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2014-01-26T17:27:20Z

79.179.197.86: /* Random search */

{{Orphan|date=July 2013}}

In [[enzymology]], a '''ceramide phosphoethanolamine synthase''' ([[Enzyme Commission number|EC]] 2.7.8.-) is an [[enzyme]] that [[Catalysis|catalyzes]] the [[chemical reaction]]

:a [[ceramide]] + a phosphoethanolamine head group donor <math>\rightleftharpoons</math> a ceramide-phosphoethanolamine + side product

[[Ceramide phosphoethanolamine]] (CPE) is a [[sphingolipid]] consisted of a [[ceramide]] and a [[phosphatidylethanolamine]]. Thus, this class of enzymes uses ceramide and a donor molecule for phosphoethanolamine as [[substrate (biochemistry)|substrate]]s to produce a ceramide phosphoethanolamine and a side product. The head group donor for phosphoethanolamine can be either [[phosphatidylethanolamine]] or [[CDP-ethanolamine]], thus the side product is either a [[1,2-diacylglycerol]] or a [[Cytidine monophosphate|CMP]], respectively.

This enzyme belongs to the family of [[transferase]]s, specifically those transferring non-standard substituted [[phosphate]] groups.

==Mammalian Ceramide Phosphoethanolamine Synthases==
In [[mammals|mammalian cells]], two [[Ceramide phosphoethanolamine|CPE]] synthase activities have been described, one resides in the [[endoplasmic reticulum]], and the other one is associated with the [[plasma membrane]].<ref>Malgat, M., Maurice, A., and Baraud, J. (1986) Sphingomyelin and ceramide-phosphoethanolamine synthesis by microsomes and plasma membranes from rat liver and brain. J. Lipid Res. 27, 251–260</ref><ref>Malgat, M., Maurice, A., and Baraud, J. (1987) Sidedness of ceramidephosphoethanolamine synthesis on rat liver and brain microsomal membranes. J. Lipid Res. 28, 138–143</ref><ref>Maurice, A., Malgat, M., and Baraud, J. (1989) Sidedness of ceramidephosphoethanolamine synthesis on rat liver plasma membrane. Biochimie 71, 373–378</ref><ref name = "Ternes2009">Ternes, P., Brouwers, J. F., van den Dikkenberg, J., and Holthuis, J. C. (2009) Sphingomyelin synthase SMS2 displays dual activity as ceramide phosphoethanolamine synthase. J. Lipid Res. 50, 2270–2277</ref><ref name = "Vacaru2009">Vacaru, A. M., Tafesse, F. G., Ternes, P., Kondylis, V., Hermansson, M., Browers, J. F. H. M., Somerharju, P., Rabouille, C., and Holthuis, J. C.(2009) Sphingomyelin synthase-related protein SMSr controls ceramide homeostasis in the ER. J. Cell. Biol. 185, 1013–1027</ref> The endoplasmic reticulum-resident CPE synthase, SMSr, is identified as a monofunctional CPE synthase produces trace amounts of CPE.<ref name="Ternes2009" /><ref name="Vacaru2009" /> On the other hand, mammalian CPE synthase that is on the [[plasma membrane]], [[Sphingomyelin synthase|SMS2]], is a bifunctional enzyme that produces both CPE and [[sphingomyelin]], thus also functioning as a [[sphingomyelin synthase]].<ref name="Ternes2009" /> Both mammalian CPE synthases, [[Sphingomyelin synthase|SMS2]] and SMSr, use [[phosphatidylethanolamine]] (PE) as head group donor and catalyzes the reaction

:a [[ceramide]] + a [[phosphatidylethanolamine]] <math>\rightleftharpoons</math> a ceramide-phosphoethanolamine + [[1,2-diacylglycerol]]

==Invertebrate Ceramide Phosphoethanolamine Synthases==
SMSr protein is found in all organisms throughout the animal kingdom as a CPE synthase, yet it produces trace amounts of CPE.<ref name="Vacaru2009" /><ref name="Vacaru2013" /> [[Drosophila]] and a group of [[invertebrates]] lack [[Sphingomyelin synthase|SMS2]] [[homology (biology)|homologues]].<ref name="Vacaru2009" /><ref name = "Vacaru2013">Vacaru AM, van den Dikkenberg J, Ternes P, Holthuis JC. Ceramide phosphoethanolamine biosynthesis in Drosophila is mediated by a unique ethanolamine phosphotransferase in the Golgi lumen. J Biol Chem. 2013 Apr 19;288(16):11520-30. doi: 10.1074/jbc.M113.460972. Epub 2013 Feb 28. PubMed PMID 23449981; PubMed Central PMCID: PMC3630839</ref> This group of invertebrates synthesizes CPE using a particular enzyme called CPES.<ref name="Vacaru2013" /> CPES uses [[CDP-ethanolamine]] rather than phosphatidylethanolamine as head group donor, thus catalyzes the reaction <ref name="Vacaru2013" />

