Metabolic Control Analysis - Control Equations

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 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.