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Parikh vectors

7PJT_1	6QGQ_1	2IDZ_1	Letter	Amino acid
4	12	17	V	Valine
0	5	1	C	Cysteine
1	8	11	E	Glutamic acid
5	18	16	S	Serine
2	3	6	Y	Tyrosine
2	18	20	I	Isoleucine
1	17	13	P	Proline
0	3	4	W	Tryptophan
1	10	8	N	Asparagine
3	11	15	D	Aspartic acid
2	11	10	Q	Glutamine
4	21	28	G	Glycine
7	7	13	R	Arginine
4	6	5	H	Histidine
3	22	27	L	Leucine
5	12	13	T	Threonine
5	21	36	A	Alanine
6	11	9	K	Lycine
2	9	9	M	Methionine
0	8	7	F	Phenylalanine

7PJT_1|Chain A[auth 0]|50S ribosomal protein L32|Escherichia coli (562)
>6QGQ_1|Chains A, C, D|Acyl-protein thioesterase 1|Homo sapiens (9606)
>2IDZ_1|Chain A|Enoyl-[acyl-carrier-protein] reductase [NADH]|Mycobacterium tuberculosis (1773)

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 \(w=f(c)\)
7PJT , Knot	34	57	0.80	34	51	55	MAVQQNKPTRSKRGMRRSHDALTAVTSLSVDKTSGEKHLRHHITADGYYRGRKVIAK
6QGQ , Knot	108	233	0.84	40	165	228	GSRMSGNNMSTPLPAIVPAARKATAAVIFLHGLGDTGHGWAEAFAGIRSSHIKYICPHAPVRPVTLNMNVAMPSWFDIIGLSPDSQEDESGIKQAAENIKALIDQEVKNGIPSNRIILGGFSQGGALSLYTALTTQQKLAGVTALSCWLPLRASFPQGPIGGANRDISILQCHGDCDPLVPLMFGSLTVEKLKTLVNPANVTFKTYEGMMHSSCQQEMMDVKQFIDKLLPPID
2IDZ , Knot	114	268	0.79	40	167	252	TGLLDGKRILVSGIITDSSIAFHIARVAQEQGAQLVLTGFDRLRLIQRITDRLPAKAPLLELDVQNEEHLASLAGRVTEAIGAGNKLDGVVHSIGFMPQTGMGINPFFDAPYADVSKGIHISAYSYASMAKALLPIMNPGGSIVGMDFDPSRAMPAYNWMTVAKSALESVNRFVAREAGKYGVRSNLVAAGPIRTLAMSAIVGGALGEEAGAQIQLLEEGWDQRAPIGWNMKDATPVAKTVCALLSDWLPATTGDIIYADGGAHTQLL

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 \(w=f(c)\)

7PJT , Knot

0.80

MAVQQNKPTRSKRGMRRSHDALTAVTSLSVDKTSGEKHLRHHITADGYYRGRKVIAK

6QGQ , Knot

108

233

0.84

165

228

GSRMSGNNMSTPLPAIVPAARKATAAVIFLHGLGDTGHGWAEAFAGIRSSHIKYICPHAPVRPVTLNMNVAMPSWFDIIGLSPDSQEDESGIKQAAENIKALIDQEVKNGIPSNRIILGGFSQGGALSLYTALTTQQKLAGVTALSCWLPLRASFPQGPIGGANRDISILQCHGDCDPLVPLMFGSLTVEKLKTLVNPANVTFKTYEGMMHSSCQQEMMDVKQFIDKLLPPID

2IDZ , Knot

114

268

0.79

167

252

TGLLDGKRILVSGIITDSSIAFHIARVAQEQGAQLVLTGFDRLRLIQRITDRLPAKAPLLELDVQNEEHLASLAGRVTEAIGAGNKLDGVVHSIGFMPQTGMGINPFFDAPYADVSKGIHISAYSYASMAKALLPIMNPGGSIVGMDFDPSRAMPAYNWMTVAKSALESVNRFVAREAGKYGVRSNLVAAGPIRTLAMSAIVGGALGEEAGAQIQLLEEGWDQRAPIGWNMKDATPVAKTVCALLSDWLPATTGDIIYADGGAHTQLL

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 \(c\).

\(|P_{f(7PJT_1)}(2) \setminus P_{f(6QGQ_1)}(2)|=32\), \(|P_{f(6QGQ_1)}(2) \setminus P_{f(7PJT_1)}(2)|=146\). 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 1:111000010000011000001101100101000010001000101010001001110

Pair	\(Z_2\)	Length of longest common subsequence
7PJT_1,6QGQ_1	178	3
7PJT_1,2IDZ_1	164	3
6QGQ_1,2IDZ_1	158	4

Newick tree

 
[
	7PJT_1:87.65,
	[
		2IDZ_1:79,6QGQ_1:79
	]:8.65
]

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 \((0.85)(0.8)(\frac{290 }{\log_{20} 290}-\frac{57}{\log_{20}57})=75.4\)

Status	Protein1	Protein2	d	d₁/2
Query variables	7PJT_1	6QGQ_1	100	62

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

Graphviz Engine:

digraph G {
							subgraph cluster_theyellow {
								G [style=filled; fillcolor=yellow];
								A [style=filled; fillcolor=yellow];
								V [style=filled; fillcolor=yellow];
								C [style=filled; fillcolor=yellow];
								P [style=filled; fillcolor=yellow];
								L [style=filled; fillcolor=yellow];
								I [style=filled; fillcolor=yellow];
								M [style=filled; fillcolor=yellow];
								W [style=filled; fillcolor=yellow];
								F [style=filled; fillcolor=yellow];
							}
							subgraph cluster_thegreen {
								K [style=filled; fillcolor=green];
								R [style=filled; fillcolor=green];
								H [style=filled; fillcolor=green];
							}
							subgraph cluster_theblue {
								S [style=filled; fillcolor=blue];
								T [style=filled; fillcolor=blue];
								Y [style=filled; fillcolor=blue];
								N [style=filled; fillcolor=blue];
								Q [style=filled; fillcolor=blue];
							}
							subgraph cluster_thepurple {
								D [style=filled; fillcolor=purple];
								E [style=filled; fillcolor=purple];
							}
		
							M->A
							A->V
							V->Q
							Q->Q
							Q->N
							N->K
							K->P
							P->T
							T->R
							R->S
							S->K
							K->R
							R->G
							G->M
							M->R
							R->R
							S->H
							H->D
							D->A
							A->L
							L->T
							T->A
							V->T
							T->S
							S->L
							L->S
							S->V
							V->D
							D->K
							K->T
							S->G
							G->E
							E->K
							K->H
							H->L
							L->R
							R->H
							H->H
							H->I
							I->T
							A->D
							D->G
							G->Y
							Y->Y
							Y->R
							G->R
							R->K
							K->V
							V->I
							I->A
							A->K
						}

Graphviz Engine:

CoV2D BrowserTM

Newick tree

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