:a [[ceramide]] + a [[CDP-ethanolamine]] <math>\rightleftharpoons</math> a ceramide-phosphoethanolamine + [[Cytidine monophosphate|CMP]]

CPES uses a different reaction mechanism than the one [[sphingomyelin synthase]] uses, but very similar to that of enzymes involved in [[Biosynthesis|synthesis]] of phosphatidyl ethanolamine ([[Enzyme Commission number|EC]] 2.7.8.1) via the [[Kennedy pathway]].<ref>KENNEDY EP, WEISS SB. The function of cytidine coenzymes in the biosynthesis of phospholipides. J Biol Chem. 1956 Sep;222(1):193-214. PubMed PMID 13366993.</ref>

==References==
{{reflist}}

[[Category:Chemical reactions]]
[[Category:EC 2.7.8]]
[[Category:Lipids]]

Solovay–Strassen primality test

2013-07-01T20:03:34Z

79.179.4.230: /* Concepts */

[[File:Heilbronn square n=6.svg|thumb|300px|Solution to the Heilbronn triangle problem for six points in the unit square. These points form triangles of four different shapes, with minimum area 1/8, as large as possible for six points in the square. This solution is an [[affine transformation]] of a [[regular hexagon]] but larger numbers of points have solutions that include interior points of the square.]]
In [[discrete geometry]] and [[discrepancy theory]], the '''Heilbronn triangle problem''' is a problem of placing points within a region in the plane, in order the avoid [[triangle]]s of small [[area]]. It is named after [[Hans Heilbronn]], who [[conjecture]]d prior to 1950 that this smallest triangle area is necessarily at most [[Proportionality (mathematics)#Inverse proportionality|inversely proportional]] to the [[Square (algebra)|square]] of the number of points. Heilbronn's conjecture was proven false, but the [[Asymptotic analysis|asymptotic growth]] rate of the minimum triangle area remains unknown.

==Definition==
The problem may be defined in terms of any [[compact space|compact]] set ''D'' in the plane with nonzero area such as the [[unit square]] or the [[unit disk]]. If ''S'' is a set of ''n'' points of ''D'', then every three points of ''S'' determine a triangle (possibly a degenerate one, with zero area). Let Δ(''S'') denote the minimum of the areas of these triangles, and let Δ(''n'') (for an integer ''n'' ≥ 3) denote the [[supremum]] of the values of Δ(''S'').

The question posed by Heilbronn was to give an expression, or matching asymptotic [[upper and lower bounds]], for Δ(''n''). That is, the goal is to find a [[function (mathematics)|function]] ''f'', described by a [[closed-form expression]], and constants ''c''1 and ''c''2, such that for all ''n'',
:<math>c_1 f(n) \le \Delta(n) \le c_2 f(n)</math>.
In terms of [[big O notation]], the left inequality may be written as Δ(''n'') = Ω(''f''(''n'')), the right inequality may be written as Δ(''n'') = ''O''(''f''(''n'')), and both of them together may be written as Δ(''n'') = Θ(''f''(''n'')). The shape and area of ''D'' may affect the exact values of Δ(''n''), but only by a constant factor, so they are unimportant for its asymptotic growth rate.

==Heilbronn's conjecture and lower bound constructions==
Heilbronn conjectured that
:<math>\Delta(n)=O\left(\frac{1}{n^2}\right).</math>
As [[Paul Erdős]] showed, no smaller bound is possible: when ''n'' is a [[prime number]], the set of ''n'' points (''i'', ''i''2 mod ''n'') on an ''n'' × ''n'' [[integer lattice|integer grid]] have [[No-three-in-line problem|no three collinear points]], and therefore by [[Pick's formula]] each of the triangles they form has area at least 1/2. When this set of grid points is scaled to a unit square, they form a set of points whose smallest triangle area is at least proportional to 1/''n''2, matching Heilbronn's conjectured upper bound.<ref name="r51">{{citation
| last = Roth
| first = K. F. | authorlink = Klaus Roth
| title = On a problem of Heilbronn
| journal = [[Journal of the London Mathematical Society]]
| volume = 26
| year = 1951
| issue = 3
| pages = 198–204
| doi = 10.1112/jlms/s1-26.3.198}}.</ref>
If ''n'' is not prime, then a similar construction using the next prime number larger than ''n'' achieves the same asymptotic lower bound.

