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Parikh vectors
3EBA_1 9CSG_1 3ACX_1 Letter Amino acid
8 20 17 Q Glutamine
5 8 27 I Isoleucine
3 24 4 P Proline
4 1 2 W Tryptophan
15 12 8 G Glycine
6 16 13 H Histidine
4 59 20 K Lycine
9 28 9 T Threonine
2 6 9 M Methionine
9 18 21 Y Tyrosine
11 61 25 A Alanine
4 24 15 R Arginine
6 35 22 D Aspartic acid
7 60 18 L Leucine
14 24 12 S Serine
9 41 18 V Valine
3 17 11 N Asparagine
2 35 3 C Cysteine
4 62 23 E Glutamic acid
2 31 16 F Phenylalanine 

3EBA_1|Chain A|CAbHul6|Camelus dromedarius (9838)
>9CSG_1|Chain A|Albumin|Homo sapiens (9606)
>3ACX_1|Chain A|Dehydrosqualene synthase|Staphylococcus aureus (1280)
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 \(w=f(c)\)
3EBA , Knot 60 127 0.76 40 101 121 
QVQLVESGGGSVQAGGSLRLSCSASGYTYISGWFRQAPGKGLEWVAAIRSSDGTTYYADSVKGRFTISQDNAKNTVYLQMNSLKPEDTAMYYCAATEVAGWPLDIGIYDYWGQGTQVTVSSHHHHHH
9CSG , Knot 228 582 0.83 40 236 527 
HKSEVAHRFKDLGEENFKALVLIAFAQYLQQCPFEDHVKLVNEVTEFAKTCVADESAENCDKSLHTLFGDKLCTVATLRETYGEMADCCAKQEPERNECFLQHKDDNPNLPRLVRPEVDVMCTAFHDNEETFLKKYLYEIARRHPYFYAPELLFFAKRYKAAFTECCQAADKAACLLPKLDELRDEGKASSAKQRLKCASLQKFGERAFKAWAVARLSQRFPKAEFAEVSKLVTDLTKVHTECCHGDLLECADDRADLAKYICENQDSISSKLKECCEKPLLEKSHCIAEVENDEMPADLPSLAADFVESKDVCKNYAEAKDVFLGMFLYEYARRHPDYSVVLLLRLAKTYETTLEKCCAAADPHECYAKVFDEFKPLVEEPQNLIKQNCELFEQLGEYKFQNALLVRYTKKVPQVSTPTLVEVSRNLGKVGSKCCKHPEAKRMPCAEDYLSVVLNQLCVLHEKTPVSDRVTKCCTESLVNRRPCFSALEVDETYVPKEFNAETFTFHADICTLSEKERQIKKQTALVELVKHKPKATKEQLKAVMDDFAAFVEKCCKADDKETCFAEEGKKLVAASQAALG
3ACX , Knot 128 293 0.82 40 182 280 
AAAAAAMTMMDMNFKYCHKIMKKHSKSFSYAFDLLPEDQRKAVWAIYAVCRKIDDSIDVYGDIQFLNQIKEDIQSIEKYPYEYHHFQSDRRIMMALQHVAQHKNIAFQSFYNLIDTVYKDQHFTMFETDAELFGYCYGVAGTVGEVLTPILSDHETHQTYDVARRLGESLQLINILRDVGEDFENERIYFSKQRLKQYEVDIAEVYQNGVNNHYIDLWEYYAAIAEKDFRDVMDQIKVFSIEAQPIIELAARIYIEILDEVRQANYTLHERVFVEKRKKAKLFHEINSKYHRI

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 \(c\).

\(|P_{f(3EBA_1)}(2) \setminus P_{f(9CSG_1)}(2)|=43\), \(|P_{f(9CSG_1)}(2) \setminus P_{f(3EBA_1)}(2)|=178\). 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 1:0101100111010111010100010100010111001110110111110000100001001010101000010001010100101000110001100111111011100011010010100000000
Pair \(Z_2\) Length of longest common subsequence
3EBA_1,9CSG_1 221 3
3EBA_1,3ACX_1 197 3
9CSG_1,3ACX_1 168 5

Newick tree

 
[
	3EBA_1:11.70,
	[
		3ACX_1:84,9CSG_1:84
	]:26.70
]

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 \((0.85)(0.8)(\frac{709 }{\log_{20} 709}-\frac{127}{\log_{20}127})=166.\)
Status Protein1 Protein2 d d1/2
Query variables 3EBA_1 9CSG_1 209 127
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]