formulasearchengine - User contributions [en]

Exponential growth

2014-02-18T05:21:56Z

71.236.213.188:

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MurmurHash

2014-01-24T08:24:03Z

71.236.219.237:

{{expert|Probability|date=November 2009}}

In the mathematics of [[free probability]] theory, the '''free Poisson distribution''' is a counterpart of the [[Poisson distribution]] in conventional probability theory.

==Definition==
The free Poisson distribution<ref>Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992</ref> with jump size <math>\alpha</math> and rate <math>\lambda</math> arises in [[free probability]] theory as the limit of repeated [[free convolution]]

: <math>
\left( \left(1-\frac{\lambda}{N}\right)\delta_0 + \frac{\lambda}{N}\delta_\alpha\right)^{\boxplus N}</math>

as ''N'' → ∞.

In other words, let <math>X_N</math> be random variables so that <math>X_N</math> has value <math>\alpha</math> with probability <math>\frac{\lambda}{N}</math> and value 0 with the remaining probability. Assume also that the family <math>X_1,X_2,\ldots</math> are [[free independence|freely independent]]. Then the limit as <math>N\to\infty</math> of the law of <math>X_1+\cdots +X_N</math>
is given by the Free Poisson law with parameters <math>\lambda,\alpha</math>.

This definition is analogous to one of the ways in which the classical [[Poisson distribution]] is obtained from a (classical) Poisson process.

The measure associated to the free Poisson law is given by

:<math>\mu=\begin{cases} (1-\lambda) \delta_0 + \lambda \nu,& \text{if } 0\leq \lambda \leq 1 \\
\nu, & \text{if }\lambda >1,
\end{cases}
</math>

where

: <math>\nu = \frac{1}{2\pi\alpha t}\sqrt{4\lambda \alpha^2 - ( t - \alpha (1+\lambda))^2} \, dt</math>

and has support <math>\alpha (1-\sqrt{\lambda})^2,\alpha (1+\sqrt{\lambda})^2]</math>.

This law also arises in [[random matrix]] theory as the [[Marchenko–Pastur law]]. Its [[free cumulants]] are all equal to <math>\lambda</math>.

==Some transforms of this law==
We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book ''Lectures on the Combinatorics of Free Probability'' by A. Nica and R. Speicher<ref>Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher, pp. 203–204, Cambridge Univ. Press 2006</ref>

The [[R-transform]] of the free Poisson law is given by

: <math>R(z)=\frac{\lambda \alpha}{1-\alpha z}. </math>

The [[Stieltjes transformation]] (also known as the Cauchy transform) is given by

: <math>
G(z) = \frac{ z + \alpha - \lambda \alpha - \sqrt{ (z-\alpha (1+\lambda))^2 - 4 \lambda \alpha^2}}{2\alpha z}
</math>

The [[S-transform]] is given by

: <math>
S(z) = \frac{1}{z+\lambda}
</math>

in the case that <math>\alpha=1</math>.

==References==
{{Reflist}}

[[Category:Probability distributions]]
[[Category:Functional analysis]]
[[Category:Free probability theory]]

Sherwood number

2013-11-23T21:50:45Z

71.236.108.121:

The '''Damköhler numbers''' ('''Da''') are [[dimensionless number]]s used in [[chemical engineering]] to relate the [[chemical reaction]] timescale ([[reaction rate]]) to the [[transport phenomena]] rate occurring in a system. It is named after German chemist [[Gerhard Damköhler]].

In its most commonly used form, the Damköhler number relates the reaction timescale to the [[convection]] times scale, [[flow rate]], through the [[reactor]] for continuous or [[Semibatch reactor|semibatch]] chemical processes:
: <math>\mathrm{Da} = \frac{ \text{reaction rate} }{ \text{convective mass transport rate} }</math>


In reacting systems that include interphase mass transport, the '''second Damköhler number''' ('''DaII''') is defined as the ratio of the chemical reaction rate to the mass transfer rate
: <math>\mathrm{Da}_{\mathrm{II}} = \frac{ \text{reaction rate} }{ \text{diffusive mass transfer rate} }</math>

Since the reaction timescale is determined by the reaction rate, the exact formula for the Damköhler number varies according to the raw law equation. For a general chemical reaction A → B of nth [[Order of reaction|order]], the Damköhler number for a convective flow system is defined as:

: <math>\mathrm{Da} = k C_0^{\ n-1}\tau</math>
where:
* ''k'' = [[chemical kinetics|kinetics]] [[reaction rate constant]]
* ''C''0 = initial concentration
* ''n'' = [[reaction order]]
* <math>\tau</math> = mean [[residence time]] or '''space time'''

