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Parikh vectors
5IDV_1 8GAI_1 8FSQ_1 Letter Amino acid
19 22 41 D Aspartic acid
21 38 25 I Isoleucine
11 32 27 T Threonine
0 7 20 W Tryptophan
31 40 55 A Alanine
16 12 37 R Arginine
16 10 4 H Histidine
14 32 38 S Serine
7 8 33 Y Tyrosine
18 38 30 V Valine
21 25 25 Q Glutamine
17 31 27 E Glutamic acid
6 6 14 M Methionine
9 11 29 F Phenylalanine
6 22 37 P Proline
18 30 26 G Glycine
6 21 33 K Lycine
29 53 49 L Leucine
14 28 35 N Asparagine
0 5 0 C Cysteine 

5IDV_1|Chain A|Lipid A export ATP-binding/permease protein MsbA|Acinetobacter baumannii AB5075 (1116234)
>8GAI_1|Chains A, C|Importin subunit alpha|Cavia porcellus (10141)
>8FSQ_1|Chain A|YejA|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 \(w=f(c)\)
5IDV , Knot 122 279 0.82 36 164 266 
MAHHHHHHGLAAAHSVFELLDLPEEQNSGELKPQLQGAIRFDHVVLNYADGTQAIKDFSLDIRPGETVALVGRSGAGKTSLVNMLVRFQEVSSGQIYLDDLPIRDIELSSLRTQIAMVNQQVVLFNRTVRENIAYGQLHNASDEDVIAAAKAAYAHDFIMNLPNGYDTVLGAQGLNLSGGQRQRIAIARAILKNAPILILDEATSALDNESEHFIQQAFDEAMQDRTTIVIAHRLSTIENADRIVVMDRGQIVEQGTHQELLAKHGAYYQLHQRNFEDH
8GAI , Knot 187 471 0.81 40 228 435 
QSTSPLQENRNNQGTVNWTVDDIVKGINSNNLENQLQATQAARKLLSREKQPPIDHIIRAGLIPKFVSFLGRTDCSPIQFESAWALTNIASGSSEQTKAVVDGGAIPALISLLASPHAHISEQAVWALGNIAGDGSVFRDLVIKYGAIDPLLALLAVPDMSSLAGGYLRNLTWTLSNLCRNKNPAPPLDAVEQILPTLVRLLHHNDLEVLADTCWAISYLTDGPNERIEMVVKTGVVPHLVKLLGSNELSIMTPALRAIGNIVTGTDEQTQIVIDAGALAVFPSLLVHPKTNIQKEATWTMSNITAGRQDQIQQVVNHGLVPILVSVLSKADFKTQKEAVWAVTNYTSGGTVEQIVYLVHCGIIEPLMNLLTAKDTKIILVILDAISNIFQAAEKLGETEKLSIMIEECGGLDKIEALQNHENESVYKASLNLIEKYFSVEEEEDQTVVPETTSEGYTFQVQDGAPGSFNF
8FSQ , Knot 235 585 0.85 38 268 561 
QAIKESYAFAVLGEPRYAFNFNHFDYVNPAAPKGGQITLSALGTFDNFNRYALRGNPGARTEQLYDTLFTTSDDEPGSYYPLIAESARYADDYSWVEVAINPRARFHDGSPITARDVEFTFQKFMTEGVPQFRLVYKGTTVKAIAPLTVRIELAKPGKEDMLSLFSLPVFPEKYWKDHKLSDPLATPPLASGPYRVTSWKMGQNIVYSRVKDYWAANLPVNRGRWNFDTIRYDYYLDDNVAFEAFKAGAFDLRMENDAKNWATRYTGKNFDKKYIIKDEQKNESAQDTRWLAFNIQRPVFSDRRVREAITLAFDFEWMNKALFYNAWSRTNSYFQNTEYAARNYPDAAELVLLAPMKKDLPSEVFTQIYQPPVSKGDGYDRDNLLKADKLLNEAGWVLKGQQRVNATEAQPLSFELLLPASSNSQWVLPFQHSLQRLGINMDIRKVDNSQITNRMRSRDYDMMPRVWRAMPWPSSDLQISWSSEYINSTYNAPGVQSPVIDSLINQIIAAQGNKEKLLPLGRALDRVLTWNYYMLPMWYMAEDRLAWWDKFSQPAVRPIYSLGIDTWWYDVNKAAKLPSASKQGE

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 \(c\).

\(|P_{f(5IDV_1)}(2) \setminus P_{f(8GAI_1)}(2)|=46\), \(|P_{f(8GAI_1)}(2) \setminus P_{f(5IDV_1)}(2)|=110\). 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 1:110000001111100110110110000010101010111010011100101001100101010110011111001110001101110100100101010011100101001000111100011110001000110101001000011111011010011101101000111101101011000011110111001111110010011000000110011001100000111100100100100111100101100100001110011000100001000
Pair \(Z_2\) Length of longest common subsequence
5IDV_1,8GAI_1 156 4
5IDV_1,8FSQ_1 166 4
8GAI_1,8FSQ_1 152 3

Newick tree

 
[
	5IDV_1:81.99,
	[
		8GAI_1:76,8FSQ_1:76
	]:5.99
]

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 \((0.85)(0.8)(\frac{750 }{\log_{20} 750}-\frac{279}{\log_{20}279})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 5IDV_1 8GAI_1 166 129.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]