Controllability coefficient

In both intermediate and maximum rates of respiration, control is distributed between several different steps, including the activity of the adenine nucleotide translocator (Groen et al., 1983). It is now recognized that the idea of a simple rate-limiting step for a metabolic pathway is simplistic and that control is shared by all steps although to different extents (Kacserand Bums, 1978 Fell, 1992). Each step in a pathway has a flux control coefficient (FCC) defined as ... 

Metabolic control analysis (MCA) assigns a flux control coefficient (FCC) to each step in the pathway and considers the sum of the coefficients. Competing pathway components may have negative FCCs. To measure FCCs, a variety of experimental techniques including radio isotopomers and pulse chase experiments are necessary in a tissue culture system. Perturbation of the system, for example, with over-expression of various genes can be applied iteratively to understand and optimize product accumulation. 

Both quantities are usually written as m x r elasticity matrices e and n, respectively. In contrast to the local elasticities, the control coefficients describe the global or systemic properties of the system, that is, the response to the perturbation after all variable shave relaxed to the new state. 

Taking into account mass conservation relationships, specified by the link matrix L defined in Eq. (13), the expressions for the control coefficient need to be modified. We obtain... 

Using matrix notation, we define Dso and Dvo to be diagonal matrices with elements S° and v° on the diagonal, respectively. The normalized (or scaled) matrices of elasticities e and control coefficients Cs are then obtained by the... 

Correspondingly, the normalized flux control coefficient CJ is defined as... 

Equation (99) implies that it is often possible to specify intervals or approximate values for the scaled elasticities in terms of relative saturation, even when detailed kinetic information is not available. For example, as a rule of thumb, the substrate concentration can often be considered to be on the order of the Km value. As the scaled elasticities, by means of the control coefficients, can be directly translated into a systemic response, it is possible to utilize such heuristic arguments to acquire an initial approximation of global network properties. 

Finally, and more profoundly, not all properties require explicit knowledge of the functional form of the rate equations. In particular, many network properties, such as control coefficients or the Jacobian matrix, only depend on the elasticities. As all rate equations discussed above yield, by definition, the assigned elasticities, a discussion which functional form is a better approximation is not necessary. In Section VIII we propose to use (variants of) the elasticities as bona fide parameters, without going the loop way via explicit auxiliary functions. 

The basic idea is very simple In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. 

For a lucid account of the kinetics of multi-enzyme systems, the reader should consult Cornish-Bowden who defines such related parameters as flux control coefficients, summation relationships, and response coefficients. 

The formalized application of metabolic control analysis deals with several parameters (a) The flux control coefficient is defined as the fractional change in pathway flux... 

CONTROL COEFFICIENT Convergence-point method, POINT-OF-INTERSECTION Conversion of pepsinogen to active pepsin, AUTOCATALYSIS Cooperative oxygen binding,... 

Ray s approach was revised by Brown and Cooper (1993) from the system control analysis point of view (see the book of Comish-Bowden and Cardenas, 1990). They stress again that there is no unique rate-limiting step specific for an enzyme, and this step, even if it exists, depends on substrate, product and effector concentrations. They also demonstrated that the control coefficients... 

The Control Coefficient Quantifies the Effect of a Change in Enzyme Activity on Metabolite Flux through a Pathway... 

Quantitative data obtained as described in Figure 15-33 can be used to calculate a flux control coefficient,... 

C, for each enzyme in a pathway. This coefficient expresses the relative contribution of each enzyme to setting the rate at which metabolites flow through the pathway—that is, the flux, J. C can have any value from 0.0 (for an enzyme with no impact on the flux) to 1.0 (for an enzyme that wholly determines the flux). An enzyme can also have a negative flux control coefficient. In a branched pathway, an enzyme in one branch, by drawing intermediates away from the other branch, can have a negative impact on the flux through that other branch (Fig. 15-34). C is not a constant, and it is not... 

FIGURE 15-34 Flux control coefficient, C, in a branched metabolic pathway. In this simple pathway, the intermediate B has two alternative fates. To the extent that reaction B —> E draws B away from the pathway A —> D, it controls that pathway, which will result in a negative flux control coefficient for the enzyme that catalyzes step B —> E. Note that the sum of all four coefficients equals 1.0, as it must. 

When real data from the experiment on glycolysis in a rat liver extract (Fig. 15-33) were subjected to this kind of analysis, investigators found flux control coefficients (for enzymes at the concentrations found in the extract) of 0.79 for hexoldnase, 0.21 for PFK-1, and 0.0 for phosphohexose isomerase. It is not just fortuitous that these values add up to 1.0 we can show that for any complete pathway, the sum of the flux control coefficients must equal unity. 

The three coefficients C, e, and R are related in a simple way the responsiveness (R) of a pathway to an outside factor that affects a certain enzyme is a function of (1) how sensitive the pathway is to changes in the activity of that enzyme (the control coefficient, C) and (2) how sensitive that specific enzyme is to changes in the outside factor (the elasticity, e) ... 

Thus the responsiveness of each enzyme in a pathway to a change in an outside controlling factor is a simple function of two things the control coefficient, a variable that expresses the extent to which that enzyme influences the flux under a given set of conditions, and the elasticity, an intrinsic property of the enzyme that reflects its sensitivity to substrate and effector concentrations. 

FIGURE 2 The flux control coefficient, (a) Typical variation of the pathway flux, )ymeasured at the step catalyzed by the enzyme ydh, as a function of the amount of the enzyme xase, xase, which catalyzes an earlier step in the pathway. The flux control coefficient at (e,j) is the slope of the product of the tangent to the curve, d/ydh/3 xase/ and the ratio (scaling factor), e/j. (b) On a double-logarithmic plot of the same curve, the flux control coefficient is the slope of the tangent to the curve. 

The flux control coefficient, C, is an experimentally determined measure of the effect of an enzyme s concentration on flux through a multienzyme pathway. It is characteristic of the whole system, not intrinsic to the enzyme. 

Report of the experimental determination of the flux control coefficients for glucokinase and the glucokinase regulatory protein in hepatocytes. 

Not only can the concentration [EJ change but also allosteric effectors can alter the activity. Kacser and Bums defined this in temis of a controllability coefficient k,... 

Control elements of metabolic reactions 536 Controllability coefficient 537 Coomassie brilliant blue 102,122s Cooperative binding 352 of protons 331 of substrate 476... 

Components are commonly represented as nodes in the metabolic network. These nodes can act as branch points if the number of input and output fluxes is not equivalent. Non-essential reactions around a node can be collected into reaction groups the coefficients of their fluxes, in general termed the metabolic flux coefficients (in analogy to rate coefficients), can be rearranged as group control coefficients. 

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