On the other hand, the second Damköhler number is defined as:
: <math>\mathrm{Da}_{\mathrm{II}} = \frac{k C_0^{n-1}}{k_g a}</math>
where
* ''kg'' is the global mass transport coefficient
* ''a'' is the interfacial area

The value of Da provides a quick estimate of the degree of [[Conversion (chemistry)|conversion]] that can be achieved. As a [[rule of thumb]], when Da is less than 0.1 a conversion of less than 10% is achieved,and when Da is greater than 10 a conversion of more than 90% is expected.<ref name="Fogler">{{cite book |last=Fogler |first=Scott |title=Elements of Chemical Reaction Engineering |location=Upper Saddle River, NJ |publisher=Pearson Education |year=2006 |edition=4th |isbn=0-13-047394-4 }}</ref>

==References==
{{reflist}}

{{NonDimFluMech}}

{{DEFAULTSORT:Damkohler numbers}}
[[Category:Catalysis]]
[[Category:Chemical engineering]]
[[Category:Dimensionless numbers of chemistry]]
[[Category:Dimensionless numbers of fluid mechanics]]
[[Category:Fluid dynamics]]

Frictionless plane

2013-10-24T22:35:59Z

71.236.145.234:

In [[computer science]], in particular in the field of [[formal language]] theory,
the term '''abstract family of languages''' refers to an abstract mathematical notion generalizing characteristics common to the [[regular language]]s, the [[context-free language]]s and the [[recursively enumerable language]]s, and other families of formal languages studied in the scientific literature.

==Formal definitions==

A ''[[formal language]]'' is a set <math>L</math> for which there exists a finite set of abstract symbols <math>\Sigma</math> such that <math>L \subseteq\Sigma^*</math>, where * is the [[Kleene star]] operation.

A ''family of languages'' is an ordered pair <math>(\Sigma,\Lambda)</math>, where
# <math>\Sigma</math> is an infinite set of symbols;
# <math>\Lambda</math> is a set of formal languages;
# For each <math>L</math> in <math>\Lambda</math> there exists a finite subset <math>\Sigma_1</math> ⊂ <math>\Sigma</math> such that <math>L</math> ⊆ <math>\Sigma_1^*</math>; and
# <math>L</math> ≠ Ø for some <math>L</math> in <math>\Lambda</math>.

A ''trio'' is a family of languages [[Closure (mathematics)|closed]] under [[e-free homomorphism]], inverse [[homomorphism]], and intersection with [[regular language]].

A ''full trio,'' also called a ''[[cone (formal languages)|cone]],'' is a trio closed under arbitrary homomorphism.

A ''(full) semi-AFL'' is a (full) trio closed under [[Union (set theory)|union]].

A ''(full) AFL'' is a ''(full) semi-AFL'' closed under [[concatenation]] and the [[Kleene plus]].

==Some families of languages==
The following are some simple results from the study of abstract families of languages.<ref name="Seymour">{{harvtxt|Ginsburg|1975}}</ref><ref name="SpringerAFL">{{SpringerEOM| title=Abstract family of languages | id=Abstract_family_of_languages | oldid=18934 | first=A. | last=Mateescu | first2=A. | last2=Salomaa }}</ref>

Within the [[Chomsky hierarchy]], the [[regular language]]s, the [[context-free language]]s, and the [[recursively enumerable language]]s are all full AFLs. However, the [[Context-sensitive language|context sensitive languages]] and the [[recursive language]]s are AFLs, but not full AFLs because they are not closed under arbitrary homomorphisms.

The family of regular languages are contained within any cone (full trio). Other categories of abstract families are identifiable by closure under other operations such as shuffle, reversal, or substitution.<ref name="SpringerOps">{{SpringerEOM| title=AFL operations | id=AFL_operations | oldid=13097 | first=Gh. | last=PÄƒun }}</ref>

==Origins==

[[Seymour Ginsburg]] of the [[University of Southern California]] and [[Sheila Greibach]] of [[Harvard University]] presented the first AFL theory paper at the IEEE Eighth Annual [[Symposium on Switching and Automata Theory]] in 1967.<ref>{{harvtxt|Ginsburg|Greibach|1967}}</ref>

