Mutational robustness and evolvability (the capacity to generate beneficial heritable variation), and two important characteristics of proteins that contribute to fitness. Using the TEM-1 beta-lactamase as a model, we show that robustness and evolvability are not fixed quantities, but instead depend on the strength of selection, with both properties collapsing as selection pressures increase. This result due to a steep non-linear relationship between fitness and catalytic efficiency that shifts as selection pressure varies. Spatially, the pattern of constraints radiates nearly uniformly outward from the active site as selection strength increases. In contrast, the pattern of adaptive mutations comprises physically contiguous “wires” that link the active site to a few distantly positioned surface sites through the protein core. In summary, robustness and evolability depend on an excess of cellular enzymatic activity relative to the selection threshold. A logical implication is that high catalytic activity in enzymes could be driven not by direct selection, but by the need to be adaptive under fluctuating enviornments.