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Parikh vectors
4YPL_1 4LZO_1 2EIA_1 Letter Amino acid
32 32 16 I Isoleucine
60 23 16 L Leucine
15 5 4 Y Tyrosine
48 17 13 G Glycine
36 16 14 R Arginine
0 9 3 C Cysteine
46 17 16 E Glutamic acid
17 9 7 M Methionine
6 9 7 F Phenylalanine
47 30 15 A Alanine
32 22 12 D Aspartic acid
22 9 15 Q Glutamine
36 10 16 P Proline
32 14 11 T Threonine
11 15 11 N Asparagine
32 20 9 K Lycine
22 24 7 S Serine
7 2 3 W Tryptophan
42 28 8 V Valine
12 15 3 H Histidine 

4YPL_1|Chains A, B, C, D, E, F|Lon protease|Meiothermus taiwanensis (172827)
>4LZO_1|Chains A, B|Ribose-phosphate pyrophosphokinase 1|Homo sapiens (9606)
>2EIA_1|Chains A, B|EIAV CAPSID PROTEIN P26|Equine infectious anemia virus (11665)
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 \(w=f(c)\)
4YPL , Knot 215 555 0.81 38 234 510 
GHMGLSDLEALRKKIEEVGMPEAVKTKALKELDRLERMQQGSPEATVARTYLDWLTEVPWSKADPEVLDINHTRQVLDEDHYGLKDVKERILEYLAVRQLTQGLDVRNKAPILVLVGPPGVGKTSLGRSIARSMNRKFHRISLGGVRDEAEIRGHRRTYIGAMPGKLIHAMKQVGVINPVILLDQIDKMSSDWRGDPASAMLEVLDPEQNNTFTDHYLDVPYDLSKVFFITTANTLQTIPRPLLDRMEVIEIPGYTNMEKQAIARQYLWPKQVRESGMEGRIEVTDAAILRVISEYTREAGVRGLERELGKIARKGAKFWLEGAWEGLRTIDASDIPTYLGIPRYRPDKAETEPQVGTAQGLAWTPVGGTLLTIEVAAVPGSGKLSLTGQLGEVMKESAQAALTYLRAHTQDYGLPEDFYNKVDLHVHVPDGATPKDGPSAGITMATAIASALSRRPARMDIAMTGEVSLRGKVMPIGGVKEKLLAAHQAGIHKIVLPKDNEAQLEELPKEVLEGLEIKLVEDVGEVLEYLLLPEPTMPPVVQPSDNRQQPGAGA
4LZO , Knot 139 326 0.82 40 207 307 
MPNIKIFSGSSHQDLSQKIADRLGLELGKVVTKKFSNQETCVEIGESVRGEDVYIVQSGCGEINDNLMELLIMINACKIASASRVTTVIPCFPYARQDKKDKSRAPISAKLVANMLSVAGADHIITMDLHASQIQGFFDIPVDNLYAEPAVLKWIRENISEWRNCTIVSPDAGGAKRVTSIADRLNVDFALIHKERKKANEVDRMVLVGDVKDRVAILVDDMADTCGTICHAADKLLSAGATRVYAILTHGIFSGPAISRINNACFEAVVVTNTIPQEDKMKHCSKIQVIDISMILAEAIRRTHNGESVSYLFSHVPLLEHHHHHH
2EIA , Knot 99 206 0.85 40 155 201 
PRGYTTWVNTIQTNGLLNEASQNLFGILSVDCTSEEMNAFLDVVPGQAGQKQILLDAIDKIADDWDNRHPLPNAPLVAPPQGPIPMTARFIRGLGVPRERQMEPAFDQFRQTYRQWIIEAMSEGIKVMIGKPKAQNIRQGAKEPYPEFVDRLLSQIKSEGHPQEISKFLTDTLTIQNANEECRNAMRHLRPEDTLEEKMYACRDIG

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 \(c\).

\(|P_{f(4YPL_1)}(2) \setminus P_{f(4LZO_1)}(2)|=91\), \(|P_{f(4LZO_1)}(2) \setminus P_{f(4YPL_1)}(2)|=64\). 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 1:101110010110001001111011000110010010010010101011000101100111001010110100000110000011001000110011100100110100011111111111110001100110010001001011110001010100000111111011011001111011111001001000101011011101101000001000010110010011110010010011011100101101110001000111000111001000110101010011110110000001110110001101100110111011101100101001100111100010010001011010111101111011010111111010101010110110001011100101000001110010001010101101101001101110110111011000110101110101010101111111000111100111001111000010100110011011010110011011001111010111110100000011111
Pair \(Z_2\) Length of longest common subsequence
4YPL_1,4LZO_1 155 4
4YPL_1,2EIA_1 175 4
4LZO_1,2EIA_1 180 4

Newick tree

 
[
	2EIA_1:92.20,
	[
		4YPL_1:77.5,4LZO_1:77.5
	]:14.70
]

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 \((0.85)(0.8)(\frac{881 }{\log_{20} 881}-\frac{326}{\log_{20}326})=149.\)
Status Protein1 Protein2 d d1/2
Query variables 4YPL_1 4LZO_1 186 146
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]