Metabolic control analysis
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.
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.
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 () of step i. The two main control coefficients are the flux and concentration control coefficients. Flux control coefficients are defined by:
and concentration control coefficients by:
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.
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.
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 is different from the local species .
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 and are fixed boundary species so that the pathway can reach a steady state. Let the first step have a rate and the second step . Focusing on the flux control coefficients, we can write one summation and one connectivity theorem for this simple pathway:
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 . In this case, the control coefficients reduce to:
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 which changes the local rate . The effect on the steady-state to a small change in is to increase the flux and concentration of S. We can express these changes locally by describing the change in and using the expressions:
The local changes in rates are equal to the global changes in flux, J. In addition if we assume that the enzyme elasticity of with respect to is unity, then
Dividing both sides by the fractional change in and taking the limit yields:
From these equations we can choose either to eliminate or 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 . Note that we have expressed the control coefficients relative to and but if we assume that then the control coefficients can be written with respect to as before.
Three Step Pathway
Consider the simple three step pathway:
where and 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.
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
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 and are determined by two enzymes and respectively. Changing either enzyme will result in a change to the steady state level of and the steady state reaction rates . Consider a small change in of magnitude . This will have a number of effects, it will increase which in turn will increase which in turn will increase . Eventually the system will settle to a new steady state. We can describe these changes by focusing on the change in and . The change in , which we designate , came about as a result of the change . Because we are only considering small changes we can express the change in terms of using the relation:
where the derivative measures how responsive is to changes in . The derivative can be computed if we know the rate law for . For example, if we assume that the rate law is then the derivative is . We can also use a similar strategy to compute the change in as a result of the change . This time the change in is a result of two changes, the change in itself and the change in . We can express these changes by summing the two individual contributions:
We have two equations, one describing the change in and the other in . Because we allowed the system to settle to a new steady state we can also state that the change in reaction rates must be the same (otherwise it wouldn't be at steady state). That is we can assert that . With this in mind we equate the two equations and write:
Solving for the ratio we obtain:
In the limit, as we make the change smaller and smaller, the left-hand side converges to the derivative :
We can go one step further and scale the derivatives to eliminate units. Multiplying both sides by and dividing both sides by $x$ yields the scaled derivatives:
The scaled derivatives on the right-hand side are the elasticities, and the scaled left-hand term is the scaled sensitivity coefficient or concentration control coefficient,
We can simplify this expression further. The reaction rate is usually a linear function of . For example in the Briggs-Haldane equation, the reaction rate is given by . Differentiating this rate law with respect to and scaling yields: .
Using this result gives:
A similar analysis can be done where is perturbed. In this case we obtain the sensitivity of with respect to :
The above expressions measure how much enzymes and control the steady state concentration of intermediate . We can also consider how the steady state reaction rates and are affected by perturbations in and . This is often of importance to metabolic engineers who are interested in increasing rates of production. At steady state the reaction rates are often called the fluxes and abbreviated to and . For a linear pathway such this example, both fluxes are equal at steady state so that the flux through the pathway is simply referred to as . Expressing the change in flux as a result of a perturbations in and taking the limit as before we obtain:
The above expressions tell us how much enzymes and control the steady state flux. The key point here is that changes in enzyme concentration, or equivalently the enzyme activity, must be brought about by an external action.
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