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Parikh vectors
3SAV_1 8SUX_1 5LBG_1 Letter Amino acid
23 37 19 A Alanine
13 22 15 I Isoleucine
25 56 7 L Leucine
5 13 7 M Methionine
17 23 20 T Threonine
28 27 10 R Arginine
9 24 18 D Aspartic acid
6 4 1 C Cysteine
4 13 17 N Asparagine
5 46 5 Q Glutamine
23 29 12 E Glutamic acid
22 23 29 G Glycine
12 29 15 S Serine
1 10 9 W Tryptophan
8 13 9 Y Tyrosine
22 25 30 V Valine
9 11 8 H Histidine
12 28 9 K Lycine
11 12 7 F Phenylalanine
18 20 15 P Proline 

3SAV_1|Chain A|DNA GLYCOSYLASE|Geobacillus stearothermophilus (1422)
>8SUX_1|Chains A, B, C, D, E, F|PtuA|Escherichia coli (562)
>5LBG_1|Chains A, B|L,D-transpeptidase 2|Mycobacterium tuberculosis (83331)
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 \(w=f(c)\)
3SAV , Knot 118 273 0.80 40 168 259 
PELPEVETIRRTLLPLIVGKTIEDVRIFWPNIIRHPRDSEAFAARMIGQTVRGLERRGKFLKFLLDRDALISHLRMEGRYAVASALEPLEPHTHVVFCFTDGSELRYRDVRKFGTMHVYAKEEADRRPPLAELGPEPLSPAFSPAVLAERAVKTKRSVKALLLDCTVVAGFGNIYVDESLFRAGILPGRPAASLSSKEIERLHEEMVATIGEAVMKGGSTVRTYVNTQGEAGTFQHHLYVYGRQGNPCKRCGTPIEKTVVAGRGTHYCPRCQR
8SUX , Knot 189 465 0.83 40 230 441 
MRIDKLSLLNFRCFKQLDITFDEHITILVAPNGAGKTTVLDAVRLALFPFIRGFDASLYVKDKSLAIRTEDLRLIYRQEALNMEMSSPAKITATGEWASGKTATWMLDKRGEQPPHEDKMAAQLTRWGEQLQKRVREEHSLQQVELPLMLYLGTARLWYQERYEKQPTEQRLDNSAFSRLSGYDDCLSATSNYKQFEQWYSWLWLSYREHQITQLESPSAKLKEGVRVQRMKEAIQAIQQAINCLTQQVTGWHDLEYSASHNQQLVMSHPQYGKIPLSQLSDGLRNAVAMVADIAFRCVKLNPHLQNDAALKTQGIVLIDEVDMFLHPAWQQQIIQSLRSAFPQIQFIVTTHSPQVLSTVKRESIRLLEQDENGNGKALMPLGATYGEPSNDVLQSVMGVDPQPAVKEKADLQKLTGWVDQGKYDEPKTQQLMVALEVALGEKHPQLQRLQRSIARQRLLKGKAQ
5LBG , Knot 119 262 0.84 40 177 254 
SMAHLTMPYVMPGDGEVVGVGEPVAIRFDENIADRGAAEKAIKITTNPPVEGAFYWLNNREVRWRPEHFWKPGTAVDVAVNTYGVDLGEGMFGEDNVQTHFTIGDEVIATADDNTKILTVRVNGEVVKSMPTSMGKDSTPTANGIYIVGSRYKHIIMDSSTYGVPVNSPNGYRTDVDWATQISYSGVFVHSAPWSVGAQGHTNTSHGCLNVSPSNAQWFYDHVKRGDIVEVVNTVGGTLPGIDGLGDWNIPWDQWRAGNAKA

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 \(c\).

\(|P_{f(3SAV_1)}(2) \setminus P_{f(8SUX_1)}(2)|=62\), \(|P_{f(8SUX_1)}(2) \setminus P_{f(3SAV_1)}(2)|=124\). 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 1:101101001000111111100100101111011001000011110111001011000101101110001110010101001110110110100011101001001000010011010101000100011110111011011101111100110000010111100011111101010001101111110111010000100100011101101110110010001000101101000101010010100001011000111101000010000
Pair \(Z_2\) Length of longest common subsequence
3SAV_1,8SUX_1 186 3
3SAV_1,5LBG_1 173 3
8SUX_1,5LBG_1 183 3

Newick tree

 
[
	8SUX_1:94.09,
	[
		3SAV_1:86.5,5LBG_1:86.5
	]:7.59
]

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 \((0.85)(0.8)(\frac{738 }{\log_{20} 738}-\frac{273}{\log_{20}273})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 3SAV_1 8SUX_1 165 129.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]