Enzyme elasticities associated with most common kinetic mechanisms are uniquely determined by the scaled metabolite concentrations, which require a complete knowledge about in vivo metabolite concentration levels and enzyme kinetic parameters. In general, such information is not available, and when available, it is subject to uncertainty due to experimental conditions as discussed above.

Furthermore, in vivo conditions are very different from the in vitro conditions under which the kinetic parameters have been estimated. To simulate these uncertainties in the elasticities, we introduced the following Monte Carlo sampling technique. Most of the enzymatic reactions initiate with the binding of the substrate or regulatory metabolite, S , on the active site, A , of the enzyme. According to this model, the free active site of the enzyme, A , binds reversibly to metabolite, S , to form the enzyme-substrate complex, AS , with an on-rate constant, k on , and an off-rate constant, k off.

The total active site concentration is the sum of the concentration of the free active site and the active site-metabolite complex,. The degree of the saturation of the active site can then be defined as. Note that the distributions of the randomized elasticities with respect to metabolites will be determined by the particular enzyme kinetic mechanisms.

For example, for enzymes that follow Michaelis-Menten kinetics, the elasticity with respect to the substrate will be uniformly distributed between 0 and 1; whereas for enzymes that follow Hill kinetics, the elasticity with respect to the substrate will be distributed between 0 and the value of the Hill coefficient, n h , following a bimodal distribution with peaks at the limiting values Fig.

These observations are in agreement with the nature of different kinetic mechanisms. Enzymes that follow Hill kinetics switch between an active form and an inactive form, depending on the saturation status.

Therefore the elasticity falls into two regimes corresponding to the different enzyme activity states. This type of information about the enzyme kinetics is captured by the Monte Carlo sampling and the uncertainty in the enzyme saturation levels, i. Uniform sampling of enzyme saturation degrees and randomization of elasticities with respect to metabolites. Within each moiety, the individual species will be interconverted to each other through reactions in the metabolic network, but the total amount of each group always remains constant Andersen and von Meyenburg, ; Ball and Atkinson, ; Reich, ; Reich and Selkov, From the definition of Q i and Q m , their elements are merely the relative ratios of concentration of the various species within the conserved moiety Table 1 :.

## Metabolic control from the back benches: Biochemistry towards Biocomplexity

Uncertainty should be considered again in the analysis since intracellular metabolite concentrations might be unknown and they usually fluctuate under different cellular states and environments. In general, each of the metabolite concentrations could be uniformly sampled from physiological ranges if experimental measurements exist Teusink et al. For the adenylates pool, in particular, the values of the weight matrix entries are directly connected with the energetic state of the living cell.

All three adenylates are important regulators of metabolic reactions and, more generally, many catabolic and anabolic processes are regulated by the energy status of the cell. An index named energy charge ec has been defined to reflect cellular energy status by quantifying the relative number of high-energy phosphate bonds in the adenylates moiety Atkinson, ,.

For a given ec value, the steady-state values of intracellular adenylates concentrations have the relations of. After substituting these relationships into Eqs. These equations suggest that the elements of the adenylates weight matrices can be calculated based on the values for the ec index and the fraction of ATP in the adenylates pool.

### Description

The viable ranges of the ec index has been experimentally accessed for different organism species and physiological conditions of the cells Ball and Atkinson, ; Chapman et al. Thus, the corresponding indices of the weight matrices are randomized based on these samples Eqs.

Applying similar concepts on the quantification of the pyridine nucleotides moiety, we start from the linear conservation relationship. If we choose NAD as the dependent metabolite, the corresponding elements in Q i and Q m are again the relative ratios of moiety concentrations,.

Taking physiological conditions into consideration, the pyridine nucleotides must be largely oxidized for the glycolytic reactions to proceed. An index called the catabolic reduction charge crc has been defined to represent the redox status of the cellular condition Andersen and von Meyenburg, ,. After the introduction of this index, Eqs.

After feasible configurations of simulation outputs are calculated and collected, statistical properties of the control coefficients are analyzed. The primary tool that we use in the studies presented here is that of the complementary cumulative distribution functions CCDFs. For a discrete sample, like the Monte Carlo simulation used in this study,. The CCDF will provide a measure of the probability that a control coefficient is greater than a certain value.

Further statistical analysis of the populations of the control coefficients could also be performed Pritchard and Kell, , but it is beyond the scope of this article. In this section, we will illustrate the application of the MCA Monte Carlo method using two case studies: a prototypical module of biosynthetic pathways; and the more complex yeast glycolytic pathway. A primary module of metabolic networks is the branched pathway which, together with the linear pathway, constitutes the major building blocks of most metabolic networks.

