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Parikh vectors
2PUW_1 4GXX_1 1NYY_1 Letter Amino acid
24 21 25 A Alanine
10 9 2 C Cysteine
14 9 8 Q Glutamine
32 17 15 I Isoleucine
13 9 5 F Phenylalanine
12 19 12 P Proline
32 33 15 S Serine
13 22 8 N Asparagine
24 26 21 G Glycine
39 28 31 L Leucine
19 27 21 T Threonine
8 15 4 Y Tyrosine
20 12 18 R Arginine
19 12 16 D Aspartic acid
24 17 20 E Glutamic acid
8 9 6 H Histidine
2 7 4 W Tryptophan
19 19 11 K Lycine
8 2 8 M Methionine
27 18 13 V Valine 

2PUW_1|Chains A, B|isomerase domain of glutamine-fructose-6-phosphate transaminase (isomerizing)|Candida albicans (237561)
>4GXX_1|Chains A, C, E|Hemagglutinin HA1 chain|Influenza A virus (88776)
>1NYY_1|Chain A|Beta-lactamase TEM|Escherichia coli (562)
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 \(w=f(c)\)
2PUW , Knot 155 367 0.83 40 205 352 
MKGPYKHFMQKEIFEQPDSAFNTMRGRIDFENCVVTLGGLKSWLSTIRRCRRIIMIACGTSYHSCLATRSIFEELTEIPVSVELASDFLDRRSPVFRDDTCVFVSQSGETADSILALQYCLERGALTVGIVNSVGSSMSRQTHCGVHINAGPEIGVASTKAYTSQYIALVMFALSLSNDSISRKGRHEEIIKGLQKIPEQIKQVLKLENKIKDLCNSSLNDQKSLLLLGRGYQFATALEGALKIKEISYMHSEGVLAGELKHGILALVDEDLPIIAFATRDSLFPKVMSAIEQVTARDGRPIVICNEGDAIISNDKVHTTLEVPETVDCLQGLLNVIPLQLISYWLAVNRGIDVDFPRNLAKSVTVE
4GXX , Knot 143 331 0.83 40 204 315 
ADPGDTICIGYHANNSTDTVDTVLEKNVTVTHSVNLLEDSHNGKLCKLKGIAPLQLGKCNIAGWLLGNPECDLLLTASSWSYIVETSNSENGTCYPGDFIDYEELREQLSSVSSFEKFEIFPKTSSWPNHETTKGVTAACSYAGASSFYRNLLWLTKKGSSYPKLSKSYVNNKGKEVLVLWGVHHPPTGTEQQSLYQNADAYVSVGSSKYNRRFTPEIAARPKVRGQAGRMNYYWTLLEPGDTITFEATGNLIAPWYAFALNRGSGSGIITSDAPVHDCNTKCQTPHGAINSSLPFQNIHPVTIGECPKYVRSTKLRMATGLRNIPSIQSR
1NYY , Knot 117 263 0.82 40 176 256 
HPETLVKVKDAEDQLGARVGYIELDLNSGKILESFRPEERFPMMSTFKVLLCGAVLSRVDAGQEQLGRRIHYSQNDLVEYSPVTEKHLTDGMTVRELCSAAITMSDNTAANLLLTTIGGPKELTAFLHNMGDHVTRLDRWEPELNEAIPNDERDTTTPAAMATTLRKLLTGELLTLASRQQLIDWMEADKVAGPLLRSALPAGWFIADKSGAGERGSRGIIAALGPDGKPSRIVVIYTTGSQATMDERNRQIAEIGASLIKHW

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 \(c\).

\(|P_{f(2PUW_1)}(2) \setminus P_{f(4GXX_1)}(2)|=84\), \(|P_{f(4GXX_1)}(2) \setminus P_{f(2PUW_1)}(2)|=83\). 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 1:1011000110001100100110010101010001101111001100100000111110100000011000110010011101011001100001110000011100010010011110001001110111100110010000001101011101111000100000111111110100001000100001101100110010011010001001000010000011111010011011011101001001000111110100111111000111111100001110110110010100101111000101110000100010110010010111011110110011110011010110011001010
Pair \(Z_2\) Length of longest common subsequence
2PUW_1,4GXX_1 167 5
2PUW_1,1NYY_1 153 3
4GXX_1,1NYY_1 170 4

Newick tree

 
[
	4GXX_1:86.68,
	[
		2PUW_1:76.5,1NYY_1:76.5
	]:10.18
]

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 \((0.85)(0.8)(\frac{698 }{\log_{20} 698}-\frac{331}{\log_{20}331})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 2PUW_1 4GXX_1 126 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]