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Parikh vectors

1KVC_1	6LVK_1	3QMD_1	Letter	Amino acid
4	12	6	S	Serine
6	5	0	W	Tryptophan
9	22	0	V	Valine
14	19	3	A	Alanine
8	13	2	N	Asparagine
12	26	5	E	Glutamic acid
14	20	5	G	Glycine
8	8	5	Q	Glutamine
5	6	7	H	Histidine
7	13	2	I	Isoleucine
5	13	3	Y	Tyrosine
5	17	2	P	Proline
10	16	11	R	Arginine
3	6	8	C	Cysteine
4	17	3	M	Methionine
2	9	4	F	Phenylalanine
6	21	3	D	Aspartic acid
12	31	3	L	Leucine
11	23	6	K	Lycine
10	16	1	T	Threonine

1KVC_1|Chain A|RIBONUCLEASE H|Escherichia coli (562)
>6LVK_1|Chains A, B|Fibroblast growth factor receptor 2|Homo sapiens (9606)
>3QMD_1|Chain A|CpG-binding protein|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 \(w=f(c)\)
1KVC , Knot	80	155	0.86	40	127	152	MLKQVEIFTDGSCLGNPGPGGYGAILRYRGREKTFSAGYTRTTNNRMELMAAIVALEALKEHCEVILSTDSQYVRQGITQWIHNWKKRGWKTADKKPVKNVDLWQRLDAALGQHQIKWEWVKGHAGHPENERCNELARAAAMNPTLEDTGYQVEV
6LVK , Knot	136	313	0.83	40	196	295	GSHMLAGVSEYELPEDPKWEFPRDKLTLGKPLGEGCFGQVVMAEAVGIDKDKPKEAVTVAVKMLKDDATEKDLSDLVSEMEMMKMIGKHKNIINLLGACTQDGPLYVIVEYASKGNLREYLRARRPPGMEYSYDINRVPEEQMTFKDLVSCTYQLARGMEYLASQKCIHRDLAARNVLVTENNVMKIADFGLARDINNIDYYKKTTNGRLPVKWMAPEALFDRVYTHQSDVWSFGVLMWEIFTLGGSPYPGIPVEELFKLLKEGHRMDKPANCTNELYMMMRDCWHAVPSQRPTFKQLVEDLDRILTLTTNEE
3QMD , Knot	42	79	0.77	36	65	74	MHHHHHHSSRENLYFQGQIKRSARMCGECEACRRTEDCGHCDFCRDMKKFGGPNKIRQKCRLRQCQLRARESYKYFPSS

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 \(w=f(c)\)

1KVC , Knot

155

0.86

127

152

MLKQVEIFTDGSCLGNPGPGGYGAILRYRGREKTFSAGYTRTTNNRMELMAAIVALEALKEHCEVILSTDSQYVRQGITQWIHNWKKRGWKTADKKPVKNVDLWQRLDAALGQHQIKWEWVKGHAGHPENERCNELARAAAMNPTLEDTGYQVEV

6LVK , Knot

136

313

0.83

196

295

GSHMLAGVSEYELPEDPKWEFPRDKLTLGKPLGEGCFGQVVMAEAVGIDKDKPKEAVTVAVKMLKDDATEKDLSDLVSEMEMMKMIGKHKNIINLLGACTQDGPLYVIVEYASKGNLREYLRARRPPGMEYSYDINRVPEEQMTFKDLVSCTYQLARGMEYLASQKCIHRDLAARNVLVTENNVMKIADFGLARDINNIDYYKKTTNGRLPVKWMAPEALFDRVYTHQSDVWSFGVLMWEIFTLGGSPYPGIPVEELFKLLKEGHRMDKPANCTNELYMMMRDCWHAVPSQRPTFKQLVEDLDRILTLTTNEE

3QMD , Knot

0.77

MHHHHHHSSRENLYFQGQIKRSARMCGECEACRRTEDCGHCDFCRDMKKFGGPNKIRQKCRLRQCQLRARESYKYFPSS

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 \(c\).

\(|P_{f(1KVC_1)}(2) \setminus P_{f(6LVK_1)}(2)|=48\), \(|P_{f(6LVK_1)}(2) \setminus P_{f(1KVC_1)}(2)|=117\). 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 1:11001011001001101111101111000100001011000000001011111111011000001110000001001100110010001100100011001011001011110001010110101101000000011011110101000100101

Pair	\(Z_2\)	Length of longest common subsequence
1KVC_1,6LVK_1	165	4
1KVC_1,3QMD_1	146	3
6LVK_1,3QMD_1	195	4

Newick tree

 
[
	6LVK_1:95.38,
	[
		1KVC_1:73,3QMD_1:73
	]:22.38
]

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 \((0.85)(0.8)(\frac{468 }{\log_{20} 468}-\frac{155}{\log_{20}155})=92.4\)

Status	Protein1	Protein2	d	d₁/2
Query variables	1KVC_1	6LVK_1	114	85.5

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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