The Michaelis constant KM describes the affinity of an enzyme for a specific substrate and is a central parameter in studies of enzyme kinetics and cellular physiology. As measurements of KM are often difficult and time-consuming, experimental estimates exist for only a minority of enzyme-substrate combinations even in model organisms. Here, we build and train an organism-independent model that successfully predicts KM values for natural enzyme-substrate combinations using machine and deep learning methods. Predictions are based on a task-specific molecular fingerprint of the substrate, generated using a graph neural network, and on a deep numerical representation of the enzyme's amino acid sequence. We provide genome-scale KM predictions for 47 model organisms, which can be used to approximately relate metabolite concentrations to cellular physiology and to aid in the parameterization of kinetic models of cellular metabolism.