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Parikh vectors
1YYG_1 3RAJ_1 2KTX_1 Letter Amino acid
15 8 2 N Asparagine
15 17 1 E Glutamic acid
8 7 1 H Histidine
20 18 1 T Threonine
23 17 2 V Valine
15 14 1 I Isoleucine
8 18 6 K Lycine
7 4 2 M Methionine
23 13 1 F Phenylalanine
30 12 3 P Proline
23 21 1 D Aspartic acid
10 12 6 C Cysteine
29 10 4 G Glycine
1 8 0 W Tryptophan
0 5 0 Y Tyrosine
43 9 1 A Alanine
12 14 2 R Arginine
18 13 1 Q Glutamine
28 17 1 L Leucine
29 20 2 S Serine 

1YYG_1|Chain A|Peroxidase manganese-dependent I|Phanerochaete chrysosporium (2822231)
>3RAJ_1|Chain A|ADP-ribosyl cyclase 1|Homo sapiens (9606)
>2KTX_1|Chain A|KALIOTOXIN|Androctonus mauretanicus mauretanicus (6860)
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 \(w=f(c)\)
1YYG , Knot 151 357 0.82 38 196 335 
AVCPDGTRVSHAACCAFIPLAQDLQETIFQNECGEDAHEVIRLTFHDAIAISRSQGPKAGGGADGSMLLFPTVEPNFSANNGIDDSVNNLIPFMQKHNTISAADLVQFAGAVALSNCPGAPRLEFLAGRPNKTIAAVDGLIPEPQDSVTKILQRFEDAGGFTPFEVVSLLASHSVARADKVDQTIDAAPFDSTPFTFDTQVFLEVLLKGVGFPGSANNTGEVASPLPLGSGSDTGEMRLQSDFALAHDPRTACIWQGFVNEQAFMAASFRAAMSKLAVLGHNRNSLIDCSDVVPVPKPATGQPAMFPASTGPQDLELSCPSERFPTLTTQPGASQSLIAHCPDGSMSCPGVQFNGPA
3RAJ , Knot 118 257 0.85 40 177 250 
EFWRQTWSGPGTTKRFPETVLARCVKYTEIHPEMRHVDCQSVWDAFKGAFISKHPCDITEEDYQPLMKLGTQTVPCNKILLWSRIKDLAHQFTQVQRDMFTLEDTLLGYLADDLTWCGEFDTSKINYQSCPDWRKDCSNNPVSVFWKTVSRRFAEAACDVVHVMLDGSRSKIFDKDSTFGSVEVHNLQPEKVQTLEAWVIHGGREDSRDLCQDPTIKELESIISKRNIQFSCKNIYRPDKFLQCVKNPEDSSCTSEI
2KTX , Knot 25 38 0.79 36 35 36 
GVEINVKCSGSPQCLKPCKDAGMRFGKCMNRKCHCTPK

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 \(c\).

\(|P_{f(1YYG_1)}(2) \setminus P_{f(3RAJ_1)}(2)|=94\), \(|P_{f(3RAJ_1)}(2) \setminus P_{f(1YYG_1)}(2)|=75\). 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 1:110101001001100111111001000110000100100110101001111000011011111010111110101010100110001001111100000101101101111111000111101011110100011110111101000100110010011110110110111000110100100010111100011010001110111011111101000101101111101000101010001111001001011011100011111010111001111100000110000111110110101111110011001010010001101000111000111001010100111010111
Pair \(Z_2\) Length of longest common subsequence
1YYG_1,3RAJ_1 169 3
1YYG_1,2KTX_1 197 3
3RAJ_1,2KTX_1 178 2

Newick tree

 
[
	2KTX_1:96.79,
	[
		1YYG_1:84.5,3RAJ_1:84.5
	]:12.29
]

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 \((0.85)(0.8)(\frac{614 }{\log_{20} 614}-\frac{257}{\log_{20}257})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 1YYG_1 3RAJ_1 124 108.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]