==Notes==
<references/>

==References==
* {{cite conference
| first1 = Seymour
| last1 = Ginsburg
| first2 = Sheila
| last2= Greibach
| title=Abstract Families of Languages
| booktitle = Conference Record of 1967 Eighth Annual Symposium on Switching and Automata Theory, 18-20 October 1967, Austin, Texas, USA
| year = 1967
| pages= 128-139
|publisher = IEEE
}}
*[[Seymour Ginsburg]], ''Algebraic and automata theoretic properties of formal languages'', North-Holland, 1975, ISBN 0-7204-2506-9.
* John E. Hopcroft and Jeffrey D. Ullman, ''[[Introduction to Automata Theory, Languages, and Computation]]'', Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 0-201-02988-X. Chapter 11: Closure properties of families of languages.
* {{cite book |last1=Mateescu | first1=Alexandru |last2=Salomaa|first2=Arto |editor1-first=Grzegorz| editor1-last=Rozenberg|editor2-first=Arto| editor2-last=Salomaa |title=Handbook of Formal Languages. Volume I: Word, language, grammar |publisher=Springer-Verlag |year=1997 |pages=175–252 |chapter=Chapter 4: Aspects of Classical Language Theory |isbn=3-540-61486-9}}

[[Category:Formal languages]]
[[Category:Applied mathematics]]

Time evolution

2013-09-14T04:56:56Z

71.236.136.184:

{{about|the scientific phenomenon of sedimentation|sedimentation in the treatment of water and wastewater|Sedimentation (water treatment)}}

'''Sedimentation''' is the tendency for [[particle (ecology)|particle]]s in [[Suspension (chemistry)|suspension]] to settle out of the fluid in which they are entrained, and come to rest against a barrier. This is due to their motion through the fluid in response to the forces acting on them: these forces can be due to [[gravitation|gravity]], [[centrifugal acceleration]] or [[electromagnetism]]. In geology sedimentation is often used as the polar opposite of erosion, i.e., the terminal end of [[sediment transport]]. In that sense it includes the termination of transport by [[Saltation (geology)|saltation]] or true [[bedload|bedload transport]]. [[Settling]] is the falling of suspended particles through the liquid, whereas sedimentation is the termination of the settling process.

Sedimentation may pertain to objects of various sizes, ranging from large rocks in flowing water to [[suspension (chemistry)|suspensions]] of dust and pollen [[particle (ecology)|particles]] to [[cell (biology)|cell]]ular suspensions to [[solution]]s of single [[molecule]]s such as [[protein]]s and [[peptide]]s. Even small molecules suly a sufficiently strong force to produce significant sedimentation.

The term is typically used in geology, to describe the [[deposition (geology)|deposition]] of [[sediment]] which results in the formation of [[sedimentary rock]], and in various chemical and environmental fields to describe the motions of often-smaller particles and molecules. Process is also used in biotech industry to separate out cells from the culture media.

==Experiments==

In a sedimentation experiment called tripothsissp?, the applied force accelerates the particles to a [[terminal velocity]] <math>v_{term}</math> at which the applied force is exactly canceled by an opposing drag force. For small enough particles (low [[Reynolds number]]), the drag force varies linearly with the [[terminal velocity]], i.e., <math>F_{drag} = f v_{term}</math> ([[Stokes flow]]) where ''f'' depends only on the properties of the particle and the surrounding fluid. Similarly, the applied force generally varies linearly with some coupling constant (denoted here as ''q'') that depends only on the properties of the particle, <math>F_\mathrm{app} = q E_\mathrm{app}</math>. Hence, it is generally possible to define a [[sedimentation coefficient]] <math>s \ \stackrel{\mathrm{def}}{=}\ q/f</math> that depends only on the properties of the particle and the surrounding fluid. Thus, measuring ''s'' can reveal underlying properties of the particle.

In many cases, the motion of the particles is blocked by a hard boundary; the resulting accumulation of particles at the boundary is called a [[sediment]]. The concentration of particles at the boundary is opposed by the [[diffusion]] of the particles.

The sedimentation of a single particle under gravity is described by the [[Mason–Weaver equation]], which has a simple exact solution. The sedimentation coefficient ''s'' in this case equals <math>m_{b}/f</math>, where <math>m_{b}</math> is the [[buoyant mass]].

The sedimentation of a single particle under [[centrifugal force]] is described by the [[Lamm equation]], which likewise has an exact solution. The sedimentation coefficient ''s'' also equals <math>m_{b}/f</math>, where <math>m_{b}</math> is the buoyant mass. However, the Lamm equation differs from the Mason–Weaver equation because the centrifugal force depends on radius from the origin of rotation, whereas in the Mason–Weaver equation gravity is constant. The Lamm equation also has extra terms, since it pertains to [[Circular sector|sector]]-shaped cells, whereas the Mason–Weaver equation is one dimensional.

