Metabolic control analysis (MCA) is a mathematical framework for describing metabolic, signaling, and genetic pathways. MCA quantifies how variables, such as fluxes and species concentrations, depend on network parameters. In particular it is able to describe how network dependent properties, called control coefficients, depend on local properties called elasticities.
Contents
- Control Coefficients
- Summation Theorems
- Elasticity Coefficients
- Connectivity Theorems
- Control Equations
- Three Step Pathway
- Derivation using Perturbations
- References
MCA was originally developed to describe the control in metabolic pathways but was subsequently extended to describe signaling and 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.
More recent work has shown that MCA can be mapped directly on to classical control theory and are as such equivalent.
Biochemical systems theory is a similar formalism, though with a rather different objectives. Both are evolutions of an earlier theoretical analysis by Joseph Higgins.
Control Coefficients
A control coefficient 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 activity or the steady-state rate (
and concentration control coefficients by:
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 implies that metabolic fluxes are systemic properties and that their control is shared by all 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.
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 Coefficients.
Connectivity Theorems
The connectivity theorems are specific relationships between elasticities and control coefficients. They are useful because they highlight the close relationship between the 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
Control Equations
It is possible to combine the summation with the connectivity theorems to obtain closed expressions that relate the control coefficients to the elasticity coefficients. For example, consider the simplest non-trivial pathway:
We assume that
Using these two equations we can solve for the flux control coefficients to yield:
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
That is all the control (or sensitivity) is on the first step. This situation represents the classic 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:
An alternative approach to deriving the control equations is to consider the perturbations explicitly. Consider making a perturbation to
The local changes in rates are equal to the global changes in flux, J. In addition if we assume that the enzyme elasticity of
Dividing both sides by the fractional change in
From these equations we can choose either to eliminate
Three Step Pathway
Consider the simple three step pathway:
where
where D the denominator is given by:
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
And for
Note that the denominators remain the same as before and behave as a normalizing factor.
Derivation using Perturbations
Control equations can also be derived by considering the effect of perturbations on the system. Consider that reaction rates
where the derivative
We have two equations, one describing the change in
Solving for the ratio
In the limit, as we make the change
We can go one step further and scale the derivatives to eliminate units. Multiplying both sides by
The scaled derivatives on the right-hand side are the elasticities,
We can simplify this expression further. The reaction rate
Using this result gives:
A similar analysis can be done where
The above expressions measure how much enzymes
The above expressions tell us how much enzymes