|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| '''Metabolic control analysis''' (MCA) is a mathematical framework for describing
| | Claude is her title and she completely digs that title. Delaware is our beginning place. The favorite pastime for him and his children is to perform badminton but he is having difficulties to discover time for it. Bookkeeping is what I do for a residing.<br><br>Here is my web blog ... car warranty ([http://Www.Skyperoom.com/blogs/post/9816 related webpage]) |
| [[Metabolic pathway|metabolic]], [[Cell signaling#Signaling pathways|signaling]] and [[genetic pathway]]s. MCA quantifies how variables,
| |
| such as [[flux]]es and [[Chemical species|species]] concentrations, depend on [[Network (mathematics)|network]] parameters.
| |
| In particular it is able to describe how network dependent properties,
| |
| called control [[coefficient]]s, depend on [[Local property|local properties]] called [[Elasticity of a function|elasticities]].<ref>Fell D., (1997) Understanding the Control of Metabolism, Portland Press.</ref><ref>Heinrich R. and Schuster S. (1996) The Regulation of Cellular Systems, Chapman and Hall.
| |
| </ref><ref>{{cite pmid|7925313}}</ref>
| |
| | |
| MCA was originally developed to describe the control in metabolic pathways
| |
| but was subsequently extended to describe signaling and [[Gene regulatory network|genetic networks]]. MCA has sometimes also been referred to as ''Metabolic Control Theory'' but this terminology was rather strongly opposed by [[Henrik Kacser]], one of the founders{{Citation needed|date=November 2010}}.
| |
| | |
| More recent work<ref>Ingalls, B. P. (2004) A Frequency Domain Approach to Sensitivity Analysis of Biochemical Systems , Journal of Physical Chemistry B, 108, 1143-1152.</ref> has shown that MCA can be [[Isomorphism|mapped directly]] on to classical [[control theory]] and are as such equivalent.
| |
| | |
| [[Biochemical systems theory]]<ref>Savageau M.A (1976) Biochemical systems analysis: a study of function and design in molecular biology, Reading, MA, Addison–Wesley.</ref> is a similar [[Formalism_(mathematics)#Formalism|formalism]], though with a rather different objectives. Both are evolutions of an earlier theoretical analysis by Joseph Higgins.<ref>{{cite pmid|13954410}}</ref>
| |
| | |
| == Control Coefficients ==
| |
| | |
| A control coefficient<ref>{{cite pmid|4148886}}</ref><ref>{{cite pmid|4830198}}</ref><ref>Burns, J.A., Cornish-Bowden, A., Groen, A.K., Heinrich, R., Kacser, H., Porteous, J.W., Rapoport, S.M., Rapoport, T.A., Stucki, J.W., Tager, J.M., Wanders, R.J.A. & Westerhoff, H.V. (1985) Control analysis of metabolic systems. Trends Biochem. Sci. 10, 16.</ref> measures the relative [[steady state]] change in a system variable, e.g. pathway flux (J) or metabolite concentration (S), in response to a relative change in a [[parameter]], e.g. [[Enzyme kinetics|enzyme activity]] or the steady-state rate (<math> v_i </math>) of step i. The two main control coefficients are the flux and concentration control coefficients. Flux control coefficients are defined by:
| |
| | |
| <math> C^J_{v_i} = \left( \frac{dJ}{dp} \frac{p}{J} \right) \bigg/ \left( \frac{\partial v_i}{\partial p}\frac{p}{v_i} \right) = \frac{d\ln J}{d\ln v_i} </math>
| |
| | |
| and concentration control coefficients by:
| |
| | |
| <math> C^S_{v_i} = \left( \frac{dS}{dp} \frac{p}{S} \right) \bigg/ \left( \frac{\partial v_i}{\partial p} \frac{p}{v_i} \right) = \frac{d\ln S}{d\ln v_i} </math>
| |
| | |
| === Summation Theorems ===
| |
| The flux control [[summation]] theorem was discovered independently by the Kacser/Burns group and the Heinrich/Rapoport group in the early 1970s and late 1960s. The flux control summation theorem [[Logical implication|implies]] that metabolic fluxes are systemic properties and that their control is shared by all [[Chemical reaction|reactions]] in the system. When a single reaction changes its control of the flux this is compensated by changes in the control of the same flux by all other reactions.
| |
| | |
| <math> \sum_i C^J_{v_i} = 1 </math>
| |
| | |
| <math> \sum_i C^S_{v_i} = 0 </math>
| |
| | |
| == Elasticity Coefficients ==
| |
| | |
| The elasticity coefficient measures the local response of an enzyme or other chemical reaction to changes in its environment. Such changes include factors such as substrates, products or effector concentrations. For further information please refer to the dedicated page at [[Elasticity Coefficient]]s.