Classification of sedimentation:{{Citation needed|date=April 2011}}

*Type 1 sedimentation is characterized by particles that settle discretely at a constant settling velocity,or by a the deposition of Iron-Rich minerals to streamlines down to the point source. They settle as individual particles and do not flocculate or stick to other during settling. Example: sand and grit material
*Type 2 sedimentation is characterized by particles that flocculate during sedimentation and because of this their size is constantly changing and therefore their settling velocity is changing. Example: alum or iron coagulation
*Type 3 sedimentation is also known as zone sedimentation. In this process the particles are at a high concentration (greater than 1000 mg/L) such that the particles tend to settle as a mass and a distinct clear zone and sludge zone are present. Zone settling occurs in lime-softening, sedimentation, active sludge sedimentation and sludge thickeners.

==Geology==
[[File:Siltation or Sedimentation.jpg|thumb|Siltation]]
In [[geology]], sedimentation is the deposition of particles carried by a fluid flow. For [[suspension (chemistry)|suspended]] load, this can be expressed mathematically by the [[Exner equation]], and results in the formation of depositional [[landform]]s and the rocks that constitute [[depositional record|sedimentary record]]. An undesired increased transport and sedimentation of suspended material is called [[siltation]], and it is a major source of pollution in waterways in some parts of the world.<ref>
{{cite web
|url=http://blackwarriorriver.org/siltation-sedimentation.html
|title=Siltation & Sedimentation
|publisher=blackwarriorriver.org
|accessdate=2009-11-16
|last=
|first=
}}
</ref><ref>
{{cite web
|url=http://malaysiadigest.blogspot.com/2009/02/siltation-killed-fish-at-batang-rajang.html
|title=Siltation killed fish at Batang Rajang - Digest on Malaysian News
|publisher=malaysiadigest.blogspot.com
|accessdate=2009-11-16
|last=
|first=
}}
</ref> [[Climate change]] also affects siltation rates.<ref>
{{cite web
|url=http://ijc.cgpublisher.com/product/pub.185/prod.38
|title=The International Journal of Climate Change: Impacts and Responses » Rate of Siltation in Wular Lake, (Jammu and Kashmir, India) with Special Emphasis on its Climate & Tectonics
|publisher=The International Journal of Climate Change: Impacts and Responses
|accessdate=2009-11-16
|last=U.D. Kulkarni
|first=et al
}}
</ref>

==Chemistry==
In chemistry, sedimentation has been used to measure the size of large molecules ([[macromolecule]]), where the force of gravity is augmented with [[centrifugal force]] in an [[ultracentrifuge]].

==See also==
*[[Coagulation (disambiguation)]]
*[[Flocculation]]

==Notes==
{{reflist}}

{{Geologic Principles}}

[[Category:Earth sciences]]
[[Category:Laboratory techniques]]
[[Category:Separation processes]]

[[it:Sedimentazione]]

Template:Circle of fifths

2013-05-16T00:14:56Z

71.236.121.101: change link

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Sigma heat

2013-04-14T16:40:33Z

71.236.121.101: fix link

In [[probability]] theory, the '''chain rule''' permits the calculation of any member of the [[joint distribution]] of a set of [[random variables]] using only [[conditional probabilities]]. The rule is useful in the study of [[Bayesian network]]s, which describe a probability distribution in terms of conditional probabilities.

Consider an indexed set of sets <math>A_1, \ldots , A_n</math>. To find the value of this member of the joint distribution, we can apply the definition of conditional probability to obtain:
::<math>\mathrm P(A_n, \ldots , A_1) = \mathrm P(A_n | A_{n-1}, \ldots , A_1) \cdot\mathrm P( A_{n-1}, \ldots , A_1)</math>
Repeating this process with each final term creates the product:
::<math>\mathrm P(\cap_{k=1}^n A_k ) = \prod_{k=1}^n \mathrm P( A_k \mid \cap_{j=1}^{k-1} A_j )</math>
With four variables, the chain rule produces this product of conditional probabilities:
::<math> \mathrm P(A_4, A_3, A_2, A_1) = \mathrm P(A_4 \mid A_3, A_2, A_1)\cdot \mathrm P(A_3 \mid A_2, A_1)\cdot \mathrm P(A_2 \mid A_1)\cdot \mathrm P(A_1)</math>

This rule is illustrated in the following example. Urn 1 has 1 black ball and 2 white balls and Urn 2 has 1 black ball and 3 white balls. Suppose we pick an urn at random and then select a ball from that urn. Let event A be choosing the first urn: P(A) = P(~A) = 1/2. Let event B be the chance we choose a white ball. The chance of choosing a white ball, given that we've chosen the first urn, is P(B|A) = 2/3. Event A, B would be their intersection; choosing the first urn and a white ball from it. The probability can be found by the chain rule for probability:
::<math> \mathrm P(A, B)=\mathrm P(B \mid A) \mathrm P(A) = 2/3 \times 1/2 = 1/3</math>.

