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Parikh vectors

3OAN_1	1SPF_1	9MQY_1	Letter	Amino acid
9	0	35	D	Aspartic acid
1	2	2	C	Cysteine
4	0	17	F	Phenylalanine
6	0	31	T	Threonine
10	13	43	V	Valine
13	8	42	L	Leucine
15	1	30	K	Lycine
6	0	32	S	Serine
2	0	11	Y	Tyrosine
6	2	28	R	Arginine
7	0	12	H	Histidine
1	0	4	W	Tryptophan
7	2	27	I	Isoleucine
4	1	8	M	Methionine
2	2	22	P	Proline
7	1	53	A	Alanine
5	1	5	N	Asparagine
4	0	18	Q	Glutamine
10	0	26	E	Glutamic acid
11	2	48	G	Glycine

3OAN_1|Chain A|ABR034Wp|Ashbya gossypii (33169)
>1SPF_1|Chain A|PULMONARY SURFACTANT-ASSOCIATED POLYPEPTIDE C|Sus scrofa (9823)
>9MQY_1|Chains A, B|NADH:ubiquinone reductase (non-electrogenic)|Mycolicibacterium smegmatis MC2 155 (246196)

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 \(w=f(c)\)
3OAN , Knot	63	130	0.78	40	98	124	MSLTNGEENEEVLFCEKAKLLIFDSDTKGYTSRGVGELKLLRKKDDKGKVRVLMRSEGMGHVLLNTSVVKSFKYQPIDADNENLIKWPIITDGKLETFIIKVKQKADGRRLVGAVADAQQAMEGHHHHHH
1SPF , Knot	17	35	0.57	22	24	26	LRIPCCPVNLKRLLVVVVVVVLVVVVIVGALLMGL
9MQY , Knot	200	494	0.83	40	233	462	MSHPGATASDRHKVVIIGSGFGGLTAAKTLKRADVDVKLIARTTHHLFQPLLYQVATGIISEGEIAPATRVILRKQKNAQVLLGDVTHIDLENKTVDSVLLGHTYSTPYDSLIIAAGAGQSYFGNDHFAEFAPGMKSIDDALELRGRILGAFEQAERSSDPVRRAKLLTFTVVGAGPTGVEMAGQIAELADQTLRGSFRHIDPTEARVILLDAAPAVLPPMGEKLGKKARARLEKMGVEVQLGAMVTDVDRNGITVKDSDGTIRRIESACKVWSAGVSASPLGKDLAEQSGVELDRAGRVKVQPDLTLPGHPNVFVVGDMAAVEGVPGVAQGAIQGGRYAAKIIKREVSGTSPKIRTPFEYFDKGSMATVSRFSAVAKVGPVEFAGFFAWLCWLVLHLVYLVGFKTKIVTLLSWGVTFLSTKRGQLTITEQQAYARTRIEELEEIAAAVQDTEKAASGLSGQPPRSPSSGSSDYKDHDGDYKDHDIDYKDDDDK

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 \(w=f(c)\)

3OAN , Knot

130

0.78

124

MSLTNGEENEEVLFCEKAKLLIFDSDTKGYTSRGVGELKLLRKKDDKGKVRVLMRSEGMGHVLLNTSVVKSFKYQPIDADNENLIKWPIITDGKLETFIIKVKQKADGRRLVGAVADAQQAMEGHHHHHH

1SPF , Knot

0.57

LRIPCCPVNLKRLLVVVVVVVLVVVVIVGALLMGL

9MQY , Knot

200

494

0.83

233

462

MSHPGATASDRHKVVIIGSGFGGLTAAKTLKRADVDVKLIARTTHHLFQPLLYQVATGIISEGEIAPATRVILRKQKNAQVLLGDVTHIDLENKTVDSVLLGHTYSTPYDSLIIAAGAGQSYFGNDHFAEFAPGMKSIDDALELRGRILGAFEQAERSSDPVRRAKLLTFTVVGAGPTGVEMAGQIAELADQTLRGSFRHIDPTEARVILLDAAPAVLPPMGEKLGKKARARLEKMGVEVQLGAMVTDVDRNGITVKDSDGTIRRIESACKVWSAGVSASPLGKDLAEQSGVELDRAGRVKVQPDLTLPGHPNVFVVGDMAAVEGVPGVAQGAIQGGRYAAKIIKREVSGTSPKIRTPFEYFDKGSMATVSRFSAVAKVGPVEFAGFFAWLCWLVLHLVYLVGFKTKIVTLLSWGVTFLSTKRGQLTITEQQAYARTRIEELEEIAAAVQDTEKAASGLSGQPPRSPSSGSSDYKDHDGDYKDHDIDYKDDDDK

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 \(c\).

\(|P_{f(3OAN_1)}(2) \setminus P_{f(1SPF_1)}(2)|=86\), \(|P_{f(1SPF_1)}(2) \setminus P_{f(3OAN_1)}(2)|=12\). 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 1:1010010000011100010111100000100001110101100000010101110001110111000110010001101000011011110010100111010001010011111101001101000000

Pair	\(Z_2\)	Length of longest common subsequence
3OAN_1,1SPF_1	98	3
3OAN_1,9MQY_1	177	4
1SPF_1,9MQY_1	223	3

Newick tree

 
[
	9MQY_1:11.73,
	[
		3OAN_1:49,1SPF_1:49
	]:63.73
]

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 \((0.85)(0.8)(\frac{165 }{\log_{20} 165}-\frac{35}{\log_{20}35})=45.7\)

Status	Protein1	Protein2	d	d₁/2
Query variables	3OAN_1	1SPF_1	57	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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