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| [[File:Killing-vector-s2.png|thumb|450 px|A Killing vector field (red) with integral curves (blue) on a sphere.]]
| | {{Infobox enzyme |
| | | Name = Phosphorylase |
| | | EC_number = 2.4.1.1 |
| | | CAS_number = 9035-74-9 |
| | | IUBMB_EC_number = 2/4/1/1 |
| | | GO_code = |
| | | image = |
| | | width = |
| | | caption = |
| | }} |
| | '''Phosphorylases''' are enzymes that catalyze the addition of a phosphate group from an inorganic phosphate (phosphate+hydrogen) to an acceptor. |
| | :A-B + P <math>\rightleftharpoons</math> A + P-B |
| | They include [[Allosteric regulation|allosteric]] [[enzyme]]s that [[catalysis|catalyze]] the production of [[glucose-1-phosphate]] from a [[glucan]] such as [[glycogen]], [[starch]] or [[maltodextrin]]. Phosphorylase is also a common name used for [[glycogen phosphorylase]] in honor of Earl W. Sutherland Jr. who in the late 1930s discovered the first phosphorylase.<ref>Lehninger Principles of Biochemistry 5th ed. pg. 603</ref> |
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| In [[mathematics]], a '''Killing vector field''' (often just '''Killing field'''), named after [[Wilhelm Killing]], is a [[vector field]] on a [[Riemannian manifold]] (or [[pseudo-Riemannian manifold]]) that preserves the [[metric tensor|metric]]. Killing fields are the [[Lie group#The Lie algebra associated to a Lie group|infinitesimal generator]]s of [[isometry|isometries]]; that is, [[flow (geometry)|flow]]s generated by Killing fields are [[Isometry (Riemannian geometry)|continuous isometries]] of the [[manifold]]. More simply, the flow generates a [[symmetry]], in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object.
| | == Function == |
| | Phosphorylases should not be confused with [[phosphatases]], which remove phosphate groups. |
| | In more general terms, phosphorylases are enzymes that catalyze the addition of a phosphate group from an [[inorganic phosphate]] (phosphate + hydrogen) to an acceptor, not to be confused with a [[phosphatase]] (a [[hydrolase]]) or a [[kinase]] (a [[phosphotransferase]]). A phosphatase removes a phosphonate group from a donor using water, whereas a kinase transfers a phosphonate group from a donor (usually ATP) to an acceptor. |
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| == Definition == | | {| class=wikitable |
| | !Enzyme name |
| | !Enzymes class |
| | !Reaction |
| | !Notes |
| | |- |
| | |Phosphorylase|| Transferase<br>(EC 2.4 and EC 2.7.7) |
| | ||A-B + H-'''OP''' <math>\rightleftharpoons</math> A-'''OP''' + H-B |
| | || transfer group = A = [[glycosyl]]- group or<br> [[nucleotidyl]]- group |
| | |-' |
| | |Phosphatase|| Hydrolase<br>(EC 3) |
| | ||'''P'''-B + H-OH <math>\rightleftharpoons</math> '''P'''-OH + H-B |
| | || |
| | |- |
| | |Kinase|| Transferase<br>(EC 2.7.1-2.7.4) |
| | ||'''P'''-B + H-A <math>\rightleftharpoons</math> '''P'''-A + H-B |
| | || transfer group = '''P''' |
| | |- |
| | |colspan=4|'''P''' = [[phosphonate]] group, '''OP''' = phosphate group, H-'''OP''' or '''P'''-OH = inorganic phosphate |
| | |} |
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| Specifically, a vector field ''X'' is a Killing field if the [[Lie derivative]] with respect to ''X'' of the metric ''g'' vanishes:
| | == Types == |
| | The phosphorylases fall into the following categories: |
| | *Glycosyltransferases (EC 2.4) |
| | **Enzymes that break down [[glucan]]s by removing a glucose residue (break ''O''-glycosidic bond) |
| | ***[[glycogen phosphorylase]] |
| | ***[[starch phosphorylase]] |
| | ***[[maltodextrin phosphorylase]] |
| | **Enzymes that break down [[nucleoside]]s into their constituent bases and sugars (break ''N''-glycosidic bond) |
| | ***[[Purine nucleoside phosphorylase]] (PNPase) |
| | *Nucleotidyltransferases (EC 2.7.7) |
| | **Enzymes that have phosphorolytic 3' to 5' exoribonuclease activity (break phosphodiester bond) |
| | ***[[RNase PH]] |
| | ***[[Polynucleotide Phosphorylase]] (PNPase) |
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| :<math>\mathcal{L}_{X} g = 0 \,.</math> | | All '''known''' phosphorylases share catalytic and structural properties [http://www.expasy.org/cgi-bin/nicedoc.pl?PDOC00095]. |
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| In terms of the [[Levi-Civita connection]], this is
| | == Activation == |
| | '''Phosphorylase a''' is the active form of glycogen phosphorylase that is derived from the phosphorylation of the inactive form, '''phosphorylase b'''. |
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| :<math>g(\nabla_{Y} X, Z) + g(Y, \nabla_{Z} X) = 0 \,</math> | | == Pathology == |
| | Some disorders are related to phosphorylases: |
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| for all vectors ''Y'' and ''Z''. In [[local coordinates]], this amounts to the Killing equation
| | * [[Glycogen storage disease type V]] - muscle glycogen |
| | * [[Glycogen storage disease type VI]] - liver glycogen |
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| :<math>\nabla_{\mu} X_{\nu} + \nabla_{\nu} X_{\mu} = 0 \,.</math>
| | ==See also== |
| | | *[[Hydrolase]] |
| This condition is expressed in covariant form. Therefore it is sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems.