== References ==

* {{Russell Norvig 2003}}, p. 496.
* [https://www.ibm.com/developerworks/mydeveloperworks/blogs/nlp/entry/the_chain_rule_of_probability "The Chain Rule of Probability"], ''[[developerWorks]]'', Nov 3, 2012.

[[Category:Probability theory]]

Butyrate kinase

2013-03-31T01:06:36Z

71.236.121.101: fix link

{{enzyme
| Name = Choline Kinase
| EC_number = 2.7.1.32
| CAS_number = 9026-67-9
| IUBMB_EC_number = 2/7/1/32
| GO_code = 0004103
| image =
| width =
| caption =
}}
'''Choline Kinase''' (also known as '''CK''','''ChoK''' and '''Choline Phosphokinase''') is an [[enzyme]] which catalyzes the first reaction in the choline pathway for [[phosphatidylcholine]] (PC) biosynthesis.This reaction involves the transfer of a phosphate group from ATP to choline in order to form [[phosphocholine]].
:ATP + choline <math>\rightleftharpoons</math> ADP + O-phosphocholine
Thus, the two [[substrate (biochemistry)|substrates]] of this enzyme are [[adenosine triphosphate|ATP]] and [[choline]], whereas its two [[product (chemistry)|products]] are [[adenosine diphosphate|ADP]] and [[O-phosphocholine]].Choline Kinase requires magnesium ions (+2) as a [[cofactor (biochemistry)|cofactor]] for this reaction.<ref>{{cite journal |doi=10.1139/O09-160 |title=Choline kinase and its function |year=2010 |last1=Wu |first1=Gengshu |last2=Vance |first2=Dennis E. |journal=Biochemistry and Cell Biology |volume=88 |issue=4 |pages=559–564 |pmid=20651826}}</ref> This enzyme belongs to the family of [[transferase]]s, specifically those transferring phosphorus-containing groups ([[phosphotransferase]]s) with an alcohol group as [[Electron acceptor|acceptor]]. The first detailed investigation of the enzyme was conducted by McCamen in 1962, where it was shown that the brain is the richest source of the enzyme in mammalian tissue. A related enzyme, [[ethanolamine kinase]] tends to co-purify with choline kinase leading to a suggestion that the two activities are mediated by two distinct [[active site]]s on a single protein.<ref>{{cite journal |pmid=208357 |year=1978 |last1=Spanner |first1=S |last2=Ansell |first2=GB |title=Choline and ethanolamine kinase activity in the cytoplasm of nerve endings from rat forebrain |volume=101 |pages=237–45 |journal=Advances in experimental medicine and biology}}</ref> The systematic name of this enzyme class is '''ATP:choline phosphotransferase'''. These enzymes participate in glycine, serine and threonine metabolism and glycerophospholipid metabolism.
In mammalian cells, the enzyme exists as three [[isoform]]s,CKα-1,CKα-2 and CKβ. These isoforms are [[encoded]] by two separate [[genes]], [[CHKA]] and [[CHKB (gene)|CHKB]] and are only active in their homodimeric,heterodimeric and oligomeric forms.<ref>{{cite journal |doi=10.1016/j.plipres.2003.12.001 |title=Structure and function of choline kinase isoforms in mammalian cells |year=2004 |last1=Aoyama |first1=C |journal=Progress in Lipid Research |volume=43 |issue=3 |pages=266–281 |pmid=15003397 |last2=Liao |first2=H |last3=Ishidate |first3=K}}</ref>

== Structural studies ==
As of late 2007, 6 [[tertiary structure|structures]] have been solved for this class of enzymes, with [[Protein Data Bank|PDB]] accession codes {{PDB link|1NW1}}, {{PDB link|2CKO}}, {{PDB link|2CKP}}, {{PDB link|2CKQ}}, {{PDB link|2I7Q}}, and {{PDB link|2IG7}}.

