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Parikh vectors
8OUE_1 1GMK_1 2EFN_1 Letter Amino acid
39 2 8 A Alanine
16 0 3 Q Glutamine
13 0 1 H Histidine
46 4 12 L Leucine
62 0 15 K Lycine
31 0 18 I Isoleucine
41 4 14 V Valine
6 4 1 W Tryptophan
30 0 8 R Arginine
9 0 7 N Asparagine
25 0 7 D Aspartic acid
8 0 0 C Cysteine
31 1 7 G Glycine
5 0 5 F Phenylalanine
30 0 5 T Threonine
43 0 12 E Glutamic acid
12 0 4 M Methionine
23 0 5 P Proline
30 0 7 S Serine
14 0 11 Y Tyrosine 

8OUE_1|Chains A[auth C], E[auth G]|H/ACA ribonucleoprotein complex subunit DKC1|Homo sapiens (9606)
>1GMK_1|Chains A, B, C, D|GRAMICIDIN A|BREVIBACILLUS BREVIS (1393)
>2EFN_1|Chain A|150aa long hypothetical transcriptional regulator|Sulfolobus tokodaii (273063)
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 \(w=f(c)\)
8OUE , Knot 203 514 0.82 40 240 466 
MADAEVIILPKKHKKKKERKSLPEEDVAEIQHAEEFLIKPESKVAKLDTSQWPLLLKNFDKLNVRTTHYTPLACGSNPLKREIGDYIRTGFINLDKPSNPSSHEVVAWIRRILRVEKTGHSGTLDPKVTGCLIVCIERATRLVKSQQSAGKEYVGIVRLHNAIEGGTQLSRALETLTGALFQRPPLIAAVKRQLRVRTIYESKMIEYDPERRLGIFWVSCEAGTYIRTLCVHLGLLLGVGGQMQELRRVRSGVMSEKDHMVTMHDVLDAQWLYDNHKDESYLRRVVYPLEKLLTSHKRLVMKDSAVNAICYGAKIMLPGVLRYEDGIEVNQEIVVITTKGEAICMAIALMTTAVISTCDHGIVAKIKRVIMERDTYPRKWGLGPKASQKKLMIKQGLLDKHGKPTDSTPATWKQEYVDYSESAKKEVVAEVVKAPQVVAEAAKTAKRKRESESESDETPPAAPQLIKKEKKKSKKDKKAKAGLESGAEPGDGDSDTTKKKKKKKKAKEVELVSE
1GMK , Knot 8 16 0.46 12 10 11 
VGALAVVVWLWLWLWX
2EFN , Knot 73 150 0.81 38 114 145 
MDEIDLRILKILQYNAKYSLDEIAREIRIPKATLSYRIKKLEKDGVIKGYYAYINPASLNLDYIVITSVKAKYGKNYHVELGNKLAQIPGVWGVYFVLGDNDFIVMARYKTREEFMEKFLERVMSIPEVERTSTQVVVKIIKESPNIVIF

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 \(c\).

\(|P_{f(8OUE_1)}(2) \setminus P_{f(1GMK_1)}(2)|=233\), \(|P_{f(1GMK_1)}(2) \setminus P_{f(8OUE_1)}(2)|=3\). 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 1:1101011111000000000001100011010010011101000110100001111100100101000000111010011000110010011101001001000011111001101000100101010101011101001001100000110001111010011011001001100101111001111111000101001000011000100011111100011001001010111111111010010010011100000110100110101100000000010011011001100000111000110110011011111110000110100011110001011011111100111000001111010011100000100111110100001110011100010100001101000010000010001110110110111011001000000000000011111011000000000000101110011011010000000000000100101100
Pair \(Z_2\) Length of longest common subsequence
8OUE_1,1GMK_1 236 3
8OUE_1,2EFN_1 192 3
1GMK_1,2EFN_1 118 3

Newick tree

 
[
	8OUE_1:11.44,
	[
		2EFN_1:59,1GMK_1:59
	]:60.44
]

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 \((0.85)(0.8)(\frac{530 }{\log_{20} 530}-\frac{16}{\log_{20}16})=160.\)
Status Protein1 Protein2 d d1/2
Query variables 8OUE_1 1GMK_1 198 101
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]