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| == Examples == | |
| * The vector field on a circle that points clockwise and has the same length at each point is a Killing vector field, since moving each point on the circle along this vector field simply rotates the circle. | |
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| * If the metric coefficients <math>g_{\mu \nu} \,</math> in some coordinate basis <math>dx^{a} \,</math> are independent of <math>x^{\kappa} \,</math>, then <math>K^{\mu} = \delta^{\mu}_{\kappa} \,</math> is automatically a Killing vector, where <math>\delta^{\mu}_{\kappa} \,</math> is the [[Kronecker delta]].<ref>{{cite book | title=Gravitation | last = Misner, Thorne, Wheeler | year=1973 | publisher = W H Freeman and Company| isbn=0-7167-0344-0}}</ref><br /> To prove this, let us assume <math> g_{\mu \nu},_0=0 \,</math> <br /> Then <math> K^\mu=\delta^{\mu}_{0} \,</math> and <math> K_{\mu}=g_{\mu \nu} K^{\nu}= g_{\mu \nu} \delta^{\nu}_{0}= g_{\mu 0} \,</math> <br /> Now let us look at the Killing condition <br /> <math> K_{\mu;\nu}+K_{\nu;\mu}=K_{\mu,\nu}+K_{\nu,\mu}-2\Gamma^{\rho}_{\mu\nu}K_{\rho} = g_{\mu 0,\nu}+g_{\nu 0,\mu}-g^{\rho\sigma}(g_{\sigma\mu,\nu}+g_{\sigma\nu,\mu}-g_{\mu\nu,\sigma})g_{\rho 0} \,</math> <br /> and from <math> g_{\rho 0}g^{\rho \sigma} = \delta_{0}^{\sigma} \,</math> <br /> The Killing condition becomes <br /> <math> g_{\mu 0,\nu}+g_{\nu 0,\mu} - ( g_{0\mu,\nu}+g_{0\nu,\mu}-g_{\mu\nu,0} ) = 0 \,</math> <br /> that is <math>g_{\mu\nu,0}= 0 </math>, which is true.
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| *: The physical meaning is, for example, that, if none of the metric coefficients is a function of time, the manifold must automatically have a time-like Killing vector.
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| *: In layman's terms, if an object doesn't transform or "evolve" in time (when time passes), time passing won't change the measures of the object. Formulated like this, the result sounds like a tautology, but one has to understand that the example is very much contrived: Killing fields apply also to much more complex and interesting cases.
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| == Properties == | | == References == |
| A Killing field is determined uniquely by a vector at some point and its gradient (i.e. all [[covariant derivative]]s of the field at the point).
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| The [[Lie bracket of vector fields|Lie bracket]] of two Killing fields is still a Killing field. The Killing fields on a manifold ''M'' thus form a [[Lie algebra|Lie subalgebra]] of vector fields on ''M''. This is the Lie algebra of the [[isometry group]] of the manifold if ''M'' is complete.
| | <references/> |
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| For [[compact space|compact]] manifolds
| | == External links == |
| * Negative [[Ricci curvature]] implies there are no nontrivial (nonzero) Killing fields.
| | *[http://mcardlesdisease.org Muscle phosphorylase deficiency - McArdle's Disease Website] |
| * Nonpositive [[Ricci curvature]] implies that any Killing field is parallel. i.e. covariant derivative along any vector j field is identically zero.
| | * {{MeshName|Phosphorylases}} |
| * If the [[sectional curvature]] is positive and the dimension of ''M'' is even, a Killing field must have a zero.
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| The divergence of every Killing vector field vanishes.
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| If <math>X</math> is a Killing vector field and <math>Y</math> is a [[Hodge theory|harmonic vector field]], then <math>g(X,Y)</math> is a [[harmonic function]].
