Physiological states of bacterial cells exhibit a wide spectrum of timescale. Under nutrient-rich conditions, most of the cells in an isogenic bacterial population grow at certain rates, while a small subpopulation sometimes falls into a dormant state where the growth rates slow down by orders of magnitude. The dormant cells have unique characteristics: The metabolic activity is quite slow, and the dormant cells typically exhibit a high tolerance for a range of stresses, such as antibiotics applications. To reveal the origins of such heterogeneity of timescales, we constructed a kinetic model of Escherichia coli central carbon metabolism, including the dynamics of the energy currency molecules, and asked if perturbations of the metabolites' concentrations lead to the distinct metabolic states. By numerically studying the relaxation dynamics, we found that the model robustly exhibits two qualitatively distinct relaxation dynamics depending on the initial conditions generated by the perturbations. In the first type, the concentrations of metabolites reach the steady state quickly, resembling the growing dynamics. On the other hand, the other type of dynamics takes a much longer time to reach the steady state, and during the relaxation, cell growth almost halts, reminding us of the dormant cells. In order to unveil the mechanism of distinct behaviors, we reduced the metabolic network model into a minimal model without losing the emergence of distinct dynamics. Analytical and numerical studies of the two-variable minimal model revealed the necessary conditions for the distinct behavior, namely, the depletion of energy due to the futile cycle and its nonuniform impact on the kinetics because of the coexistence of the energy currency-coupled and uncoupled reactions as well as branching of the network. The result is consistent with the experimental reports that the dormant cells commonly exhibit low ATP levels and provides a possible explanation for the appearance of dormant cells that causes antibiotic persistence.