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Parikh vectors
5DZD_1 8AQH_1 8OIC_1 Letter Amino acid
4 6 21 P Proline
5 10 24 T Threonine
2 9 14 N Asparagine
0 7 6 Q Glutamine
2 16 28 L Leucine
2 8 16 F Phenylalanine
2 7 19 S Serine
2 5 13 Y Tyrosine
1 10 18 H Histidine
2 18 30 I Isoleucine
1 4 11 M Methionine
3 18 21 V Valine
3 13 27 D Aspartic acid
2 7 26 K Lycine
0 1 8 C Cysteine
4 8 21 E Glutamic acid
4 20 22 G Glycine
1 3 4 W Tryptophan
1 4 25 A Alanine
6 7 13 R Arginine 

5DZD_1|Chains A, B|E3 ubiquitin-protein ligase Itchy homolog|Homo sapiens (9606)
>8AQH_1|Chains A, B|NanoLuc luciferase|Oplophorus gracilirostris (727944)
>8OIC_1|Chains A, B, C, D, E, F, G, H|Inosine-uridine preferring nucleoside hydrolase family protein|Trichomonas vaginalis (5722)
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 \(w=f(c)\)
5DZD , Knot 28 47 0.76 36 42 45 
GRRASVELNEKPLPEGWEMRFTVDGIPYFVDHNRRTTTYIDPRTGKS
8AQH , Knot 83 181 0.79 40 125 167 
MHHHHHHSDNMVFTLEDFVGDWRQTAGYNLDQVLEQGGVSSLFQNLGVSVTPIQRIVLSGENGLKIDIHVIIPYEGLSGDQMGQIEKIFKVVYPVDDHHFKVILHAGTLVIDGVTPNMIDYFGRPYEGIAVFDGKKITVTGTLWNGNKIIDERLINPDGSLLFRVTINGVTGWRLCERILA
8OIC , Knot 155 367 0.83 40 222 351 
MGSSHHHHHHSSGLVPRGSHMSIKCALDCDPGHDDLAMIMLAVYSPKLDVQYISTTHGNQTVNKTYQNARRTLNLIKRADKIPVYRGYSKPLTRESVACPEIHGESGLGGVDWSEIDRTMPRNPALDILGYKDESELRPDDFFKHLHRLVSAAEDKFDIISTGSETNIAQYLLAYPEDAKKIRMTTMAGNFMIVGNIMPFAEFNVLIDPEAISNILQSGVDYTFAAPLDITHTVLVTEKVINDIKAATEPYSPKFTEMIIKLLFFFKDTYRDVFGFIDPPLHDPVAAFHLIAPEWFEHVRCHVDIETKGEYTYGCCCTNLILKKKDPTKIVKPDNATVCLKLKEGGHDAFWNQMITVWGEIAKEIGK

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 \(c\).

\(|P_{f(5DZD_1)}(2) \setminus P_{f(8AQH_1)}(2)|=24\), \(|P_{f(8AQH_1)}(2) \setminus P_{f(5DZD_1)}(2)|=107\). 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 1:10010101000111011010101011101100000000010100100
Pair \(Z_2\) Length of longest common subsequence
5DZD_1,8AQH_1 131 3
5DZD_1,8OIC_1 214 3
8AQH_1,8OIC_1 191 7

Newick tree

 
[
	8OIC_1:11.82,
	[
		5DZD_1:65.5,8AQH_1:65.5
	]:45.32
]

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 \((0.85)(0.8)(\frac{228 }{\log_{20} 228}-\frac{47}{\log_{20}47})=60.6\)
Status Protein1 Protein2 d d1/2
Query variables 5DZD_1 8AQH_1 74 46
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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