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Parikh vectors

3EIC_1	2LVH_1	5XSQ_1	Letter	Amino acid
16	5	23	I	Isoleucine
4	2	1	M	Methionine
5	1	9	P	Proline
4	4	10	Y	Tyrosine
4	3	16	Q	Glutamine
12	3	17	E	Glutamic acid
7	1	23	G	Glycine
12	6	23	S	Serine
8	0	22	A	Alanine
1	2	3	C	Cysteine
9	6	41	L	Leucine
11	5	17	K	Lycine
6	3	18	T	Threonine
2	0	1	W	Tryptophan
8	4	15	R	Arginine
10	2	15	D	Aspartic acid
6	2	14	F	Phenylalanine
6	6	27	V	Valine
8	3	17	N	Asparagine
3	1	15	H	Histidine

3EIC_1|Chains A, B, C, D, E, F|Nucleoside diphosphate kinase|Acanthamoeba polyphaga mimivirus (212035)
>2LVH_1|Chain A|Putative zinc finger protein ORF59a|Acidianus filamentous virus 1 (654909)
>5XSQ_1|Chains A, C, E|Nucleoprotein|Lake Victoria marburgvirus (strain Ozolin-75) (482820)

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 \(w=f(c)\)
3EIC , Knot	71	142	0.82	40	113	138	YKKAGLQRTLVLIKPDAFERSLVAEIMGRIEKKNFKIVSMKFWSKAPRNLIEQHYKEHSEQSYFNDNCDFMVSGPIISIVYEGTDAISKIRRLQGNILTPGTIRGDLANDIRENLIHASDSEDSAVDEISIWFPETKMETDN
2LVH , Knot	34	59	0.78	36	53	57	MIEVSSMERVYQCLRCGLTFRTKKQLIRHLVNTEKVNPLSIDYYYQSFSVSLKDVNKII
5XSQ , Knot	142	327	0.83	40	192	314	VRNKKVILFDTNHQVSICNQIIDAINSGIDLGDLLEGGLLTLCVEHYYNSDKDKFNTSPIAKYLRDAGYEFDVIKNADATRFLDVIPNEPHYSPLILALKTLESTESQRGRIGLFLSFCSLFLPKLVVGDRASIEKALRQVTVHQEQGIVTYPNHWLTTGHMKVIFGILRSSFILKFVLIHQGVNLVTGHDAYDSIISNSVGQTRFSGLLIVKTVLEFILQKTDSGVTLHPLVRTSKVKNEVASFKQALSNLARHGEYAPFARVLNLSGINNLEHGLYPQLSAIALGVATAHGSTLAGVNVGEQYQQLREAAHDAEVKLQRRHEHQE

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 \(w=f(c)\)

3EIC , Knot

142

0.82

113

138

YKKAGLQRTLVLIKPDAFERSLVAEIMGRIEKKNFKIVSMKFWSKAPRNLIEQHYKEHSEQSYFNDNCDFMVSGPIISIVYEGTDAISKIRRLQGNILTPGTIRGDLANDIRENLIHASDSEDSAVDEISIWFPETKMETDN

2LVH , Knot

0.78

MIEVSSMERVYQCLRCGLTFRTKKQLIRHLVNTEKVNPLSIDYYYQSFSVSLKDVNKII

5XSQ , Knot

142

327

0.83

192

314

VRNKKVILFDTNHQVSICNQIIDAINSGIDLGDLLEGGLLTLCVEHYYNSDKDKFNTSPIAKYLRDAGYEFDVIKNADATRFLDVIPNEPHYSPLILALKTLESTESQRGRIGLFLSFCSLFLPKLVVGDRASIEKALRQVTVHQEQGIVTYPNHWLTTGHMKVIFGILRSSFILKFVLIHQGVNLVTGHDAYDSIISNSVGQTRFSGLLIVKTVLEFILQKTDSGVTLHPLVRTSKVKNEVASFKQALSNLARHGEYAPFARVLNLSGINNLEHGLYPQLSAIALGVATAHGSTLAGVNVGEQYQQLREAAHDAEVKLQRRHEHQE

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 \(c\).

\(|P_{f(3EIC_1)}(2) \setminus P_{f(2LVH_1)}(2)|=93\), \(|P_{f(2LVH_1)}(2) \setminus P_{f(3EIC_1)}(2)|=33\). 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 1:0001110001111010110001110111010000101101011001100110000000000001000001110111101100100110010010101101101010110010001101000000110010111100010000

Pair	\(Z_2\)	Length of longest common subsequence
3EIC_1,2LVH_1	126	2
3EIC_1,5XSQ_1	189	3
2LVH_1,5XSQ_1	175	2

Newick tree

 
[
	5XSQ_1:98.66,
	[
		3EIC_1:63,2LVH_1:63
	]:35.66
]

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 \((0.85)(0.8)(\frac{201 }{\log_{20} 201}-\frac{59}{\log_{20}59})=47.7\)

Status	Protein1	Protein2	d	d₁/2
Query variables	3EIC_1	2LVH_1	62	43

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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