEKSKGIALLQGSLKPVRSCLQRREGEEVTGSMIRAILEIIVGGVASTSQDMHTYAACTFLAASMKEGKQGIQRNQESVQLGAIEACVMWLLENEFIQSTEASDGTEGKVYHPTHLGSATLSSSLSPADTLDIFADLQRAMKGFVLENDLHILYLVTPMFEDWTTIDWYRFFCLWEKLPTSMKRVAELVGVEEGFLARCVKGKVVARTERQHRQMAIHKRFFTSLVLLDLISEVPLREINQKYGCNRGQIQSLQQSAAVYAGMITVFSNRLGWHNMELLLSQFQKRLTFGIQRELCDLVRVSLLNAQRARVLYASGFHTVADLARANIVEVEVILKNAVPFKSARKAVDEEEEAVEERRNMRTIWVTGRKGLTEREAAALIVEEARMILQQDLVEM
6BQH , Knot 184 452 0.83 40 236 421 
MKTIIALSYIFCLVFADYKDDDDGAPTSDGGRFKFPDGVQNWPALSIVIIIIMTIGGNILVIMAVSMEKKLHNATNYFLMSLAIADMLVGLLVMPLSLLAILYDYVWPLPRYLCPVWISLDVLFSTASIMHLCAISLDRYVAIRNPIEHSRFNSRTKAIMKIAIVWAISIGVSVPIPVIGLRDEEKVFVNNTTCVLNDPNFVLIGSFVAFFIPLTIMVITYCLTIYVLRRQAADLEDNWETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMKDFRHGFDILVGQIDDALKLANEGKVKEAQAAAEQLKTTRNAYIQKYLQAINNERKASKVLGIVFFVFLIMWCPFFITNILSVLCEKSCNQKLMEKLLNVFVWIGYVNSGINPLVYTLFNKIYRRAFSNYLRCNYKVEKKPGRPLEVLFQGPHHHHHHHHHH

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 \(c\).

\(|P_{f(5MIL_1)}(2) \setminus P_{f(9ASL_1)}(2)|=22\), \(|P_{f(9ASL_1)}(2) \setminus P_{f(5MIL_1)}(2)|=190\). 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 1:1100000000001111010011010110011010100010100100110100010001101010000110110110110010010010001101101010011011111101100011010010011100110101000011001100110111100001011101100100100110100001000000001010110101
Pair \(Z_2\) Length of longest common subsequence
5MIL_1,9ASL_1 212 4
5MIL_1,6BQH_1 168 6
9ASL_1,6BQH_1 156 4

Newick tree

 
[
	5MIL_1:10.82,
	[
		6BQH_1:78,9ASL_1:78
	]:22.82
]

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 \((0.85)(0.8)(\frac{1096 }{\log_{20} 1096}-\frac{202}{\log_{20}202})=241.\)
Status Protein1 Protein2 d d1/2
Query variables 5MIL_1 9ASL_1 302 184
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]