Understanding the metabolic function of microorganisms has recently received a lot of attention due to its importance in fields such as health and industry. Metabolism in microorganisms is a sophisticated process comprised of several thousands of different components with intricate interactions between them. This characteristic is translated into complex dynamical behavior that such systems can exhibit e.g., multiple steady states, hysteresis, or oscillations. Kinetic models of the processes occurring in the cells allow us not just to determine optimal conditions for a given objective but also to assess their stability properties. In this work we evaluate the stability of the central carbon metabolism of E. coli, using the Chassagnole et al. model, at optimal enzyme levels as determined by  for the production of serine. To accomplish this, we construct bifurcation diagrams considering the level of one, two and three enzymes as bifurcation parameters. We determine that the system goes through a Hopf bifurcation and/or a limit point for certain parameter values. This is achieved for a 68% change in the enzyme level, implying that the system is robust to perturbations on the process parameters at the optimal enzyme levels for the employed model description.