The splitting ratios of branching fluxes at key nodes essentially determine the overall distribution of metabolic fluxes in most biological systems. We chose the branched pathway presented in Fig. Branched pathway models. Assuming irreversible Michaelis-Menten kinetics for all five fluxes, we derived an analytical expression for the flux control coefficients Table 2. The expressions suggest that the flux control coefficients depend on the enzyme elasticities and on the splitting ratios alone. We used the sampling procedure described above and we generated populations of control coefficients for different splitting ratios of the flux through the pathway to study the distribution of the control coefficient of flux through enzyme 2, v 2 , with respect to the activity of enzyme 2,.

MCA elements and analytical results for the branched pathway model with irreversible kinetics. Statistical analysis of the simulated results suggests that the splitting ratio has a critical impact on the control coefficients distributions. When the flux through enzyme 2 is a small portion of the overall flux, i.

## Technological and Medical Implications of Metabolic Control Analysis

Effects of splitting ratio on the flux control coefficients in branched pathway. Reversibility also affects the distributions of control coefficients. If v 2 is assumed to be a reversible reaction Fig. The closer the flux v 2 is to equilibrium, the smaller the control coefficient Fig. This illustrates the well-known postulate that enzymes that operate near thermodynamic equilibrium do not have significant control over metabolic fluxes.

In addition, this result also captures the fact that reversible reactions can have a significant control of metabolic fluxes if they do not operate near equilibrium, contrary to the common misunderstanding that reversible enzymes do not have control over metabolic fluxes regardless of their operating state with respect to the equilibrium conditions. Another critical feature of enzyme-catalyzed reactions is their capacity to adjust according to the metabolic requirements.

Such control sometimes is accomplished at the transcription level by enzyme synthesis through induction and repression, which can involve complicated molecular mechanisms. We studied how the distributions of the control coefficients of the unregulated branched pathway change when we consider two different regulatory structures: 1 , a product competitive inhibition scheme Fig. When product competitive inhibition is present, the kinetic expressions for v 2 and v 3 are considered to be.

## Metabolic control analysis in biotechnology and medicine | Nature Biotechnology

When crossover cooperative activation is included with competitive inhibition, the kinetics of v 2 and v 3 are expressed as As indicated by the results of Fig. This observation is in agreement with the metabolic rigidity concept, which states that flux alterations at branching points will be largely constrained by control architectures of the network Stephanopoulos and Vallino, Impacts of enzyme regulation on the control coefficients distributions.

No enzyme regulation solid line ; product competitive inhibition dashed line ; and product competitive inhibition and crossover cooperative activation dotted line. Different enzyme regulation mechanisms also influence the distributions of control coefficients. To illustrate this, we compared two common types of activations in the inhibition-activation branched pathway model of Fig. Despite the differences in the enzyme regulation mechanisms, the differences in the distribution of the control coefficients are not dramatic, although the difference of the means of the two populations of the control coefficients is statistically significant Fig.

These results also illustrate that in the absence of knowledge about the kinetic mechanisms we can assume alternative mechanisms, perform the Monte Carlo studies, and evaluate the significance of the alternative mechanisms on the calculation of the control coefficients. As shown by curve 3 in Fig. The proposed computational methodology offers the possibility to computationally investigate the effects of different enzyme kinetics by assuming alternative kinetics expressions, or by explicit randomization of the elasticities, and comparing the distributions of the control coefficients.

Such studies will further provide suggestions on the importance of knowledge of enzyme kinetics in deriving conclusions about rate-limiting enzymes and metabolic system responses to changes in system parameters. The glycolysis pathway is the main pathway of glucose catabolism in most of the organisms, and is one of the most complex metabolic pathways.

In glycolysis, a molecule of glucose is degraded to two molecules of pyruvate via a series of enzyme-catalyzed reactions. Under anaerobic conditions, pyruvate is further transformed into reduced products such as ethanol and lactic acid. This fermentation process is one of the major energy sources for living organisms under respiration-limited conditions. We studied the anaerobic metabolism in yeast to illustrate the application of our framework in such a complex metabolic network.

We based our study on a kinetic model of glycolysis of nongrowing, anaerobic S. The model, represented by the pathway shown in Fig. In addition to the primary product of ethanol, byproducts such as glycerol and succinate are also considered in the system see Supplementary Material for model details.