Biology: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.

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 ($\displaystyle{ v_i }$) of step i. The two main control coefficients are the flux and concentration control coefficients. Flux control coefficients are defined by

$\displaystyle{ C^J_{v_i} = \left( \frac{dJ}{dp} \frac{p}{J} \right) \bigg/ \left( \frac{\partial v_i}{\partial p}\frac{p}{v_i} \right) = \frac{d\ln J}{d\ln v_i} }$

and concentration control coefficients by

$\displaystyle{ C^S_{v_i} = \left( \frac{dS}{dp} \frac{p}{S} \right) \bigg/ \left( \frac{\partial v_i}{\partial p} \frac{p}{v_i} \right) = \frac{d\ln S}{d\ln v_i} }$

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.

$\displaystyle{ \sum_i C^J_{v_i} = 1 }$
$\displaystyle{ \sum_i C^S_{v_i} = 0 }$

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 $\displaystyle{ S_n }$ is different from the local species $\displaystyle{ S_m }$.

$\displaystyle{ \sum_i C^J_i \varepsilon^i_S = 0 }$
$\displaystyle{ \sum_i C^{S_n}_i \varepsilon^i_{S_m} = 0 \quad n \neq m }$
$\displaystyle{ \sum_i C^{S_n}_i \varepsilon^i_{S_m} = -1 \quad n = m }$

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:

$\displaystyle{ X_o \rightarrow S \rightarrow X_1 }$

We assume that $\displaystyle{ X_o }$ and $\displaystyle{ X_1 }$ are fixed boundary species so that the pathway can reach a steady state. Let the first step have a rate $\displaystyle{ v_1 }$ and the second step $\displaystyle{ v_2 }$. Focusing on the flux control coefficients, we can write one summation and one connectivity theorem for this simple pathway:

$\displaystyle{ C^J_{v_1} + C^J_{v_2} = 1 }$
$\displaystyle{ C^J_{v_1} \varepsilon^{v_1}_S + C^J_{v_2} \varepsilon^{v_2}_S = 0 }$

Using these two equations we can solve for the flux control coefficients to yield

$\displaystyle{ C^J_{v_1} = \frac{\varepsilon^{2}_S}{\varepsilon^{2}_S - \varepsilon^{1}_S} }$
$\displaystyle{ C^J_{v_2} = \frac{-\varepsilon^{1}_S}{\varepsilon^{2}_S - \varepsilon^{1}_S} }$

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 $\displaystyle{ \varepsilon^{v_1}_S = 0 }$. In this case, the control coefficients reduce to

$\displaystyle{ C^J_{v_1} = 1 }$
$\displaystyle{ C^J_{v_2} = 0 }$

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:

$\displaystyle{ C^S_{v_1} = \frac{1}{\varepsilon^{2}_S - \varepsilon^{1}_S} }$
$\displaystyle{ C^S_{v_2} = \frac{-1}{\varepsilon^{2}_S - \varepsilon^{1}_S} }$

Three step pathway

Consider the simple three step pathway:

$\displaystyle{ X_o \rightarrow S_1 \rightarrow S_2 \rightarrow X_1 }$

where $\displaystyle{ X_o }$ and $\displaystyle{ X_1 }$ 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.

$\displaystyle{ C^J_{e_1} = \varepsilon^{2}_1 \varepsilon^{3}_2 / D }$
$\displaystyle{ C^J_{e_2} = -\varepsilon^{1}_1 \varepsilon^{3}_2 / D }$
$\displaystyle{ C^J_{e_3} = \varepsilon^{1}_1 \varepsilon^{2}_2 / D }$

where D the denominator is given by

$\displaystyle{ D = \varepsilon^{2}_1 \varepsilon^{3}_2 -\varepsilon^{1}_1 \varepsilon^{3}_2 + \varepsilon^{1}_1 \varepsilon^{2}_2 }$

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 $\displaystyle{ S_1 }$

$\displaystyle{ C^{S_1}_{e_1} = (\varepsilon^{3}_2 - \varepsilon^{2}_2) / D }$
$\displaystyle{ C^{S_1}_{e_2} = - \varepsilon^{3}_2 / D }$
$\displaystyle{ C^{S_1}_{e_3} = \varepsilon^{2}_2 / D }$

And for $\displaystyle{ S_2 }$

$\displaystyle{ C^{S_2}_{e_1} = \varepsilon^{2}_1 / D }$
$\displaystyle{ C^{S_2}_{e_2} = -\varepsilon^{1}_1 / D }$
$\displaystyle{ C^{S_2}_{e_3} = (\varepsilon^{1}_1 - \varepsilon^{2}_1) / D }$

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 $\displaystyle{ v_1 }$ and $\displaystyle{ v_2 }$ are determined by two enzymes $\displaystyle{ e_1 }$ and $\displaystyle{ e_2 }$ respectively. Changing either enzyme will result in a change to the steady state level of $\displaystyle{ x }$ and the steady state reaction rates $\displaystyle{ v }$. Consider a small change in $\displaystyle{ e_1 }$ of magnitude $\displaystyle{ \delta e_1 }$. This will have a number of effects, it will increase $\displaystyle{ v_1 }$ which in turn will increase $\displaystyle{ x }$ which in turn will increase $\displaystyle{ v_2 }$. Eventually the system will settle to a new steady state. We can describe these changes by focusing on the change in $\displaystyle{ v_1 }$ and $\displaystyle{ v_2 }$. The change in $\displaystyle{ v_2 }$, which we designate $\displaystyle{ \delta v_2 }$, came about as a result of the change $\displaystyle{ \delta x }$. Because we are only considering small changes we can express the change $\displaystyle{ \delta v_2 }$ in terms of $\displaystyle{ \delta x }$ using the relation

