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Parikh vectors
2GBQ_1 1WZO_1 4DKC_1 Letter Amino acid
3 19 9 P Proline
4 6 16 S Serine
1 0 6 C Cysteine
1 4 11 Q Glutamine
7 25 15 E Glutamic acid
1 6 2 H Histidine
4 9 4 F Phenylalanine
4 6 11 N Asparagine
2 5 3 M Methionine
1 3 4 W Tryptophan
6 18 6 A Alanine
7 12 10 D Aspartic acid
5 20 8 G Glycine
2 11 7 T Threonine
3 23 19 V Valine
3 18 10 R Arginine
5 7 3 I Isoleucine
4 34 27 L Leucine
8 10 10 K Lycine
3 10 9 Y Tyrosine 

2GBQ_1|Chain A|GRB2|Mus musculus (10090)
>1WZO_1|Chains A, B, C, D|HpcE|Thermus thermophilus (300852)
>4DKC_1|Chains A, B|Interleukin-34|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 \(w=f(c)\)
2GBQ , Knot 41 74 0.79 40 67 72 
GSRRASVGSMEAIAKYDFKATADDELSFKRGDILKVLNEECDQNWYKAELNGKDGFIPKNYIEMKPHPEFIVTD
1WZO , Knot 106 246 0.79 38 147 233 
MKLARFLAKGRVHQGVYREGLLLDEAGEAHRPEDVTWLLPFTPGKILGVALNYADHAEELGLSRPEEPALFWKPNTSLLPHKGVVLYPKGARFVHYEVELAVVVGRPMKRVRAKDALDYVLGYTIANDLVARDYVTNTFRPPIRAKGRDTFLPLGPFLVVEEVEDPQDLWLRAYVNGELRQEGHTSRMLYSVAELLEFISEFMTLEPYDVLLTGTPKGISQVRPGDVMRLEIEGLGALENPIEEEP
4DKC , Knot 88 190 0.81 40 139 181 
AGSNEPLEMWPLTQNEECTVTGFLRDKLQYRSRLQYMKHYFPINYKISVPYEGVFRIANVTRLQRAQVSERELRYLWVLVSLSATESVQDVLLEGHPSWKYLQEVETLLLNVQQGLTDVEVSPKVESVLSLLNAPGPNLKLVRPKALLDNCFRVMELLYCSCCKQSSVLNWQDCEVGNSGNSDYKDDDDK

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 \(c\).

\(|P_{f(2GBQ_1)}(2) \setminus P_{f(1WZO_1)}(2)|=35\), \(|P_{f(1WZO_1)}(2) \setminus P_{f(2GBQ_1)}(2)|=115\). 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 1:10001011010111000101010001010010110110000000100101010011110001010101011100
Pair \(Z_2\) Length of longest common subsequence
2GBQ_1,1WZO_1 150 3
2GBQ_1,4DKC_1 142 4
1WZO_1,4DKC_1 156 4

Newick tree

 
[
	1WZO_1:78.26,
	[
		2GBQ_1:71,4DKC_1:71
	]:7.26
]

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 \((0.85)(0.8)(\frac{320 }{\log_{20} 320}-\frac{74}{\log_{20}74})=77.9\)
Status Protein1 Protein2 d d1/2
Query variables 2GBQ_1 1WZO_1 94 62
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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