| |
| | |
| === Connectivity Theorems ===
| |
| | |
| The [[Connectivity (graph theory)|connectivity]] theorems are specific relationships between elasticities and control coefficients. They are useful because they highlight the close relationship between the [[Chemical kinetics|kinetic]] properties of individual reactions and the system properties of a pathway. Two basic sets of theorems exists, one for flux and another for concentrations. The concentration connectivity theorems are divided again depending on whether the system species <math> S_n </math> is different from the local species <math> S_m </math>.
| |
| | |
| <math> \sum_i C^J_i \varepsilon^i_S = 0 </math>
| |
| | |
| <math> \sum_i C^{S_n}_i \varepsilon^i_{S_m} = 0 \quad n \neq m </math>
| |
| | |
| <math> \sum_i C^{S_n}_i \varepsilon^i_{S_m} = -1 \quad n = m </math>
| |
| | |
| == Control Equations ==
| |
| | |
| It is possible to combine the summation with the connectivity theorems to obtain [[Closed-form expression|closed expressions]] that relate the control coefficients to the elasticity coefficients. For example, consider the simplest [[Trivial (mathematics)|non-trivial]] pathway:
| |
| | |
| <center> <math> X_o \rightarrow S \rightarrow X_1 </math> </center>
| |
| | |
| We assume that <math> X_o </math> and <math> X_1 </math> are [[Boundary (thermodynamic)|fixed boundary]] species so that the pathway can reach a steady state. Let the first step have a rate <math> v_1 </math> and the second step <math> v_2 </math>. Focusing on the flux control coefficients, we can write one summation and one connectivity theorem for this simple pathway:
| |
| | |
| <center>
| |
| <math> C^J_{v_1} + C^J_{v_2} = 1 </math>
| |
| | |
| <math> C^J_{v_1} \varepsilon^{v_1}_S + C^J_{v_2} \varepsilon^{v_2}_S = 0 </math>
| |
| </center>
| |
| | |
| Using these two equations we can solve for the flux control coefficients to yield:
| |
| | |
| <center>
| |
| <math> C^J_{v_1} = \frac{\varepsilon^{2}_S}{\varepsilon^{2}_S - \varepsilon^{1}_S} </math>
| |
| | |
| <math> C^J_{v_2} = \frac{-\varepsilon^{1}_S}{\varepsilon^{2}_S - \varepsilon^{1}_S} </math>
| |
| </center>
| |
| | |
| Using these equations we can look at some simple extreme behaviors. For example, let us assume that the first step is completely insensitive to its product (i.e. not reacting with it), S, then <math> \varepsilon^{v_1}_S = 0 </math>. In this case, the control coefficients reduce to:
| |
| | |
| <center>
| |
| <math> C^J_{v_1} = 1 </math>
| |
| | |
| <math> C^J_{v_2} = 0 </math>
| |
| </center>
| |
| | |
| That is all the control (or sensitivity) is on the first step. This situation represents the classic [[Rate-determining step|rate-limiting step]] that is frequently mentioned in text books. The flux through the pathway is completely dependent on the first step. Under these conditions, no other step in the pathway can affect the flux. The effect is however dependent on the complete insensitivity of the first step to its product. Such a situation is likely to be rare in real pathways. In fact the classic rate limiting step has almost never been observed experimentally. Instead, a range of limitingness is observed, with some steps having more limitingness (control) than others.