CKα-2 originating from [[Caenorhabditis elegans|C. elegans]], is a dimeric enzyme with each monomer being composed of two domains.The active site is located between the two domains. (See figure below) Its overall structure is similar to members of the [[eukaryotic]] [[protein kinase]] family. Mammalian choline kinases exists in either dimeric or tetrameric forms in solution.<ref>{{cite journal |pmid=2152925 |year=1990 |last1=Porter |first1=TJ |last2=Kent |first2=C |title=Purification and characterization of choline/ethanolamine kinase from rat liver |volume=265 |issue=1 |pages=414–22 |journal=The Journal of Biological Chemistry}}</ref><ref>{{cite journal |pmid=1577786 |year=1992 |last1=Uchida |first1=T |last2=Yamashita |first2=S |title=Molecular cloning, characterization, and expression in Escherichia coli of a cDNA encoding mammalian choline kinase |volume=267 |issue=14 |pages=10156–62 |journal=The Journal of Biological Chemistry}}</ref> Structural studies carried out on CKα-2, have implied that the conserved residues in the CK family of enzymes could possible play a vital role in substrate binding as well as in the stabilization of catalytically important residues.<ref name="Peisach" />

An enlarged view of the residues involved in the dimer interface between the S-shaped loop of the yellow subunit and the loop following helix A and strand 4 of the cyan subunit. Only residues that are involved in direct salt bridges, hydrogen bonds, or van der Waals interactions are shown. Salt bridges and hydrogen bonds, dashed lines; labels of residues from the yellow subunit, red; labels of residues from the cyan subunit, blue.<ref name="Peisach">{{cite journal |doi=10.1016/S0969-2126(03)00094-7 |title=The Crystal Structure of Choline Kinase Reveals a Eukaryotic Protein Kinase Fold |year=2003 |last1=Peisach |first1=D |last2=Gee |first2=P |last3=Kent |first3=C |last4=Xu |first4=Z |journal=Structure |volume=11 |issue=6 |pages=703–713 |pmid=12791258}}</ref>

== Mechanism ==
Although not much is known about the mechanism by which choline kinase reacts, the recent advancement in the elucidation of the structure of the enzyme has provided scientists with much more insight than they had previously.Since the structure of CK is very similar to that of the eukaryotic protein kinase family,the location of ATP and choline binding pockets have been proposed. These are shown in the figures below.{{Citation needed|date=July 2011}}

=== Proposed ATP binding site ===
In this figure, we see a striking similarity between APH(3′)-IIIa, an [[aminoglycoside]] phosphotransferase and CK.{{Citation needed|date=July 2011}}

=== Proposed Choline binding site ===
[[File:Proposed Choline Binding Site.png|thumb|center|450px|Proposed Choline Binding Site-The loops are colored as follows: the ATP binding loop (residues 81–88), red; residue Asp301, magenta; choline pocket loop 1 (residues 322–343), green;choline pocket loop 2 (residues 396–405), blue. On the right, the surface is colored on the basis of the electrostatic potential of the molecule.<ref name="Peisach" />]]

Propositions for this mechanism have been made based on mechanistic studies done on eukaryotic protein kinases. It has been proposed that in the CKα-2 mechanism,ATP binds first, followed by Choline, and then the transfer of the phosphoryl group takes place. The product O-phosphocholine is then released, followed by the release of ADP.<ref>{{cite journal |doi=10.1074/jbc.270.42.24686 |title=Kinetic Mechanism of Aminoglycoside Phosphotransferase Type IIIa |year=1995 |last1=Wright |first1=G. D. |journal=Journal of Biological Chemistry |volume=270 |issue=42 |pages=24686–24692 |pmid=7559583 |last2=Wright |first2=GD }}</ref>

== Evolution ==
After closely studying the structurally similar enzymes,CKα-2, APH(3′)-IIIa, and [[Protein kinase A|PKA]],researchers observed that PKA had less insertions to its structural core compared to the other enzymes. Against this background,it is believed that CKα-2 have evolved from PKA to have more structural elements attached to it.<ref>{{cite journal |doi=10.1016/S0092-8674(00)80274-3 |title=Structure of an Enzyme Required for Aminoglycoside Antibiotic Resistance Reveals Homology to Eukaryotic Protein Kinases |year=1997 |last1=Hon |first1=W |journal=Cell |volume=89 |issue=6 |pages=887–895 |pmid=9200607 |last2=McKay |first2=GA |last3=Thompson |first3=PR |last4=Sweet |first4=RM |last5=Yang |first5=DS |last6=Wright |first6=GD |last7=Berghuis |first7=AM}}</ref>

== Biological function ==
Choline Kinase catalyzes the formation of phoshocholine,the committed step in phosphatidylcholine biosynthesis. Phosphatidylcholine is the major [[phospholipid]] in eukaryotic membranes. Phosphatidylcholine is important for a variety of function in eukaryotes such as facilitating the transport of [[cholesterol]] through the organism, acting as a substrate for the production of second messengers and as a cofactor for the activity of several membrane-related enzymes.<ref>{{cite journal |doi=10.1016/0163-7827(90)90010-I |title=Regulation of phosphatidylcholine biosynthesis |year=1990 |last1=Kent |first1=C |journal=Progress in Lipid Research |volume=29 |issue=2 |pages=87–105 |pmid=1965552}}</ref> CK also plays a vital role in the production of [[sphingomyelin]],another important membrane phospholipid and in the regulation of cell growth.<ref>{{cite journal |doi=10.2174/092986706776360923 |title=Choline Kinase: An Important Target for Cancer |year=2006 |last1=Janardhan |first1=S. |last2=Srivani |first2=P. |last3=Sastry |first3=G. N. |journal=Current Medicinal Chemistry |volume=13 |issue=10 |pages=1169–1186 |pmid=16719778}}</ref>

