Microbes in fragmented environments profit from yield-efficient metabolic strategies, which allow for a maximal number of cells. In contrast, cells in well-mixed, nutrient-rich environments need to grow and divide fast to out-compete others. Paradoxically, a fast growth can entail wasteful, yield-inefficient modes of metabolism and smaller cell numbers. Therefore, general trade-offs between biomass yield and growth rate have been hypothesized. To study the conditions for such rate-yield trade-offs, we considered a kinetic model of E. coli central metabolism and determined flux distributions that provide maximal growth rates or maximal biomass yields. In the model, maximal growth rates or yields are achieved by sparse flux distributions called elementary flux modes (EFMs). We screened all EFMs in the network model and computed the biomass yields and growth rates enabled by these EFMs. Growth rates were computed from the amount of protein required to sustain a given biomass production, computed from the kinetic model by enzyme cost minimization (ECM). In a scatter plot between the growth rates and yields of all EFMs, a trade-off shows up as a Pareto front. At reference glucose and oxygen levels, we find that the rate-yield trade-off is almost negligible. However, in low-oxygen environments, a clear trade-off emerges: low-yield fermentation EFMs allow for a growth 2-3 times faster than the maximal-yield EFM. The trade-off is therefore strongly condition-dependent and should be almost unnoticeable at high oxygen and glucose levels, the typical conditions in laboratory experiments. Our public web service www.neos-guide.org/content/enzyme-cost-minimization allows users to run ECM to compute enzyme costs for metabolic models flux distributions of their choice.