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Parikh vectors

2JLJ_1	4EMG_1	7LZB_1	Letter	Amino acid
3	3	15	N	Asparagine
4	3	7	M	Methionine
2	2	12	F	Phenylalanine
6	3	10	P	Proline
6	3	11	T	Threonine
2	0	3	W	Tryptophan
5	3	10	Y	Tyrosine
4	10	14	D	Aspartic acid
0	0	17	C	Cysteine
4	3	5	H	Histidine
12	7	15	A	Alanine
9	1	12	Q	Glutamine
18	10	23	E	Glutamic acid
4	4	21	G	Glycine
11	12	17	L	Leucine
12	5	26	K	Lycine
10	8	13	V	Valine
12	7	22	R	Arginine
13	6	9	I	Isoleucine
7	3	16	S	Serine

2JLJ_1|Chain A|YOP PROTEINS TRANSLOCATION PROTEIN U|YERSINIA PESTIS (632)
>4EMG_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N|Probable U6 snRNA-associated Sm-like protein LSm3|Schizosaccharomyces pombe (284812)
>7LZB_1|Chain A|Histone-lysine N-methyltransferase SETD2|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 \(w=f(c)\)
2JLJ , Knot	68	144	0.78	38	113	141	IKELKMSKDEIKREYKEMEGSPEIKSKRRQFHQEIQSRNMRENVKRSSVVVAAATHIAIGILYKRGETPLPLVTFKYTDAQVQTVRKIAEEEGVPILQRIPLARALYWDALVDHYIPAEQIEATAEVLRWLERQNIEKQHSEML
4EMG , Knot	48	93	0.78	36	77	89	MESAQAVAEPLDLVRLSLDEIVYVKLRGDRELNGRLHAYDEHLNMVLGDAEEIVTIFDDEETDKDKALKTIRKHYEMLFVRGDSVILIAPPRN
7LZB , Knot	126	278	0.85	40	189	269	GETSVPPGSALVGPSCVMDDFRDPQRWKECAKQGKMPCYFDLIEENVYLTERKKNKSHRDIKRMQCECTPLSKDERAQGEIACGEDCLNRLLMIECSSRCPNGDYCSNRRFQRKQHADVEVILTEKKGWGLRAAKDLPSNTFVLEYCGEVLDHKEFKARVKEYARNKNIHYYFMALKNDEIIDATQKGNCSRFMNHSCEPNCETQKWTVNGQLRVGFFTTKLVPSGSELTFDYQFQRYGKEAQKCFCGSANCRGYLGGENRVSIRAAGGKMKKERSRK

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 \(w=f(c)\)

2JLJ , Knot

144

0.78

113

141

IKELKMSKDEIKREYKEMEGSPEIKSKRRQFHQEIQSRNMRENVKRSSVVVAAATHIAIGILYKRGETPLPLVTFKYTDAQVQTVRKIAEEEGVPILQRIPLARALYWDALVDHYIPAEQIEATAEVLRWLERQNIEKQHSEML

4EMG , Knot

0.78

MESAQAVAEPLDLVRLSLDEIVYVKLRGDRELNGRLHAYDEHLNMVLGDAEEIVTIFDDEETDKDKALKTIRKHYEMLFVRGDSVILIAPPRN

7LZB , Knot

126

278

0.85

189

269

GETSVPPGSALVGPSCVMDDFRDPQRWKECAKQGKMPCYFDLIEENVYLTERKKNKSHRDIKRMQCECTPLSKDERAQGEIACGEDCLNRLLMIECSSRCPNGDYCSNRRFQRKQHADVEVILTEKKGWGLRAAKDLPSNTFVLEYCGEVLDHKEFKARVKEYARNKNIHYYFMALKNDEIIDATQKGNCSRFMNHSCEPNCETQKWTVNGQLRVGFFTTKLVPSGSELTFDYQFQRYGKEAQKCFCGSANCRGYLGGENRVSIRAAGGKMKKERSRK

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 \(c\).

\(|P_{f(2JLJ_1)}(2) \setminus P_{f(4EMG_1)}(2)|=81\), \(|P_{f(4EMG_1)}(2) \setminus P_{f(2JLJ_1)}(2)|=45\). 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 1:100101000010000001010101000000100010000100010000111111001111110001001111101000010100100110001111100111101101011100011100101010110110000100000011

Pair	\(Z_2\)	Length of longest common subsequence
2JLJ_1,4EMG_1	126	3
2JLJ_1,7LZB_1	178	4
4EMG_1,7LZB_1	180	3

Newick tree

 
[
	7LZB_1:96.73,
	[
		2JLJ_1:63,4EMG_1:63
	]:33.73
]

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 \((0.85)(0.8)(\frac{237 }{\log_{20} 237}-\frac{93}{\log_{20}93})=46.4\)

Status	Protein1	Protein2	d	d₁/2
Query variables	2JLJ_1	4EMG_1	58	47.5

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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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