The production of phosphocholine from CK is necessary for the [[signal transduction]] pathways related to [[mitogenesis]]. It has also been found that CK plays a critical role in the proliferation of human mammary [[epithelial]] cells.<ref>{{cite journal |doi=10.1158/0008-5472.CAN-04-0489 |title=Choline Kinase Activation Is a Critical Requirement for the Proliferation of Primary Human Mammary Epithelial Cells and Breast Tumor Progression |year=2004 |last1=De Molina |first1=A. R. |journal=Cancer Research |volume=64 |issue=18 |pages=6732–6739 |pmid=15374991 |last2=Báñez-Coronel |first2=M |last3=Gutiérrez |first3=R |last4=Rodríguez-González |first4=A |last5=Olmeda |first5=D |last6=Megías |first6=D |last7=Lacal |first7=JC}}</ref>

[[In vivo]] studies carried out using CKα-1 and CKβ isoforms suggest that each isoform might be involved in different biochemical pathways.CKβ plays a major role in the catalysis of the phosphorylation of [[ethanolamine]] while CKα-1 catalyzes the phosphorylation of both choline and ethanolamine.<ref>{{cite journal |doi=10.1016/j.advenzreg.2010.09.010 |title=Involvement of human choline kinase alpha and beta in carcinogenesis: A different role in lipid metabolism and biological functions |year=2011 |last1=Gallego-Ortega |first1=David |last2=Gómez Del Pulgar |first2=Teresa |last3=Valdés-Mora |first3=FáTima |last4=Cebrián |first4=Arancha |last5=Lacal |first5=Juan Carlos |journal=Advances in Enzyme Regulation |volume=51 |pages=183–194 |pmid=21035492 |issue=1}}</ref>

== Disease relevance ==

=== Oncogenic activity and CKα-1 ===
[[Overexpression]] of CKα-1 has been found to be associated with cancer.Recent studies carried out on cancer cell lines have shown that CKα-1 is overexpressed in breast cancer cells. This leads to an accumulation of phosphocholine in the breast and causes malignancy.<ref>{{cite journal |doi=10.1002/ijc.22293 |title=Phosphocholine as a biomarker of breast cancer: Molecular and biochemical studies |year=2007 |last1=Eliyahu |first1=Galit |last2=Kreizman |first2=Tamar |last3=Degani |first3=Hadassa |journal=International Journal of Cancer |volume=120 |issue=8 |pages=1721–1730}}</ref>

Studies using colon, human lung and prostate carcinomas also revealed that CK is [[upregulated]] by overexpression of CKα-1 in these cells compared to the normal, non-cancerous cells.<ref name="Ramirez de Molina 580–3">{{cite journal |doi=10.1016/S0006-291X(02)00920-8 |title=Overexpression of choline kinase is a frequent feature in human tumor-derived cell lines and in lung, prostate, and colorectal human cancers |year=2002 |last1=Ramírez De Molina |first1=A |journal=Biochemical and Biophysical Research Communications |volume=296 |issue=3 |pages=580–583 |pmid=12176020 |last2=Rodríguez-González |first2=A |last3=Gutiérrez |first3=R |last4=Martínez-Piñeiro |first4=L |last5=Sánchez |first5=J |last6=Bonilla |first6=F |last7=Rosell |first7=R |last8=Lacal |first8=J}}</ref>

One possible explanation for this is that CKα-1 aids in the regulation of [[Protein Kinase B]] phoshorylation, particularly at the Serine-473 end.Consequently, high levels of expression and activity of CKα-1 promotes cell growth and survival.<ref>{{cite journal |doi=10.1186/1476-4598-8-131 |title=Regulation of Akt(ser473) phosphorylation by Choline kinase in breast carcinoma cells |year=2009 |last1=Chua |first1=Boon |last2=Gallego-Ortega |first2=David |last3=De Molina |first3=Ana |last4=Ullrich |first4=Axel |last5=Lacal |first5=Juan |last6=Downward |first6=Julian |journal=Molecular Cancer |volume=8 |pages=131 |pmid=20042122 |pmc=2806310}}</ref> Based on the observation that increased activity of CKα-1 is related to cancer,CKα-1 has promising use as a tumor [[biomarker]] and in diagnosing and following the progression of tumors. Even more interesting is the critical role that the discovery of inhibitors of CKα-1 could play in the development of novel drugs for cancer treatment. This is key point, given the fact that all human cancer cells have shown increased levels of this particular enzyme.<ref name="Ramirez de Molina 580–3"/>

