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Parikh vectors
1BYC_1 5HYB_1 1CGJ_1 Letter Amino acid
30 1 14 N Asparagine
6 0 10 C Cysteine
11 1 2 M Methionine
21 4 6 F Phenylalanine
25 8 28 S Serine
32 6 23 V Valine
21 6 10 Q Glutamine
47 11 19 L Leucine
30 7 14 K Lycine
11 3 8 W Tryptophan
24 2 4 Y Tyrosine
33 2 9 D Aspartic acid
11 2 2 H Histidine
27 5 10 I Isoleucine
18 2 23 T Threonine
32 5 22 A Alanine
19 6 4 R Arginine
29 8 5 E Glutamic acid
38 8 23 G Glycine
30 4 9 P Proline 

1BYC_1|Chain A|BETA-AMYLASE|Glycine max (3847)
>5HYB_1|Chains A, B|Matrix Protein|Mouse mammary tumor virus (strain BR6) (11758)
>1CGJ_1|Chain A[auth E]|ALPHA-CHYMOTRYPSINOGEN|Bos taurus (9913)
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 \(w=f(c)\)
1BYC , Knot 205 495 0.85 40 256 478 
ATSDSNMLLNYVPVYVMLPLGVVNVDNVFEDPDGLKEQLLQLRAAGVDGVMVDVWWGIIELKGPKQYDWRAYRSLFQLVQECGLTLQAIMSFHQCGGNVGDIVNIPIPQWVLDIGESNHDIFYTNRSGTRNKEYLTVGVDNEPIFHGRTAIEIYSDYMKSFRENMSDFLESGLIIDIEVGLGPAGELRYPSYPQSQGWEFPRIGEFQCYDKYLKADFKAAVARAGHPEWELPDDAGKYNDVPESTGFFKSNGTYVTEKGKFFLTWYSNKLLNHGDQILDEANKAFLGCKVKLAIKVSGIHWWYKVENHAAELTAGYYNLNDRDGYRPIARMLSRHHAILNFTCLEMRDSEQPSDAKSGPQELVQQVLSGGWREDIRVAGENALPRYDATAYNQIILNAKPQGVNNNGPPKLSMFGVTYLRLSDDLLQKSNFNIFKKFVLKMHADQDYCANPQKYNHAITPLKPSAPKIPIEVLLEATKPTLPFPWLPETDMKVDG
5HYB , Knot 47 91 0.77 38 82 89 
GVSGSKGQKLFVSVLQRLLSERGLHVKESSAIEFYQFLIKVSPWFPEEGGLNLQDWKRVGREMKRYAAEHGTDSIPKQAAPIWLQLREILT
1CGJ , Knot 111 245 0.83 40 154 236 
CGVPAIQPVLSGLSRIVNGEEAVPGSWPWQVSLQDKTGFHFCGGSLINENWVVTAAHCGVTTSDVVVAGEFDQGSSSEKIQKLKIAKVFKNSKYNSLTINNDITLLKLSTAASFSQTVSAVCLPSASDDFAAGTTCVTTGWGLTRYTNANTPDRLQQASLPLLSNTNCKKYWGTKIKDAMICAGASGVSSCMGDSGGPLVCKKNGAWTLVGIVSWGSSTCSTSTPGVYARVTALVNWVQQTLAAN

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 \(c\).

\(|P_{f(1BYC_1)}(2) \setminus P_{f(5HYB_1)}(2)|=196\), \(|P_{f(5HYB_1)}(2) \setminus P_{f(1BYC_1)}(2)|=22\). 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 1:100000111001110111111110100110010110001101011110111101111110101100001010001101100011010111010001101101101111011101100000110000010000001011100011101001101000010010001001100111101011111110100100100011011011010000001010101111011010101100110000110001110001001000101110100001100100110010011110010111010110110010001101011000100001001110110000111010010100000100100110011001101110001011100111000101000111010101100011101011110010100011000010110011101010000010100000110110101101110111010010111111100010101
Pair \(Z_2\) Length of longest common subsequence
1BYC_1,5HYB_1 218 4
1BYC_1,1CGJ_1 186 4
5HYB_1,1CGJ_1 168 3

Newick tree

 
[
	1BYC_1:10.46,
	[
		1CGJ_1:84,5HYB_1:84
	]:22.46
]

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 \((0.85)(0.8)(\frac{586 }{\log_{20} 586}-\frac{91}{\log_{20}91})=146.\)
Status Protein1 Protein2 d d1/2
Query variables 1BYC_1 5HYB_1 190 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]