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Parikh vectors
6NMO_1 4YLU_1 5DLR_1 Letter Amino acid
7 12 9 P Proline
0 5 3 W Tryptophan
1 10 4 C Cysteine
6 9 15 E Glutamic acid
5 11 14 K Lycine
5 8 13 H Histidine
17 24 48 L Leucine
5 26 11 T Threonine
10 32 13 V Valine
27 23 15 A Alanine
4 16 9 N Asparagine
14 12 13 D Aspartic acid
2 16 15 M Methionine
5 14 7 F Phenylalanine
6 20 20 S Serine
3 11 4 Y Tyrosine
5 17 9 Q Glutamine
16 26 10 G Glycine
12 9 13 I Isoleucine
13 5 12 R Arginine 

6NMO_1|Chains A, B, C|2-C-methyl-D-erythritol 2,4-cyclodiphosphate synthase|Burkholderia pseudomallei (strain 1710b) (320372)
>4YLU_1|Chains A, B, C, D|ORF1a protein|Middle East respiratory syndrome coronavirus (1335626)
>5DLR_1|Chains A, B|Estrogen receptor|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 \(w=f(c)\)
6NMO , Knot 80 163 0.83 38 113 156 
SMDFRIGQGYDVHQLVPGRPLIIGGVTIPYERGLLGHSDADVLLHAITDALFGAAALGDIGRHFSDTDPRFKGADSRALLRECASRVAQAGFAIRNVDSTIIAQAPKLAPHIDAMRANIAADLDLPLDRVNVKAKTNEKLGYLGRGEGIEAQAAALVVREAAA
4YLU , Knot 132 306 0.82 40 189 292 
SGLVKMSHPSGDVEACMVQVTCGSMTLNGLWLDNTVWCPRHVMCPADQLSDPNYDALLISMTNHSFSVQKHIGAPANLRVVGHAMQGTLLKLTVDVANPSTPAYTFTTVKPGAAFSVLACYNGRPTGTFTVVMRPNYTIKGSFLCGSCGSVGYTKEGSVINFCYMHQMELANGTHTGSAFDGTMYGAFMDKQVHQVQLTDKYCSVNVVAWLYAAILNGCAWFVKPNRTSVVSFNEWALANQFTEFVGTQSVDMLAVKTGVAIEQLLYAIQQLYTGFQGKQILGSTMLEDEFTPEDVNMQIMGVVMQ
5DLR , Knot 117 257 0.84 40 165 246 
IKRSKKNSLALSLTADQMVSALLDAEPPILYSEYDPTRPFSEASMMGLLTNLADRELVHMINWAKRVPGFVDLTLHDQVHLLECAWLEILMIGLVWRSMEHPGKLLFAPNLLLDRNQGKCVEGMVEIFDMLLATSSRFRMMNLQGEEFVCLKSIILLNSGVYTFLSSTLKSLEEKDHIHRVLDKITDTLIHLMAKAGLTLQQQHQRLAQLLLILSHIRHMSNKGMEHLYSMKCKNVVPLSDLLLEMLDAHRLHAPTS

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 \(c\).

\(|P_{f(6NMO_1)}(2) \setminus P_{f(4YLU_1)}(2)|=49\), \(|P_{f(4YLU_1)}(2) \setminus P_{f(6NMO_1)}(2)|=125\). 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 1:0101011010010011110111111101100011110001011101100111111111011001000010101100011100010011011111001000111011011101011010111010111001010100000110110101101011111100111
Pair \(Z_2\) Length of longest common subsequence
6NMO_1,4YLU_1 174 3
6NMO_1,5DLR_1 150 3
4YLU_1,5DLR_1 186 3

Newick tree

 
[
	4YLU_1:94.53,
	[
		6NMO_1:75,5DLR_1:75
	]:19.53
]

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 \((0.85)(0.8)(\frac{469 }{\log_{20} 469}-\frac{163}{\log_{20}163})=90.1\)
Status Protein1 Protein2 d d1/2
Query variables 6NMO_1 4YLU_1 115 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]