$\displaystyle{ \delta v_2 = \frac{\partial v_2}{\partial x} \delta x }$

where the derivative $\displaystyle{ \partial v_2/\partial x }$ measures how responsive $\displaystyle{ v_2 }$ is to changes in $\displaystyle{ x }$. The derivative can be computed if we know the rate law for $\displaystyle{ v_2 }$. For example, if we assume that the rate law is $\displaystyle{ v_2 = k_2 x }$ then the derivative is $\displaystyle{ k_2 }$. We can also use a similar strategy to compute the change in $\displaystyle{ v_1 }$ as a result of the change $\displaystyle{ \delta e_1 }$. This time the change in $\displaystyle{ v_1 }$ is a result of two changes, the change in $\displaystyle{ e_1 }$ itself and the change in $\displaystyle{ x }$. We can express these changes by summing the two individual contributions:

$\displaystyle{ \delta v_1 = \frac{\partial v_1}{\partial e_1} \delta e_1 + \frac{\partial v_1}{\partial x} \delta x }$

We have two equations, one describing the change in $\displaystyle{ v_1 }$ and the other in $\displaystyle{ v_2 }$. 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 $\displaystyle{ \delta v_1 = \delta v_2 }$. With this in mind we equate the two equations and write

$\displaystyle{ \frac{\partial v_2}{\partial x} \delta x = \frac{\partial v_1}{\partial e_1} \delta e_1 + \frac{\partial v_1}{\partial x} \delta x }$

Solving for the ratio $\displaystyle{ \delta x/\delta e_1 }$ we obtain:

$\displaystyle{ \frac{\delta x}{\delta e_1} = \dfrac{-\dfrac{\partial v_1}{\partial e_1}}{\dfrac{\partial v_2}{\partial x} - \dfrac{\partial v_1}{\partial x}} }$

In the limit, as we make the change $\displaystyle{ \delta e_1 }$ smaller and smaller, the left-hand side converges to the derivative $\displaystyle{ dx/de_1 }$:

$\displaystyle{ \lim_{\delta e_1 \rightarrow 0} \frac{\delta x}{\delta e_1} = \frac{dx}{de_1} = \dfrac{-\dfrac{\partial v_1}{\partial e_1}}{\dfrac{\partial v_2}{\partial x} - \dfrac{\partial v_1}{\partial x}} }$

We can go one step further and scale the derivatives to eliminate units. Multiplying both sides by $\displaystyle{ e_1 }$ and dividing both sides by $\displaystyle{ x }$ yields the scaled derivatives:

$\displaystyle{ \frac{dx}{de_1} \frac{e_1}{x}= \frac{-\dfrac{\partial v_1}{\partial e_1}\dfrac{e_1}{v_1}} {\dfrac{\partial v_2}{\partial x} \dfrac{x}{v_2} - \dfrac{\partial v_1}{\partial x} \dfrac{x}{v_1}} }$

The scaled derivatives on the right-hand side are the elasticities, $\displaystyle{ \varepsilon^v_x }$ and the scaled left-hand term is the scaled sensitivity coefficient or concentration control coefficient, $\displaystyle{ C^x_{e} }$

$\displaystyle{ C^x_{e_1} = \frac{\varepsilon^1_{e_1}}{\varepsilon^2_x - \varepsilon^1_x} }$

We can simplify this expression further. The reaction rate $\displaystyle{ v_1 }$ is usually a linear function of $\displaystyle{ e_1 }$. For example, in the Briggs–Haldane equation, the reaction rate is given by $\displaystyle{ v= e_1 k_{cat} x/(K_m + x) }$. Differentiating this rate law with respect to $\displaystyle{ e_1 }$ and scaling yields $\displaystyle{ \varepsilon^{v_{1}}_{e_1} = 1 }$.

Using this result gives:

$\displaystyle{ C^x_{e_1} = \frac{1}{\varepsilon^2_x - \varepsilon^1_x} }$

A similar analysis can be done where $\displaystyle{ e_2 }$ is perturbed. In this case we obtain the sensitivity of $\displaystyle{ x }$ with respect to $\displaystyle{ e_2 }$:

$\displaystyle{ C^x_{e_2} = -\frac{1}{\varepsilon^2_x - \varepsilon^1_x} }$

The above expressions measure how much enzymes $\displaystyle{ e_1 }$ and $\displaystyle{ e_2 }$ control the steady state concentration of intermediate $\displaystyle{ x }$. We can also consider how the steady state reaction rates $\displaystyle{ v_1 }$ and $\displaystyle{ v_2 }$ are affected by perturbations in $\displaystyle{ e_1 }$ and $\displaystyle{ e_2 }$. 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 $\displaystyle{ J_1 }$ and $\displaystyle{ J_2 }$. 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 $\displaystyle{ J }$. Expressing the change in flux as a result of a perturbations in $\displaystyle{ e_1 }$ and taking the limit as before we obtain

$\displaystyle{ C^J_{e_1} = \frac{\varepsilon^1_x}{\varepsilon^2_x - \varepsilon^1_x}, \quad C^J_{e_2} = \frac{-\varepsilon^1_x}{\varepsilon^2_x - \varepsilon^1_x} }$

The above expressions tell us how much enzymes $\displaystyle{ e_1 }$ and $\displaystyle{ e_2 }$ 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.