{{harvtxt|Komlós|Pintz|Szemerédi|1982}} eventually disproved Heilbronn's conjecture, by finding sets of points whose smallest triangle area grows asymptotically as
:<math>\Delta(n)=\Omega\left(\frac{\log n}{n^2}\right).</math><ref>{{citation|last1=Komlós|first1=J.|author1-link=János Komlós (mathematician)|author2-link=János Pintz|last2=Pintz|first2=J.|last3=Szemerédi|first3=E.|author3-link=Endre Szemerédi|title=A lower bound for Heilbronn's problem|journal=[[Journal of the London Mathematical Society]]|volume=25|issue=1|pages=13–24|year=1982|mr=0645860|doi=10.1112/jlms/s2-25.1.13}}.</ref>

==Upper bounds==
Trivially, either by [[Point set triangulation|triangulating]] the [[convex hull]] of the given point set ''S'' or by choosing consecutive triples of points in the sorted order of their ''x''-coordinates, it is possible to show that every point set contains a small triangle, whose area is at most inversely proportional to ''n''. {{harvtxt|Roth|1951}} was the first to prove a nontrivial upper bound on Δ(''n''), of the form<ref name="r51"/>
:<math>\Delta(n)=O\left(\frac{1}{n\sqrt{\log\log n}}\right).</math>
The best bound known to date is of the form
:<math>\Delta(n)\leq\frac{\exp{\left(c\sqrt{\log n}\right)}}{n^{8/7}},</math>
for some constant ''c'', proven by {{harvtxt|Komlós|Pintz|Szemerédi|1981}}.<ref>{{citation|last1=Komlós|first1=J.|author1-link=János Komlós (mathematician)|author2-link=János Pintz|last2=Pintz|first2=J.|last3=Szemerédi|first3=E.|author3-link=Endre Szemerédi|title=On Heilbronn's triangle problem|journal=[[Journal of the London Mathematical Society]]|volume=24|issue=3|pages=385–396|year=1981|mr=0635870|doi=10.1112/jlms/s2-24.3.385}}.</ref>

==Specific shapes and numbers==
{{harvtxt|Goldberg|1972}} has investigated the optimal arrangements of ''n'' points in a square, for ''n'' up to 16.<ref>{{citation
| last = Goldberg | first = Michael
| journal = [[Mathematics Magazine]]
| jstor = 2687869
| mr = 0296816
| pages = 135–144
| title = Maximizing the smallest triangle made by ''n'' points in a square
| volume = 45
| year = 1972}}.</ref> Goldberg's constructions for up to six points lie on the boundary of the square, and are placed to form an [[affine transformation]] of the vertices of a [[regular polygon]]. For larger values of ''n'', {{harvtxt|Comellas|Yebra|2002}} improved Goldberg's bounds, and for these values the solutions include points interior to the square.<ref>{{citation
| last1 = Comellas | first1 = Francesc
| last2 = Yebra | first2 = J. Luis A.
| issue = 1
| journal = [[Electronic Journal of Combinatorics]]
| mr = 1887087
| page = R6
| title = New lower bounds for Heilbronn numbers
| url = http://www.combinatorics.org/Volume_9/Abstracts/v9i1r6.html
| volume = 9
| year = 2002}}.</ref> These constructions have been proven optimal for up to seven points.<ref>{{citation
| last1 = Zeng | first1 = Zhenbing
| last2 = Chen | first2 = Liangyu
| contribution = On the Heilbronn optimal configuration of seven points in the square
| doi = 10.1007/978-3-642-21046-4_11
| location = Heidelberg
| mr = 2805061
| pages = 196–224
| publisher = Springer
| series = Lecture Notes in Comput. Sci.
| title = Automated deduction in geometry
| volume = 6301
| year = 2011}}.</ref>

==Variations==
There have been many variations of this problem
including the case of a uniformly random set of points, for which an argument based on [[Kolmogorov complexity]] shows that the [[expected value]] of the minimum area is inversely proportional to the cube of the number of points.<ref>{{citation
| last1 = Jiang | first1 = Tao
| last2 = Li | first2 = Ming
| last3 = Vitányi | first3 = Paul | author3-link = Paul Vitanyi
| doi = 10.1002/rsa.10024
| issue = 2
| journal = Random Structures & Algorithms
| mr = 1884433
| pages = 206–219
| title = The average-case area of Heilbronn-type triangles
| volume = 20
| year = 2002}}.</ref> Variations involving the volume of higher dimensional [[simplex|simplices]] have also been studied.<ref>{{citation
| last = Lefmann | first = Hanno
| doi = 10.1007/s00454-007-9041-y
| issue = 3
| journal = [[Discrete and Computational Geometry]]
| mr = 2443292
| pages = 401–413
| title = Distributions of points in ''d'' dimensions and large ''k''-point simplices
| volume = 40
| year = 2008}}.</ref>

==References==
{{reflist}}

==External links==
*{{mathworld|id=HeilbronnTriangleProblem|title=Heilbronn Triangle Problem}}
*[http://www2.stetson.edu/~efriedma/packing.html Erich's Packing Center], by Erich Friedman, including the best known solutions to the Heilbronn problem for small values of ''n'' for squares, circles, equilateral triangles, and convex regions of variable shape but fixed area

[[Category:Discrete geometry]]
[[Category:Mathematical problems]]
[[Category:Triangle geometry]]
[[Category:Area]]

1929–30 Beşiktaş JK season

2011-08-17T15:24:36Z

79.179.194.199:

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