=== Muscular Dystrophy and CKβ ===
It has been shown, using CKβ [[knockout]] mice models,that a defect in the CKβ activity leads to a decrease in the Phosphatidylcholine(PC) content in the hindlimb muscle. This, however, does not affect the phoshoethanolamine (PE) content.<ref name="Wu 347–356">{{cite journal |doi=10.1016/j.bbalip.2009.02.006 |title=Understanding the muscular dystrophy caused by deletion of choline kinase beta in mice |year=2009 |last1=Wu |first1=Gengshu |last2=Sher |first2=Roger B. |last3=Cox |first3=Gregory A. |last4=Vance |first4=Dennis E. |journal=Biochimica et Biophysica Acta (BBA) - Molecular and Cell Biology of Lipids |volume=1791 |issue=5 |pages=347–356}}</ref>

The net effect is then that the PC/PE ratio decreases and this is leads to impaired membrane integrity in the liver.<ref>{{cite journal |doi=10.1016/j.cmet.2006.03.007 |title=The ratio of phosphatidylcholine to phosphatidylethanolamine influences membrane integrity and steatohepatitis |year=2006 |last1=Li |first1=Zhaoyu |last2=Agellon |first2=Luis B. |last3=Allen |first3=Theresa M. |last4=Umeda |first4=Masato |last5=Jewell |first5=Larry |last6=Mason |first6=Andrew |last7=Vance |first7=Dennis E. |journal=Cell Metabolism |volume=3 |issue=5 |pages=321–331 |pmid=16679290}}</ref> This compromised membrane potential leads to malfunctioning of the [[mitochondria]]. Although CK is required for the biosynthesis of PC,CK is normally present in excess and so is not generally considered the [[rate-limiting step]].<ref>{{cite book |last1=Vance |first1=Dennis |last2=Vance |first2=Jean |title=Biochemistry of Lipids, Lipoproteins and Membranes |chapter=Phospholipid biosynthesis in eukaryotes |editor1-first=Dennis E. |editor1-last=Vance |editor2-first=Jean E. |editor2-last=Vance |publisher=Elsevier |year=2008 |pages=213–244 |doi=10.1016/B978-044453219-0.50010-6 |accessdate=2011-05-16 |isbn=978-0-444-53219-0}}</ref> Researchers have concluded, however, that due to the reduced activity of CK seen in the hindlimb muscle of the CKβ knockout mice model,CK is probably the rate-limiting enzyme in skeletal muscles.This suggests that defect in CKβ may lead to a decrease in PC synthesis in the muscles resulting in [[muscular dystrophy]].<ref name="Wu 347–356"/> These results suggest that CK could possibly play a vital role in the [[homeostasis]] of PC.<ref>{{cite journal |doi=10.1158/0008-5472.CAN-07-2728 |title=Choline Kinase Down-regulation Increases the Effect of 5-Fluorouracil in Breast Cancer Cells |year=2007 |last1=Mori |first1=N. |last2=Glunde |first2=K. |last3=Takagi |first3=T. |last4=Raman |first4=V. |last5=Bhujwalla |first5=Z. M. |journal=Cancer Research |volume=67 |issue=23 |pages=11284–11290 |pmid=18056454}}</ref>

== References ==

{{Reflist}}

==Further reading==
{{reflist|1}}
* {{cite journal |pmid=5335908 |year=1967 |last1=Hayashi |first1=SI |last2=Lin |first2=EC |title=Purification and properties of glycerol kinase from Escherichia coli |volume=242 |issue=5 |pages=1030–5 |journal=The Journal of Biological Chemistry}}
* {{cite journal |pmid=13061469 |year=1953 |last1=Wittenberg |first1=J |last2=Kornberg |first2=A |title=Choline phosphokinase |volume=202 |issue=1 |pages=431–44 |journal=The Journal of Biological Chemistry}}

{{DEFAULTSORT:Choline Kinase}}
[[Category:Articles created via the Article Wizard]]
[[Category:EC 2.7.1]]
[[Category:Enzymes of known structure]]

Talk:Alderson disk

2012-03-22T02:55:16Z

71.236.2.96: /* The point of an Alderson disk? */

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