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Parikh vectors

6EWL_1	9KAD_1	4UXS_1	Letter	Amino acid
10	0	21	R	Arginine
7	0	30	T	Threonine
11	0	38	E	Glutamic acid
5	0	13	H	Histidine
7	0	26	I	Isoleucine
3	0	20	F	Phenylalanine
3	0	19	Y	Tyrosine
2	0	16	N	Asparagine
9	0	15	Q	Glutamine
8	0	19	K	Lycine
2	0	10	M	Methionine
13	0	22	S	Serine
6	0	27	D	Aspartic acid
19	0	31	L	Leucine
5	5	38	G	Glycine
9	0	20	P	Proline
2	0	4	W	Tryptophan
9	0	34	V	Valine
7	4	36	A	Alanine
2	5	12	C	Cysteine

6EWL_1|Chain A|Centrosomal protein 120|Danio rerio (7955)
>9KAD_1|Chains A, B|RNA (5'-R(*GP*CP*AP*GP*CP*AP*GP*CP*AP*GP*CP*AP*GP*C)-3')|Homo sapiens (9606)
>4UXS_1|Chain A|TUBULIN ALPHA-1B CHAIN|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 \(w=f(c)\)
6EWL , Knot	67	139	0.79	40	111	135	GPHMASKSEQLLIVVSILEGRQFPRSPRLSLVVEARFDGETLSTDPVEHKEQPQFCTELAWELDRRTLHQHRLQRTPIKLQCYAVDSSTSARESVGYIVLDLRSVQEIKQAPKWHPLLSSKYTKLKPALLIGMILENDN
9KAD , Knot	4	14	0.25	6	3	3	GCAGCAGCAGCAGC
4UXS , Knot	191	451	0.86	40	254	425	MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFSVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLIGQIVSSITASLRFDGALNVDLTEFQTNLVPYPRGHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRTIQFVDWCPTGFKVGINYEPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEEY

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 \(w=f(c)\)

6EWL , Knot

139

0.79

111

135

GPHMASKSEQLLIVVSILEGRQFPRSPRLSLVVEARFDGETLSTDPVEHKEQPQFCTELAWELDRRTLHQHRLQRTPIKLQCYAVDSSTSARESVGYIVLDLRSVQEIKQAPKWHPLLSSKYTKLKPALLIGMILENDN

9KAD , Knot

0.25

GCAGCAGCAGCAGC

4UXS , Knot

191

451

0.86

254

425

MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFSVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLIGQIVSSITASLRFDGALNVDLTEFQTNLVPYPRGHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRTIQFVDWCPTGFKVGINYEPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEEY

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 \(c\).

\(|P_{f(6EWL_1)}(2) \setminus P_{f(9KAD_1)}(2)|=111\), \(|P_{f(9KAD_1)}(2) \setminus P_{f(6EWL_1)}(2)|=3\). 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 1:1101100000111110110100110010101110101010010001100000101000111010000100001000110100011000001000110111010010010011010111000000101111111110000

Pair	\(Z_2\)	Length of longest common subsequence
6EWL_1,9KAD_1	114	1
6EWL_1,4UXS_1	197	4
9KAD_1,4UXS_1	253	2

Newick tree

 
[
	4UXS_1:12.70,
	[
		6EWL_1:57,9KAD_1:57
	]:69.70
]

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 \((0.85)(0.8)(\frac{153 }{\log_{20} 153}-\frac{14}{\log_{20}14})=51.1\)

Status	Protein1	Protein2	d	d₁/2
Query variables	6EWL_1	9KAD_1	65	34

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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