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| If <math>X</math> is a Killing vector field and <math>\omega</math> is a [[Hodge_theory|harmonic p-form]], then <math>\mathcal{L}_{X} \omega = 0 \,.</math>
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| === Geodesics ===
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| Each Killing vector corresponds to a quantity which is conserved along [[Geodesics as Hamiltonian flows|geodesics]]. This conserved quantity is the metric product between the Killing vector and the geodesic tangent vector. That is, along a geodesic with some affine parameter <math>\lambda
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| </math>, the equation <math>\frac d {d\lambda} ( K_\mu \frac{dx^\mu}{d\lambda} ) = 0</math> is satisfied. This aids in analytically studying motions in a [[spacetime]] with symmetries.<ref>{{Cite book|title = An Introduction to General Relativity Spacetime and Geometry|last = Carrol|first = Sean|publisher = Addison Wesley|year = 2004|isbn = |location = |pages = 133-139}}</ref>
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| == Generalizations ==
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| * Killing vector fields can be generalized to [[Conformal vector field|''conformal'' Killing vector fields]] defined by <math>\mathcal{L}_{X} g = \lambda g \,</math> for some scalar <math>\lambda \,.</math> The derivatives of one parameter families of [[conformal map]]s are conformal Killing fields.
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| * Killing ''tensor ''fields are symmetric [[tensor]] fields ''T'' such that the trace-free part of the symmetrization of <math>\nabla T \,</math> vanishes. Examples of manifolds with Killing tensors include the [[Kerr spacetime|rotating black hole]] and the [[FRW cosmology]].<ref>{{Cite book|title = An Introduction to General Relativity Spacetime and Geometry|last = Carrol|first = Sean|publisher = Addison Wesley|year = 2004|isbn = |location = |pages = 263,344}}</ref>
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| * Killing vector fields can also be defined on any (possibly nonmetric) manifold M if we take any Lie group G [[group action|acting]] on it instead of the group of isometries.<ref>{{citation | |
| |last = Choquet-Bruhat
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| |first = Yvonne
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| |authorlink = Yvonne Choquet-Bruhat
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| |first2 = Cécile |last2=DeWitt-Morette|authorlink2 = Cécile DeWitt-Morette| title = Analysis, Manifolds and Physics| publisher = Elsevier| year= 1977| location = Amsterdam |isbn = 978-0-7204-0494-4}}</ref> In this broader sense, a Killing vector field is the pushforward of a right invariant vector field on G by the group action. If the group action is effective, then the space of the Killing vector fields is isomorphic to the Lie algebra <math>\mathfrak{g}</math> of G.
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| ==See also==
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| * [[Affine vector field]]
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| * [[Curvature collineation]]
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| * [[Homothetic vector field]]
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| * [[Killing form]]
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| * [[Killing horizon]]
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| * [[Killing spinor]]
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| * [[Killing tensor]]
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| * [[Matter collineation]]
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| * [[Spacetime symmetries]]
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| ==Notes==
| | {{Glycosyltransferases}} |
| {{Reflist}} | | {{Kinases}} |
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| ==References==
| | [[Category:Transferases]] |
| * {{cite book | author=Jost, Jurgen| title= Riemannian Geometry and Geometric Analysis| location=Berlin | publisher=Springer-Verlag | year=2002 | isbn=3-540-42627-2}}.
| | [[Category:EC 2.4.1]] |
| * {{cite book | author=Adler, Ronald; Bazin, Maurice & Schiffer, Menahem| title= Introduction to General Relativity (Second Edition)| location=New York | publisher=McGraw-Hill | year=1975 | isbn=0-07-000423-4}}. ''See chapters 3,9''
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| {{DEFAULTSORT:Killing Vector Field}} | | {{transferase-stub}} |
| [[Category:Riemannian geometry]]
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Template:Infobox enzyme
Phosphorylases are enzymes that catalyze the addition of a phosphate group from an inorganic phosphate (phosphate+hydrogen) to an acceptor.
- A-B + P
A + P-B
They include allosteric enzymes that catalyze the production of glucose-1-phosphate from a glucan such as glycogen, starch or maltodextrin. Phosphorylase is also a common name used for glycogen phosphorylase in honor of Earl W. Sutherland Jr. who in the late 1930s discovered the first phosphorylase.[1]
Function
Phosphorylases should not be confused with phosphatases, which remove phosphate groups.
In more general terms, phosphorylases are enzymes that catalyze the addition of a phosphate group from an inorganic phosphate (phosphate + hydrogen) to an acceptor, not to be confused with a phosphatase (a hydrolase) or a kinase (a phosphotransferase). A phosphatase removes a phosphonate group from a donor using water, whereas a kinase transfers a phosphonate group from a donor (usually ATP) to an acceptor.
Types
The phosphorylases fall into the following categories:
- Glycosyltransferases (EC 2.4)
- Enzymes that break down glucans by removing a glucose residue (break O-glycosidic bond)
- Enzymes that break down nucleosides into their constituent bases and sugars (break N-glycosidic bond)
- Nucleotidyltransferases (EC 2.7.7)
- Enzymes that have phosphorolytic 3' to 5' exoribonuclease activity (break phosphodiester bond)
All known phosphorylases share catalytic and structural properties [1].
Activation
Phosphorylase a is the active form of glycogen phosphorylase that is derived from the phosphorylation of the inactive form, phosphorylase b.
Pathology
Some disorders are related to phosphorylases:
See also
References
- ↑ Lehninger Principles of Biochemistry 5th ed. pg. 603
External links
Template:Glycosyltransferases
Template:Kinases
Template:Transferase-stub