| |
| | |
| We can also derive the concentration control coefficients for the simple two step pathway:
| |
| | |
| <center>
| |
| <math> C^S_{v_1} = \frac{1}{\varepsilon^{2}_S - \varepsilon^{1}_S} </math>
| |
| | |
| <math> C^S_{v_2} = \frac{-1}{\varepsilon^{2}_S - \varepsilon^{1}_S} </math>
| |
| </center>
| |
| | |
| An alternative approach to deriving the control equations is to consider the [[Perturbation theory|perturbations]] explicitly. Consider making a perturbation to <math> E_1 </math> which changes the local rate <math> v_1 </math>. The effect on the steady-state to a small change in <math> E_1 </math> is to increase the flux and concentration of S. We can express these changes locally by describing the change in <math> v_1 </math> and <math> v_2 </math> using the expressions:
| |
| | |
| <center><math> \frac{\delta v_1}{v_1} = \varepsilon^{1}_{E_1} \frac{\delta E_1}{E_1} + \varepsilon^{1}_S \frac{\delta S}{S} </math>
| |
| | |
| <math> \frac{\delta v_2}{v_2} = \varepsilon^{2}_S \frac{\delta S}{S} </math>
| |
| </center>
| |
| | |
| The local changes in rates are equal to the global changes in flux, J. In addition if we assume that the [[enzyme]] elasticity of <math> v_1 </math> with respect to <math> E_1 </math> is unity, then
| |
| | |
| <center><math> \frac{\delta J}{J} = \frac{\delta E_1}{E_1} + \varepsilon^{1}_S \frac{\delta S}{S} </math>
| |
| | |
| <math> \frac{\delta J}{J} = \varepsilon^{2}_S \frac{\delta S}{S} </math>
| |
| </center>
| |
| | |
| Dividing both sides by the fractional change in <math> E_1 </math> and taking the [[Limit (mathematics)|limit]] <math> \delta E_1 \rightarrow 0 </math> yields:
| |
| | |
| <br> | |
| <center><math> C^J_{E_1} = 1 + \varepsilon^{1}_S C^S_{E_1} </math>
| |
| | |
| <math> C^J_{E_1} = \varepsilon^{2}_S C^S_{E_1} </math>
| |
| </center>
| |
| <br> | |
| From these equations we can choose either to eliminate <math> C^J_{E_1} </math> or <math> C^S_{E_1} </math> to yield the control equations given earlier. We can do the same kind of analysis for the second step to obtain the flux control coefficient for <math> E_2 </math>. Note that we have expressed the control coefficients relative to <math> E_1 </math> and <math> E_2 </math> but if we assume that <math> \delta v_i/v_i = \delta E_i/E_i </math> then the control coefficients can be written with respect to <math> v_i </math> as before.
| |
| | |
| == Three Step Pathway ==
| |
| | |
| Consider the simple three step pathway:
| |
| | |
| <center> <math> X_o \rightarrow S_1 \rightarrow S_2 \rightarrow X_1 </math> </center>
| |
| | |
| where <math> X_o </math> and <math> X_1 </math> are fixed boundary species, the control equations for this pathway can be derived in a similar manner to the simple two step pathway although it is somewhat more tedious.
| |
| | |
| <math>
| |
| C^J_{E_1} = \varepsilon^{2}_1 \varepsilon^{3}_2 / D
| |
| </math>
| |
| | |
| <br>
| |
| <math>
| |
| C^J_{E_2} = -\varepsilon^{1}_1 \varepsilon^{3}_2 / D
| |
| </math>
| |
| | |
| <br>
| |
| <math>
| |
| C^J_{E_3} = \varepsilon^{1}_1 \varepsilon^{2}_2 / D
| |
| </math>
| |
| | |
| <br>
| |
| where D the denominator is given by:
| |
| | |
| <br>
| |
| <math>
| |
| D = \varepsilon^{2}_1 \varepsilon^{3}_2 -\varepsilon^{1}_1 \varepsilon^{3}_2 + \varepsilon^{1}_1 \varepsilon^{2}_2
| |
| </math>
| |
| | |
| Note that every term in the numerator appears in the denominator, this ensures that the flux control coefficient summation theorem is satisfied.
| |
| | |
| Likewise the concentration control coefficients can also be derived, for <math> S_1 </math>
| |
| | |
| <math>
| |
| C^{S_1}_{E_1} = (\varepsilon^{3}_2 - \varepsilon^{2}_2) / D
| |
| </math>
| |
| | |
| <br>
| |
| <math>
| |
| C^{S_1}_{E_2} = - \varepsilon^{3}_2 / D
| |
| </math>
| |
| | |
| <br>
| |
| <math>
| |
| C^{S_1}_{E_3} = \varepsilon^{2}_2 / D
| |
| </math>
| |
| | |
| <br>
| |
| And for <math> S_2 </math>
| |
| | |
| <br>
| |
| <math>
| |
| C^{S_2}_{E_1} = \varepsilon^{2}_1 / D
| |
| </math>
| |
| | |
| <br>
| |
| <math>
| |
| C^{S_2}_{E_2} = -\varepsilon^{1}_1 / D
| |
| </math>
| |
| | |
| <br>
| |
| <math>
| |
| C^{S_2}_{E_3} = (\varepsilon^{1}_1 - \varepsilon^{2}_1) / D
| |
| </math>
| |
| | |
| Note that the denominators remain the same as before and behave as a [[Normalizing constant|normalizing]] factor.
| |
| | |
| == References ==
| |
| <references/>
| |
| | |
| == External links ==
| |
| *[http://dbkgroup.org/mca_home.htm The Metabolic Control Analysis Web]
| |
| | |
| {{DEFAULTSORT:Metabolic Control Analysis}}
| |
| [[Category:Biochemistry methods]]
| |
| [[Category:Metabolism]]
| |
| [[Category:Mathematical and theoretical biology]]
| |
| [[Category:Systems biology]]
| |