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Parikh vectors
4GEZ_1 3MDB_1 5AOJ_1 Letter Amino acid
26 5 16 T Threonine
13 3 9 A Alanine
13 6 9 D Aspartic acid
30 15 17 L Leucine
38 9 20 S Serine
24 9 6 I Isoleucine
19 5 5 F Phenylalanine
14 3 11 C Cysteine
16 6 7 Q Glutamine
26 5 13 E Glutamic acid
3 11 9 H Histidine
17 4 19 R Arginine
19 5 18 P Proline
16 6 14 V Valine
10 1 1 W Tryptophan
10 3 8 Y Tyrosine
22 10 9 N Asparagine
38 11 15 G Glycine
18 4 8 K Lycine
6 3 5 M Methionine 

4GEZ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Neuraminidase|Influenza A virus (1129346)
>3MDB_1|Chains A, B|Kinesin-like protein KIF13B|Homo sapiens (9606)
>5AOJ_1|Chains A, B|CELLULAR TUMOR ANTIGEN P53|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 \(w=f(c)\)
4GEZ , Knot 164 378 0.85 40 225 363 
GSGDSGSPGEKFYWRAKSQMCEVKGWVPTHRGFPWGPELPGDLILSRRAYVSCDLTSCFKFFIAYGLSANQHLLNTSMEWEESLYKTPIGSASTLSTSEMILPGRSSSACFDGLKWTVLVANGRDRNSFIMIKYGEEVTDTFSASRGGPLRLPNSECICIEGSCFVIVSDGPNVNQSVHRIYELQNGTVQRWKQLNTTGINFEYSTCYTINNLIKCTGTNLWNDAKRPLLRFTKELNYQIVEPCNGAPTDFPRGGLTTPSCKMAQEKGEGGIQGFILDEKPAWTSKTKAESSQNGFVLEQIPNGIESEGTVSLSYELFSNKRTGRSGFFQPKGDLISGCQRICFWLEIEDQTVGLGMIQELSTFCGINSPVQNINWDS
3MDB , Knot 63 124 0.81 40 101 118 
MHHHHHHSSGRENLYFQGGIKVGDDKCFLVNLNADPALNELLVYYLKEHTLIGSANSQDIQLCGMGILPEHCIIDITSEGQVMLTPQKNTRTFVNGSSVSSPIQLHHGDRILWGNNHFFRLNLP
5AOJ , Knot 102 219 0.83 40 150 211 
SSSVPSQKTYQGSYGFRLGFLHSGTAKSVTCTYSPALNKLFCQLAKTCPVQLWVDSTPPPGTRVRAMAIYKQSQHMTEVVRRCPHHERCSDSDGLAPPQHLIRVEGNLRAEYLDDRNTFRHSVVVPCEPPEVGSDCTTIHYNYMCYSSCMGGMNRRPILTIITLEDSSGNLLGRDSFEVRVCACPGRDRRTEEENLRKKGEPHHELPPGSTKRALPNNT

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 \(c\).

\(|P_{f(4GEZ_1)}(2) \setminus P_{f(3MDB_1)}(2)|=166\), \(|P_{f(3MDB_1)}(2) \setminus P_{f(4GEZ_1)}(2)|=42\). 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 1:101001011001010100010010111100011111101110111000101000100010111101101000110001010001000111010010000111110000101011010111101000001111001001000101001111011000010101001111001101000100100100101001001000110100000001001100010011001001110100010001101001110011011100100011000101110111100011100000100000111100110110001010100011000001001110101011010001011101000011111100100101100110010100
Pair \(Z_2\) Length of longest common subsequence
4GEZ_1,3MDB_1 208 4
4GEZ_1,5AOJ_1 175 4
3MDB_1,5AOJ_1 153 4

Newick tree

 
[
	4GEZ_1:10.80,
	[
		5AOJ_1:76.5,3MDB_1:76.5
	]:25.30
]

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 \((0.85)(0.8)(\frac{502 }{\log_{20} 502}-\frac{124}{\log_{20}124})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 4GEZ_1 3MDB_1 143 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]