Metabolic Control Analysis

Metabolic Control Analysis (MCA) is a calculation method that can be used to analyze the concentrations and flows of the various intermediate products that arise in a network of metabolic processes from ingestion to excretion of the end product. In particular, the effect (“control”) of changes in the activity of enzymes on concentrations and flows is quantified.

History of origin

One of the first publications by Reinhart Heinrich and Tom Rapoport on metabolic control theory ( Eur J Biochem , 1974)

For a long time, the general belief in biochemistry was that in every metabolic pathway, a single enzyme, the slowest one, would control the flow. This enzyme has been called the pacemaker enzyme or rate limiting step. By “controlling the flow” it is meant that activation or inhibition of this enzyme increases or decreases the flow. To test this view, one needs quantitative methods. In biochemistry, control and regulation are often spoken of in a qualitative manner. For example: "When this hormone acts, this reaction is activated, which leads to an increase in the concentration of this substance, which in turn inhibits this or that reaction, etc." Metabolic systems are i. a. so complex that it is difficult to describe their behavior with purely qualitative arguments.

The basics of Metabolic Control Analysis were developed and published independently by Reinhart Heinrich and Tom Rapoport at the Humboldt University in Berlin and Henrik Kacser and Jim Burns at the University of Edinburgh in the early 1970s . Since then it has been further developed in a variety of ways.

Applications

Biotechnology: Which enzyme has to be strengthened in its effect in order to achieve the greatest possible increase in the synthesis rate of a desired product?
Medicine: Estimation of the impact of enzyme defects (these must, however, be relatively small, e.g. that the activity of an enzyme is reduced by 10% because control coefficients are defined as differential quotients)
Pharmacology: Which enzyme in pathogenic microorganisms should be inhibited so that the viability of these microorganisms is restricted as much as possible, but the lowest possible side effects in humans or in useful for humans, e.g. B. symbiotic bacteria occur. The control coefficient of this enzyme on an important flow should then be as large as possible in the microorganism, but as small as possible in humans.
Determination of the site of action of effectors (e.g. hormones, drugs). This requires special test planning and evaluation.

Basic definitions

Metabolic control analysis defines parameters with which the control properties of enzymes can be quantified. These are in particular the control coefficients and the elasticities. The analysis is limited i. a. on steady state equilibria , d. H. the transition from one stationary state to another after activation or inhibition of one or more enzymes is considered.

Control coefficients

Kacser and Burns define the control coefficient for the flow as follows:

{\ displaystyle C_ {E_ {k}} ^ {J_ {j}} = \ left ({\ frac {E_ {k}} {J_ {j}}} {\ frac {\ Delta J_ {j}} {\ Delta E_ {k}}} \ right) _ {\ Delta E_ {k} \ rightarrow 0} = {\ frac {E_ {k}} {J_ {j}}} {\ frac {\ partial J_ {j}} {\ partial E_ {k}}}}

The (total) concentration of an enzyme is changed and the relative effect on a flow is related to this relative change. An index j is used because there can be several flows (e.g. in branched systems). The concentrations of the internal metabolites level off again after the change in the enzyme concentration, which i. a. partially counteracts the increase in the river. For example, a control coefficient of 0.2 means that an increase in the concentration of the enzyme k by e.g. B. 10% increases the flow by 2%. However, this only applies approximately because the control coefficient is defined for infinitesimally small changes.

In the case of an enzyme, not only the total enzyme concentration but also other kinetic parameters can be changed. That is why Reinhart Heinrich and Tom Rapoport proposed a more general definition:

{\ displaystyle C_ {jk} ^ {J} = \ left ({\ frac {v_ {k}} {J_ {j}}} {\ frac {\ Delta J_ {j}} {\ Delta v_ {k}} } \ right) _ {\ Delta v_ {k} \ rightarrow 0} = {\ frac {v_ {k}} {J_ {j}}} {\ frac {\ partial J_ {j}} {\ partial v_ {k }}}}

Consider the change in the reaction rate of reaction k if the corresponding enzyme were to operate “in isolation”. This change can e.g. B. by an activator, a genetic change or by changing the total concentration of this enzyme. One determines the effect on the flow in the overall system and, on the other hand, the effect on the speed of the individual enzyme. In the latter case, all substrates and products of this enzyme are regarded as external metabolites, i.e. their concentration is kept constant.

