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Parikh vectors
3PAG_1 3QHD_1 4IHC_1 Letter Amino acid
7 13 20 R Arginine
7 14 27 D Aspartic acid
21 7 26 E Glutamic acid
16 12 32 I Isoleucine
8 5 12 F Phenylalanine
8 8 30 P Proline
6 0 7 W Tryptophan
7 3 19 Y Tyrosine
14 10 25 V Valine
17 4 19 N Asparagine
11 7 13 Q Glutamine
20 19 41 G Glycine
24 5 22 K Lycine
12 6 21 S Serine
16 29 30 A Alanine
0 1 8 C Cysteine
15 7 24 T Threonine
4 11 13 H Histidine
22 18 35 L Leucine
10 4 16 M Methionine 

3PAG_1|Chains A, B|Beta-lactamase|Acinetobacter baumannii (470)
>3QHD_1|Chains A, B, C|2-C-methyl-D-erythritol 2,4-cyclodiphosphate synthase|Burkholderia pseudomallei (320372)
>4IHC_1|Chains A, B, C, D, E, F, G, H|Mandelate racemase/muconate lactonizing protein|Dickeya dadantii (579405)
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 \(w=f(c)\)
3PAG , Knot 112 245 0.83 38 161 230 
MHISSQQHEKAIKSYFDEAQTQGVIIIKEGKNLSTYGNALARANKEYVPASTFKMLNALIGLENHKATTNEIFKWDGKKRTYPMWEKDMTLGEAMALSADPVYQELARRTGLELMQKEVKRVNFGNTNIGTQVDNFWLVGPLKITPVQEVNFADDLAHNRLPFKLETQEEVKKMLLIKEVNGSKIYAKSGWGMGVTPQVGWLTGWVEQANGKKIPFSLNLEMKEGMSGSIRNEITYKSLENLGII
3QHD , Knot 86 183 0.81 38 125 172 
MAHHHHHHMGTLEAQTQGPGSMDFRIGQGYDVHQLVPGRPLIIGGVTIPYERGLLGHSDADVLLHAITDALFGAAALGDIGRHFSDTDPRFKGADSRALLRECASRVAQAGFAIRNVDSTIIAQAPKLAPHIDAMRANIAADLDLPLDRVNVKAKTNEKLGYLGRGEGIEAQAAALVVREAAA
4IHC , Knot 184 440 0.84 40 244 417 
MHHHHHHSSGVDLGTENLYFQSMMSKLKITNVKTILTAPGGIDLAVVKVETNEPGLYGLGCATFTQRIFAVKSAIDEYMAPFLIGKDPTRIEDIWQSAAVSGYWRNGPIMNNALSGVDMALWDIKGKLAGMPVYELLGGKCRDGIPLYCHTDGGDEVEVEDNIRARMEEGYQYVRCQMGMYGGAGTDDLKLIATQLARAKNIQPKRSPRSKTPGIYFDPEAYAKSVPRLFEHLRNKLGFGIEFIHDVHERVTPVTAIQLAKTLEPYQLFYLEDPVAPENIDWLRMLRQQSSTPISMGELFVNINEWKPLIDNKLIDYIRCHVSTIGGITPAKKLAVYSELNGVRTAWHGPGDISPVGVCANMHLDMSSPNFGIQEYTPMNDALREVFPGCPEIDQGYAYVNDKPGLGIDINETLAEKYPCDGGIPSWTMARTPDGTASRP

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 \(c\).

\(|P_{f(3PAG_1)}(2) \setminus P_{f(3QHD_1)}(2)|=101\), \(|P_{f(3QHD_1)}(2) \setminus P_{f(3PAG_1)}(2)|=65\). 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 1:10100000001100010010001111100100100010111010000111001011011111000010000110101000001110001011011110101100011000110110001001011000110010011111110101100101100110001110100000100111100101001010011111101011110111001010011101010100110101000100001001111
Pair \(Z_2\) Length of longest common subsequence
3PAG_1,3QHD_1 166 6
3PAG_1,4IHC_1 183 4
3QHD_1,4IHC_1 193 6

Newick tree

 
[
	4IHC_1:97.43,
	[
		3PAG_1:83,3QHD_1:83
	]:14.43
]

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 \((0.85)(0.8)(\frac{428 }{\log_{20} 428}-\frac{183}{\log_{20}183})=72.3\)
Status Protein1 Protein2 d d1/2
Query variables 3PAG_1 3QHD_1 90 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]