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Parikh vectors
6EUE_1 5ANS_1 3GAX_1 Letter Amino acid
34 11 5 F Phenylalanine
24 9 8 R Arginine
44 19 7 G Glycine
14 10 3 H Histidine
49 17 7 L Leucine
25 9 7 K Lycine
34 12 5 E Glutamic acid
43 10 8 S Serine
17 4 4 Y Tyrosine
40 14 10 V Valine
7 2 6 C Cysteine
16 11 7 Q Glutamine
21 5 2 I Isoleucine
15 5 3 M Methionine
22 8 7 T Threonine
23 4 10 A Alanine
35 1 5 N Asparagine
24 12 7 D Aspartic acid
32 8 8 P Proline
14 4 1 W Tryptophan 

6EUE_1|Chain A|Acetylcholinesterase|Tetronarce californica (7787)
>5ANS_1|Chain A|7,8-DIHYDRO-8-OXOGUANINE TRIPHOSPHATASE|HOMO SAPIENS (9606)
>3GAX_1|Chains A, B|Cystatin-C|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 \(w=f(c)\)
6EUE , Knot 214 533 0.84 40 265 512 
HSELLVNTKSGKVMGTRVPVLSSHISAFLGIPFAEPPVGNMRFRRPEPKKPWSGVWNASTYPNNCQQYVDEQFPGFSGSEMWNPNREMSEDCLYLNIWVPSPRPKSTTVMVWIYGGGFYSGSSTLDVYNGKYLAYTEEVVLVSLSYRVGAFGFLALHGSQEAPGNVGLLDQRMALQWVHDNIQFFGGDPKTVTIFGESAGGASVGMHILSPGSRDLFRRAILQSGSPNCPWASVSVAEGRRRAVELGRNLNCNLNSDEELIHCLREKKPQELIDVEWNVLPFDSIFRFSFVPVIDGEFFPTSLESMLNSGNFKKTQILLGVNKDEGSFFLLYGAPGFSKDSESKISREDFMSGVKLSVPHANDLGLDAVTLQYTDWMDDNNGIKNRDGLDDIVGDHNVICPLMHFVNKYTKFGNGTYLYFFNHRASNLVWPEWMGVIHGYEIEFVFGLPLVKELNYTAEEEALSRRIMHYWATFAKTGNPNEPHSQESKWPLFTTKEQKFIDLNTEPMKVHQRLRVQMCVFWNQFLPKLLNAT
5ANS , Knot 81 175 0.79 40 127 166 
GSSHHHHHHSSGLVPRGSHMGASRLYTLVLVLQPQRVLLGMKKRGFGAGRWNGFGGKVQEGETIEDGARRELQEESGLTVDALHKVGQIVFEFVGEPELMDVHVFCTDSIQGTPVESDEMRPCWFQLDQIPFKDMWPDDSYWFPLLLQKKKFHGYFKFQGQDTILDYTLREVDTV
3GAX , Knot 64 120 0.85 40 101 118 
SSPGKPPRLVGGPMDASVEEEGVRRALDFAVGEYNKASNDMYHSRACQVVRARKQIVAGVNYFLDVELCRTTCTKTQPNLDNCPFHDQPHLKRKAFCSFQIYAVPWQGTMTLSKSTCQDA

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 \(c\).

\(|P_{f(6EUE_1)}(2) \setminus P_{f(5ANS_1)}(2)|=166\), \(|P_{f(5ANS_1)}(2) \setminus P_{f(6EUE_1)}(2)|=28\). 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 1:00011100001011100111100010111111110111101010010100110111010001000000100011110100110100010000101011110101000011111011110010001010010011000011110100011111111101000111011110001110110001011110100101110011110111011011000110011100101001110101101000110110010001000001100100001001101010111100110101111101011100100110010100001111100001011110111110000000100001101101011010011101101000011000011000011001110001101110110000011010010110001001111011111010010111111110010001000110001100110110010100100000011110000001101000110100010101011100111011010
Pair \(Z_2\) Length of longest common subsequence
6EUE_1,5ANS_1 194 3
6EUE_1,3GAX_1 226 3
5ANS_1,3GAX_1 148 3

Newick tree

 
[
	6EUE_1:11.84,
	[
		5ANS_1:74,3GAX_1:74
	]:39.84
]

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 \((0.85)(0.8)(\frac{708 }{\log_{20} 708}-\frac{175}{\log_{20}175})=150.\)
Status Protein1 Protein2 d d1/2
Query variables 6EUE_1 5ANS_1 193 125.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]