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Parikh vectors
6NIS_1 3RSG_1 2QGZ_1 Letter Amino acid
11 28 26 A Alanine
3 21 14 R Arginine
8 25 15 D Aspartic acid
2 13 13 H Histidine
4 31 20 I Isoleucine
19 57 32 L Leucine
14 35 19 K Lycine
6 0 3 W Tryptophan
6 20 11 F Phenylalanine
12 20 8 P Proline
39 29 25 S Serine
21 54 19 V Valine
22 27 11 T Threonine
9 12 13 Y Tyrosine
9 22 12 N Asparagine
6 4 0 C Cysteine
8 6 19 Q Glutamine
6 38 27 E Glutamic acid
19 52 13 G Glycine
1 8 8 M Methionine 

6NIS_1|Chain A[auth H]|MZ24 antibody heavy chain|Homo sapiens (9606)
>3RSG_1|Chain A|Putative uncharacterized protein|Thermotoga maritima (243274)
>2QGZ_1|Chain A|Putative primosome component|Streptococcus pyogenes serotype M3 (301448)
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 \(w=f(c)\)
6NIS , Knot 98 225 0.78 40 140 213 
QVQLQESGPGLVKSSETLSLSCTVSGASISSYYWSWVRQSPGKGLEWMGSIYYTGTTTYNPSLKSRLTISVDTSKNQFSLKLNSVTAADTAVYSCAGLDRFNWNDEGDCWGQGTLVTVFSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCDK
3RSG , Knot 198 502 0.81 38 224 454 
MGSDKIHHHHHHMKEIDELTIKEYGVDSRILMERAGISVVLAMEEELGNLSDYRFLVLCGGGNNGGDGFVVARNLLGVVKDVLVVFLGKKKTPDCEYNYGLYKKFGGKVVEQFEPSILNEFDVVVDAIFGTGLRGEITGEYAEIINLVNKSGKVVVSVDVPSGIDSNTGKVLRTAVKADLTVTFGVPKIGHILFPGRDLTGKLKVANIGHPVHLINSINRYVITREMVRSLLPERPRDSHKGTYGKVLIIAGSRLYSGAPVLSGMGSLKVGTGLVKLAVPFPQNLIATSRFPELISVPIDTEKGFFSLQNLQECLELSKDVDVVAIGPGLGNNEHVREFVNEFLKTLEKPAVIDADAINVLDTSVLKERKSPAVLTPHPGEMARLVKKTVGDVKYNYELAEEFAKENDCVLVLKSATTIVTDGEKTLFNITGNTGLSKGGSGDVLTGMIAGFIAQGLSPLEASTVSVYLHGFAAELFEQDERGLTASELLRLIPEAIRRLKE
2QGZ , Knot 130 308 0.80 38 184 292 
MEKIGETMAKLGQNTRVNSDQLIQTILADPEVASFISQHHLSQEQINLSLSKFNQFLVERQKYQLKDPSYIAKGYQPILAMNEGYADVSYLETKELVEAQKQAAISERIQLVSLPKSYRHIHLSDIDVNNASRMEAFSAILDFVEQYPSAEQKGLYLYGDMGIGKSYLLAAMAHELSEKKGVSTTLLHFPSFAIDVKNAISNGSVKEEIDAVKNVPVLILDDIGAEQATSWVRDEVLQVILQYRMLEELPTFFTSNYSFADLERKWATIKGSDETWQAKRVMERVRYLAREFHLEGANRRLEHHHHHH

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 \(c\).

\(|P_{f(6NIS_1)}(2) \setminus P_{f(3RSG_1)}(2)|=43\), \(|P_{f(3RSG_1)}(2) \setminus P_{f(6NIS_1)}(2)|=127\). 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 1:010100011111000001010001011010000101100011011011101000100000101000101010000001010100101100110001110010100010011010110110100011011111100000011011110110001101101010011100110011111000110010011011000110000100100010000100010100000
Pair \(Z_2\) Length of longest common subsequence
6NIS_1,3RSG_1 170 4
6NIS_1,2QGZ_1 174 4
3RSG_1,2QGZ_1 148 6

Newick tree

 
[
	6NIS_1:89.65,
	[
		3RSG_1:74,2QGZ_1:74
	]:15.65
]

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 \((0.85)(0.8)(\frac{727 }{\log_{20} 727}-\frac{225}{\log_{20}225})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 6NIS_1 3RSG_1 175 126
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]