Coefficient of elasticity

Control coefficients are global properties of the system. The properties of the individual enzymes, on the other hand, are described by so-called elasticity coefficients:

{\ displaystyle \ epsilon _ {ij} = {\ frac {S_ {j}} {v_ {i}}} {\ frac {\ partial v_ {i}} {\ partial S_ {j}}}}

This can be substrate, product or effector. An elasticity coefficient (or elasticity for short) expresses how much the speed of the enzyme isolated from the rest of the system changes when a concentration is changed. Here, too, the non-standardized form is sometimes used. Both can be represented as a matrix. ${\ displaystyle {S_ {j}}}$

Example: Henri-Michaelis-Menten kinetics :

{\ displaystyle v_ {i} = {\ frac {V _ {\ max, i} S_ {j}} {K_ {m, i} + S_ {j}}}}

Unnormalized elasticity:

{\ displaystyle \ epsilon _ {ij} '= {\ frac {V _ {\ max, i} K_ {m, i}} {(K_ {m, i} + S_ {j}) ^ {2}}}}

Normalized elasticity:

{\ displaystyle \ epsilon _ {ij} = {\ frac {K_ {m, i}} {K_ {m, i} + S_ {j}}}}

.

In addition to the elasticities with regard to concentrations, such parameters have been defined:

{\ displaystyle \ pi _ {kl} = {\ frac {p_ {l}} {v_ {k}}} {\ frac {\ partial v_ {k}} {\ partial p_ {l}}}}

To distinguish them, these two quantities are also referred to as - and - elasticities. ${\ displaystyle \ epsilon}$ ${\ displaystyle \ pi}$

The above definitions of the control coefficients use enzyme-specific parameters. However, these coefficients can also be used to express the influence of non-specific parameters, e.g. B. Temperature, pH and effectors that act on several enzymes. Be such a parameter. Then ${\ displaystyle {p_ {m}}}$

{\ displaystyle \ mathrm {d} J_ {j} = {\ frac {\ partial J_ {j}} {\ partial p_ {m}}} \ mathrm {d} p_ {m}}

This results in

{\ displaystyle \ mathrm {d} J_ {j} = \ sum _ {k} {C '} _ {jk} ^ {J} \ pi' _ {km} \ mathrm {d} p_ {m}}

This equation can be understood as a chain rule: A parameter change initially has an influence on some or all reactions k and these in turn have an effect on the flow . These influences must be multiplied with one another and added up across all reactions. This formula shows once again that it makes sense to define control coefficients. The effect of parameters on steady flow or other system variables is broken down into control coefficients and elasticities. The former have the advantage of forming a well-defined set (since the number of reactions is better defined than the number of parameters) and the latter that they are local quantities, i.e. are only defined for one reaction, and can therefore be calculated more easily. ${\ displaystyle \ mathrm {d} p_ {m}}$ ${\ displaystyle J_ {j}}$ ${\ displaystyle \ pi}$

General calculation of the control coefficients from the elasticities

An essential goal in metabolic control theory is to calculate the global properties of a biochemical reaction system expressed by control coefficients from the local properties expressed by elasticities. This can be done using very general equations that can be written down in matrix notation. Its derivation is based on the stationarity equation, since the theory assumes that the biochemical system is in a stable steady state. The equations can be found e.g. B. in the monograph Heinrich and Schuster (1996). From these equations it follows that the values of the control coefficients are independent of the choice of the perturbation parameters. z. B. the same value results if you change the Michaelis-Menten constant or if you change the enzyme concentration.

credentials

Fell D., Understanding the Control of Metabolism, Portland Press, 1997.
Heinrich R. and Schuster S., The Regulation of Cellular Systems, Chapman and Hall, 1996.

↑ H. Kacser, JA Burns, The control of flux, Symp. Soc. Exp. Biol. 27 (1973) 65-104
^ R. Heinrich, TA Rapoport: A linear steady-state treatment of enzymatic chains. General properties, control and effector strength. Eur. J. Biochem. 42 (1974) 89-95.

[1] H. Kacser, JA Burns, The control of flux, Symp. Soc. Exp. Biol. 27 (1973) 65-104

[2] R. Heinrich, TA Rapoport: A linear steady-state treatment of enzymatic chains. General properties, control and effector strength. Eur. J. Biochem. 42 (1974) 89-95.